Theoretical Studies of Surface Phase Transitions
A thesis submitted to the University of Cambridge
for the degree of Doctor of Philosophy
Lev Gelb
Clare College
August 1995
Declaration of Originality
This thesis describes research done in the Theoretical Division of the Chemistry Department between October 1992 and August 1995. The contents are the original work
of the author, except where indicated otherwise. No part of this thesis has been previously submitted for any other degree or qualification at any academic institution. The
length of this thesis does not exceed the maximum permitted length of 60000 words.
Lev Gelb
i
ii
Acknowledgements
I thank my supervisor, Prof. Ruth Lynden-Bell for her help and encouragement, Dr
Ali Alavi and Dr Anthony Stone for good advice, and the many past and present
occupants of Rooms 246 and 248 for their company and good humour.
Many thanks to my family and friends for all of their encouragement and support.
My studies have been supported by a 1992 British Marshall Scholarship from the
Marshall Aid Commemoration Commission of the Association of Commonwealth
Universities. In my third year I have been supported by a Student Fellowship from
the National Science Foundation, with additional aid from the Cambridge Commonwealth Trust. This support is gratefully appreciated.
Thanks also to Dean Kathleen McDermott for her help and encouragement in
getting me here.
iii
iv
So they rolled up their sleeves and sat down to experiment — by simulation,
that is, mathematically and all on paper. And the mathematical models of King
Krool and the beast did such fierce battle across the equation-covered table
that the constructors’ pencils kept snapping. Furious, the beast writhed
and wriggled its iterated integrals beneath the King’s polynomial blows,
collapsed into an infinite series of indeterminate terms, then got back up
by raising itself to the nth power, but the King so belaboured it with
differentials and partial derivatives that its Fourier coefficients all
cancelled out (see Riemann’s Lemma), and in the ensuing confusion
the constructors completely lost sight of both King and beast.
So, they took a break, stretched their legs, had a swig from the
Leyden jug to bolster their strength, then went back to work
and tried it again from the beginning, this time unleashing
their entire arsenal of tensor matrices and grand canonical
ensembles, attacking the problem with such fervor that the
very paper began to smoke. The King rushed forward
with all his cruel coordinates and mean values, stumbled
into a dark forest of roots and logarithms, had to
backtrack, then encountered the beast on a field of
irrational numbers (Fi ) and smote it so grievously
that it fell two decimal places and lost an epsilon,
but the beast slid around an asymptote and hid
in an n-dimensional orthogonal phase space,
underwent expansion and came out, fuming
factorially, and fell on the King and hurt
him passing sore. But the King, nothing
daunted, put on his Markov chain mail
and all his impervious parameters, took
his increment ∆k to infinity and dealt
the beast a truly Boolean blow, sent
it reeling through an x-axis and
several brackets — but the beast,
prepared for this, lowered its
horns and — wham!! — the
pencils flew like mad through
transcendental
functions
and double eigentransformations, and when at
last the beast closed
in and the King was
down and out for
the count, the constructors jumped
up, danced a
jig, laughed
and sang as
they tore all
their papers to
shreds . . .
Stanislaw Lem, The Cyberiad [1]
v
vi
Abstract
This thesis is composed of three separate studies. The first study is of a model
Atomic Force Microscope system, and the second two studies are of statisticalmechanical models of surface phase transitions.
An Atomic Force Microscope tip can be used to measure the structure of a liquid
near a flat surface by bringing the tip down to the surface and measuring the force on
it due to the liquid. This is a sensitive probe of the ordered structure that the liquid
assumes near the surface. I have performed molecular-dynamics simulations of a
model of this system in which the wall and tip are smooth and structureless, and the
liquid is described by the Lennard-Jones potential. I have measured the structure of
the liquid for different sizes of tip and different tip–surface separations, and found that
these density profiles correlate well with features in the force-distance curve measured
by the tip. I have also modelled this system with an integral equation theory based
on approximate closures of the Ornstein-Zernike relation and found that this theory
agrees well with simulations at all but very small tip–surface separations.
The Blume-Emery-Griffiths model is the simplest lattice model that can be used
to describe a system with solid, liquid and vapour phases. I have used this model
(within the mean-field approximation) to look at the surface phase transitions of such
a material. I have measured both the previously seen surface melting of this model
and also a series of single-layer transitions at lower temperatures. This was done
as a preliminary study before developing a more realistic model for surface phase
transitions.
Lastly, I have worked with a model applicable to the low-index crystal faces of a
material given by a general potential which can include a degree of anisotropy. This
is a lattice-model closely related to that used by Trayanov and Tosatti to study surface
melting, but I have treated certain aspects differently, and introduced additional order
parameters. This results in a better phase diagram. I have studied the behaviour
of the (100), (110) and (111) faces of the Lennard-Jones crystal all along the solidvapour phase coexistence line. Although attempts to apply this model to a realistic
anisotropic potential have failed, I was able to study the surface behaviour of materials
given by the Lennard-Jones potential plus a simple anisotropic term similar to that
used in Maier-Saupe theory. All of these calculations have more or less supported
conclusions drawn from adsorbed-film experiments, and have helped to explain the
processes that occur in these films.
vii
viii
Contents
I Studies of Atomic Force Microscopy
1
1 Studies of Atomic Force Microscopy
1.1 Introduction . . . . . . . . . . . . . . . . . . . .
1.2 Molecular Dynamics Simulations . . . . . . . . .
1.2.1 The Model System . . . . . . . . . . . .
1.2.2 Results from Molecular Dynamics . . . .
1.3 Integral Equation Methods . . . . . . . . . . .
1.3.1 Theory . . . . . . . . . . . . . . . . . .
1.3.2 Integral Equation Results . . . . . . . . .
1.3.3 Integral Equation Results - Hard Walls . .
1.3.4 Integral Equation Results - “Weak” Walls
1.4 Comparison With Experiment . . . . . . . . . .
1.5 Conclusions . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
6
6
8
23
23
31
35
38
40
40
II Studies of Surface Phase Transitions
43
2 Introduction
2.1 Classification of Surface Phase Transitions
2.2 Review of Experimental Work . . . . . .
2.3 Review of Molecular Simulation . . . . .
2.4 Review of Theoretical Approaches . . . .
2.5 This Work . . . . . . . . . . . . . . . . .
.
.
.
.
.
45
45
49
51
54
58
.
.
.
.
61
62
62
63
66
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 The Blume-Emery-Griffiths Model
3.1 The Bulk System . . . . . . . . . . . . . . . .
3.1.1 Definition of the Model . . . . . . . . .
3.1.2 Mean-Field Solution of the Bulk Model
3.1.3 Properties of the Bulk Model . . . . . .
ix
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
x
Contents
3.2
3.3
3.4
Calculations on Surfaces . . . . . . . . . . .
3.2.1 Mean-Field Theory for a Surface . . .
3.2.2 Technical Details . . . . . . . . . . .
Results for the Surface System . . . . . . . .
3.3.1 Roughening and Layering Transitions
3.3.2 Surface Melting . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 A Better Theory
4.1 Bulk Systems . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Recasting the Partition Function . . . . . . . . . .
4.1.2 Mean-Field Solution of the Lattice Model . . . . .
4.1.3 Solution of the Mean-Field Orientational Problem
4.1.4 Determination of the Configuration Factor . . . . .
4.1.5 Calculating the Free Volume . . . . . . . . . . . .
4.1.6 Using Different Ensembles . . . . . . . . . . . . .
4.2 Theorems, Identities and Relations . . . . . . . . . . . . .
4.2.1 Bulk Thermodynamics . . . . . . . . . . . . . . .
4.2.2 Surface Thermodynamics . . . . . . . . . . . . .
4.3 Modelling the Surface . . . . . . . . . . . . . . . . . . . .
4.3.1 γ Function for the Slab System . . . . . . . . . . .
4.3.2 Free Volume in the Slab System . . . . . . . . . .
4.3.3 Internal Energy in the Slab System . . . . . . . . .
4.3.4 Orientational Free Energy in the Slab System . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Technical Details and Algorithms
5.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Definition of the Lattice Spacings . . . . . . . . . . . . . .
5.1.2 Performing Surface Calculations . . . . . . . . . . . . . . .
5.1.3 Calculation of the Gibbs Dividing Surface and Surface Excess Properties . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Calculation of Vcore . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Finding Coexistence Lines . . . . . . . . . . . . . . . . . .
5.1.6 Finding the Triple Point . . . . . . . . . . . . . . . . . . .
5.1.7 Finding the Critical Point . . . . . . . . . . . . . . . . . . .
5.1.8 Polishing a Solution . . . . . . . . . . . . . . . . . . . . .
5.2 Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
69
69
69
71
71
74
76
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
79
79
79
81
82
83
85
88
88
88
90
93
94
95
95
96
97
. 97
. 97
. 98
.
.
.
.
.
.
.
.
99
100
100
101
102
102
103
103
Contents
xi
5.2.2
5.2.3
5.2.4
5.2.5
5.2.6
5.2.7
5.2.8
Units . . . . . . . . . . . . . . . . . . . . . . . . . .
Constraints on the Order Parameters . . . . . . . . . .
Handling the Orientational Integrals . . . . . . . . . .
Self-Consistent Solution for the Orientational Problem
False p-Potential . . . . . . . . . . . . . . . . . . . .
Edge Detection and the Dummy Function . . . . . . .
False s-Potential . . . . . . . . . . . . . . . . . . . .
6 Bulk Studies
6.1 Isotropic Potentials . . . . . . . . . . . . . . . . . .
6.2 Sensitivity of the Model . . . . . . . . . . . . . . . .
6.3 Anisotropic Potentials . . . . . . . . . . . . . . . . .
6.3.1 Relaxed-Lattice Anisotropic Phase Diagrams
6.3.2 Frozen-Lattice Anisotropic Phase Diagrams .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7 Surfaces of the Lennard-Jones Crystal I. Behaviour of The (100),
and (111) Faces
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Results for the (100) Surface . . . . . . . . . . . . . . . . . . .
7.3 Results for the (110) Surface . . . . . . . . . . . . . . . . . . .
7.4 Results for the (111) Surface . . . . . . . . . . . . . . . . . . .
7.5 Surface Melting . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Comparison With Previous Lattice Models . . . . . . . . . . . .
7.7 Comparison with Simulations and Experiments . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
103
104
105
105
106
107
108
.
.
.
.
.
109
109
114
120
122
126
(110)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
129
129
132
138
142
147
149
150
8 Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests and Constrained
Melting Behaviour
153
8.1 Sensitivity of the Surface Behaviour to Input Parameters . . . . . . . 153
8.2 Calculation of the Surface Tension at Fixed Liquid Layer Thickness . 159
8.2.1 Calculation Technique . . . . . . . . . . . . . . . . . . . . . 160
8.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9 Relaxed-Lattice Studies of the Surface
171
9.1 How to Do the Calculation . . . . . . . . . . . . . . . . . . . . . . . 171
9.2 Tests of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
xii
Contents
10 Studies of Surfaces with Anisotropic Potentials
10.1 Surfaces of the δ02 = −0.30 Crystal . . . . .
10.1.1 (100) Surface . . . . . . . . . . . .
10.1.2 (110) and (111) Surfaces . . . . . .
10.2 Surfaces of the δ02 = −0.40 Crystal . . . . .
10.2.1 (100) Surface . . . . . . . . . . . .
10.2.2 (110) and (111) Surfaces . . . . . .
10.3 Conclusions . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
175
175
175
177
182
182
185
188
11 Conclusions
191
A A Pathological Minimisation
195
B Programs
B.0.1
B.0.2
B.0.3
B.0.4
B.0.5
199
199
199
200
200
200
References
AFM - Simulations . . . . . . .
AFM - Integral Equations . . .
BEG Calculations . . . . . . .
“A Better Theory” Calculations
This Thesis . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
201
Part I
STUDIES OF ATOMIC FORCE
MICROSCOPY
Chapter 1
STUDIES OF ATOMIC FORCE
MICROSCOPY
1.1 Introduction
Atomic Force Microscopy is a very powerful tool for the study of surfaces. In this
technique, a sharp tip is slowly moved towards a surface and the force it encounters is
measured as a function of height. The tip is mounted on a cantilever of known spring
constant so by measuring the deflection angle of the tip with optical techniques this
force may be determined. Images of the surface can be obtained by rastering the tip
relative to the surface either in constant height mode or constant force mode, and the
atomic structure of the surface can often be resolved. One advantage of the Atomic
Force Microscope (AFM) as a tool for surface science is that it need not be used
under conditions of ultra-high vacuum, so that the structures of adsorbed layers and
covered surfaces can be probed. Recently [2, 3], O’Shea, Welland and Rayment have
investigated the force-distance curves of a tip being brought near a surface under a
number of liquids. Under Octamethylcyclotetrasiloxane (OMCTS), a liquid made up
of roughly spherical, non-polar molecules which are not strongly hydrogen-bonded,
they observed as many as six oscillations in the force-distance curve (see Figure 1.1
on page 4). When these experiments were performed under water, these oscillations
were absent; within a certain distance the tip was drawn quickly into contact with the
surface, excluding the solvent. The existence of force oscillations as a function of the
distance between two crossed cylinders was first demonstrated in experiments by Israelachvili and Adams [4] using what is known as a Surface Force Apparatus (SFA).
Horn and Israelachvili [5, 6] have studied the OMCTS system using this device, and
observed behaviour similar to that observed by O’Shea, Welland and Rayment, although Horn and Israelachvili measured more oscillations (see Figure 1.2).
The interpretation of these oscillations is straightforward. A liquid near a surface
3
4
Studies of Atomic Force Microscopy
Figure 1.1: (From O’Shea, Welland and Rayment [2]) The variation of the
normal force acting on the tip as a function of the tip-sample distance for
HOPG immersed in OMCTS. The dashed line represents the discontinuous jumps in the force. The sample-tip approach speed is 5 nm/s.
Figure 1.2: (From Horn and Israelachvili [5]) Experimental results of measurements of force F as a function of separation D between two curved
mica surfaces of radius R = 0.74 cm in OMCTS at 22◦ C.
1.1 Introduction
5
shows layering on a molecular scale. The density distribution of molecules has a
maximum at the liquid-wall potential well minimum, and this layer of molecules at
the wall causes the formation of a second layer at roughly one molecular diameter
further away, and so on. In simulations, depending on the wall-solvent interaction
potential, the third, fourth and even fifth such layers may be observed [7]. These
oscillations are analogous to those in the molecular pair distribution function g(r),
but are larger in magnitude, since a wall is more confining than a single particle held
fixed. As two surfaces approach each other, favourable packing of solvent molecules
occurs when these maxima coincide, and unfavourable packing occurs when they are
out of step. Maxima and minima in the force curve will occur between these positions
of favourable and unfavourable packing, as has been shown in numerous simulations
of fluids confined between two parallel walls [8–10]. It should be emphasised that the
force measured at constant temperature is a derivative of the free energy rather than
the internal energy, and so there is a large entropic contribution. Oscillations occur
even in a hard-sphere fluid.
It is not as clear what happens when a “sharp” AFM tip approaches a surface, as
the solvent layers around it have a different geometry than those parallel to the surface
and so will be “in phase” in some places and “out of phase” in others. The fact that
the geometry of an AFM tip is not usually known and also subject to change via
interaction with the surface makes it even less clear. We investigate this by studying
a simple model. We represent the tip with a smooth sphere of variable diameter held
in a liquid of spherical particles near a smooth surface, and study the force curve
and the liquid structure as a function of separation. We expect that even a sphere the
size of a single particle should experience oscillations, since for a fluid in which all
potentials are pairwise additive we may express the average force felt between two
particles (or a particle and the surface) as a function of the total correlation function
h(r) = g(r) − 1;
∂
(1.1)
βFi j (r) = log hi j (r) + 1
∂r
so that for an oscillatory density profile (and thus an oscillatory h(r)) we should see an
oscillatory average force between a single fluid particle and the surface. Furthermore,
we expect the geometry of the tip to influence the magnitude of these oscillations;
differently shaped tips should show very different force-distance curves.
We have studied this system by two methods. The first is the technique of molecular dynamics, in which we directly measured via computer “experiment” the average
total force on the tip at different tip-wall separations. This method also generated (by
accumulating onto a histogram) density profiles of the liquid near the tip and wall.
The second method that we used to look at this system is one based on an approx-
6
Studies of Atomic Force Microscopy
imate liquid theory. Although approximate theories tend to break down in highly
anisotropic situations and are often in quantitative disagreement with simulations of
bulk fluids, for all but the smallest tip-wall separations this theory agrees reasonably
well with the simulation results. Many of these results have been published [11, 12].
1.2 Molecular Dynamics Simulations
1.2.1 The Model System
The first method that we used to study this model system is that of molecular dynamics (MD). In an MD simulation one integrates the equations of motion for all of the
particles in the system using a finite difference algorithm, first over a long enough
time period to insure that the system has equilibrated, and then long enough to accumulate good data about its behaviour. (For a detailed discussion of MD techniques,
refer to Allen and Tildesley, Computer Simulation of Liquids [13])
We have simulated three systems with this method, with spherical tip diameters
of 11σ, 5σ and 3σ.1 For the 5σ diameter tip, the simulation model consisted of a box
bounded top and bottom by smooth walls at z = 0 and z = 16σ with periodic boundary
conditions of box length 10σ in the x and y directions. The box contained 1024 atoms
and a structureless, infinitely massive (that is, immobile) sphere representing the tip.
The atoms interact with each other via a cut-and-shifted Lennard-Jones potential [14,
15]
 12 6
σ
σ
 4ε
− ri j
−V (rmax ) ri j ≤ rmax
ri j
Vi j ri j =
(1.2)

0
ri j > rmax
and with the two walls via the uncorrugated Steele potential for a (100) face of a
face-centred cubic lattice [16]


√
10 4
2
σ

2 σ
−
− V (z) = 2πε 
(1.3)
3  .
5 z
z
√ z
3 σz + 0.61
σ
2
2
It should be mentioned that the cut-and-shifted Lennard-Jones potential has a discontinuity in the force (but not the potential) at the radial cutoff distance r max but that
this will not alter the dynamics of our system much. No cutoffs were imposed on the
wall-particle or sphere-particle interactions. The interaction with the sphere is given
by

!12
!6 
σ
σ
,
−
(1.4)
V (r) = 3 × 4ε 
r − rsphere
r − rsphere
1 The
tip diameter is defined by the van der Waals radius of the tip considered as a single molecule.
1.2 Molecular Dynamics Simulations
7
where rsphere is the radius of the large sphere, and the multiplicative factor of three is
added to make the well depth approximately equal to that of the Steele potential. With
this potential, for rsphere = 0 the tip behaves like a “strong” Lennard-Jones particle,
so that the effective diameter of the tip is 2rsphere + σ. Simulations were carried out
at T ∗ = 1.0 using a Berendsen [17] thermostat to readjust the temperature during the
equilibration periods, and rmax in the Lennard-Jones potential was taken as 3.5σ. This
temperature is roughly between the triple point and critical point of the Lennard-Jones
fluid. The equations of motion were integrated with the Verlet algorithm, with a timestep of 10 reduced time units (t ∗ ). While performing the simulations, the density of
liquid in the volume at least 2.0σ removed from all surfaces was measured and taken
to be an estimate of the density of the bulk fluid in equilibrium with our confined fluid.
With these system parameters, this density was ρ∗ ' 0.72. In the simulations, the
sphere was moved slowly (0.25σ over 250 time-steps) from one fixed position to the
next and the system was then equilibrated for at least 2000 steps. Data concerning the
molecular positions and forces on the sphere were then collected over runs of at least
6000 time-steps. It was found that the fluctuations in the forces on the sphere were
quite large and slow to decay. Errors in the accumulated quantities were estimated by
averaging separately the first and second halves of each run.
For the smaller (3σ diameter) tip simulation, the box measured 12σ in height
by 8σ in width, and contained only 512 particles. The “bulk” density, estimated
as above, was the same as in the 5σ sphere simulation. All molecular dynamics
simulations were performed on a DAP 600 computer, which has a Single Instruction,
Multiple Data (SIMD) parallel architecture, with 4096 individual 1-bit processors.
These force-distance simulations are computationally expensive, since an entire MD
run must be performed at each tip-wall separation, and many such separations were
required in order to see the oscillatory behaviour of the solvation force.
For the largest (11σ diameter) tip simulation, the box measured 22σ in height by
17σ in width and contained 4096 particles. Again, these parameters were chosen so
that the “bulk density” would be comparable with the other simulations. Only a few
separations were simulated using this system, as runs with this many particles took
approximately 10 times longer than runs with the smaller (5σ) system.
Due to the confined environment of this model, it was difficult to properly characterise the state point of the liquid. The systems were large enough that the volumes
of liquid not near any surfaces behaved like bulk fluids at the proscribed temperature
and density, and so the liquid in the region of interest (under the tip) was in equilibrium with a bulk fluid at a well-defined state point. Since we did not compare systems
at different state points it was not necessary to precisely characterise the state points
8
Studies of Atomic Force Microscopy
in these systems; it was sufficient (for our purposes) to know them approximately.
Should precise information about the state point be required, a different simulation
technique, such as Grand Canonical Monte Carlo or Gibbs Ensemble Monte Carlo
would have to be employed.
In testing our computer code we attempted to reproduce the results obtained by
Magda, Tirrell and Davis [8], who used molecular dynamics to simulate a LennardJones fluid in an infinite planar slit with walls given by the Steele potential. We were
successful in this effort, but it should be mentioned that the pressure normal to the
wall (the main quantity of interest in our simulations with the sphere) was extremely
sensitive to such parameters as the density and radial cutoff distances, and that similar looking density profiles could give rise to very different values for the solvation
force. The qualitative behaviour of the system was insensitive to these parameters, but
this is important because it means quantitative comparisons between slightly different
computer simulations (let alone with experiment) may be quite difficult.
1.2.2 Results from Molecular Dynamics
Figure 1.3 shows the force versus distance curve measured in the 5σ sphere simulation. In this figure a positive force represents a repulsion between tip and surface. This
data does not include any direct contribution from the tip-surface interaction, which
would provide a steadily decreasing (attractive) background; one can show using the
Derjaguin approximation (see Israelachvili, Intermolecular and Surface Forces [18])
that for a large sphere near a surface (in a vacuum), the van der Waals interaction
energy is given by
−π2Cρ1 ρ2
W=
,
(1.5)
6D
where C is determined by properties of the material and D is the separation. When
the separation is 1σ the sphere and wall are in contact. That is, we are defining the
separation as the distance from the atomic centres in the outermost layer of the wall
to the corresponding atomic centres in the sphere. At contact the solvent exerts a
net upwards force on the tip which in a physical system would be balanced by the
adhesion force between the tip and wall. As the distance between the tip and the wall
increases oscillations in the solvation force appear. The separations where the force is
zero and its derivative is negative correspond to stable positions of the tip where the
free energy is a minimum, while positions where the force is zero and its derivative
is positive correspond to unstable positions of the tip where the free energy of the
system is a maximum.
1.2 Molecular Dynamics Simulations
9
Total Force on Tip vs. Separation
for the 5σ Sphere (Molecular Dynamics Results)
90.0
Force on Tip (reduced units)
60.0
30.0
0.0
-30.0
1.0
2.0
3.0
Separation (σ)
4.0
5.0
Figure 1.3: Total force on tip due to solvent in the direction normal to the
surface versus sphere-wall separation, for a tip 5σ in diameter. A positive
force indicates repulsion.
10
Studies of Atomic Force Microscopy
this page intentionally left blank
1.2 Molecular Dynamics Simulations
11
x
x
z
z
x
z
x
z
Figure 1.4: Liquid density profiles from the 5σ diameter sphere simulations. The profiles are taken at 4.0σ (top left), 3.375σ (top right), 2.875σ
(bottom left) and 2.375σ (bottom right) separations of sphere and surface.
The peaks are truncated for clarity.
12
Studies of Atomic Force Microscopy
this page intentionally left blank
1.2 Molecular Dynamics Simulations
13
x
z
Figure 1.5: Liquid density profile from the 5σ diameter sphere simulation
at 1.50σ separation.
Figures 1.4 (page 11) and 1.5 show the number density distribution of the centres
of the liquid particles near the sphere and lower wall for the 5σ diameter sphere system. As the system is cylindrically symmetrical the data have been averaged over all
angles around a vertical axis through the sphere and surface, but for clarity are shown
in the figures as complete cross-sections obtained by reflecting the data over itself.
Figure 1.4 shows the outermost two (measured) free energy maxima and minima.
The solvation layers of the tip and wall are seen to coincide favourably at 4.0σ, while
at 3.375σ the maxima are out of phase directly under the sphere and the peaks in this
region are broadened. The liquid is not as closely packed under the sphere for this
separation as for the larger one, even though the energetic benefit of a particle being
directly under the sphere is greater for the smaller separation. That is, as we narrow
the separation between sphere and surface the potential well between them becomes
deeper, but packing considerations can prevent this from being exploited by the liquid
molecules. At 2.875σ separation we have a situation similar to the 4.0σ separation,
where geometrical constraints are such that dense layering of the liquid is possible
14
Studies of Atomic Force Microscopy
under the sphere (and certainly energetically favoured.) At 2.375σ (a maximum) the
layer nearest the wall has been pulled towards the sphere and widened, due to the
near-coincidence of the potential minima of the sphere and wall at this separation.
A remarkable feature of these density plots is that at small separations structure
begins to appear in the crevices between the sphere and the wall. This is especially
noticeable in Figure 1.5 where the tip and wall are close enough to entirely exclude
solvent. These peaks can be thought of as resulting from the interference of the solvation shells of the tip and wall, which could lead to crystallisation of the liquid around
the tip at small separations. From these plots it also seems that the tip has a strongly
associated solvent structure, so that if the tip were moved parallel to the surface, a
large volume of liquid would move with it and could certainly influence measurements of surface properties performed in this way.
Qualitative information about the dynamics of the solvent molecules in these simulations may be obtained from their velocity autocorrelation functions. We define the
volume under the tip as the cylinder of 3σ radius centred under the sphere, extending
from the surface to the equator of the sphere. We then measure the velocity autocorrelation functions Cv (t) = hvi (t) · vi(0)i of the particles in this volume, in the x, y and
z directions. (Since Cvx (t) and Cvy (t) are equivalent, we average them to obtain better
statistics.) Figure 1.6 shows these functions while Figure 1.7 shows the curves of the
integrals of these functions as defined by
t
Cvαα (t 0 )dt 0.
Iαα (t) =
(1.6)
0
At long times this is related to the diffusion constant by [19]
Dαα = lim
t→∞
kT
Iαα (t).
m
(1.7)
While it is not possible to carry out these integrations for long times as the molecules
diffuse away, the plots of I(t) out to t ∗ = 0.5 show that there is a definite reduction
in the diffusion constant in the x and y directions as the tip-wall separation decreases.
Diffusion in the z direction is restricted directly by the presence of the sphere and
wall, while diffusion in the x and y directions is reduced indirectly via the increased
structure of the solvent. Looking at the Cv (t) functions themselves we see that the
depth of the first minima in the z-function is increased for decreasing separation,
indicating an increasingly confined environment, and in the (x, y) directions the first
minima is moved in slightly as the sphere is lowered, indicating decreased mobility.
Figure 1.8 and Figure 1.9 show the force-distance data for all three sphere sizes
studied. In Figure 1.8 the data is not normalised, while in Figure 1.9 we show
1.2 Molecular Dynamics Simulations
15
Z Functions
(X,Y) Functions
1.0
1.0
1.5 σ
2.0 σ
3.0 σ
1.5 σ
2.0 σ
3.0 σ
0.5
0.0
0.0
Arbitrary units
0.5
-0.5
-0.5
0.00 0.10 0.20 0.30 0.40 0.50
0.00 0.10 0.20 0.30 0.40 0.50
*
*
t (reduced time units)
t (reduced time units)
Figure 1.6: Velocity autocorrelation functions in the (x, y) and z directions
Z Integrated Functions
0.08
Arbitrary units
0.06
1.5 σ
2.0 σ
3.0 σ
3.375 σ
(X,Y) Integrated Functions
0.08
0.06
0.04
0.04
0.02
0.02
1.5 σ
2.0 σ
3.0 σ
3.375 σ
0.00
0.00
0.00 0.10 0.20 0.30 0.40 0.50
0.00 0.10 0.20 0.30 0.40 0.50
*
*
t (reduced time units)
t (reduced time units)
Figure 1.7: Integrated velocity autocorrelation functions in the (x, y) and z
directions.
16
Studies of Atomic Force Microscopy
Force vs. Distance for Different Sphere Sizes
Total Force on Sphere
100.0
3σ
5σ
11 σ
50.0
0.0
-50.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.8: Force-distance data for all three spheres from the molecular
dynamics simulations. These curves are not normalised.
Force/Radius for Different Sphere Sizes
40.0
3σ
5σ
11 σ
F/Rsphere (reduced units)
30.0
20.0
10.0
0.0
-10.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.9: Force-distance data for all three spheres from the molecular
dynamics simulations, normalised by sphere radius.
1.2 Molecular Dynamics Simulations
17
F(r)/Rsphere against r following Horn and Israelachvili [5]. We see that the magnitude of the force on the tip varies linearly with the radius of the tip. This is quite
surprising; one might expect the magnitude of the solvation force to vary with R 2
since the surface area of the sphere goes as R2 , and the surface area should principally
determine the magnitude of interaction with the wall.
R
dz
2C
R2
R2
C2
C
=
=
=
'
C 2 + (R − dz)2
C 2 + R2 − 2Rdz + dz2
2Rdz + dz2
√
2Rdz
(1.8)
Figure 1.10: Geometric construction of the “flat” part of the tip.
Consider the force generated by the solvent in the cylindrical volume centred under the sphere, where the sphere “looks” flat (for the 5σ diameter tip, this would be
a cylinder of approximately 1σ radius). This force should vary linearly with the area
of the top of the cylinder. We must determine the area of this “flat” part as a function
of R. We see from the geometric construction in Figure 1.10 that the radius of this
√
cylinder is proportional to R, and so the area will be proportional to R. Thus, the
contributions to the force from the overlap of the spherical solvation shells “under”
the tip with the planar solvation layers of the wall will vary linearly with R. If we
look at the limit of large sphere size this becomes even clearer. As R → ∞ we require that the force per unit area should go to a constant value, which is exactly the
18
Studies of Atomic Force Microscopy
behaviour predicted by our simple construction. This indicates that the magnitude of
the force oscillations is principally determined by the interactions “under” the tip, and
interactions in the crevices around the tip do not contribute much to the force curve.
Below 2σ separation this linear behaviour is not observed; the larger spheres experience much larger repulsions at 1.75σ separation. At this separation, solvent is
excluded from the volume immediately under the tip, and so our previous argument
will not be valid. Instead, the large repulsive force observed may be due to molecules
“wedged” into the crevice created by the tip and the surface. Figures 1.11 (page 19)
and 1.12 (page 21) show representative configurations from the MD simulations for
all three sphere sizes. We see that molecules penetrate quite close to the central axis at
this separation, and only a few are in the region expected to contribute very strongly
to the repulsive force. Under the largest sphere, we observe two rings of single atoms,
which may explain the particularly large repulsive force observed in this system. The
ringed structures in these figures are interesting but not surprising; at relatively low
temperatures we should expect the liquid to pack closely into such a deep potential
well.
It is interesting to note the discrepancy in the normalised curves of Figure 1.9
for the 5σ sphere and 3σ sphere that occurs at a tip-wall separation of 1.75σ. The
larger sphere shows a maximum at this point, while the smaller sphere shows a much
smoother curve. At this separation, all solvent should be excluded from the region
immediately under the tip. The force on the tip should be very dependent on the ability of the solvent particles to penetrate near this forbidden region, and it may be that
particles are more mobile under the “flatter” 5σ sphere. We have looked at the density
profiles for these two spheres, and it seems that the particles penetrate further under
the larger sphere at this separation. The fact that the particles are mostly excluded
from the central volume makes it impossible to look at velocity autocorrelation functions at these separations, since it would be extremely difficult to accumulate enough
data due to the lack of liquid in the region of interest.
1.2 Molecular Dynamics Simulations
Figure 1.11: Representative molecular dynamics configurations from the
3σ (top) and 5σ (bottom) diameter sphere simulations, both at a tip-wall
separation of 1.75σ. The viewpoint is from the centre of the bottom surface, looking up along the z axis. In other words, the topmost layer of
visible atoms in each configuration is the first solvation layer alongside
the flat surface.
19
20
Studies of Atomic Force Microscopy
this page intentionally left blank
1.2 Molecular Dynamics Simulations
Figure 1.12: Representative molecular dynamics configuration from the
11σ diameter sphere simulation, at a tip-wall separation of 1.75σ.
21
22
Studies of Atomic Force Microscopy
this page intentionally left blank
1.3 Integral Equation Methods
23
1.3 Integral Equation Methods
1.3.1 Theory
As we have seen, molecular dynamics techniques successfully predict the shape of
the tip-wall force curve and yield density profiles which can be sensibly explained
in terms of this force curve. Molecular dynamics could be used to study this system
at many different state points (although there are certain problems in characterising
them) and for many different sizes of tip. The only difficulty with pursuing this line of
investigation is that MD calculations are very time consuming; to complete the 11σ
sphere study would take on the order of two hundred CPU hours on the DAP computer, which is expensive for a solvation force curve. Generating many such curves
would be prohibitively expensive. To obtain a more attractive method of generating
these curves we turn to approximate theories.
Integral equation methods allow us to (approximately) calculate the pair distribution functions for homogeneous and inhomogeneous liquids and mixtures of liquids.
There are several different integral equation theories; the one that we will use is based
on the use of the Ornstein-Zernike [20] (OZ) relation:
h (r1 , r2 ) = c (r1 , r2 ) + ρ3
h (r1 , r3 ) c (r3 , r2 ) dr3 .
(1.9)
For a homogeneous liquid governed by an isotropic pair potential, h (r 1 , r2 ) and
c (r1 , r2 ) are functions of r = |r1 − r2 | and the OZ relation becomes
h(r) = c(r) + ρ
h(r0 )c(|r − r0 |)d 3 r0 .
(1.10)
This equation may be regarded as a definition of the function c(r) and so is exact.
The function c(r) is called the direct correlation function and may be interpreted
as the part of the total correlation function h(r) that arises from direct interactions
between particles 1 and 3, and 2 and 3. h(r) itself contains all (direct and indirect)
correlations between particles 1 and 2 that arise from the entire structure of the liquid.
Because the OZ relation is open, we must apply an approximate “closure relation” to
obtain a system of two non-linear equations in two unknowns, which is (numerically)
solvable. There are several such closures available to us; all of them may be expressed
as different truncations of the diagrammatic expansions for h(r) and c(r). (For a
discussion of diagrammatic expansions, refer to Hansen and McDonald, Theory of
Simple Liquids [19]). The closure relations that we used in this work are the PercusYevick [21, 22] (PY) relation (here written for an isotropic homogeneous fluid)
c(r) = (exp(βV (r)) − 1) (h(r) + 1),
(1.11)
24
Studies of Atomic Force Microscopy
where V (r) is the intermolecular potential, the “Hypernetted Chain” (HNC) closure
[23–27],
c(r) = exp (t(r) − βV (r)) − t(r) − 1
(1.12)
(t(r) = h(r) − c(r))
(1.13)
(it will prove convenient to work in terms of c and t rather than c and h) and the
Soft-core Mean Spherical Approximation (SMSA) closure [28, 29], given by
c(r) = exp −βV1 (r) (1 + t(r) − βV2 (r)) − t(r) − 1,
(1.14)
where we have divided the potential into a short-ranged repulsive part V1 (r) and a
long-ranged attractive part V2 (r). There are several other closure relations available,
some of which are interpolations between two of these closures. Examples are the
Rogers-Young (RY) closure [30], which mixes the HNC and PY closures, and the
Zerah-Hansen (ZH) closure [31], which mixes the HNC and SMSA closures. The ZH
and RY closures are so far the only closures developed which yield thermodynamically consistent correlation functions so that the pressure route to the equation of state
and the compressibility route to the equation of state give the same answer. However,
this is only achieved by an empirical optimisation of the parameter that controls the
way the two closures are mixed, and different values of the parameter are required for
different state points. Other attempts at improving these approximations often resort
to including diagrammatic terms from the (nearly exactly solvable) hard-sphere liquid, as in the RHNC closure of Foiles, Ashcroft and Reatto [32]. There are still other
formulations of these theories, such as the Born-Green integral equations [33, 34] and
the Born-Green-Yvon equations [35], but none of these have any particular advantage
over those already described.
There are a number of possible approaches to applying these equations to our
problem. One would be to follow the example of Kjellander and Sarman [36,37], who
very successfully treated the infinite planar slit system via an anisotropic solution to
these equations. Applying their technique here would be difficult because we would
have to discretise over two dimensions, while they needed only one. A somewhat
simpler approach to the same problem is given by Attard et al [38].
Another method, and the one we will use, is an adaptation of the method used by
Henderson and Plischke [39] to study a colloidal suspension in a Lennard-Jones fluid.
We regard the system as an isotropic trinary mixture, where the density of two of the
species approaches zero, and one of these species approaches infinite size (giving a
wall). Specifically, we would have a liquid with three species present: species 1 (the
fluid) of mole fraction X1 = 1, species 2 (the tip) of finite radius R and mole fraction
1.3 Integral Equation Methods
25
X2 → 0 and species 3 (the wall) of infinite radius, and mole fraction X3 → 0. Since
all three of these species are spherical, we see that all pair correlation functions are
functions of r only. Thus, the numerical solution of this system will require discretisation of only one variable, and will be quite tractable. This method will allow us
to calculate force-distance curves, but not to generate the density plots obtained from
MD simulations without making further (large) approximations. Specifically, to make
such a plot we would need to know the function g123 (|r2 − r3 |, r1 ), which is only approximately calculable given the three gi j (r) functions. We must now solve the OZ
equation for an isotropic mixture. In general, for a mixture of N different species this
is
N
(1.15)
hi j (r) = ci j (r) + ρ ∑ Xk hik (r0 )ck j |r − r0 | d 3 r0 ,
k=1
where i and j index the different species. For the system described above, this reduces
to
h11 (r) = c11 (r) + ρ
h11 (r0 )c11 (|r − r0 |)d 3 r0 ,
(1.16)
h12 (r) = c12 (r) + ρ
h11 (r0 )c12 (|r − r0 |)d 3 r0 ,
(1.17)
h13 (r) = c13 (r) + ρ
h11 (r0 )c13 (|r − r0 |)d 3 r0 ,
(1.18)
h23 (r) = c23 (r) + ρ
h21 (r0 )c13 (|r − r0 |)d 3 r0 .
(1.19)
(There are, of course, relations for the 22 and 33 functions, and equivalent relations
for the 21, 31, and 32 functions, which are identical with the 12, 13 and 23 functions,
but they are of no interest at present.) Our goal is to obtain the 23 functions, since
from h23 (r) we may obtain the force curve between the wall and the sphere. To solve
this system of equations we must apply an approximate closure to each of these four
relations. The choice of these closures is somewhat arbitrary, and we expect to see
different behaviours for different closures.
To solve these equations, we must first obtain the 11 functions (by solving the
integral equations for the pure Lennard-Jones fluid) which we can then use as input
into the equations for the 12 and 13 functions. The 23 functions can be obtained by a
single integration once we have the 12 and 13 functions. In order to solve the 11 and
12 equations, we first apply a fourier transform to the OZ relation, which yields
ĥ11 (k) − ĉ11 (k) = ρĥ11 (k)ĉ11 (k),
ĥ12 (k) − ĉ12 (k) = ρĥ11 (k)ĉ12 (k),
(1.20)
(1.21)
which replaces the use of numerical integrations (an O(N 2 ) process) with fast fourier
transforms (an O(N log N) process). There are several numerical methods available
26
Studies of Atomic Force Microscopy
for solving integral equations [40]. The traditional method is called Picard iteration,
wherein, given a starting guess at the solution (h0 (r)), we use one of our two nonlinear equations to solve for c0 (r) and then use this function and the other equation to
construct h1 (r). We iterate this procedure until the functions are no longer changing,
that is, hn−1 (r) ' hn (r). Picard iteration is divergent for strongly coupled (in this case,
high density) equations, and must be forced to converge by mixing several previous
hi (r) in constructing each new function [41]. Alternatively, solving the integral equations can be cast as a root-finding problem in many dimensions. If we write the OZ
relation and a closure (say, the HNC closure) as functions F1 and F2 :
F1 (t, c) = exp (t(r) − βV (r)) − t(r) − c(r) − 1,
F2 (t, c) = ρ (tˆ(k) + ĉ(k)) ĉ(k) − tˆ(k),
(1.22)
(1.23)
then we see that we are looking for t and c which are common zeros of F1 and F2 ; if the
t and c functions are discretised over M points, then this is just a 2M-dimensional rootfinding problem. An efficient algorithm for finding the roots of non-linear equations
is the Newton-Raphson method [42], which is a multidimensional form of the familiar
Newton’s method of root-finding. For a vector x and a vector equation F (x) = 0, we
write
F (x + δx) = F (x) + J δx + O(δx2 ),
(1.24)
where J is the Jacobian of F . Then to find the root (to first order) we solve a linear
set of equations for δx:
J δx = −F (x).
(1.25)
This is a quite powerful method of root-finding; provided that our initial guess x is
within the radius of convergence of the method, by iterating the above equations we
get quadratic convergence towards the root. The only difficulty is that matrix inversion
is an expensive process for large dimensional x.
We solved the 11 and 12 relations and their corresponding closure relations using
Zerah’s method [43]. This is essentially a Newton-Raphson iteration where we use
a conjugate gradient procedure [44] to iteratively invert the matrix at each step. The
advantage of using the conjugate gradient procedure over more direct methods of
matrix inversion (like QR decomposition) is that it does not require actual calculation
or storage of the Jacobian matrix, and although it is only guaranteed to converge
within N iterations (where N is the dimensionality of the problem) it often converges
much faster.
When applying either of the above methods, it is important to have a reasonably
accurate first guess at the solution, or either method will diverge quickly. Since approximations to the 12 and 13 functions at the state points studied were not available,
1.3 Integral Equation Methods
27
we seeded the program with the low density (high temperature) limits of these functions (which are known analytically — see Hansen and McDonald) and gradually
reduced the temperature and increased the density, using the functions from the last
state point as the first approximations to the functions at each new state point. To
improve the efficiency of this method, after two or more state points had been solved,
we linearly extrapolated a first guess at the next one based on the previous two. This
procedure was found to be satisfactory for the entire range of state points studied, although at low temperatures or high densities only very small “moves” could be made
without over-stepping the radius of convergence of these methods. In future work it
may be advantageous to use a globally convergent (but less efficient) root-finding algorithm, such as the back-stepping Newton’s method suggested in Numerical Recipes.
The equations involving the wall cannot be solved using Zerah’s method, since
the infinite radius of one particle would require discretisation over an infinite range of
the variable of integration. To make them tractable, we follow Henderson, Abraham
and Barker [45] and change coordinates (via a bipolar coordinate transformation [46])
to the surface of the large sphere before allowing its radius to become infinite. This
results in the following form of the OZ relation for the 13 functions:
h13 (z) = c13 (z) + 2πρ
∞
0
z+t
dt tc11 (t)
z−t
ds h13 (s),
(1.26)
where all coordinates are zero at the surface of the infinite sphere. A similar equation
can be written for the 23 relation:
h23 (z) = c23 (z) + 2πρ
∞
0
z+t
dt tc21 (t)
z−t
ds h13 (s).
(1.27)
Although a Hankel transform could be used to avoid the integrations, in the work
that follows we solve these equations via Picard iteration and numerical integration
using the trapezoid rule. Picard iteration is slow to converge, and has a smaller radius
of convergence that the Newton-Raphson technique described previously, but it is
sufficient for our purposes.
As mentioned previously, we must choose approximate closure relations for each
of the four OZ relations. We need not use the same closure relations everywhere, and
we expect that some will be more appropriate for certain systems than others. We tried
several combinations of closures, and decided to use the SMSA closure for the liquidliquid 11 functions, and the HNC closure for the 12, 13 and 23 functions. The SMSA
closure is generally acknowledged to be very good at describing liquids with a longranged attractive potential, such as the Lennard-Jones fluid. In Figure 1.13 (page 28)
we show h11 (r) as generated with the three different closures described above. Al-
28
Studies of Atomic Force Microscopy
h11 Functions in the PY, HNC and SMSA Closures
2.0
HNC
PY
SMSA
Number Density
1.0
0.0
-1.0
0.0
1.0
2.0
3.0
Radius (σ)
Figure 1.13: h11 (r) as generated by using the HNC, PY, and SMSA closures in the approximate theory.
1.3 Integral Equation Methods
29
SMSA, HNC and MD Wall Functions - ρ13(z)
Number Density
4.0
MD
HNC
SMSA
3.0
2.0
1.0
0.0
0.0
1.0
2.0
3.0
z (σ)
Figure 1.14: SMSA, HNC and molecular dynamics results for the density
profile of the liquid against the wall.
though the different closures yield similar results for h11 (r), they do have different
peak heights, and differ in the exact location of their maxima and minima.
By comparison with simulation results, shown in Figure 1.14, we see that the
HNC closure applied to the 13 equations gives a better fit (especially at longer range)
to the simulation results for the density profile of the liquid near the wall (and away
from the sphere) than does the SMSA closure. Also, by applying the HNC closure to
the 23 functions, we may eliminate the actual calculation of h23 (r) and c23 (r) from
our solution by combining Equation 1.1 with the HNC closure itself, yielding
∂
log (h23 (r) + 1)
∂r
∂
=
log (exp (t23 (r) − βV23 (r)))
∂r
∂
t23 (r).
=
∂r
βF23 (r) =
(1.28)
Applying this shortcut was found to reduce numerical errors in the determination
of F23 (r) considerably. The PY closure was found to yield very poor results when
applied to the 12 or 13 functions, and was rejected.
30
Studies of Atomic Force Microscopy
Molecular Dynamics vs. Theory, 5σ Sphere
Total Force on Sphere
100.0
Theory
MD
50.0
0.0
-50.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.15: Integral equation prediction of the force curve for the 5σ
diameter sphere compared to molecular dynamics results.
Molecular Dynamics vs. Theory, 3σ Sphere
Total Force on Sphere
40.0
Theory
MD
20.0
0.0
-20.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.16: Integral equation prediction of the force curve for the 3σ
diameter sphere compared to molecular dynamics results.
1.3 Integral Equation Methods
31
1.3.2 Integral Equation Results
In solving these equations, all functions were discretised over a grid of 0.0469σ step
size. The 11 functions were obtained between r = 0 and r = 12σ, the 12 functions
were obtained between r = 0 and r = 24σ and the 13 functions were obtained between r = −12σ and r = 12σ, following Henderson and Plischke. We observed no
quantitative differences between using grids of size ∼ 0.10σ and 0.0469σ, and are
confident that these solutions are accurate. The Lennard-Jones potential in these calculations was cut off at 12σ. All of the following results are quoted in reduced units,
omitting the (∗ ) notation. The numerical solutions were obtained on a Silicon Graphics R3000 Indigo workstation. As hoped for, these results were not computationally
expensive; solving for the 11 and 12 functions (starting from the low density limit)
only took a few minutes of CPU time, and the Picard iterations required to get the
wall functions took five to ten times longer.
Figure 1.15 shows the force curve predicted by this theory for a sphere of diameter
5σ, (at T = 1.0, ρ = 0.73) compared with the simulation results. Temperature and
density of the “bulk” fluid are the same for the two curves. We see that the sharp
peak at 1.75σ is absent, and that the first minimum is somewhat too deep. For larger
separations the agreement is better.
Figure 1.16 shows the same comparison for the 3σ sphere. Again, for separations
greater than 2.5σ the agreement is quite good, while within this region we see that the
approximate method falls somewhat short. This failure is expected and is explained
by the fact that the approximate theory treats the liquid as a continuous density distribution, while it is within this very confined region that the discrete molecular nature
of the liquid becomes most important. Furthermore, one of the approximations inherent in our treatment of the system is that the 12 functions are not changed by the
presence of the wall, and that the 13 functions are not changed by the presence of
the sphere. As we bring the sphere and wall closer together this becomes a more
and more severe approximation. In general, we can expect the approximate theory to
be qualitatively correct for all but the smallest separations, and to be quantitatively
correct for separations greater than about 2.5σ.
We first apply this theory to the comparison of different size spheres. Figure 1.17
(page 32) shows the force curves generated by the approximate theory, for a much
larger range of sphere sizes that we were able to simulate by MD. In Figure 1.18
(page 33) we plot the magnitudes of the first three force oscillations against the radius
of the sphere to show that the linear behaviour observed earlier holds (quite well!) for
the approximate theory. Again we see that the “range” of the oscillations is relatively
unaffected by the size of the sphere. Note that the lines in Figure 1.18 have positive
32
Studies of Atomic Force Microscopy
Force on Sphere (reduced units)
Force Curves for Different Size Spheres
150.0
2.5 σ radius
3.5 σ
4.5 σ
5.5 σ
6.5 σ
7.5 σ
100.0
50.0
0.0
-50.0
1.0
2.0
3.0
4.0
5.0
6.0
Separation (σ)
Figure 1.17: Integral equation results for the force curves for different
sized spheres, at a bulk liquid density of ρ = 0.73 and a reduced temperature of T = 1.0.
y-intercepts, supporting our earlier assertion that any size of probe brought near a
liquid-solid interface should experience oscillations in solvation force. (We cannot
include such a plot for our MD results because we have only two complete force
curves to draw data from, and so would necessarily see linear behaviour.)
Figure 1.19 shows the results of varying the temperature of the system; the bulk
fluid density is constant at ρ = 0.73 for these curves. We see that decreasing the
temperature increases the magnitude of the force oscillations, but has no effect on
their periodicity and only a small effect on their range. This temperature dependence
can be explained as follows. Since the force is a (negative) derivative of the free
energy A = E − T S, we may write
∂S ∂E
−
(1.29)
∂r ∂r
so that by decreasing the temperature, we decrease the entropic contribution to the
force, which acts in the opposite direction to the (dominant) energetic contribution to
the force. That is, energetically favourable (i.e., close-packed) liquid structures are
entropically unfavourable, and as we lower the temperature these effects are diminished. For much higher temperatures we might expect to see a change in the “phase”
of the oscillations as the entropic contribution to the free energy begins to outweigh
the energetic one. This would certainly only occur well above the critical point, in the
F(r) = T
1.3 Integral Equation Methods
33
Magnitude of Force Oscillations vs. Radius of Sphere
Force (reduced units)
100.0
First Oscillation
Second Oscillation
Third Oscillation
80.0
60.0
40.0
20.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Radius of Sphere
Figure 1.18: Magnitudes of the first three force oscillations versus radius
of the sphere, for different spheres, in the approximate theory at a bulk
liquid density of ρ = 0.73 and a reduced temperature of T = 1.0. A single
oscillation is defined as the difference between the height of a force maximum and the height of the following minimum as the tip-wall separation
is increased.
Force Curves for Different Temperatures
Force (reduced units)
150.0
T=0.60
T=0.70
T=0.80
T=0.90
T=1.00
100.0
50.0
0.0
-50.0
1.0
2.0
3.0
4.0
5.0
6.0
Separation (σ)
Figure 1.19: Effects of varying temperature on force curves, in the approximate theory, at constant liquid density of ρ = 0.73, for the 5σ diameter
sphere.
34
Studies of Atomic Force Microscopy
Force Curves for Different Liquid Densities
Force (reduced units)
60.0
ρ = 0.60
ρ = 0.70
ρ = 0.775
30.0
0.0
-30.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.20: Effects of varying density on force curves, in the approximate
theory, at constant temperature of T = 1.0, for the 5σ diameter sphere.
supercritical fluid region of the phase diagram.
Figure 1.20 shows the results of varying the liquid density at constant temperature. We see that increasing the density both increases the magnitude of the force
oscillations and decreases their period; such behaviour is to be expected considering
our previous packing-oriented explanation of the force oscillations. As we increase
the density, we increase the packing fraction and thus decrease the average interparticle distance, which would account for the decreased period of the oscillations.
This is a situation where the approximate theory and MD calculations may very well
show somewhat different trends; this behaviour should be very sensitive to the approximation mentioned earlier, that the 12 functions are calculated independent of
the presence of the wall, etc., but we have not yet simulated different liquid densities
by molecular dynamics to see if this is true.
Figure 1.21 shows the decay rate of the normalised oscillations of the force curves
for spheres of diameter 5σ and 11σ as predicted by integral equation theory. We see
a simple exponential decay, with a slight deviation in the first point, which is the
difference between the first maxima and the first minima. Figure 1.21 also shows
the decay rate for the MD simulation of the 3σ diameter sphere system. This data
is not as clean as the approximate theory results, but within errors it too is linear.
Horn and Israelachvili have experimentally observed such behaviour, and it seems to
be generally characteristic of the fluid structure near a wall. These oscillations (and
1.3 Integral Equation Methods
35
Decay Rate of Force Oscillations for MD and Theory
Magnitude (reduced units)
100
5σ sphere (theory)
11σ sphere (theory)
3σ sphere (MD)
10
1
1.0
2.0
3.0
4.0
5.0
Oscillation Number
Figure 1.21: Decay rate of the force oscillations for both MD and approximate theory force curves.
their period and decay lengths) have been predicted using asymptotic theories and
density functional theories in the context of a liquid layer adsorbed at a wall [47, 48]
or between macroscopic particles [38, 49].
1.3.3 Integral Equation Results - Hard Walls
Recognising that our choice of potentials for the fluid-wall and fluid-sphere interactions is quite arbitrary, we have also used the approximate theory to obtain force
curves for a system composed of a Lennard-Jones fluid, a hard wall and a hard sphere.
That is, the potentials for the liquid-wall interaction and the liquid-sphere interaction
are
n
∞ z≤0
Vwall (z) =
,
(1.30)
0 z > 0
∞ r ≤ rsphere
Vsphere (r) =
.
(1.31)
0 r > rsphere
This is easily done; even when doing calculations on the “soft” wall or sphere one
must divide space into regions inside the surface and outside the surface (since the
potential is infinite at the surface) and treat them separately. To simulate hard walls,
we need only turn off the potentials themselves, so that the discontinuous surfaces
remain. Essentially, within the Steele wall or our tip model is a hard wall located at
36
Studies of Atomic Force Microscopy
h13(z) Functions for Steele and Hard Walls
5.0
Number Density
Steele Wall
Hard Wall
3.0
1.0
-1.0
0.0
1.0
2.0
3.0
4.0
z (σ)
Figure 1.22: h13 (r) functions for Steele potential wall and hard wall.
the centre of the outer layer of atoms, and we can do calculations on the hard wall
by itself. The same closures were used in these calculations as were used above,
although the discretisation was not as fine (∼ 0.10σ). For purposes of comparison,
these calculations were also done at conditions of T = 1.0 and ρ = 0.73. Figure 1.22
shows the h13 (r) functions for the Steele potential wall and the hard wall. We see
that the magnitude of the oscillations in the correlation function near the hard wall
is smaller, but that their frequency is the same. The hard wall peaks are also shifted
somewhat from the Steele potential peaks, but this is in part due to difficulties in
defining exactly where the hard wall (or the Steele wall) is.2
In Figure 1.23 we show the force curves between hard spheres of several different
radii and the hard wall. The most striking thing about these curves in comparison
with our previous results is that they are (almost) always negative; the tip is pulled
towards the wall by an oscillatory but increasing (in magnitude) force as it is brought
nearer. This can be explained by considering the energetics of a Lennard-Jones fluid
near a hard surface. A molecule very near the surface (within 1σ of contact) has
fewer than its usual twelve nearest neighbours because the wall excludes them (for a
close packed solid near a wall, it would have exactly nine neighbours) and so pays
2 Because
we are using a finite discretisation of the potential in these calculations, it is not possible
to specify the position of the (infinitely sharp) boundary of the hard wall to a precision greater than the
discretisation size!
1.3 Integral Equation Methods
37
Force Curves for Hard Surfaces
Force (reduced units)
50.0
0.0
2.5 σ
3.5 σ
4.5 σ
5.5 σ
-50.0
-100.0
-150.0
1.0
2.0
3.0
4.0
5.0
Separation (σ)
Figure 1.23: Force-distance curves for the hard-wall/hard-sphere system,
for several different sizes of sphere.
an energetic penalty for being near the wall. We can expect to see “hydrophobic”
behaviour in the fluid because of this. In our hard-surface system, when we create
a highly constrained environment (under the tip) by moving the tip near the surface,
the liquid should evacuate this region, and the pressure of the liquid on the top of
the sphere should push it towards the surface. In a dense liquid the volume under
the tip will still be occupied and the packing behaviour mentioned earlier will still
be observed, but the oscillations will be superimposed on a net attractive force curve.
As one might expect from the smaller density oscillations near the hard wall, not
as many force oscillations are visible in the hard-wall/hard-sphere system as in the
Steele system. Figure 1.24 (page 38) shows the magnitudes of the force oscillations
as a function of sphere size for this system, and we see that even with hard surfaces
the force oscillations vary linearly with sphere size.
This kind of behaviour would be difficult to observe experimentally. In a physical
system with similar potentials, if we tried to slowly lower the tip towards the surface
it would be pulled towards contact as soon as it “sensed” the surface (with increasing
force) and only a single jump would be observed. The tip should jump from one
force-distance peak to the next (and we would see a stepped curve) but if the noise
in the system is of sufficient magnitude, (or if the fluctuations in the liquid phase are
large) these individual jumps would not be detected. Likewise, if we tried to slowly
38
Studies of Atomic Force Microscopy
Magnitude of Force Oscillations vs. Radius of Hard Sphere
Force (reduced units)
40.0
30.0
first oscillation
second oscillation
third oscillation
20.0
10.0
0.0
2.0
3.0
4.0
5.0
6.0
Radius of Sphere (σ)
Figure 1.24: Magnitudes of the first three force oscillations for different
sphere sizes in the hard-wall/hard-sphere system. The definition of the
oscillation is the same as that used in Figure 1.18.
pull the tip away from contact with the surface, by applying enough force to remove it
a small distance, we would (without incredibly fine control) immediately pull it quite
far away from the surface. Thus, we could measure the adhesion force induced by
the liquid between the two solids, but not the oscillatory force at longer ranges. This
may explain the AFM experiments under water mentioned earlier. Since graphite
and silicon nitride (the tip material) are reasonably hydrophobic materials (at least,
the water-water interactions are much stronger than the water-surface interactions)
the water system is roughly similar to our hard-surface system, and so the oscillatory
force curve may still be there, but invisible to these measurements.
1.3.4 Integral Equation Results - “Weak” Walls
Hard walls are not physically realistic, except for very large-scale systems. We have
also done these calculations for a “weak-walled” system where the wall potential
and sphere potential are as they were for the MD simulations and the first group
of approximate theory calculations, but scaled down by a factor of ten. That is, the
potential well depth near the wall or sphere is only about 0.3ε rather than 3ε, so that in
this system the liquid-liquid interactions are stronger than the liquid-wall interactions.
1.3 Integral Equation Methods
39
Force (reduced units)
Force vs. Distance for ‘Weak Wall’ System
0.0
2.5 σ
3.5 σ
4.5 σ
5.5 σ
6.5 σ
-20.0
-40.0
-60.0
1.0
2.0
3.0
4.0
5.0
6.0
Separation (σ)
Figure 1.25: Force-distance curves for the “weak-wall” system for several
different sizes of sphere.
These calculations were performed using the same parameters used in the hard-wall
calculations. The results for several different radii spheres are shown in Figure 1.25.
The results are quite similar to those of the hard-wall system; the force curve
is principally negative, and the same number of oscillations are visible. Comparing
Figure 1.25 with Figure 1.23 (the hard-wall force curves) we see that the weak-wall
curves are shifted up from the hard-wall ones, and that the oscillations are of very
similar magnitude. Essentially, the weak-wall force curves interpolate between the
hard-wall curves and our original system’s force curves. Such comparisons allow
us to resolve the force curve into a sum of two parts; an oscillatory force that arises
entirely from the liquid structure and a monotonically increasing force that arises from
surface-liquid interactions. The stronger the surface-liquid interactions, the more the
force curve is shifted up at small separations, since for a deeper potential well the
system should favour a larger tip-wall separation because more liquid could occupy
the well. A continuum-liquid model should show a force curve that increases with
decreasing separation (up to some contact value), with magnitude dependent on the
surface-liquid interaction strength.
40
Studies of Atomic Force Microscopy
1.4 Comparison With Experiment
We may compare the results of both MD simulations and integral equation theory with
the results of the experiments mentioned earlier. Looking at the AFM experimental
force curve in Figure 1.1, we see five or six oscillations superimposed on a net attractive force. We cannot really make judgements as to the decay rate of these oscillations
since the spread in the data is quite large. If we attempt to scale our reduced LennardJones parameters to the OMCTS system, from the heat of vaporisation of OMCTS
∆Hvap = 48.179 kJ/mol [50] and taking ∆Hvap /molecule = 8.61ε (the lattice energy
for the perfect fcc Lennard-Jones crystal [51]) we get ε = 673.3K = 9.29 × 10 −21 J
and σ ' 8Å. From the MD simulation of the 5σ sphere system, our first oscillation
(the difference between the second maxima and the first minima) is approximately
31.5σ/ε which becomes 3.65 × 10−10 N or 0.365nN. This is very nearly the same
size oscillation as seen in the experiment, and certainly of the correct order of magnitude. If we convert back to reduced units, we find that the AFM measurements were
performed at the state point T = 0.44 and ρ = 0.994, (which is a cooler, denser fluid
than ours) so that we should not expect quantitatively similar force data. The close
agreement found is probably coincidental.
If we look at the force curves of Horn and Israelachvili, we see as many as eight
discernible oscillations and a very well defined simple exponential decay, both features of which are qualitatively very similar to our simulated curves. Furthermore,
if we normalise our previous “measurement” of 0.365nN by the radius of the sphere
(2.5σ = 20Å) we get a value of ∆F/R = 0.183N/m which is approximately one order of magnitude greater than the experimentally observed oscillation of 10mN/m.
Of course, these are systems of dramatically different geometry, and we also expect
mica-OMCTS interactions to be different from graphite-OMCTS interactions, and so
are satisfied with only qualitative agreement in this case.
1.5 Conclusions
We have seen that by performing molecular dynamics simulations on a very simple
model of an atomic force microscope tip in a liquid we obtain force versus distance
curves that are in good qualitative agreement with experiment. Also, we can generate
such curves by the much less computationally expensive route of applying integral
equation theory to our model, and get reasonable agreement between these two methods. We have successfully explained these oscillatory force curves in terms of the
density profiles of the solvent in the region under the tip, and have demonstrated that
1.5 Conclusions
41
the observed liquid structures at various tip-wall separations correlate well with the
features in the force curve.
We have also observed the effect that tip size has on these force curves. The
magnitude of the force oscillations is found to vary linearly with the tip radius, and
the number of oscillations present in the force curve is not particularly sensitive to
tip size. This is because a larger tip does not have a much longer-ranged effect on
solvent structure than a smaller one. This linearity is not entirely surprising; Horn
and Israelachvili have observed it in SFA experiments. In those experiments this
dependence can be explained by assuming that each surface as seen by the other is
effectively flat and integrating over the surfaces. Here, the much larger curvature of
one of the surfaces means that replacing the surfaces by flat walls is a very severe
approximation.
We have also used the approximate theory to look at the force-distance behaviour
for different state points of the liquid. We find that increasing temperature causes a
decrease in the magnitude of the oscillations, which can be explained by considering
the entropic contributions to the free energy. Also, increasing the liquid density is
found to increase the magnitude of the oscillations, as well as slightly decrease their
period.
Applying the approximate theory to systems with different surface-liquid potentials has allowed us to see the way that liquid-surface interactions and liquid-liquid
interactions are separately related to the total force felt by the tip. The oscillations are
a result of the liquid structure and are mostly unaffected by varying the liquid-surface
potential well depth, while the curve upon which these oscillations are imposed is
very dependent on the liquid-surface potential. It may also be very dependent on the
liquid-liquid potential, but we have not yet studied systems with liquid-liquid potentials other than the Lennard-Jones, and the scaling properties of that potential make
considerations of absolute well depth inconsequential. Looking at a hard-surface system with a hard-sphere liquid would be another logical step, and would also allow us
to measure the purely entropic contributions to the force curve, since the behaviour of
such a system would be entirely determined by packing considerations.
In comparison with experiment, we point out that our observed force oscillations
are of the same order of magnitude as those seen by O’Shea, Welland, and Rayment (once translated from reduced units, of course) so that we would expect their
AFM tip to have an effective radius of curvature of about 40Å. The fact that we have
nearly quantitative agreement is surprising using an arbitrary choice of potentials and
tip size, and should not be taken too seriously. We also have reasonable qualitative
agreement with the results of Horn and Israelachvili. Our hard-surface simulations
42
Studies of Atomic Force Microscopy
provide a possible explanation for the failure of the AFM experiments to detect any
structure at the water-graphite interface.
As with many systems of interest, this study has provided many new questions to
answer and suggested many other systems to simulate. At this point, the two most
interesting directions to explore are (i) changing the tip geometry and (ii) looking at
a molecular solvent. It may happen that a sharper tip (conical geometry) may yield a
much different force curve, and also a different size/force dependence. This system
can probably be studied using the approximate theory, by making some approximations to yield new h12 (r) functions for odd-geometry colloidal particles. Simulations
of molecular solvents could also be quite interesting. O’Shea, Welland and Rayment have performed their measurements on 1-Dodecanol as well [2], and observed a
stepped force curve, indicating a liquid-crystal–like layered structure on the surface.
Simulating a simple model such as dipolar rods could yield interesting results, especially since the liquid structure would be more dramatically influenced by the tip
because the molecules would experience orientational packing problems. Simulations
on water in this system have been suggested; the open structure of liquid water should
be very interesting in the partially confined environment near and under the tip, and it
would be a test of our hypothesis that hydrophobicity (rather than liquid structure) is
the cause of the jumps observed in AFM experiments.
Part II
STUDIES OF SURFACE PHASE
TRANSITIONS
Chapter 2
INTRODUCTION
2.1 Classification of Surface Phase Transitions
Before we begin the study on surface phase transitions, we will review the nature and
expected behaviour of these systems and define the terms that we will use to describe
them.
A phase transition occurs where there is a singularity in the free energy or one of its derivatives. What is often visible is a sharp change in
the properties of a substance. The transitions from liquid to gas, from a
normal conductor to a superconductor, or from paramagnet to ferromagnet are common examples.
J. M. Yeomans, Statistical Mechanics of Phase Transitions [52]
Phase transitions are classified by the type of this singularity. Before modern
critical theory, phase transitions were classified using the method of Ehrenfest [53].
Ehrenfest classified transitions where there was a discontinuity in the first derivative
of the free energy as first-order. In a first-order phase transition the free energy is a
continuous function that has a kink at the transition point, so that the first (and higher)
derivatives are discontinuous. Examples of first-order phase transitions are melting,
freezing, vaporisation and the transition between different crystal structures of a solid.
In the same way, transitions where the first derivative is continuous but the second
derivative is discontinuous are called second-order. In a second-order transition, there
is a kink in the first derivative of the free energy. Typical examples of a second-order
transition are the critical vaporisation of a liquid and the phase transition of an Ising
model. Third-order phase transitions are identified by a discontinuity in the third
derivative of the free energy, and so on.
Fisher showed that this was an inappropriate scheme of classification, because in
these higher order phase transitions what identifies the transition is a divergence in
45
46
Introduction
one or more of the derivatives, rather than a discontinuity [52]. Keeping this in mind,
although first-order transitions continue to be classified as “first-order”, higher order
transitions are nowadays best termed continuous, though they are often called critical
transitions. Also, the term second-order transition still appears quite often, and is
taken to be synonymous with “continuous transition”. In this work we will use the
modern nomenclature.
It is also possible for a system to change smoothly, without any divergences at all,
between two distinct “states”. While this is not a phase transition, it is a transition
in the non-thermodynamic sense, and we will refer to these situations as “smooth
transitions”. An example is the deformation of a crystal lattice with a change in
temperature.
Wetting phenomena occur when in a system of many phases one entirely covers an
interface between some of the others. For instance, in a closed container containing
a vapour, some of the vapour may condense to form a liquid which adheres to the
container. In this case, we would say that the liquid wets the container. If two partially
miscible liquids a and b are in a container, one of the two phases (say, the a-rich
phase) might wet the wall of the container. In addition, in an open container, one of
the phases might cover the surface of the liquid; we would then say that the surface is
wet by this phase1 . After all, the atmosphere above the liquid is a third phase, and the
surface is a liquid-vapour interface. If a crystal melts from its surface inwards, then
near the melting point the crystal-vapour interface is wet by the liquid phase.
There are several different types of wetting, some of which go by several different
names. Complete wetting, which is the same as perfect wetting, occurs when the layer
of material wetting the interface is infinitely thick in an infinite system. In the case of
single-component adsorption, this just means that the thickness of the adsorbed layer
is macroscopic. In the case of wetting by one component of a binary mixture, the
thickness of the layer of the a (for example) component between the wall and the b
component is macroscopically thick. These cases are distinct from partial wetting,
incomplete wetting, and pseudowetting. Partial wetting is the term given to a threephase (or three-component) system where γac < γab + γbc , where γ is a surface tension,
which is equal to the partial derivative of the energy of the system with respect to the
surface area A; γ = (∂U /∂A)N,V,T . Phase c can either be a third liquid phase, or a solid
wall. In this case, both the a and b phases (or components) will be in contact with
the wall, which is partially wet by each of them [54]. Incomplete wetting describes
a case where the thickness of the wetting layer is finite. Pseudowetting is a term
1 This is
only true if the densities of the two phases are equal, so that we are not observing a gravitydriven process.
2.1 Classification of Surface Phase Transitions
47
used to describe cases where the thickness of the wetting layer appears to increase
divergently, but the divergence terminates at some maximum thickness [55]. This
occurs for some systems in which the free energy as a function of fluid thickness
(γ(l)) has a significant oscillatory character.
In an adsorption system, where a vapour phase is in contact with a wall, the quantity of interest is the surface excess density (nex ) which is the amount of material adsorbed onto the wall in excess of the vapour density in the same volume. A first-order
wetting transition occurs when at a certain pressure the value of n ex jumps discontinuously from a finite value to an infinite value, indicating that the surface is wet by a
macroscopically thick layer of the adsorbed material. In contrast, a critical wetting
transition occurs when the value of nex changes from a finite value to an infinite value
via a continuous divergence as the temperature is increased, so that at any intermediate temperature up to the wetting temperature TW , the surface excess thermodynamic
quantities are well-defined [56]. In other words, the liquid layer grows faster and
faster, but there are never two different states (thin and thick layers) in coexistence.
Another term often used in this field is critical point wetting, which was first
described by Cahn [57]. This describes a situation in which a two-component mixture
(liquid) is in contact with a third (solid) phase. For materials where neither of the
liquid components completely wets the solid, Cahn showed that for temperatures near
(but slightly below) the critical point of the mixture, one of the near-critical phases
must completely wet the third phase. This is also true if the third phase is a liquid,
which makes the problem entirely symmetrical. These systems display a complex
wetting behaviour, where (upon cooling) critical wetting will end at some temperature
below the critical point in a first-order transition to a system where both phases are in
contact with the third.
A prewetting transition occurs when the thickness of the adsorbed layer (or n ex )
increases discontinuously from a finite value to another finite value; in this case, as
the system is heated through the prewetting temperature a transition still takes place,
but the wall is only incompletely wet above this point.
Premelting is a term used to describe solid surfaces that are in some way partially
melted, so that there is a disordered layer (which may not be “liquid”) at their surface.
Premelting phenomena include surface disordering and surface melting.
The surface roughening transition is distinct from the different kinds of wetting
transitions. At the surface roughening temperature, the free-energy cost of creating a
surface-step vanishes, which leads to unconstrained fluctuations in the height of the
surface [58]. This is signalled by a divergence at long ranges of the height-height correlation function [59]. Surface roughening occurs in directions parallel to the surface,
48
Introduction
and is not described well by theories concerned with the density profile perpendicular
to the surface. Surface roughening strongly influences surface phase behaviour because at temperatures below the roughening temperature (TR ) layer-wise transitions
are possible; at temperatures about TR the surface becomes corrugated due to this
roughening, so that phase transitions in layers parallel to the surface do not occur
since the layers are no longer well-defined. The roughening transition is technically
an infinite-order transition because all of the derivatives of the free energy vanish at
the transition point [60]. There is no heat capacity feature visible at TR , but there
should be a broad peak in the heat capacity at temperatures slightly below TR [58].
Pandit, Schick and Wortis [61] have reviewed the different kinds of phase behaviour that can occur for the Ising lattice gas in contact with an attractive substrate,
and classify the resulting transitions by the relative positions of the roughening temperature TR and the wetting temperature TW and the strength of the (short-ranged)
adsorbate-substrate interaction. For very strong substrate fields, at liquid-vapour coexistence we expect to see complete wetting. The buildup to this macroscopically
thick layer of adsorbed material begins with a series of discrete layering transitions,
where the film gets thicker by one layer in a discontinuous fashion. If the temperature is raised past the roughening temperature TR , the surface of the film is no longer
“clean”, so that layer-wise transitions are not possible, and the film grows in thickness smoothly. For strong substrate fields, there is no wetting temperature; at any
temperature at liquid-vapour coexistence the adsorbed film will be infinitely thick.
Substrates of intermediate strength give rise to a wetting temperature TW . Above this
temperature, the surface will be completely wet by the lattice gas, while below it, only
incomplete wetting can occur. The relative positions of TW and TR determine whether
or not the buildup of the liquid layer occurs in a step-wise or continuous way. Weakly
attractive substrates have no wetting temperature, so that the surface film will be finite
in thickness at all temperatures. In fact, for sufficiently weak substrates, the lattice gas
dries the surface, so that for values of the pressure where the liquid phase is thermodynamically stable, the liquid–substrate interface will be wet by a layer of the vapour
phase. This system is symmetric with the strong-substrate system, except that the liquid and vapour phases have been exchanged. Of course, the Ising lattice gas has only
two phases; in a system with liquid, vapour and also solid phases the adsorption behaviour can be a great deal more complex, since the melting and sublimation lines of
the adsorbate can intersect the adsorption layer transition lines, leading to the growth
of films which are either entirely solid, entirely liquid, or a mixture of both. Because
there are several more “dimensions” in the parameter space than just the substrate
strength in these systems, full characterisations are difficult. Some systems have been
2.2 Review of Experimental Work
49
extensively studied, such as the six-state Potts model [62, 63].
A crystal surface in equilibrium with its own vapour is essentially a special case
of these wetting systems. We can think of this situation as a substrate (the crystal)
being wet or dried by its own liquid and vapour. These systems generally fall into the
“intermediate substrate” region, since the interaction between vapour particles is the
same as that between vapour and substrate particles. Because the substrate itself may
undergo phase transitions, the phase diagram becomes more complicated. The most
studied transition of crystal surfaces is surface melting, in which a liquid layer grows
at the surface of the solid as it is brought near its bulk melting temperature. This
growth can be either complete or incomplete, in analogy with wetting transitions.
Surface melting can only occur when the condition γsl + γlv < γsv is satisfied, so that
the free energy change for introducing a liquid layer between the solid and vapour
phases is negative. This condition is balanced by the (positive) free energy cost of
melting some of the solid at a temperature below the triple point. It is the interplay
between these two quantities that determines the thickness of the melted layer.
2.2 Review of Experimental Work
The first observation of surface melting was made by Faraday [64] in 1860, who
determined that ice blocks froze together (regelation) because ice was covered in
a very thin layer of liquid water which itself froze when two pieces of ice were
brought into contact, acting like glue. Since then, an enormous number of experimental techniques have been brought to bear on the surface melting of ice, including
X-ray scattering [65], measurements of the conductance of ice/metal interfaces [66],
measurement of the thermal expansion of ice surfaces [67], quasi-elastic neutron scattering [68], optical reflection measurements [69, 70], adsorption isotherms [71, 72],
NMR [73–75], photo-emission spectroscopy [76], proton backscattering [77], and ellipsometric techniques [78, 79]. All of these measurements have confirmed that at
temperatures above about −30◦ C a liquid-like layer is present at the surface of ice,
and with increasing temperature the thickness of this layer increases logarithmically;
that is, as − log(TM − T ), where TM is the melting temperature of ice. Some experiments [70] have determined that the surface melting of ice is incomplete, and also that
this behaviour is strongly crystal-face dependent; some exposed lattice planes do not
surface-melt, while others do [70, 78]. Ellipsometric measurements [79] have pointed
to roughening of the solid-liquid interface at −2◦ C for some ice surfaces.
The second-most popular system for studies of melting and roughening of molecular crystals is that of the inert gases adsorbed onto graphite, and occasionally onto
50
Introduction
magnesium oxide (MgO). Thick, smooth films of argon, particularly, can be deposited
on graphite and then studied by a variety of methods. This allows the determination
of the behaviour of argon crystal surfaces. Some studies [80–82] have also focussed
on thin adsorbed layers, where the substrate effects are very important. The methods
of choice for studying these systems are quasi-elastic neutron scattering [83, 84], adsorption isotherm measurements [85], and calorimetry [86, 87], though high-energy
electron-diffraction [88], NMR [89], X-ray scattering [85], and ellipsometry [90] have
also been used. The evolution from thin-layer to thick-layer behaviour has been extensively studied for both krypton [91] and argon [85, 87]. Phillips, Zhang and Larese
have investigated the disappearance and re-appearance of vertical steps in the adsorption isotherms [90, 92] and determined that this behaviour is caused by layer freezing
transitions occurring near the top of the layer; if the second-to-top layer is disordered,
increased coverage can cause it to freeze into a solid which causes a discontinuity
in the heat capacity. Argon and krypton are the most studied elements in this context because neon (and many more complex molecules) does not completely wet the
graphite surface, so that thick neon films cannot be prepared on graphite under any
conditions [88]. In general, the inert gases only partially wet metal surfaces [93, 94].
This has been exploited by Maruyama [95], who grew small crystals of krypton and
xenon on a copper substrate and measured their surface melting with a microscope.
Methane adsorbed onto MgO has been extensively studied [96]. Some studies
have focussed on monolayer coverages [97], but many have considered thicker films
and found evidence for premelting phenomena [98,99] and surface melting and freezing [100]. Coulomb et al [101] have studied the evolution from thin layers to thick
layers in this system, and found that it behaves much like inert gas systems. Methane
on graphite has been less extensively studied [102, 103] and has also been found to
surface melt. These studies have measured the compression of the first and second
adlayers and have determined that the first two layers are slightly incommensurate in
two and three layer films.
The surfaces of only a few other non-metallic substances besides ice have been
studied using single crystal techniques, due to the difficulty in preparing the samples
involved. These studies include observation of a prewetting transition in the surface of
caprolactam [104] and characterisation of the roughening transition of n-paraffin crystals [105]. Many other molecules besides those already mentioned have been studied
in adsorbed systems; these include OMCTS [106] which is found to undergo at least
one layering transition near room temperature, ethane and carbon tetrafluoride on
graphite [107] and ethylene monolayers on graphite [108–110], though these are usually approached as a model system for 2D melting. Nham and Hess [111] have mea-
2.3 Review of Molecular Simulation
51
sured the layer critical points of ethane on graphite using ellipsometry, and neutron
scattering has been used to probe the structure of thick films of HD [112,113]. Oxygen
completely wets graphite, and these films have been found to surface melt [114–116].
Crystalline metal surfaces are qualitatively different from molecular surfaces, and
have been extensively studied. Although the majority of metal surface studies have
been focussed on reconstructions of the surface, many have considered thermal roughening and surface melting as well. The most studied metal surface in this context is
the (110) face of lead. This has been found to surface melt [117, 118], and has also
been characterised by its direction-dependent diffusion constants [119] and its thermal
expansion [120]. These studies have indicated that the growth of the surface melted
layer near TM first proceeds logarithmically, as in the case of molecular crystals, but
then crosses over to a power-law form. This crossover has been predicted for surface
melting of materials with long-ranged potentials [121]. Some other crystal faces of
lead have also been studied [122], and lead exhibits strong surface anisotropy: the
(111) and (100) faces remain ordered and stable up to the triple point, while the (110)
and most higher-index faces all disorder and surface melt. Many other metal surfaces
have been examined for surface melting and roughening, including Ag(115) [123],
Ag(110) [124], and surfaces of aluminium [121] and copper [125]. Because of the
high temperatures involved, calorimetry and many other techniques are not easily
applicable to these surfaces. The large masses of metal atoms permit studies by ionblocking and shadowing, and the neutron scattering techniques used to study many
molecular systems are also used on metal surfaces.
There have been many excellent review articles written about surface melting and
crystal growth; see, for instance, van der Veen, Pluis and Denier van der Gon [126],
van der Veen and Frenken [127], Weeks and Gilmer [60], and Dash [128].
2.3 Review of Molecular Simulation
The definitive study of the solid surface and solid-liquid interface of the LennardJones crystal is the series of six papers of Broughton and Gilmer [129–134]. These
consider all aspects of the crystal interfaces of a potential like the Lennard-Jones potential, but slightly modified so that it has no long tail. (This modification is done not
by cutting and shifting the potential, but by adding an extra term on to it at longer
ranges that makes it die off faster. The resulting potential has (practically) the same
well depth of the Lennard-Jones form, but decays to zero at 2.5σ and is zero further away.) The surfaces are simulated in a two-faced slab geometry with periodic
boundary conditions, and thermodynamic data are extracted from the simulations us-
52
Introduction
ing a variety of methods. As explained in [132], these simulations are not capable
of describing such phenomena as surface melting (either complete or incomplete),
roughening or faceting.
A contemporary study by Rosato, Ciccotti and Pontikis [135] of the (110) face
of the Lennard-Jones crystal using somewhat different potential parameters (the full
Lennard-Jones potential) and considerably larger system sizes found similar results.
These authors also explicitly consider the effect of the finite size (in terms of the
surface-to-volume energy ratio) of the simulation, asserting that their own simulations
are of sufficient size that these effects can be neglected. Both of these sets of simulations extend to temperatures within about 4% of the triple point, but no higher. Rosato,
Ciccotti and Pontikis conclude that the (110) face of the Lennard-Jones crystal does
not show any appreciable melting behaviour near the triple point, and interpret the
disorder found at the surface as roughening. Although Broughton and Gilmer do not
observe surface melting in their simulations, they do feel that temperature trends in
their results indicate that this should happen at high enough temperatures. Rosato, Ciccotti and Pontikis also study the concentration of adatom-vacancy pairs and defects
formed in their simulations and identify a temperature for the roughening transition
of 0.8TM , in good agreement with the experimental results of Zhu and Dash [87].
Lynden-Bell [136] simulated the disordering of surfaces of adsorbed multilayers of Lennard-Jones particles. In these simulations of four-layer adsorbed films the
surfaces were always found to disorder into liquid-like layers smoothly, so that no
roughening or premelting transitions occurred. The (111) face of the crystal disorders
much more sharply than do the (100) or (110) faces. The second layer of each system
disorders more smoothly than does the first. The substrate-particle interactions were
modelled using a 10-4 potential, which falls into Pandit and Wortis’s “strong” substrate category. There have been many other simulations of Lennard-Jones particles
adsorbed onto both smooth and corrugated substrates. Attempts at using the Monte
Carlo method in the Grand Canonical Ensemble [137] have proved difficult for coverages greater than one monolayer [138], although by greatly increasing the number
of trial creation and destruction moves this difficulty has been overcome [139]. Most
calculations of adsorption phase diagrams have simply performed many different MC
or MD runs at different fractional coverages [140–142]. These calculations have revealed yet another layer of complexity in these systems, measuring commensurateincommensurate transitions between the first and second adsorbed layers [143, 144]
due to substrate-induced compression of the first adlayer [145].
There have been relatively few simulations of the surfaces of molecular crystals,
probably because these simulations are more difficult and more expensive than those
2.3 Review of Molecular Simulation
53
of spherical particles, which are already quite expensive and difficult. Alavi and Chan¯ and (0001) faces of Lennard-Jones diatomic
davarkar [146] simulated the (1010)
¯ surface, molecules lie flat in their
molecules fit to nitrogen and ethane. In the (1010)
lattice planes, while in the (0001) surface they stand perpendicular to the surface,
in a hexagonal array. Alavi and Chandavarkar found that for the “ethane” system,
the close-packed (0001) face remained stable and orientationally ordered up to the
¯ face undergoes a surface melting (or at least
bulk melting transition, while the (1010)
extreme disordering) transition, so that the surface layer of the crystal loses all translational and orientational order just below the melting point. These calculations do
not make any prediction as to whether or not this surface completely melts, or only
incompletely melts. Lennard-Jones dimers fit to the shape of nitrogen molecules, on
the other hand, orientationally disorder at temperatures significantly below the melting point; near the melting point they behave much like the spherical Lennard-Jones
system.
Boutin, Rousseau and Fuchs [147, 148] have performed simulations of the surfaces of sulfur hexafluoride using a potential function consisting entirely of fluorinefluorine terms modelled by Lennard-Jones potentials. SF 6 has an orientationally disordered bcc crystal structure near its melting point, so that on the basis of Alavi and
Chandavarkar’s work, we would expect it to show surface behaviour much like that of
the spherical Lennard-Jones potential. Boutin, Rousseau and Fuchs have performed
the longest simulations on surface melting to date, with simulation lengths extending
past 1.6 × 106 time-steps. They observe surface melting behaviour in accord with
the predictions of the (mean-field) Landau theory of Lipowsky and Speth [149]. As
before, these simulations cannot measure the growth rate of the surface melted layer
or determine whether this surface melting is complete or incomplete. (Boutin and
Fuchs feel that the next generation of parallel computers will be sufficiently powerful to answer these questions [148].) Thus, these simulations show considerably
different results than those of Rosato, Ciccotti, and Pontikis. This potential for SF 6 ,
although composed of Lennard-Jones terms, is shorter-ranged (relative to the molecular diameter) than the spherical Lennard-Jones potential, which could explain the
greatly increased surface mobility and disorder found in these simulations. That is,
the weaker periodic field induced by sub-surface ordered layers allows for greater
disorder on the surface, so that melting behaviour occurs at lower temperatures.
Kroes [150] has used molecular dynamics to study the most famous of surface
melting systems, water. Using a modification of the TIP4P potential, Kroes simulated
the (0001) plane of the ice surface over the temperature range T = 190K − 250K, in
which the surface structure first disorders. These simulations show that for T > 230K
54
Introduction
the top layer of the crystal is liquid-like, but that the influence of the next lowest
layer causes the “liquid” molecules to point their protons down. Kroes calculated
the polarisation of the surface, self-diffusion coefficients and orientational correlation
functions, and found reasonable agreement with experimental NMR data. It is not
clear whether or not this behaviour constitutes a phase transition; because the temperature of the transition is not precisely located and heat capacity data are not available,
we cannot tell. Kroes performed additional simulations using different starting data
which indicate that at higher temperatures (where surface disordering is significant)
the relaxation times of these systems can be at least 100 ps, so that two runs can appear to have equilibrated but still have very different properties. This was observed
by Boutin and Fuchs, who found that the two surfaces in their system could exhibit
different behaviour in the same run, and also by Alavi and Chandavarkar, who found
that the (0001) face of their ethane crystal remained stable for 60000 time-steps at the
melting temperature, at which point the entire slab system melted abruptly. Broughton
and Gilmer predict that the free energy “driving force” in systems exhibiting surfacemelting will become smaller as the liquid layer thickness grows (this is supported by
our own calculations in Chapter 8) so that the equilibration time should grow accordingly. This bodes badly for any attempt at simulations of thick liquid layer systems,
regardless of the amount of computing power available.
There have also been some simulation studies done of the surface melting behaviour of finite systems. Weber and Stillinger performed MD calculations on ice
crystallites; (H2 O)250 . Rose and Berry [151] have studied the behaviour of (KCl)32 ,
and characterised it as displaying “nonwetting”, so that melted parts of the cluster
would clump together rather than coat the solid parts. Finite systems have also been
studied with phenomenological models [68, 152], and the important phenomenon of
frost heave is caused by these effects [128]. In general, curvature is found to lower
transition temperatures and accentuate disordered behaviour in both finite and adsorbed systems [152, 153].
2.4 Review of Theoretical Approaches
Many theoretical studies of surface phenomena approximate the free energy of a
model system as a simple function of any number of order parameters which are
chosen to describe the physically important aspects of the system. The order parameter generally chosen in magnetic systems such as the simple Ising model is the
average magnetisation M. In surface systems this can have a different value in each
layer parallel to the surface, so the system is described by a magnetisation profile
2.4 Review of Theoretical Approaches
55
M(z). In the corresponding lattice gas, the density per layer is the quantity of interest,
so we consider the free energy as a function of ρ(z). In lattice-model systems, these
quantities are discrete, and the system is described by a collection of ρ i , the density
in the ith layer. For continuum models, the density profile is a continuous function
which must be finely discretised in any computational study. “Higher-order” order
parameters may also be used, providing better approximations to a complete description of the system. For instance, using only the average density as an order parameter
cannot distiguish between solid and liquid phases. This corresponds to a truncation
of the k-space expansion of the density at 0th order. If the first-order term is retained,
(which complicates the theory) the order in the solid phase and the disorder in the
liquid phase is recovered.
One of the first studies to use the mean-field approximation to calculate the surface
phase behaviour of a lattice model is that of de Oliveira and Griffiths [154], who used
the simple lattice gas (isomorphic with the Ising model) to study adsorption on a
structureless substrate. De Oliveira and Griffiths found that material built up on the
surface in first-order layer-wise transitions at low temperatures. That is, at a fixed low
temperature, if the pressure (or chemical potential) was increased, the adsorbed layer
became thicker in discrete steps of one layer. At higher temperatures these first-order
transitions persisted for a while, and then the first-order lines ended in critical points
above which the buildup was smooth. De Oliveira and Griffiths predicted that this
sequence of critical points converges to the bulk phase critical point as the chemical
potential nears its coexistence value. Saam [155] has applied the Migdal-Kadanoff
renormalisation group technique [156, 157] to model the in-plane fluctuations of the
Ising lattice gas on a hexagonal lattice, with the out-of-plane interactions treated in the
mean-field approximation. This treatment indicates that the layering transitions are all
of the 2D Ising model universality class (not surprisingly!) and that the layer critical
points converge not to the critical temperature of the adsorbate, but to its roughening
temperature, as predicted by Pandit, Schick and Wortis [61]. The simple lattice gas
has since been studied in the context of adsorption by many others, using both Monte
Carlo methods [63,158–161] and mean-field approaches [63], as well as more realistic
potentials [162] and dynamical interpretations of Monte Carlo data [163].
Without a doubt, the most complete Monte Carlo studies of the surface behaviour
of the simple lattice gas have been done on the isomorphic Ising model of magnetism.
Binder and Landau [164, 165] have studied extremely large Ising systems influenced
by both surface and bulk applied fields, and have characterised wetting, layering and
roughening transitions, and the more exotic prewetting transitions which can be induced by variation of the surface fields. They have also measured scaling and critical
56
Introduction
behaviour in surface transitions [166], and used the lattice gas interpretation of the
Ising model to consider capillary condensation in pores [167].
In the simple lattice gas model, the number of available sites in each layer does
not depend on the number of occupied sites in the layer underneath, which distinguishes it from “Solid-On-Solid” models. SOS models are of great interest, because they accurately describe roughening transitions [168, 169], and under some circumstances can be solved analytically [170–173] or by renormalisation-group methods [155,174,175]. In any case, these models are essentially restrictions of the simple
lattice gas, so can also be treated with Monte Carlo methods [176] or in the mean-field
approximation [177]. Much of our knowledge about the roughening transition comes
from studies of these models.
The mean-field approach has been used to study planar interfaces in the q-state
Potts model in physisorbed systems [62,63] and to study the surface melting of 2-state
Potts model crystals [178]. A model isomorphic with the 3-state Potts model has been
used by Teraoka and co-workers to study surface segregation in binary alloys [179],
adsorbed layers [180] and the superheating of a crystalline surface [181].
Also using the mean-field approach, Dietrich and Schick [56] studied adsorption
using the simple lattice gas model but with long-ranged (that is, varying as r −3 ) forces,
concluding that for this potential critical wetting of the surface replaced the first-order
wetting observed in models using the Lennard-Jones potential [182]. In these calculations the excess adsorbed mass diverged as (TW − T )−1 , where TW is the wetting
temperature. This mass increased by a series of layer-wise transitions which the authors attributed to the lattice structure of the model.
Yashonath and Sarma [183] repeated the study of de Oliveira and Griffiths using a
different numerical technique, wherein an iterative process was used to determine the
total free energy as a function of the density profile, by determining self-consistent
density profiles at a given T and µ. This allowed for the identification of metastable
states in the system and the precise calculation of first-order transition temperatures.
Trayanov and Tosatti [184, 185] developed a more realistic lattice model of the
Lennard-Jones system. As in more primitive models, the order parameters used to
describe the surface are essentially the same as those used in Jayanthi’s study of the
2-state Potts model [178], with considerable improvements to the rest of the model.
Unlike most lattice models, particles interact through the full, un-truncated LennardJones potential. Also, the contribution of the “free volume” to the free energy has
been included explicitly. The free volume is the part of the partition function that
comes from the particles moving within their lattice cells; it is effectively ignored
in all simpler lattice gas models, yet is of similar magnitude to the internal energy.
2.4 Review of Theoretical Approaches
57
Trayanov and Tosatti explicitly perform this integral in the solid phase, and make an
approximation to it in the liquid phase. Partially ordered configurations are handled
by an interpolation recipe based on the theory of freezing developed by Ramakrishnan
and Yussouff [186, 187]. This results in a phase diagram which, although not entirely
satisfying, is improved over those of more primitive models. This model is then
used to study surface melting behaviour near the triple point, with the result that the
(100) and (110) faces of the crystal both show smooth surface-melting, obeying the
logarithmic growth law predicted for short-ranged potentials and thin liquid layers.
The (111) face does not surface melt, which Trayanov and Tosatti feel is an artifact
of the lattice model.
The other popular theoretical approach to surface melting, wetting, and layering
phenomena is to use classical density functional theory to determine the equilibrium
density profile of the interface. In these calculations, the free energy of the system is
written as
Ω[ρ] = Fexc [ρ] +
d~r ρ(~r) Vext (~r) − µ + kB T log Λ3 ρ(~r) − kB T
(2.1)
The density ρ(~r) is described by a basis set (usually plane waves), and is minimised
by one of several recipes. The term Fexc [ρ] contains all of the correlations between
particles and is not known exactly. Several methods have been used to approximate it,
the most popular being the “Weighted Density Approximation” or WDA [188, 189],
in which
¯ r))
(2.2)
Fexc [ρ] = d~r ρ(~r)Ψ (ρ(~
¯ r) determined from
with the weighted density ρ(~
¯ r) =
ρ(~
¯ r) × ρ(~r).
d~r0 w |~r −~r0 |, ρ(~
(2.3)
The weighting function w(r, ρ) is chosen so that the structure factor of the homogeneous liquid in the approximation is correct. If the liquid is composed of hard-spheres,
then analytic (though thermodynamically inconsistent) expressions for the excess free
energy per particle Ψ(ρ) may be derived using the solution to the Percus-Yevick integral equation theory of the liquid state [190–192]. Alternatively, the CarnahanStarling equation of state for hard-spheres [193] can be used. Other, softer interatomic
potentials can be treated using the perturbation theory of Weeks, Chandler and Andersen [194, 195]. After the introduction of a few small approximations, these theories
result in a massive minimisation problem which can be implemented on a large computer. For instance, the calculations of Löwen, Ohnesorge, and Wagner [196], which
are probably the most successful calculations in this area, used 5 × 10 5 independent
58
Introduction
variables in their minimisation, requiring between 4 and 40 hours of Cray YMP CPUtime per temperature considered; the exact amount of time taken depends, as always,
on the initial density profile. Density functional theories reproduce good phase diagrams for the Lennard-Jones and hard-sphere system (not at all surprising, since a
reference equation of state is implicitly included through the weighting function) and
also yield good results on interfacial tensions, though there is some doubt as to the
accuracy of the DFT values for γlv [196]. These theories predict that surface melting
will occur for all three crystal faces of the Lennard-Jones solid [197], although the
“completeness” of this melting has not been determined.
Surface phase transitions have also been modelled using an empirical Landau free
energy by Pluis, Frenkel, and van der Veen [198] and Lipowsky and Speth [149].
In these theories, the object to be minimised is the free energy calculated from the
continuous profile of the order parameter (density) M:
"
#
1
dM 2
.
(2.4)
F (M) = f1 (Ms ) + dz f (M) + J
2
dz
f (M) is the free energy in the bulk phase, and its form is empirically chosen to give
phase behaviour appropriate to the system of interest. f 1 (Ms ) is a short-ranged surface
term, and the (∇)2 term describes the next-to-lowest term in a Taylor series expansion of the true free energy of a system described by a density profile M(z). Pluis,
Frenkel, and van der Veen used this model to describe the surface behaviour for many
different values of input parameters, and constructed a phase diagram in which the
different phase behaviours (surface melting/non-melting, and surface freezing/nonfreezing) are expressed in terms of parameters controlling interfacial tensions. While
this theory is very satisfying and simple to work with, it does not take into account
any aspect of the microscopic length scales involved in the problem and so cannot
describe any layer-wise transitions or similar phenomena.
2.5 This Work
Based on a long list of reasons why it was impractical or impossible to study the
systematics of surface phase transitions using simulation techniques, we decided that
theoretical approaches, for all their faults, are the most powerful means currently
available for studying these systems. These methods fall roughly into two groups,
density functional methods and lattice gas methods. Density-functional calculations
are expensive in terms of computer time; complete studies of the sort presented in
the following chapters would have taken thousands of hours of supercomputer CPU
2.5 This Work
59
time using the methods of Löwen, Ohnesorge, and Wagner. Both because of this
and because we wished to eventually add anisotropy into the studies, the lattice gas
methods are more appropriate. In addition, it is possible to perform Monte-Carlo
calculations on lattice gas models, so that if necessary our results could be checked
at certain points without extreme effort. In fact, once free volume approximations
have been included in the system à la Trayanov and Tosatti, this becomes much more
difficult.
As a preliminary study, we considered the surface behaviour of the Blume-EmeryGriffiths (BEG) model. These results are presented in Chapter 3. This is isomorphic
with the 2-state Potts model, the surface behaviour of which was studied by Jayanthi.
We determined the phase behaviour of the free surface via direct numerical minimisation of the free energy and numerical root-finding on the gradient of the free energy,
rather than with the Picard iteration technique frequently seen in studies of this kind
of model (and used by Jayanthi). This study found a sequence of layer transitions of
the solid surface, and also surface melting behaviour all along the solid-liquid line.
Although these transitions may be more meaningful in the corresponding magnetic
systems than in the lattice gas interpretation of this model, this study introduced all of
the necessary numerical techniques and theoretical concepts used in the development
of a more realistic model.
Because the BEG model does not describe a real substance very well, we extended
the basic ideas of the model of Lennard-Jones and Devonshire [199,200] and the bulk
lattice gas model developed by Mori, Okamoto and Isa [201] to consider a lattice gas
in which the particles interact via a Lennard-Jones potential plus some short-ranged
anisotropic terms. Using a free volume approximation modelled on that of Trayanov
and Tosatti results in a model that can be used to study the surface transitions of any
crystal face of the fcc crystal. Without the free volume approximation, which contains
most of the entropic difference between the solid and liquid phases, the phase diagram
of this and all similar lattice gas models is too poor to be used in surface studies
because the solid-liquid coexistence line is moved to temperatures higher than the
critical point of the model, so that the triple point disappears.
Although we have developed the necessary tools and ideas to allow the lattice
spacings in the model to relax, which is necessary for almost all studies of anisotropic
potentials, our free volume approximation turned out to be insufficiently robust to be
used in calculations on surfaces in the relaxed-lattice model. As a result, although
we were able to perform bulk calculations using these improvements, our surface
calculations have only used fixed lattice spacings.
We have performed very precise calculations on the low-index faces of the
60
Introduction
Lennard-Jones crystal, over a wide range of temperature. We have characterised
the behaviour and surface phase transitions of this system over this range, from the
low-temperature clean crystal faces to triple point surface melting. We have also
performed a long series of auxiliary calculations to determine the effects of the approximations we have made in developing the model. We conclude that while these
approximations do have significant effects on the surface phase behaviour, these effects are well-defined and can be traced to fundamental properties of the lattice model.
In these calculations we have determined that the reference lattice itself perturbs the
liquid-vapour interface in these systems and that this perturbation, which is unavoidable in this kind of model, can dramatically influence the phase behaviour of the
system near surface melting.
We have also done a series of calculations on the Lennard-Jones potential plus a
th
0 order anisotropic term. This has allowed us to characterise a model surface orientational disordering transition, as well as to look at the influence of orientational
degrees of freedom on other surface phenomena. Although this is only a crude approximation, it shows some very interesting behaviour and is comparable with studies
of nematic-isotropic transitions at surfaces [202].
Chapter 3
THE BLUME-EMERY-GRIFFITHS
MODEL
One of the earliest theories of melting and disordering is due to Lennard-Jones and
Devonshire [199, 200]. In this theory, particles occupy either α sites or β sites. Particles on adjacent “like” sites interact with some attractive energy, and particles on adjacent “unlike” sites interact with some repulsive energy. This theory could be solved
in the mean-field approximation, and could successfully describe the first-order transition of melting. The LJD theory, as it is called, is based upon a model due to Bragg
and Williams [203] used to describe order-disorder transitions in alloys [204]. The
LJD theory was extended by Pople and Karasz [205, 206] and others [207] so that
molecules could take up one of two orientations as well as lattice positions, so that
the theory could describe both solid state rotational transitions and melting transitions.
The Pople and Karasz model was used in a modified form to study liquid crystals by
Chandrasekhar, Shashidar and Tara [208, 209], and extended to more orientational
states by Amzel and Becka [210]. Since then, a number of more elaborate theories
for nematic and smectic liquid crystals have been developed [211, 212]. The original
LJD model is in fact a particular case of several more general lattice models which
have been studied in different contexts. Particularly, it can be mapped onto the 2-state
Potts model [213], which can be mapped onto the Blume-Emery-Griffiths model. In
this chapter we consider this model as a lattice gas, and use the mean-field approximation to solve for the phase behaviour of both bulk and surface systems [214]. We
identify several first-order layering phase transitions in the surface system, and also
characterise its surface melting behaviour.
61
62
The Blume-Emery-Griffiths Model
3.1 The Bulk System
3.1.1 Definition of the Model
The Blume-Emery-Griffiths model [215] is a general spin-1 Ising model given by
H = −J ∑ si s j − K ∑ s2i s2j − ∆∑ s2i − H ∑ si,
(i, j)
i
(i, j)
(3.1)
i
where the spins take the values si = 0, ±1, and the first two sums are performed over
all nearest neighbour pairs. J and K are the bilinear and biquadratic couplings, and
∆ is the crystal field term. H is an (optional) external field favouring one orientation
of the spins. This Hamiltonian can be used to describe the liquid, solid and vapour
phases of a lattice gas. It corresponds to a physical system where in each lattice cell
a particle may occupy one of two sublattice sites, identified with the s i = ±1 spin
directions. That is, zero spontaneous magnetisation corresponds to complete spatial
disorder, while states with hsi > 0 are interpreted as ordered. Nearest neighbours
interact with potential energy V = −(K + J) if they are on the same sublattice, and
V = −(K − J) if not.
Change the variables in the above equation from {si } to {ni , si }, where sold
=
i
new
new
ni si , with ni = 0, 1 and si = ±1. This results in the expression
H = −K ∑ nin j − H ∑ ni si − J ∑ ni n j sis j ,
(i, j)
i
(3.2)
(i, j)
where the sums are taken over nearest-neighbour pairs. In this formulation of the
model, J is the spin–spin coupling energy, K is the nearest-neighbour interaction for
the corresponding simple lattice gas and H is an external field favouring one sublattice. We have removed the term containing ∆, but it will reappear as the chemical
potential µ in the expression for the grand partition function appropriate to this system. It is convenient to work in reduced units where we take J = 1 and k B = 1 so that
all energies are in units of J.
The phase diagram of this system has been obtained in the mean-field approximation for many different values of J/K [216]. These calculations were all done
without the change of variables presented above, and the results were interpreted in
terms of the magnetic phases of a solid material. In our new variables, this model
is isomorphic with the 2–state Potts model [213]. Jayanthi [178] studied the surface
melting behaviour of the Potts model and found that it did completely surface melt,
and supported predictions of a logarithmic dependence of the thickness of the liquid
layer on temperature. These calculations were all performed along the solid-vapour
3.1 The Bulk System
63
coexistence line of the model, as is appropriate. We have considered a slightly different system, where the surface of the model is “free”, so that above the top lattice plane
is nothing, rather than the vapour phase. This can be interpreted as either the surface
of a magnetic system, or else as a lattice model of a solid surface slightly away from
phase equilibrium. In this system, we can calculate the behaviour of the surface off
the coexistence line, which has not been done before. Our study has identified several
first-order layer transition lines, and we have characterised the surface melting of the
system all along the solid-liquid coexistence line.
3.1.2 Mean-Field Solution of the Bulk Model
We must first determine the bulk properties of this model. In order to make the analysis tractable, we solve the model within the mean-field approximation (MFA) as
follows. Since this is an open system we consider the grand partition function, which
is
Ξ = ∑ exp (βµN0 )Z (N0 , N, T ) ,
(3.3)
N0
where Z (N0 , N, T ) is the canonical partition function for N0 particles on a lattice of
N sites. β = 1/kB T . This is the same as
Ξ=
∑ exp (βµN0) exp (−βH ),
(3.4)
ni ,si
where the sum is taken over all possible arrangements and values of the n i and si
and H is the function given in equation 3.2. µ is the chemical potential, and N0 is
the number of occupied lattice sites: N0 = ∑ ni . The grand partition function is the
appropriate thermodynamic ensemble to use in a system of constant β, µ, and V ; in
the following studies the volume can be ignored because the system is homogeneous,
leaving β and µ as the control parameters. The mean-field approximation consists (in
one interpretation) of setting the exponent in the expression for Ξ equal to its average
value. This is equivalent to saying that each lattice point sees only the average of the
rest of the lattice, or that the ni have been set equal to hni and the si have been set
equal to hsi. In this approximation, the grand partition function reduces to
Ξ=
∑
hni,hsi
γ(hni, hsi)exp (−βhH i) exp (βµNhni),
(3.5)
where γ(hni, hsi) is the number of possible ways to arrange occupied lattice sites and
spins for the specified values of hni and hsi, hH i is the Hamiltonian evaluated at those
average values and N is the number of lattice sites. (Henceforth we shall omit the hi
64
The Blume-Emery-Griffiths Model
notation; n shall represent the average over all ni , etc.) This is the Bragg-Williams
formulation of the mean-field equations [203, 217]. It is easy to show that
1
nN
(1
+
s)nN
2
γ(n, s) =
,
(3.6)
N
nN
where the first binomial coefficient determines how many ways there are to arrange
the nN particles on the N lattice sites, and the second determines how many ways
there are to arrange the (1 + s)nN/2 positive spins among the occupied positions. In
factorial notation this is simply
N!
.
(3.7)
γ(n, s) =
1
(N − nN)! 2 (1 + s)nN ! 21 (1 − s)nN !
For a lattice where each of N lattice sites has z neighbours, the mean-field energy is
given by
1
hH i = −NHns − zNn2 K + Js2 .
(3.8)
2
We are now in a position to solve for the average density and magnetisation at equilibrium. In a real system, the density of states is very sharply peaked around the average
energy, so that we can determine the equilibrium position of the system by finding this
peak. This occurs at the values of n and s that maximise the grand partition function.
In order to make the solution easier, we work with the quantity (1/N) log(Ξ) rather
than Ξ itself. In the mean-field approximation the system is homogeneous, so that
all of its extensive properties can be calculated from X = N × X0 , where X0 is a ‘per
lattice site’ quantity.
β
1
log Ξ = − hH i + log N − (1 − n) log(N − nN)
N
N
1
1
−
(1 + s)n log
(1 + s)nN
2
2
1
1
(1 − s)n log
(1 − s)nN .
−
2
2
For a maximum value we require that
∂ 1
∂ 1
log Ξ =
log Ξ = 0.
∂s N
∂n N
(3.9)
(3.10)
By taking derivatives we find that the equilibrium values of n and s will be the simultaneous solutions of the following two equations:
1
1+s
2
β Hn + zJn s − n log
= 0,
(3.11)
2
1−s
β µ + Hs + zn K + Js2 +
1+s
(1 − n)
1
log 2 √
− s log
= 0.
(3.12)
2
1−s
n 1 − s2
3.1 The Bulk System
65
These equations can have several solutions, and the one we want is the one that maximises the grand partition function, or equivalently, minimises the grand potential per
lattice site, defined by
1
log Ξ.
(3.13)
Ω=−
βN
The grand potential defined in this way is simply related to the Helmholtz free energy
A by
1
A = βµ + Ω,
(3.14)
N
so that for fixed µ, finding the fixed points of one quantity is equivalent to finding
the fixed points of the other. We solve this system numerically to obtain the meanfield phase diagram of the model, either by performing iterative root-finding on the
above system of equations or by using minimisation techniques applied to the grand
potential. In either case, there will be significant difficulties due to the fact that on
the border of the parameter space, defined by n = (1 or 0) and s = ±1, both Ω and its
derivatives are undefined, so that our numerical algorithms must be smart enough to
avoid choosing points in this region.
66
The Blume-Emery-Griffiths Model
Positional Order, (s)
Density, (n)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
3.5
4.0
4.5
0.0
3.5
Temperature
Upper branch
Lower branch
4.0
4.5
Temperature
Figure 3.1: Density and positional order versus temperature. These profiles are for K = 3.0, µ = −12.0, H = 0. Units are chosen so that J = 1.
There is a first-order phase transition in both the high and low density
branches at T ' 4.328.
3.1.3 Properties of the Bulk Model
We have determined the bulk phase behaviour of this model over a range of parameters and organised these results into a single phase diagram. In the work that follows,
all units have been scaled so that J = kB = 1. In Figure 3.1 are plots of the density and
positional order for both the high-density and low-density solutions (recall the simple lattice gas model) as a function of temperature, for parameter values of K = 3.0,
µ = −12.0, H = 0. The positional order parameter is just the average value of the
spins and is similar to the spontaneous magnetisation of an Ising magnet. We see that
at T ' 4.328 there is a first-order transition in both the high and low density “phases”
to the supercritical fluid, which for this model is characterised by n = 2/3 and s = 0.
It is simple to show (by taking β → 0) that this phase is the only solution to the model
for very high temperatures. For this choice of parameters, the grand potential Ω has
the same values in both local minima for all temperatures; by increasing the temperature we are moving along the solid-vapour coexistence line, so that the observed
melting transition occurs at the triple point.
By measuring many more such profiles we may construct a phase diagram in the
(µ, T ) plane at fixed K, shown in Figure 3.2. The transitions (points) in this diagram
have been located by fixing µ and varying T , and are accurate to within ±0.001T . This
3.1 The Bulk System
67
-11.5
Chemical Potential
-11.6
-11.7
-11.8
‘SOLID’ PHASE
(high density)
-11.9
‘LIQUID’ PHASE
(critical fluid)
-12.0
‘GAS’ PHASE
(low density)
-12.1
-12.2
-12.3
3.9
4.0
4.1
4.2
4.3 4.4 4.5
Temperature
4.6
4.7
4.8
4.9
Figure 3.2: (µ, T ) phase diagram for the BEG model at K = 3.0. For
µ chosen sufficiently high, the solid-fluid transition becomes continuous,
and for µ chosen sufficiently low, the gas-fluid transition becomes continuous.
Figure has also been obtained by both Hoston and Berker [216], and Jayanthi [178] in
studies of isomorphic models. The phase diagram for K/J = 3 is qualitatively similar
to that of an inert gas, except for the presence of a tricritical point. That is, at high
values of the chemical potential (or pressure), the melting transition becomes continuous, and the point where this occurs is called a tricritical point1 . We have measured
several other phase diagrams for different values of K/J. In Figure 3.3 (page 68)
we combine these data into a schematic global phase diagram by placing these plots
one on top of another and connecting points. We see that only for a small region of
the phase space is the model a reasonable representation of a 3-phase system, and
that only for K/J in this region should it be used to consider melting behaviour. It
may prove interesting to better characterise the bulk critical behaviour of this model
(especially the endpoints of the solid–liquid and gas–liquid critical lines, as they occur at the intersection of three different transition surfaces) but this would require a
much higher level of theory, and would not be particularly relevant to this study. It
is well known that in only three dimensions mean-field theory consistently overes1A
tricritical point occurs at the end of a triple line, where three phases become critical simultaneously. This can only occur in higher-than-two dimensional phase diagrams; by considering the (µ, T )
part of the phase diagram and not looking at the full (µ, T, H, K, J) phase diagram we are only seeing a
cross-section of the full phase behaviour of the system.
68
The Blume-Emery-Griffiths Model
Chemical
potential
C
A
Temperature
B
K/J
Figure 3.3: Possible bulk phase diagram for the BEG model, showing only
first-order transition surfaces. A is the “solid–vapour” transition surface,
B is the “vapour–fluid” transition surface, and C is the “solid–fluid” transition surface. The dashed line represents the edge of the B surface below
the C surface, and the two dotted lines are possible continuations of the
triple point line found for K ' 3.
timates critical exponents, and state-of-the-art critical theory is well past such crude
approximations. However, mean-field theory also reproduces the essential physics of
many complicated systems, and so we are confident of the shape and general characteristics of this phase diagram. Furthermore, we can apply mean-field theory to
much more complicated systems than we can renormalisation group theory, which is
usually only tractable for very simple lattice models.2
2 Of
course, the whole point of the theory of universality is that all systems near a critical point
behave in one of only a few well-defined ways, labelled ‘universality classes’. While this allows us to
characterise all systems’ critical behaviour, it does not help in finding the structure of the rest of the
phase diagram, which is not so well-behaved.
3.2 Calculations on Surfaces
69
3.2 Calculations on Surfaces
3.2.1 Mean-Field Theory for a Surface
Having characterised the bulk solid, we go on to consider properties of the free surface. In order to do this, we recast the Hamiltonian of the system in a form which
accounts for the one-dimensional broken symmetry of a bulk system with a planar
surface. Consider a slab of material, of some finite thickness (say, L layers) in the
z direction, infinite in the x and y directions. We describe such a system within the
mean-field approximation by specifying average values of the order parameters within
each of the L layers of the system. That is, instead of describing the system with a
single pair n and s, we describe it with sets of {ni } and {si }, where the individual ni is
the average density of the ith layer, etc. In order to write the Hamiltonian of this system, we need to specify how many nearest neighbours a given lattice site has within
its own layer (denoted by a) and how many it has in each of the neighbouring layers
(denoted by b). (For longer ranged interactions, we would need parameters describing
the second-nearest-layer interactions.) Lastly, we now take N to mean the number of
lattice points within an individual layer. In this case, the energy of the system is given
by
L
L
1
a L
hH i = − ∑ Hni si − ∑ n2i K + Js2i − b ∑ ni ni+1 (K + Jsi si+1 ),
N
2 i
i
i
(3.15)
where nL+1 = 0. As before, we make the mean-field approximation of setting the
exponent in the definition of the grand partition function equal to its average value.
This results in the expression
!
Ξ=
∑ ∏ γ(si, ni)
{si ,ni }
i
exp −βhH i,
(3.16)
where the γ function is defined exactly as it was in Equation 3.7 , due to the redefinition
of the variable N. The grand partition function is now a function of 2L variables, the
ni and si . As before, we take the logarithm of this function and look for its maximum.
3.2.2 Technical Details
In order to solve this system of equations and obtain the surface phase behaviour of the
model, we can use either a minimisation algorithm applied directly to the free energy
or we can perform root-finding on the derivatives of the free energy with respect to the
order parameters. Because the γ function is singular along the edges of the variable
70
The Blume-Emery-Griffiths Model
space (where si = ±1 or ni = 1, 0) we must be careful that our algorithm avoids these
areas. Except for near first-order transitions, the position of the global minimum
will be a continuous function of the model parameters (µ, K, T , etc.), so that if we
have a solution at one set of parameters, by slowly varying the parameters we can
“follow” that solution through any parameter changes. For a small perturbation of
the parameters (for instance, a change of the temperature), the previous solution will
still be within the radius of convergence of the new solution, and can be used as an
initial guess for the algorithm. This procedure is much like the one we used in the
integral equation study of Chapter 1, although in these calculations we did not attempt
to extrapolate starting guesses at each new state point.
We have used two numerical techniques to solve these systems. The first is the
Newton-Raphson technique described in Section 1.3; in order to use this method, we
must differentiate Equation 3.16 with respect to all of the ni and si , and use the NR
technique to find the common roots of the resulting 2L equations. The root-finding
method has several shortcomings. The first is that it scales badly with increasing slab
thickness, because the matrix inversions require of O(L3 ) operations. The second is
that in situations where a succession of first-order transitions occur (such as layering
phenomena) the algorithm often gets stuck in non-convergent cycles. We have found
that for reasonably small slabs, with L ≤ 64, the best way to find the stable configurations is to use a minimisation algorithm applied to the fourier transforms of the
spin and density profiles. That is, we use fast fourier transforms to project the density
profile onto a sum of even-numbered cosine functions, and then minimise the grand
potential with respect to the fourier coefficients using a conjugate gradient procedure
taken from Press, et al [42]. Only even-wavenumber functions are required because
the slab system, with a free surface on either side, has a plane of symmetry at the L/2th
layer. This method is quite successful because most of the high-wavenumber coefficients are near zero, so that the effective dimensionality of the problem is reduced.
For example, if there were no surface effects, the entire slab would be of uniform
density, and would be entirely described by the first (k = 0) coefficient. If the surface
profiles differ from those in the bulk, more functions would be required, but many
fewer than the 2L real-space variables needed to describe the system. Unfortunately,
for large slabs this method is often unstable and the solutions are contaminated by
high-frequency oscillations. This may be due to inappropriate choices of the several
tolerances used in the minimisations. In these cases, we have used a real-space minimisation to find the solutions, since the root-finding algorithm is extremely slow for
these larger systems. Most of the data that follows was gathered using the conjugate gradient procedure, with the minimisation terminated when several successive
3.3 Results for the Surface System
71
T
Chemical Potential, µ
-11.4
‘Solid’ Phase
first layer
transition
-11.6
-11.8
‘Liquid’
Phase
nd
2
rd
3
-12.0
‘Vapor’ Phase
-12.2
1.4
2.4
C
3.4
4.4
5.4
Temperature
Figure 3.4: Surface phase diagram of the semi-infinite slab system. The
bulk transitions are shown as solid lines, with critical point (C), triple
point, and tricritical point (T ). The bulk phase diagram is similar to that
of an inert gas. The first-order surface transitions are marked by dashed
lines. All the data points used are shown explicitly, with circles marking
bulk transitions and triangles marking surface transitions. The last points
along the surface transition lines are near to their tricritical endpoints.
evaluations of the grand potential differed by less that one part in 10 −14 . Also, we
compared these results against those obtained with larger slabs to insure that the two
free surfaces were far enough apart to behave independently. All of the following
calculations have used a simple cubic lattice geometry, with a = 4 and b = 1.
3.3 Results for the Surface System
3.3.1 Roughening and Layering Transitions
We have used these techniques to measure the equilibrium state of the slab surface
over the entire solid part of the phase diagram, and found many interesting features.
Figure 3.4 shows the first three observed first-order surface transitions of the slab
system superimposed on its bulk phase diagram. When the system moves across each
of these lines, successive layers undergo transitions from the high-density ordered
(solid) state to a low-density disordered (quasi-vapour) state.
72
The Blume-Emery-Griffiths Model
1.0
Third layer
Second layer
Density, ni
0.8
0.6
0.4
First (surface)
layer
0.2
0.0
2.5
Global Minimum
Metastable state
3.0
3.5
4.0
Temperature
Figure 3.5: Layer densities through the surface layer transition. These are
plots of the densities of the top three layers of the slab versus temperature
at constant µ = −11.80 through the surface layer transition. The data are
taken at temperature intervals of 0.01. The dashed lines show the position
of the metastable state, which eventually collapses.
Density profiles taken from a crossing of the first transition line are shown in Figure 3.5 . This transition is reminiscent of the surface disordering behaviour observed
in both simulations [136] and experiments [105] in which the creation of adatomvacancy pairs causes the formation of a low-density superlayer and a reduction in
the crystalline character of the surface layer. In Figure 3.6 we show the density crosssections of the slab near the surface to illustrate this. In this model, these quasi-vapour
layers are always of significantly higher density than the bulk gas phase at the same
temperature at coexistence. The surface-layer transition line extends nearly to the
tricritical value of µ, so that first-order surface roughening is observed over nearly
the same range of chemical potential as is first-order bulk melting. These lines are
terminated by tricritical points, above which the layer-wise transitions are continuous. The critical temperatures of these transition lines converge towards a roughening
temperature TR ' 3.90 above which all interfaces are rough on a microscopic scale.
Consequently, above TR no first-order layer-wise transitions may be observed because
they are characterised by the sudden motion of a sharp interface. It is noteworthy that
the surface layer transition occurs at temperatures quite far below the triple point; in
real materials such as metals, surface disordering occurs nearer to bulk melting. The
3.3 Results for the Surface System
73
1.0
Density, ni
0.8
0.6
0.4
0.2
0.0
61.0
62.0
63.0
64.0
65.0
Layer Number
Figure 3.6: Density profiles near the surface layer transition. These are
plots of the densities of the top several layers of the slab at a series of
temperatures, from T = 2.30 to T = 3.30 in steps of 0.10; the transition
occurs at T = 2.80. These data were taken at µ = −11.80, with at 64 layer
slab.
low transition temperatures in the model are probably due to the lack of long-ranged
interactions and the use of a simple cubic lattice, in which neighbouring layers are
not as strongly coupled as in a close-packed lattice.
Above the endpoint of the first line (and even above the tricritical point) the solid
surface continuously disorders. We have also observed smooth transitions in the second and third layers; for constant µ ' −11.90 (the endpoint of the second transition
line) the system undergoes a first-order surface transition followed by a continuous
transition in the second layer. For µ very near to coexistence several first-order transitions occur, followed by a continuous penetration of the quasi-vapour phase into the
bulk solid and a broadening of the solid-vapour interface.
These layer transition lines all approach the solid-vapour coexistence line asymptotically as the temperature is lowered, indicating that at solid-vapour coexistence the
free surface is wet by a macroscopic layer of the low-density disordered phase. Although systems at coexistence which have undergone the first, second, third, etc. layer
transitions are local minima of the free energy, they are metastable; the free energy
of the slab converges downwards with successive transitions, at least up to the fifth.
(Below the fifth transition, our calculations cannot distinguish between the successive
74
The Blume-Emery-Griffiths Model
minima.) For any µ greater than that at coexistence the thickness of the disordered
layer at the free surface is quite small away from the bulk melting temperature.
Figure 3.5 also shows the behaviour with temperature of the metastable unroughened surface. This superheated solution collapses to the global minimum at
3.28 < T < 3.29. We have found similar metastable solutions, both superheated
and supercooled, in the neighbourhood of all of the first-order transitions studied,
including the bulk transitions. The metastable surface states are present over much
larger ranges of temperature than the metastable bulk states. The persistence of these
metastable states makes identification of the transition temperatures difficult. We use
the method of Yashonath and Sarma [183], which is to plot the grand potential of each
local minimum as a function of temperature and then find the intersections of smooth
curves drawn through those points.
3.3.2 Surface Melting
Along the entire length of the first-order melting line, as the temperature is increased
towards the bulk melting point a quasi-liquid layer appears at the surface of the solid
and its thickness diverges logarithmically at TM . An example of this behaviour for
µ = −11.90 is shown in Figure 3.7 with the thickness of the quasi-liquid layer shown
in Figure 3.8. The thickness of the quasi-liquid layer is a logarithmic function of the
reduced temperature. This logarithmic growth behaviour is characteristic of systems
governed by short-ranged potentials, and was noted by Jayanthi for this system at
solid-vapour coexistence. In general, all systems where at least part of the potential
(the repulsive wall) is short-ranged are expected to show this kind of logarithmic
growth behaviour for thin liquid layers, and potentials with long tails will exhibit a
crossover to a power-law dependence when the liquid layer becomes thicker.
The quasi-liquid layer that grows on the solid surface has the same density as the
liquid phase, but nonzero magnetisation (spatial order) which varies smoothly from
the solid value at the solid-liquid interface to zero at the liquid-surface interface. As
the tricritical point is approached, the growth of this layer becomes more pronounced,
and it is visible further away from the melting temperature. This occurs because the
difference in density of the solid and liquid phases decreases as the chemical potential
is raised, so that the interfacial tension between these two phases decreases. In all
cases, however, the liquid layer only appears when the temperature is raised to well
within 1% of the melting temperature.
Above the tricritical point, instead of forming a liquid “shoulder” the width of the
solid-surface interface increases divergently without the intrusion of a distinct third
phase. An example of this behaviour is shown in Figure 3.9. Note that if the system
3.3 Results for the Surface System
75
Density, ni
0.81
T=4.380
0.76
T=4.3938
0.71
0.66
38.0
42.0
46.0
50.0
54.0
58.0
62.0
Layer Number
Figure 3.7: Surface density profiles near melting. These are the density
profiles of the top several layers of the slab at a range of temperatures just
below the bulk melting point, for µ = −11.90. The curves are taken at
intervals of δT = 0.001, and are superimposed to shown the progression
of the quasi-liquid “shoulder” into the slab. In this figure, L = 64 so that
the free surface is at layer number 65.
Thickness of Liquid Layer
12.0
10.0
8.0
6.0
-9.0
-8.0
-7.0
-6.0
-5.0
-4.0
log10(TM-T)
Figure 3.8: Thickness of liquid layer near TM with µ = −11.90. The thickness of the liquid layer is defined by the distance from the free surface to
the point along the density profiles with n = 0.75. The absolute vertical
scale is immaterial, as it depends on the thickness of the liquid-wall interface.
76
The Blume-Emery-Griffiths Model
T=4.700
Density, ni
0.80
0.79
T=4.708
0.78
64.0
80.0
96.0
112.0
128.0
Layer Number
Figure 3.9: Surface density profiles through continuous bulk melting.
These are the density profiles of the top 64 layers of a 128-layer slab at
a range of temperatures around the bulk melting point, for µ = −11.30.
The data are taken in temperature intervals of 0.001 from T = 4.700 to
T = 4.708, when the entire slab melted. (The melting point of the 128layer slab system is between T = 4.707 and T = 4.708.) Note that the
surface of the slab is wet by the quasi-vapour phase, but no discernible
quasi-liquid layer appears.
does surface melt, the quasi-liquid layer does not grow at the free surface but at the
surface of the solid, so that very near the triple point the free surface is wet by the
vapour phase and the solid-vapour interface is wet by the liquid phase. In general,
because the solid always undergoes a roughening-like transition (either first-order or
continuous) for µ below the tricritical point, there is always a low-density disordered
layer covering the free surface below which we may observe wetting by the liquid
phase.
3.4 Conclusions
Studies of models such as this one are useful for several reasons. Because of the
simplicity of these models, the mathematical analysis of their behaviour is easily
managed, and so they are a very good starting point for a more detailed study. Furthermore, considering the crudeness of the model and the severity of the approximations used to study it, they can give surprisingly good qualitative results that can be
helpful in the analysis of more complicated systems. Renormalisation group theory
3.4 Conclusions
77
has shown that for the determination of critical exponents and the characterisation of
divergent behaviour it is sufficient to study simple lattice models, because these quantities are not dependent on the finer aspects of the model used, so that the analysis of
materials interacting through realistic potentials is not necessary.
Of course, the BEG model is unsatisfying in many ways. The incorrect behaviour
of the solid-liquid coexistence line at high pressures (chemical potentials) is worrisome, and indicates that we are not describing the differences between two phases
well. Also, the transition between the solid and liquid phases near the triple point is
generally considered to be “weaker” than that in a realistic model, in that the changes
in entropy and energy at melting, ∆H f usion and ∆S f usion , are not large enough. Furthermore, virtually all molecular systems interact through potentials which are much
longer ranged than the one used in this model, so that we cannot correctly describe
the behaviour of thick quasi-liquid layers of these systems using this model.
In this study, we have determined the phase diagram of the free solid surface of the
model over the entire phase diagram. We have found all of the first-order transitions
that it exhibits, and also characterised its melting behaviour along the solid-vapour
coexistence line. Although these results are not applicable to any specific system,
their generality provides a framework for further studies on more complex models.
In this study, we have introduced and developed most of the numerical and analytical
techniques that will be necessary for our study of a more complex model. These
include the introduction of order parameters to simplify the partition function and
the application of the mean-field approximation to make the problem solvable. We
have applied several different minimisation and root-finding techniques in order to do
this. We have determined that, although we can solve these problems in k-space this
is not particularly advantageous, and also that minimisation techniques are generally
preferable to root-finding strategies for solving these systems of equations. Lastly, we
have familiarised ourselves with the procedures used to map out a phase diagram and
to identify phase transitions in surface systems.
78
Chapter 4
A BETTER THEORY
This is a derivation of a mean-field approach, based on a lattice model, to the thermodynamics of a simple material. A study has already been made of the surface
thermodynamics of the Lennard-Jones system by Trayanov and Tosatti, using the
same approach. We will handle the free volume terms in a way similar to theirs, but
somewhat simplified and hopefully more robust. Also, we will introduce anisotropy
and orientational degrees of freedom into the system at a mean-field level. Lastly, we
will derive the necessary thermodynamics to allow the lattice constants to vary as part
of the calculation. This is an attempt to make the model more realistic, as there is no
reason why different phases should all have the same characteristic length scales. In a
lattice-based calculation of the partition function which is ‘exact’, the sizes and shapes
of the lattice cells would be immaterial; in an approximate treatment they are crucial,
especially when dealing with order-disorder transitions and anisotropic potentials.
4.1 Bulk Systems
4.1.1 Recasting the Partition Function
Instead of beginning with a model Hamiltonian as we did in Chapter 3, we will begin
with the correct partition function for a real system and make approximations until
it is simple enough to study by analytic methods. We will consider only particles
which act via pairwise-additive forces. For N particles interacting via some potential
U (ri , r j , Ωi , Ω j ), where ri and r j are the positions of the centres of the particles and Ωi
and Ω j are the angular orientations of the two particles relative to some fixed direction
n̂, the configurational partition function is
79
80
A Better Theory
Z (N,V, T ) =
"
Ω
V
dr1 . . .
dΩ1 . . .
Ω
V
drN ×
dΩN exp −β ∑ U (ri , r j , Ωi , Ω j )
i, j
!#
.
(4.1)
That is, we must simply (!) integrate the exponential term over all possible angles and
positions of all of the particles. In a lattice model, we will divide the volume V up into
M small cells of volume v0 , chosen small enough to permit only single occupancy. We
then assign a number ni , equal to 1 or 0 if the cell is occupied or unoccupied. The
partition function Z can then be rewritten
Z (N,V, T ) =
"
Ω
∑
{ni } v0
dΩ1 . . .
Ω
dr1 . . .
v0
drM ×
dΩM exp −β ∑ ni n jU (ri , r j , Ωi , Ω j )
i, j
!#
,
(4.2)
where ∑i ni = N. By repartitioning the space this way, we have actually excluded a
large volume from the integral, corresponding to all the configurations where cells
were multiply occupied. However, the integrand is nearly zero for these configurations, so that they contribute nothing. We then separate out what we will call the free
volume term from the equations, which is the integral “within” the lattice cell. First
replace the exponent with
−β ∑ ni n j U (ri , r j , Ωi , Ω j ) −U (r0i , r0j , Ωi , Ω j ) +U (r0i , r0j , Ωi , Ω j ) ,
and then separate terms:
Z (N,V, T ) =
∑
(4.3)
{ni }
Ω
dΩ1 . . .
Ω
"
#
F
(Ωi ) × exp −β ∑ ni n jU (r0i , r0j , Ωi , Ω j ) ,
dΩM Z{n
i}
i, j
where r0i and r0j are the space-fixed positions of the centres of the ith and jth lattice
F (Ω ) is the free volume term, given by
cells, and Z{n
i
i}
F
(Ωi ) =
Z{n
i}
"
v0
dr1 . . .
v0
drM ×
exp −β ∑ ni n j U (ri , r j , Ωi , Ω j ) −U (r0i , r0j , Ωi , Ω j )
i, j
#
.
(4.4)
The free volume term is dependent on the values of the {ni }, and describes the collective, correlated motions of the particles around their lattice sites [218, 219]. The
4.1 Bulk Systems
81
F (Ω ) is meant to indicate that the free volume term should be thought
notation Z{n
i
i}
of as an implicit function of the ni and an explicit function of the orientations of
the molecules. By separating out this term we are left with the more familiar lattice
model of a collection of rotors sitting at well-defined lattice sites, interacting via the
potential U . This model is similar to the Lebwohl-Lasher model often used to study
liquid crystals [220, 221]. We can reduce this problem to a solvable one by applying
a mean-field approximation; all we will then need is some way to handle the free
volume term.
4.1.2 Mean-Field Solution of the Lattice Model
As in the study of the Blume-Emery-Griffiths model, we characterise the spatial order
in this system by two order parameters, which are again called n and s. If we choose
the lattice to be simple cubic, then we can consider it as composed of two interpenetrating face-centred-cubic (fcc) sublattices. (Consider, for instance, the structure
of NaCl.) Define each of the M lattice cells to contain two neighbouring sublattice cells, one from each sublattice. In that case, ni is the occupation number of the
cell, and si is the sublattice position: si = 1 for the Na sites, and si = 0 for the Cl
sites. These two order parameters are both allowed to be equal to zero or one. Obviously, this choice of order parameters constitutes a restriction on the summations
given above, as the lattice points in a single cell cannot both be occupied. If the cells
are chosen sufficiently small, these configurations contribute nothing to the partition
function, and so nothing is lost. The total potential interaction between two lattice
cells is then given by the sum of the interactions between the four lattice sites they
contain:


si s jU (r1i , r1j , Ωi , Ω j )+


 s (1 − s )U (r1 , r0 , Ω , Ω )+
i
j
i
j
i
j
Ucell (ni , n j , si , s j , Ωi , Ω j ) = ni n j ×
0 , r1 , Ω , Ω )+
(1
−
s
)s
U
(r

i
j
i j
j
i


 (1 − s )(1 − s )U (r0 , r0 , Ω , Ω )
i
j
i
j
i
j





.
(4.5)




(The notation here is that r1i is the position of the “1” sublattice site in the ith cell, r0j
is the position of the “0” sublattice site in the jth lattice cell, etc.) Then we apply the
mean-field approximation, which consists of replacing the exponents in the partition
function by their average values, and arrive at
F
rot
ZMF (N,V, T ) = ∑ Z{n}
Ω × γ (n, s) × ZMF
.
n,s
(4.6)
82
A Better Theory
F
rot is the meanZ{n}
Ω is the mean-field approximation to the free volume term. ZMF
field partition function for the orientational degrees of freedom, given by
"
#
rot
ZMF
M
=
Ω
dΩ1 . . .
Ω
dΩM exp −β ∑ Ucell (n, s, Ω, Ωi ) ,
(4.7)
i
where Ω is the ensemble-averaged orientation of the rotors. That is, each rotor sees
the average orientations of those around it, rather than their instantaneous values. As
usual, we expect that ZMF (N,V, T ) is sharply peaked at the equilibrium values of the
order parameters, and approximate it by its largest value
F
rot
ZMF (N,V, T ) = max Z{n}
Ω × γ (n, s) × ZMF
.
(4.8)
n,s,Ω
As in the analysis of the Blume-Emery-Griffiths model, we have arrived at what
amounts to a numerical minimisation problem, but have not yet specified any details
of the potential. Although I will perform all of the following calculations using an fcc
lattice, this derivation is by no means restricted to that geometry and could be applied
to other lattice geometries as well.
4.1.3 Solution of the Mean-Field Orientational Problem
The mean-field equations for rotors on a lattice can be approached in many ways,
some of which are more appropriate to certain potential functions than others. We
will consider intermolecular potentials of the form U (pi , p j ), where pi = P2 (cos θi ) =
(3 cos2 θi − 1)/2 and θi is the angle that one axis of the particle makes with some
direction vector n̂. That is, the potential is a function of the principle order parameter
familiar from liquid-crystal theory [222–225]. (This is the same as specifying that
the spherical harmonic expansion of the potential in space-fixed coordinates has only
l = 0 and the first few l = 2 components [226].) In this case, the partition function of
a single rotor in the mean field of its neighbours is
rot
dω exp −βU (Ωave , ω)
(4.9)
ZMF =
Ω
1
= 2π
d (cos θi ) exp −βU (pave , pi ) .
(4.10)
0
The equilibrium value of pave will just be given by the Boltzmann average of pi , so
that the equation
1
2π
d (cos θi ) exp −βU peq , pi × P2 (cos θi )
(4.11)
peq = rot
ZMF
0
4.1 Bulk Systems
83
could be solved for the ‘self-consistent’ value of peq . This would require some sort
of iterative procedure. However, we also note that the equilibrium state of the system
must be a minimum of the free energy with respect to perturbations of p ave , which
provides a second route to the solution. By considering the partition function as a
function of pave we can transform the problem of finding the self-consistent value of
peq into a minimisation problem. This minimisation could then be done at the same
time as the minimisations of the n and s order parameters. The internal energy per
particle of the system can then be easily calculated using
hU i =
N 2π
rot
2 ZMF
1
0
d (cos θi ) exp −βU peq , pi
×U peq , pi .
(4.12)
For the case of a potential that is a linear function of pi 1 , this reduces to the simple
form
hU i = U peq , peq .
(4.13)
The entropy can then be determined from this and the Helmholtz free energy
A = −(1/β) log Z .
4.1.4 Determination of the Configuration Factor
We have specified that the lattice cells must be chosen small enough that they only
be singly occupied, and also small enough that particles in adjoining cells repel each
other strongly. For example, we might choose the lattice constants so that the distance
between adjacent sites on either the Na or Cl lattice is at the potential-well minimum.
This would imply that the distance between a Na site and the nearest Cl site is well
within the repulsive part of the potential. If we count the number of possible configurations on this lattice corresponding to a given n and s as we did in Chapter 3,
we would include many configurations which had very high potential energies caused
by the occupation of neighbouring Na and Cl sites, which are at a distance much
smaller than the van der Waals radius of the particles. This would lead to a meanfield energy which was extremely positive for any disordered dense phase, which is
not realistic. To avoid this, we need a more complicated expression for γ(n, s), that
takes this problem into account. Takagi [204] originally developed such an expression for use in quasi-chemical calculations on binary alloys, which are actually a very
similar problem to this one. The idea is that we can count the number of “bonds” (or
pairs) between lattice sites; since the energy is a function of the “bonds” rather than
the sites themselves, this allows for a more accurate mean-field theory. The problem
1 That
is, it includes terms like Api and Bpi p j , but not Cp2i
84
A Better Theory
is that calculating the number of ways to arrange these “bonds” on a 3D lattice can
only be done approximately. Without deriving it here, Takagi’s expression is
(zM)!
×
(4.14)
NNa−Cl !NNa−h !Nh−Cl !Nh−h !
z−1 z−1
M!
M!
,
(Mns)! (M − Mns)!
(Mn(1 − s))! (M − Mn(1 − s))!
γ(n, s) '
where z is the number of nearest neighbours for each lattice site (z = 6, for the NaCl
structure), and there are M lattice cells each containing two sites. NNa−Cl is the number of nearest-neighbour pairs both occupied by particles, NNa−h is the number of site
pairs where the Na site is occupied and the Cl site is vacant (h is for “hole”), and
so on. In the binary alloy problem, these Nxx are order parameters to be solved for.
In this case, our packing restrictions imply that NNa−Cl = 0, which then determines
the values of the other quantities. This was first determined by Mori, Okamoto and
Isa [201]. Because each site has z neighbours, it is obvious that
NNa−Cl + NNa−h = zMns,
(4.15)
which is just the total number of “bonds” to an occupied Na site, and there are Mns
occupied Na sites. Likewise, we have
NNa−Cl + Nh−Cl = zMn(1 − s),
Nh−h + NNa−h + Nh−Cl + NNa−Cl = zM.
(4.16)
(4.17)
If NNa−Cl = 0, then it follows immediately that
NNa−h = zMns,
(4.18)
Nh−Cl = zMn(1 − s),
(4.19)
Nh−h = zM − NNa−Cl − NNa−h − Nh−Cl
= zM(1 − n).
(4.20)
As before, we will calculate the quantity (1/M) logγ(n, s), rather than γ(n, s) itself. A
large amount of algebra yields
1
log γ(n, s) =
M
s
n(1 − s)
z −ns log
− n log
− log (1 − n)
1−s
1−n
n(1 − s)
ns
− n(1 − s) log
− (z − 1) −ns log
1 − ns
1 − n(1 − s)
− (z − 1) log[(1 − ns) (1 − n − ns)] .
(4.21)
(4.22)
4.1 Bulk Systems
85
Using these equations is straightforward. We must take care in any optimisation routine to avoid the singularities that occur in this function for n = 0, 1 and s = 0, 1.
When we evaluate the mean-field energy, we must exclude the (highly repulsive) NaCl nearest neighbour pairs from the sums, since they have already been removed from
the problem by this choice of the γ function.
4.1.5 Calculating the Free Volume
In repartitioning the partition function, all that we really accomplished was to move
all of the translation-rotation coupling and correlation between particles into a single
F (Ω ). What remains is to make an approximation
complex term which we called Z{n
i
i}
to this term that is easy to calculate. This is not a trivial problem [227]. We will follow
the general approach used by Trayanov and Tosatti in their analysis of a somewhat
simpler model. The idea is that we can make good approximations to the free volume
in both the fluid and crystalline phases, and so an interpolation between these two
extremes would then give the free volume as a well-behaved function of the order parameters. We will not implement the same approximation because their interpolation
is based on the use of hard-sphere reference functions and equations of state; because
our particles are not necessarily round this will not be a good approximation. In a
lattice of fixed geometry, the free volume of the crystal phase is most easily obtained
via numerical evaluation of the integral, since it only needs to be done once. We
plan to allow the lattice constants to relax as part of the optimisations, which would
require many re-evaluations of the integral, which is computationally expensive. Instead, we will generate an approximation to the integral which we expect to be valid
for “reasonable” lattice deformations.
Removing the anisotropy
As mentioned before, we cannot handle the translation-rotation coupling in the entropy of a condensed phase in as simple a model as this one. In what follows, the
orientational entropy and the translational entropy have been entirely decoupled by
assuming that when calculating the free volume the particles are all aligned to a spacefixed axis, and when calculating the orientational entropy they are all sitting on lattice
sites. This means that the free volume is that of the system in a strong ordering field,
so that p ' 1. This is a harsh approximation, and we expect it to over-estimate the
total entropy.
86
A Better Theory
Free volume of the liquid
We have already removed all the correlations from the thermodynamics, and so we
will approximate the free volume of the liquid phase by the amount of space not taken
up by the particles themselves. We get
f
Vliq = v0 − (n/2)Vcore ,
(4.23)
where Vcore is the core volume of a particle, which we will take to be the volume
enclosed by the potential-minimum radius. This should be an increasingly good approximation as the density is lowered and correlations between particles become less
important; particles in the vapour phase will move freely within a given lattice cell.
Free volume of the solid
The mean-field approximation to the free volume is just
F
=
V f = Z{n}
v0
dr exp −βU (n, s, (r − r0 )),
(4.24)
which is the partition function for a particle moving within a mean potential. The
dependence on the orientation of the particles has been removed from this expression.
In this model, the solid phase are characterised by n ' 1 and s ' 1. Each particle in the
solid sits in a deep potential well centred on its lattice site and oscillates around that
site. The classical entropy of the system is then given by an integral over the “visited”
part of the potential well. We can fit a harmonic potential to the field felt by a particle
in a perfect crystal. For this type of potential, we can do the integral using an error
function. That is, the potential relative to the centre of the well is approximated by
βV (x, y, z) = Ax2 + By2 +Cz2 .
(4.25)
x, y and z are coordinates relative to the lattice point in the centre of the cell. Then the
crystalline free volume of a particle in a tetragonal cell with sides of lengths a, b and
c in the x, y and z directions is
r
π √ f
Vc =
erf a A ×
(4.26)
A
r
r
π √ π √ erf b B ×
erf c C .
B
C
Now, in order to determine the values of A, B and C we will assume that the potential
well is of the same “shape” as the unit cell, which tells us the ratios of A to B and A
4.1 Bulk Systems
87
to C. Define br as the ratio b/a in the isotropic crystal, and cr as the ratio c/a in the
p
p
same way. Then B/A = br a/b and C/A = cr a/c. Then the potential is
b r a 2 2 cr a 2 2
β
2
V (x, y, z) = x +
y +
z ,
A
b
c
(4.27)
where all of the quantities are known except for the scaling parameter A. We may fit
this equation by choosing a single point (x0 , y0 , z0 ) away from the origin, calculating
the potential V (x0 , y0 , z0 ), and calculating A from
b r a 2 2 c r a 2 2
β
2
×V (x0 , y0 , z0 ) = x0 +
y0 +
z0 .
A
b
c
(4.28)
B and C may then be determined from A and the ratios mentioned earlier. This turns
out to be a fairly robust recipe for a crystalline free volume in bulk calculations. Also
note that the choice of a one-point fitting procedure means that it is easy to differentiate all of these relations, which we require in order to do efficient minimisations.
Furthermore, the choice of the fitting point will influence the resulting values of A, B
and C, unless the true potential is actually ellipsoidal. In the implementation of this
theory, in order to somewhat relieve this problem we chose six different fitting points
and used the average V (x0 , y0 , z0 ) and r02 = x20 + y20 + z20 to do the fitting. The points
were chosen in the directions of the six nearest Oh holes in the lattice, each at an extension of 1/2 times the spacing between neighbours on one sublattice. r 0 is defined
in lattice spacings rather than σ, so that we do not encounter difficulties with potential
cutoffs in relaxed-lattice calculations. For the bulk phases of an isotropic crystal this
procedure is redundant, because the six sampled points all return the same values, but
for an anisotropic potential or an inhomogeneous system this sampling should be a
considerable improvement.
Interpolating between solid and fluid free volumes
Trayanov and Tosatti manage this by application of a theory of bulk melting due to
Ramakrishnan and Yussouff [186] to the problem, and use of the hard-sphere equation
of state. We shall apply a considerably simpler approximation, while maintaining
the qualitative aspects of the more complex theory. We have tried several different
interpolation functions; what follows is the simplest one that gives realistic behaviour.
Trayanov and Tosatti observe that, for small ‘crystallinity’ c, defined as c = |2s − 1|,
the free volume is fit well by a gaussian function of c. A simpler choice would be to
choose the width so that the value of the free volume evaluated at c ' 1 would just be
88
A Better Theory
f
Vc . This completely determines the width, and the resulting approximation is
! #
"
f
Vc
f
c2 ,
V f = Vliq × exp log
(4.29)
f
Vliq
which is the same as a linear interpolation between the logarithms of the volumes
based on c2 :
f
logV f = 1 − c2 logVliq + c2 logVcf .
(4.30)
4.1.6 Using Different Ensembles
Lattice gas calculations are usually performed in the grand canonical ensemble, where
the external fields (or “control parameters”) are the temperature T and chemical potential µ. This derivation was done in the canonical ensemble, but the conversion
between the two is simple. The grand partition function is just
Ξ(µ,V, T ) = ∑ eβµN Z (N,V, T ).
(4.31)
N
In the mean-field approximation, as we discovered in the analysis of the BlumeEmery-Griffiths model, this just reduces to
ΞMF (µ,V, T ) = eβµnM Z (nM,V, T ).
(4.32)
For a given state point (T, µ), the equilibrium state of the system will be the one that
minimises the grand potential per unit volume of the system, provided that the system
is homogeneous. In the case of a rigid-lattice model, minimising the grand potential
per unit volume is effectively the same as minimising the grand potential per lattice
cell; if the lattice constants are allowed to vary, the volume of the cell must be taken
into account. In the implementation of this theory, the configurational entropy and
energy are calculated on a per unit cell basis, while the orientational free energy and
the free volume are calculated per particle; these differences must be accounted for
in the construction of our thermodynamic potential.
4.2 Theorems, Identities and Relations
4.2.1 Bulk Thermodynamics
The following theorems show the equivalence of performing our calculations using
the Gibbs free energy G or the grand potential Ω as the quantity to be minimised.
4.2 Theorems, Identities and Relations
89
Theorem 1 For a homogeneous system, G/N = µ and Ω/V = −P.
Proof:
First we define the per-particle and per-cell quantities:
G(N, P, T ) = E + PV − T S
E
S
+ P/ρ − T
= N × G0 (P, T ).
= N×
N
N
Ω(µ,V, T ) = E − T S − µN
E
S
= V×
−T
− µρ = V × Ω0 (µ, T ).
V
V
(4.33)
(4.34)
If (S/N) and (E/N) are independent of N, as in a homogeneous system, then
∂G
µ≡
= G0 (P, T ).
(4.35)
∂N P,T
Therefore,
µ = (E/N) + P/ρ − T (S/N),
ρµ = (E/V ) + P − T (S/V ),
−P = Ω0 (µ, T ).
(4.36)
(Alternatively, we could have used the relation
1
PV = − log Ξ(µ,V, T ) = Ω(µ,V, T )
β
(4.37)
to finish the proof.)
We also need to show that if a set of general order parameters {ni } minimises one
of these potentials, then it also minimises the other, at the corresponding state point.
Theorem 2 If a set of order parameters {ni } is a minimum of Ω0 at constant T and
µ, then they are also a minimum of G0 at constant P and T .
Proof:
0
∂G0
∂G0
∂G
∂Ω0
=
=−
,
(4.38)
×
∂{ni } P,T
∂{ni } −Ω0 ,T
∂Ω0 {ni },T
∂{ni } G0 ,T
but we have already shown that G0 = µ, so that if the {ni } are a minimum of Ω0 , then
(∂Ω0 /∂{ni })G0 = 0, so
∂G0
∂Ω0
=
= 0.
(4.39)
∂{ni } µ,T
∂{ni } P,T
90
A Better Theory
The following theorem demonstrates one of the drawbacks of the theory that we
have derived; that some of the thermodynamics are lost in the mean-field approximation.
Theorem 3 All phases in this model have the same isothermal compressibility, that
of the ideal gas.
Proof:
(All derivatives are at constant T .) The isothermal compressibility is
∂P −1
.
κT = V −
∂V
We have already showed that P = −Ω/V , so that
1 ∂Ω
∂P
1
= 2Ω−
.
∂V
V
V ∂V
(4.40)
(4.41)
The first term, V12 Ω, is just the ideal gas contribution. The second term contains the
non-ideal parts. In the mean-field lattice model, V = M × (abc/n), where a, b and c
are the lattice cell edge lengths. Therefore,
∂n ∂Ω
∂a ∂Ω
∂b ∂Ω
∂c ∂Ω
∂Ω
=
×
+
×
+
×
+
×
.
(4.42)
∂V
∂V
∂n
∂V
∂a
∂V
∂b
∂V
∂c
However, the equilibrium condition for the grand potential implies that the partial
derivatives of Ω with respect to a, b, c, and n are all zero, so that this term contributes
nothing, and we are left with only the ideal gas result.
4.2.2 Surface Thermodynamics
In general, for a (one-component) system composed of two phases separated by an
interface of area σ, the total derivative of the energy is
dE = T dS − PdV + µdn + γdσ,
(4.43)
which defines the surface tension γ as γ = dE/dσ. We can integrate this [228] and get
E = T S − PV + µn + γσ.
(4.44)
In order to define surface excess quantities, construct an artificial system with an infinitely sharp boundary located at some position zd , and say that each phase maintains
4.2 Theorems, Identities and Relations
91
n(z)
α
β
z
zd
Figure 4.1: Picture of the dividing surface and density profiles. The dashed
lines are the density profile of the artificial construction, and the dotted line
shows the position of the dividing surface.
its bulk properties right up to zd . This construction is illustrated in Figure 4.1. Then
for any extensive state function X we can define the surface excess X as
Xex = X − Xα − Xβ ,
(4.45)
where X is the value of the quantity for the real system. For instance, the excess
energy of the interface is
Eex = E − Eα − Eβ
= T S − Sα − Sβ − P V −Vα −Vβ
−µ n − nα − nβ + γσ
= T Sex − µnex + γσ.
(4.46)
Then the surface tension is
γ=
1
(Eex − T Sex + µnex ) .
σ
(4.47)
If we choose zd such that nex = 0, then the surface tension is equal to the Helmholtz
free energy per unit area, Aex /σ. This can always be done, and the solution to this
problem is called the Gibbs dividing surface [229, 230]. The Gibbs dividing surface
(zd ) is defined as the solution to the equation
zd
−∞
dz (n(z) − nα ) +
∞
zd
dz n(z) − nβ = 0,
(4.48)
92
A Better Theory
where n(z) is the density profile of the real system and nα and nβ are the bulk densities
of the α and β phases. A solution always exists for this problem. We can also calculate
the excess grand potential of the system:
Ωex = Ω − Ωα − Ωβ
= Eex + µnex − T Sex
= (T Sex − µnex + γσ) + µnex − T Sex
= γσ.
(4.49)
This is a remarkable identity2 . At the coexistence of the α and β phases, the grand
potentials per unit volume Ω0 of the two phases are equal, so that
Ωex = Ω − Ωα − Ωβ
= Ω −Vα Ω0α −Vβ Ω0β
= Ω −V Ω0α ,
(4.50)
and we can determine the surface tension without knowing the position of the dividing surface. Once the surface tension is known, we may calculate the other surface
thermodynamic properties from its derivatives. This is a somewhat subtle problem,
because we must take all of these derivatives along the phase coexistence line, upon
which the temperature and chemical potential are not independent variables. That is,
the surface tension is a function of only one variable, and we may parameterise it by
either T or µ. I shall follow Landau and Lifshitz’s treatment [231, pages 517-519]. In
addition, we shall always be using the Gibbs dividing surface.
At constant volume, the derivative of the total grand potential of the system is
dΩ = SdT + Ndµ + γdσ,
(4.51)
from which we determine that the surface tension at constant T and µ is
2 Alternatively, we
dΩ = γdσ,
(4.52)
Ωex = γσ.
(4.53)
could have started from Equation 4.46:
Eex
Eex − T Sex
Aex − µnex
Ωex
= T Sex + µnex + γσ
= µnex + γσ
= γσ
= γσ
4.3 Modelling the Surface
93
We may also calculate nex from this expression, by saying
∂Ωex
,
(4.54)
nex = −
∂µ T
∂γ
= 0 and applied Equation 4.51. The
where we have implicitly assumed that ∂µ
T
entropy of the whole system can be obtained from Equation 4.51, and is
∂Ω
S=−
.
(4.55)
∂T µ,σ
Its surface part is
∂Ωex
Sex = −
∂T
µ,σ
= −σ
dγ
,
dT
(4.56)
where we have replaced a partial derivative by an exact one.3 Once the entropy is
known, the excess energy of the interface may be determined from A ex = γσ = Eex −
T Sex :
dγ
.
(4.57)
Eex = σ γ − T
dT
The heat capacity at constant volume may be determined from either the energy or
the entropy:
d2γ
Cex = −σT 2 .
(4.58)
dT
4.3 Modelling the Surface
Deriving the theory for a surface proceeds along exactly the same lines as the bulk
theory. We partition the space into a lattice of cells, define order parameters, and make
the mean-field approximation to obtain a minimisable recipe for the free energy. The
only difference is that, just as we did in the BEG model, we define order parameters
for each layer of lattice cells perpendicular to some chosen direction so that we may
use the resulting system of equations to study planar interfaces along arbitrary crystal
indices.
3 because
because
γ is formally a function of T and µ, write
∂γ
∂γ
dγ(T, µ)
∂µ
=
+
dT
∂T µ
∂µ T ∂T at
∂γ
,
=
∂T µ
∂γ
∂µ T
= 0 as mentioned earlier.
coexistence
94
A Better Theory
There are some differences between the extension to surfaces of this model and of
the BEG model. The principal one is that in this model the γ(n, s) function is considerably more complex, and the restriction that no two adjacent sublattice sites both be
occupied implies a “connection” between adjacent layers of the surface system which
was not present before. This leads to additional terms in the γ function which involve
the densities and crystallinities of adjacent layers. The second major difference is that
the potential in this model is not necessarily short-ranged, so that the energetic terms
will connect non-adjacent layers. Recall that we have not yet specified any details of
the potential function.
4.3.1 γ Function for the Slab System
The configurational function γ(n, s) is now transformed into a new function, γ({n i , si })
where ni and si are the equilibrium average of the density and crystallinity in the ith
layer. This new expression can be derived in several ways, either by the argument of
Trayanov and Tosatti or as a special case of the method used by Asada [177, 232] in
his study of multi-layer lattice gases in the quasi-chemical approximation. The latter
is the more satisfying derivation. In either case, the resulting expression is:
γ({ni , si }) = ∏ γi ({ni , si }),
(4.59)
i
where
"
#
(zm M)!
γi ({ni , si }) '
×
(4.60)
∏ m
m
m
m
m=−1,0,1 NNa−Cl !NNa−h !Nh−Cl !Nh−h !
z−1 z−1
M!
M!
,
(Mns)! (M − Mns)!
(Mn(1 − s))! (M − Mn(1 − s))!
which looks very much like equation 4.15, except for the product over m. In this expression z1 is the number of neighbouring sites of a given lattice in the lattice plane
over its own, z0 are the number of neighbours in its own plane, and z−1 are the number
in the plane below it. z remains the total number of lattice sites, equal to the sum of
1
is the number of occupied Na − Cl pairs where the
the zm . In this expression, NNa−Cl
th
Na is in the i layer of the system and the Cl site is in the (i + 1)th layer. In this way,
all sites in all layers are connected to all of their neighbouring sites in all layers. As
m
m
m
before, we require that NNa−Cl
= 0 for all m, so that the NNa−h
and Nh−Cl
quantities
m
are determined by the ni and si , and this determines Nh−h . We then continue by generously applying Stirling’s approximation. In order to simplify the final expression,
4.3 Modelling the Surface
95
we introduce the following notation:
A0 = n i s i ,
A−1 = ni−1 si−1 ,
A1 = ni+1 si+1 ,
B0 = ni (1 − si ),
B−1 = ni−1 (1 − si−1 ),
B1 = ni+1 (1 − si+1 ).
(4.61)
The resulting formula for the combinatorial entropy is:
1
log γi ({ni , si }) =
M
+
+
−
−
(4.62)
1 − A0 − B−1
1 − A0 − B−1
z−1 A0 log
+ B−1 log
− log (1 − A0 − B−1 )
A0
B−1
1 − A 0 − B0
1 − A 0 − B0
z0 A0 log
+ B0 log
− log (1 − A0 − B0 )
A0
B0
1 − A 0 − B1
1 − A 0 − B1
+ B1 log
− log (1 − A0 − B1 )
z1 A0 log
A0
B1
n i si
ni (1 − si )
(z − 1) −ni si log
− ni (1 − si ) log
1 − n i si
1 − ni (1 − si )
(z − 1) log [(1 − ni si ) (1 − ni − ni si )] .
(4.63)
These equations introduce additional problems to be overcome in the minimisation,
in that we must take care that the terms like (1 − A0 − B−1 ) are always positive. These
constraints are a major factor in determining the structure of the solid-liquid interface.
4.3.2 Free Volume in the Slab System
The free volume in the slab system is handled in exactly the same way as it was in
the bulk system; in each layer, the free volume is calculated as a function of s i and ni ,
with the lattice spacings in that layer and nearby ones entering indirectly through the
energy terms and v0 . The ni and si of neighbouring layers do not contribute to the free
volume of a given layer, since they do not directly affect the density or crystallinity
in that layer. Because the crystalline free volume is only a function of the lattice
spacings, only these variables in neighbouring layers have any effect.
4.3.3 Internal Energy in the Slab System
The internal energy in the slab system is calculated exactly as one would expect; a
double sum is required, connecting a single cell to every other cell in every other
layer. This is a straightforward procedure.
96
A Better Theory
4.3.4 Orientational Free Energy in the Slab System
We can apply the same theory as before to solve for the equilibrium values of the orientational order parameters. In the slab system, the orientational state is characterised
by the L values of the pi , where pi is the average value of p in the ith layer. In this case
the orientational potential felt by a single particle in the ith layer becomes a function
of pi and the p j in all of the neighbouring layers. Because pi describes the average
orientation in the ith layer, the solution to the orientational problem consists of finding
a set of self-consistent pi , where the self-consistency problem must be solved within
each layer with nearby layers entering as an external field. More specifically, we are
looking for self-consistent solutions to the L coupled equations of the form
eq
pi =
1
1
rot
ZMF
0
eq
d (cos θi ) exp −βU pi , {p j6=i }
× P2 (cos θi ) .
(4.64)
This can be accomplished either by a multidimensional root-finding or by including
the pi in the minimisation problem as explained earlier.
Chapter 5
TECHNICAL DETAILS AND
ALGORITHMS
This chapter contains some of the more important technical details of the “A Better
Theory” project, as well as some of the algorithms used in the computational side of
the study. Refer to Appendix B for further details of the programs and computers
used in this study.
5.1 Algorithms
5.1.1 Definition of the Lattice Spacings
In the actual implementation of the relaxed-lattice model for the surface calculations,
we must define the lattice spacings a, b and c in terms of some unit cell. We have used
different definitions of the three lengths for the different crystal faces, so that in each
system a and b are “in-plane” measures, and c is a vertical distance. Determining
appropriate cells for each system is trivial. In all cases, the ci are defined as the
distance between the ith and (i + 1)th lattice planes, as shown in Figure 5.1 (page 98).
In the (111) geometry system, the ‘other’ sublattice planes are defined to be halfway
between these planes, so that if the ith Na plane has a z coordinate of zi , then the ith Cl
plane has a z coordinate of zi + ci /2. This definition is unnecessary for the (110) and
(100) surfaces. (Structures of the different crystal faces are presented in Chapter 7.)
The volume of a single lattice cell is (a × b × c)/n f ace , where a, b and c are
the edge lengths of a unit cell and n f ace is the number of lattice points (Na and Cl)
contained within it. These values will all be different for different lattice geometries,
but in such a way that they describe the same bulk system, and the volume of the
lattice cell is the same.
f
In the calculation of the free volume of the liquid phase, Vliq , this definition
97
98
Technical Details and Algorithms
1
2
3
4
y
y
y
y
c1
-
c2
-
c3
-
Figure 5.1: This is a picture of the slab geometry used in the surface calculations. The value of c1 describes the distance between the lattice plane
“1” and the lattice plane “2”.
presents a small problem. The value of ci cannot alone determine the volume available to a particle in the ith layer, because it is the distance between the ith and (i + 1)th
layers, and the volume available to a particle should be centred on its lattice site, and
so extend down towards the (i − 1)th layer. In this case, we take the v0 in the free
volume expression to be a × b × (ci−1 + ci )/2n f ace , which is an average value.
Lastly, we must choose edge-lengths of a sublattice cell in order to do the calculation of the crystalline free volume. However, the gaussian that we integrate over in
this expression has a very narrow width, much smaller than the lattice spacings a, b
and c, so that the error functions have flattened out long before we get to the edge of
the cell. This means that the crystalline free volume is effectively independent of our
choice of the side lengths of the lattice cell, so that we do not need to worry about
them.
5.1.2 Performing Surface Calculations
The system that we are interested in when we do surface calculations is the interfacial
region between two bulk phases. In order to perform calculations on this region, we
set up a slab system and hold the order parameters in the top and bottom k max layers
fixed at their bulk values, where kmax is at least as large as the cutoff of the potential in
lattice-spacings. So, in order to obtain a single solution for a slab of L layers thickness
we first perform two bulk-phase minimisations, one for each phase. These solutions
are loaded into the top and bottom parts of the L-element vectors that describe the
surface system, and then we minimise the surface tension. These layers are frozen
during the minimisation.
This procedure imposes a maximum width of the interface of L − 2k max , which
5.1 Algorithms
99
constitutes a perturbation to the surface thermodynamics. The system must be chosen
large enough that this perturbation is small. Since almost all of the interfaces we
have studied (except for surface-melted systems and the liquid-vapour interface near
the critical point) have a width of only a few layers, using L = 32 and k max = 4 has
been sufficient. Many of the calculations were repeated with 64-layer slabs in order
to verify that the smaller systems were correctly behaved. Generally, the difference
between the 32-layer and 64-layer systems is a very small (10−6 ) shift in the surface
tension (down, for larger systems), with the phase behaviour unaffected.
In any study of an interface between two coexisting phases, we must insure that
the state point we are considering is on the coexistence line. The coexistence lines
themselves are found by a procedure described below. These data must be input to the
program and used to generate the bulk solutions necessary for the surface calculation.
This can be easily done with an interpolation routine that calculates T as a function
of µ (or vice-versa) from a table of (T, µ) data.
5.1.3 Calculation of the Gibbs Dividing Surface and Surface Excess Properties
In the computer program that does all of these calculations, we determine various
quantities such as the free volume per particle, the configurational γ function, and
the energy U of the system. In an interfacial system, it may be useful to determine
the surface excess values of these quantities. This requires calculation of the Gibbs
dividing surface. Recall that the Gibbs dividing surface is the solution to the equation
∞
−∞
dz (n(z) − nd (z)) = 0,
(5.1)
where nd (z) is the density profile of an infinitely sharp interface located at zd . In terms
of lattice spacings and densities per layer, this becomes:
∑ ni =
i
zd
∑i ci − z d
nα +
nβ .
cα n c
cβ n c
(5.2)
In this equation, nα and cα are the bulk values of n and c in the α phase, as usual. nc
is the height, in lattice spacings, of one lattice cell; it is dependent on the definition of
the ci used. This equation can be solved for zd :
zd =
nα
cα n c
nβ
−
cβ n c
−1
"
#
n β ∑i ci
× ∑ ni −
.
cβ n c
i
(5.3)
100
Technical Details and Algorithms
This yields a zd in units of σ, measured from the bottom of the slab. Once we have
this, any excess quantity can be calculated from its “per lattice cell” quantity through
!
" !#
0
0
X
Xα
β
Acell Xex = ∑ Xi − zd
+ ∑ ci − z d
,
(5.4)
cα n c
cβ n c
i
i
where Acell is the area of a single lattice cell on the surface, and Xi are the values, per
cell, of X in the slab. Note that we cannot calculate the thermodynamic excess entropy
and energy this way, because parts of the energy are contained in the free volume
approximation, and we cannot separate them. These quantities are best determined
by differentiation of the γ(T ) data, as explained earlier.
5.1.4 Calculation of Vcore
We have chosen, in most of these studies, to use the well-minimum radius of the
potential to determine the core volume of the particles. This is trivial for spherical
particles; just find the well-minimum, and use that radius to calculate the volume of
a sphere. For non-spherical particles, we can calculate the well-minimum distances
for placing particles side-by-side (rss ) and end-to-end (ree ); if we assume that the
2.
particles are ellipsoids of revolution, then the volume is just given by (4/3)πr eerss
5.1.5 Finding Coexistence Lines
Finding a coexistence line amounts to finding a point in (µ, T ) space where the two
phases (say, α and β) have the same free energy. This is equivalent to finding the
zeros of the function
∆Ω (µ, T ) = Ω0α (µ, T ) − Ω0β (µ, T ) .
(5.5)
Provided that we hold one of T or µ fixed, this is a one-dimensional root-finding
procedure. Since each evaluation of ∆Ω requires two separate minimisations, this
root-finding must be function-value based, as we have no derivatives. In our programs, this is done with the van Winjgaarden-Dekker-Brent method implemented in
the Numerical Recipes routine ZBRENT [42]. Once a coexistence point is known, we
can increment either the temperature or chemical potential by a small amount, and
repeat the whole root-finding procedure, finding another point.
This procedure is fairly delicate, for several reasons. The principal difficulty is that
the minimiser may, in the course of obtaining either Ω0α or Ω0β , jump from the radius of
convergence from one solution to the other and return ∆Ω ≡ 0 because the two phases
(solutions) are the same one. This can be avoided sometimes by choosing very small
5.1 Algorithms
101
step-lengths and tolerances for use in the minimisers, but even this is insufficient if the
two solutions are very close in the parameter space. In addition, if we choose a very
fine grid in T , then the bracket around the root can be taken smaller, and the procedure
is better behaved. Practically, this means that this procedure crashes a lot towards the
high-temperature end of the liquid-vapour coexistence line as the two phases become
more and more alike.
If the two phases are sufficiently similar that this automatic procedure fails a lot,
then an alternative is to attempt to find the intersection of the Ω0α (µ) and Ω0β (µ) lines at
a given temperature. This method tends to be better behaved, principally because less
of the metastable line is necessary in order to find the intersection graphically than to
find it via a minimisation. It is difficult to program this as an automatic procedure,
and it is best done “by hand”.
5.1.6 Finding the Triple Point
The triple point occurs at (µ, T ) satisfying
Ω0α (µ, T ) = Ω0β (µ, T ) = Ω0γ (µ, T ) ,
(5.6)
which completely determines Tt and µt , an example of the Gibbs Phase Rule. This
system could be solved in one of several ways; we use a minimisation procedure. (The
reason is that there are no good non-derivative-based algorithms for multidimensional
root-finding.) If we define the function
 2
Ω0α (µ, T ) − Ω0β (µ, T )
 12
+ 
 2


0 (µ, T ) − Ω0 (µ, T )
 ,
∆T Ω (µ, T ) = 
+
Ω


γ
α

 2
0
0
Ωβ (µ, T ) − Ωγ (µ, T )
(5.7)
then the minimum of this function obviously occurs at the triple point, with a function
value of zero. This is by no means a unique property of this particular function, but
it provides a nice form for the minimisation algorithm to follow. Because we have no
derivatives available for this function, the minimisation is done using the implementation of Powell’s method in the Numerical Recipes routine POWELL. Another small
advantage of this kind of procedure is that we do not need to give the routine a starting
bracket for the position of the triple point. A disadvantage of this method, and another
reason why the root-finding method is preferable for finding coexistence lines, is that
it does not converge as quickly as the root-finders.
102
Technical Details and Algorithms
5.1.7 Finding the Critical Point
In this work, since we are not particularly concerned with the critical fluid (except as a
demonstration of the quality of a phase diagram) we only determine approximately the
position of the critical point. It is easy to find supercritical temperatures because the
metastable solutions of the liquid and vapour phases disappear and there is only one
solution at any value of µ; this can be checked by restarting the minimiser from many
different points. In this way we can approximately bracket the critical temperature
between the last known point on the liquid-vapour coexistence line and the lowest
known supercritical temperature. The critical pressure (or chemical potential) can be
estimated by extrapolating the vaporisation line. If the exact location of the critical
point is necessary it can be found by looking for a point where [233]
2 ∂ P
∂P
=
= 0.
(5.8)
∂V T
∂V 2 T
This could be done with a numerical search procedure similar to the others described
in this chapter.
5.1.8 Polishing a Solution
As we demonstrate in Appendix A, it is entirely possible that the gradient of a function
can vary over a much larger range than the value of the function itself. If we minimise
numerically such a function to the limit of machine precision, then the gradient can
still have a significantly large value and cannot be used to check the quality of the
solution. What is needed is a way to polish the solution so that the elements of the
gradient are required to go to zero, rather than requiring the function value to continue
decreasing.
In the variable metric method [42] (or the conjugate gradient procedure), we construct a sequence of directions, and perform one-dimensional numerical minimisations along those directions. This procedure has to finish when two successive “lineminimisations” return essentially the same value, because we must be at the minimum. The conjugate gradient procedure actually crashes if it is forced past this point,
because the construction of the next direction involves the rate of function decrease.
In this work we use the variable metric method, which is somewhat better behaved.
Furthermore, if we are near to having precision problems, then forcing either algorithm to continue generally results in garbage answers, because the computer can’t
find the minimum within the numerical noise.
A solution is to discard the functional minimisation, and replace it with a more
accurate procedure. The idea is, when we get close to the minimum, we look for
5.2 Technical Details
103
zeros in the norm of the gradient vector along the line-minimisation directions using
the ZBRENT root-finding procedure described earlier. This will allow us to ignore the
noise in the function value and continue the minimisation. The routine should quit
when the norm of the gradient is sufficiently close to zero.
Of course, we could do all the calculations to quadruple-precision, which would
alleviate the problem, but this is costly in computer time, only available on some
makes of computer, and unnecessarily precise in the earlier parts of the minimisation
procedure.
5.2 Technical Details
5.2.1 Tolerances
As we have explained, numerical optimisation techniques yield results that are only
accurate to within some tolerance parameter. These are the tolerances that were used
in the studies in the next several chapters. All calculations of the equilibrium structure
of surface systems or bulk systems were obtained to a relative functional tolerance of
10−10 , and the norm of the gradient was required to be less than 10−4 . For surface calculations, the polishing algorithm described earlier was used to improve the gradient
tolerance to 10−5 . Triple points were usually located to a relative functional tolerance of 10−6 of the function ∆T Ω(µ, T ); see Section 5.1.6. The points on coexistence
lines were generally found to a relative tolerance in the difference in grand potential
of the coexisting phases of 10−6 . These tolerances insure that our calculations on
surfaces are extremely accurate, so that numerical differentiation of our data should
be well-behaved. Even with these requirements, calculations on surface systems generally took less that five CPU minutes per (T, µ) point on a Silicon Graphics R8000
processor, so that heating/cooling runs on a surface each ran in one or two hours.
5.2.2 Units
In all of the following calculations we will use Lennard-Jones reduced units. In some
cases we specify that certain quantities have units of σ, the Lennard-Jones length.
This is meant to distinguish these quantities from ones with units of lattice spacings.
A complete review of Lennard-Jones reduced units (and the necessary conversions to
real units) can be found in Allen and Tildesley, Computer Simulation of Liquids [13].
104
Technical Details and Algorithms
5.2.3 Constraints on the Order Parameters
There are many situations where we want to be able to constrain one or more of the
order parameters to a certain range of values, or simply to hold them fixed. In order
to constrain them, we can use Lagrange’s method of undetermined multipliers, while
to hold one or more variables fixed is a trivial problem, due to the way that the minimisation algorithms work. By setting the ith component of the gradient equal to zero,
the gradient-based minimisation algorithms “ignore” that direction completely, and
do not attempt moves along it. So in order to hold any subset of the order parameters fixed (for instance, the lattice spacings), all we need to do is reset the gradient
components for those variables to zero whenever they are recalculated. In a Powell’s
method minimisation we would remove these variables by insuring that the starting
direction set has all-zero rows for those elements. These features of the minimisation
algorithms make it very convenient to test computer programs by turning different
variables “on” and “off”.
In the discussion of surface excess quantities, we determined that any extensive
state function had an associated surface excess value. In a calculation on the surface
where the lattice spacings are allowed to relax, the surface excess of the number of
lattice points is
Mex = Msys − Mα − Mβ .
(5.9)
0 , we get
If we consider the excess number of lattice sites per area Mex
0
Acell × Mex
= L−
hα hβ
− ,
cα cβ
(5.10)
where hα is the height of the α phase (usually zd ), and cα is the height of a unit cell
in the α phase, etc. It makes intuitive sense that this quantity be constrained to zero
in any calculation we do, and that the value to constrain it to is that of the un-relaxed
surface. This is equivalent to constraining ∑i ci to a constant value ctotal . This can be
done by choosing some c f and setting, at each function evaluation,
c f = ctotal − ∑ ci .
(5.11)
i6= f
The necessary derivatives can be easily obtained from the above equation:
∂Ω
∂Ω
∂Ω
=
−
,
∂ci
∂ci unconstrained
∂c f unconstrained
where we have just used the chain rule.
(5.12)
5.2 Technical Details
105
5.2.4 Handling the Orientational Integrals
The potential that we have chosen to use can be written in the form
βU (pi ) = F0 + pi × F1 ,
(5.13)
where F0 contains all of the terms in the potential that do not contain pi (but may
contain p j ) and F1 contains all of the terms that multiply pi . (In this case, pi and p j
are the instantaneous values of the order parameters in the ith and jth cells, rather than
the average order parameters in the ith and jth layers of the slab system.) In order to
calculate the thermodynamic properties, we have to evaluate the mean-field partition
function, which means evaluating the integral
1
Ω
dΩ exp (−βU (Ω)) = 2π
0
d(cos θ) exp (−βU (cos θ)) .
(5.14)
If we write τ = cos θ, then this becomes
1
exp (−F0 ) × 2π
which is the same as
0
1
2
dτ exp −F1 3τ − 1 ,
2
F1
2π exp
− F0 ×
2
1
0
3 2
dτ exp − F1 τ .
2
(5.15)
(5.16)
The last integral is an error function. Although we could use a library routine to
evaluate this, its value is frequently very large (which confuses the library routines)
so that we have found it more convenient to use a one-dimensional lookup table 1 .
The table is constructed by a supplementary program that performs many numerical
evaluations of the integral. This is an expensive procedure, but it only needs to be
done once.
5.2.5 Self-Consistent Solution for the Orientational Problem
We have already pointed out that it is possible to determine the (bulk) equilibrium
value of p by solving the self-consistent problem, which requires solving a highly
1 There
are other reasons for using a lookup table. Earlier in this project we considered using two
order parameters to handle the orientational degrees of freedom, the other one being sin 2 θ cos 2φ.
While the mathematics for this problem are only slightly more difficult, the resulting integral cannot
be done analytically, and so we used a lookup routine. However, many other difficulties ensued, so that
this second parameter was dropped. Also, the lookup functions can be used to calculate the integral
tables for the self-consistent value of the order parameter, which requires a numerical integration as it
cannot be done analytically.
106
Technical Details and Algorithms
nonlinear equation in one variable. This can be done in a number of ways; while
testing the programs, we simply found the zero of the function
Fsc (pave ) = pave − hpi
dΩi exp (−βU (Ωi, pave )) × pi
= pave − Ω
,
Ω dΩi exp (−βU (Ωi , pave ))
(5.17)
(5.18)
which has a value of zero at pave = hpi = peq . This root-finding was done with
the RTSAFE routine from Numerical Recipes. Although this removes the quantity
p from the general minimisation problem, because we cannot differentiate through
this procedure we would have to sacrifice the use of the gradient in the free energy
minimisation in order to use the self-consistent p values at each step. Because use of
the gradient greatly speeds up the minimisation, allows for greater precision and also
allows us to easily remove degrees of freedom from the problem, this is a heavy price
to pay and it is much better to just minimise p along with everything else.
The value of hpi is useful in that we can check the minimised p against it to insure
that the solution is a good one. Also, it is necessary if we use the false p-potential described below. This quantity must be determined numerically, as it requires evaluation
of an integral of the form
1
3 2
1
2
(5.19)
dτF1 3τ − 1 exp − F1 τ .
2
2
0
This can be done using exactly the same method used to evaluate the mean-field
orientational partition function described in the previous section; a short program can
generate a table of this integral using numerical integrations, and then we can look
up the value of the integral as a function of F1 . All of these equations can easily be
extended to the multi-dimensional case.
5.2.6 False p-Potential
Although the equilibrium value of p (or of each pi , in the slab system) must occur
at the value of pave that minimises the free energy, it is entirely possible that there
are other values of pave which allow for lower (and unphysical) values of the free
energy to be found through the coupling to other variables. We must insure that any
minimisation procedure which includes the orientational order parameter must end
at a value of p which is self-consistent. It is sometimes necessary to push p in the
right direction with a false potential that is zero at the self-consistent point and large
everywhere else. A good choice is
F(p) = A × (p − hpi)2 ,
(5.20)
5.2 Technical Details
107
where hpi is defined in Equation 5.18 and is determined numerically. A is a constant.
We have found that using A between 1 and 10 works well. For larger values the
minimiser gets stuck because the potential walls are too steep. This function is equal
to zero at the self-consistent point, where p = hpi, and gets larger as we get farther
away. At the end of the minimisation, this function must be checked; if it is not equal
to zero, we have found an unphysical solution, and the process should be restarted
from a different place. In practice, either we find a good solution (F(p) = 0) or the
minimiser crashes.
5.2.7 Edge Detection and the Dummy Function
The equations that we derived earlier for the grand potential as a function of the order
parameters contain many poles and singularities which are all outside the “allowed”
parameter space. For instance, if γ(n, s) is evaluated for n > 1 or n < 0, it has a
complex value. These parts of the parameter space must be avoided smoothly, or else
the minimiser will crash. More generally, the space of {ni , si } is constrained by the
following:
0 < ni < 1,
(5.21)
0 < si < 1,
(5.22)
0 < (1 − ni si − ni+1 (1 − si+1 )) ,
(5.23)
0 < (1 − ni si − ni−1 (1 − si−1 )) ,
(5.24)
where the second two expressions come from the terms connecting the various layers
of the slab system and are not necessary in the bulk system.
For values of p greater than 1 or less that -0.5, the calculation of the orientational
entropy is meaningless, and these regions must be avoided. Lastly, the calculation of
the crystalline free volume is problematic for far-from-equilibrium extensions of the
lattice spacings, and so these must also be avoided.
In order to do all of this, we must test all of these conditions for each new evaluation of the free energy. If any of them are not satisfied, then a dummy function
is substituted for the real one. The dummy function must be chosen so as to guide
the minimiser back into the allowed regions; it should be strongly repulsive, and increasingly so with increasing violation of these conditions. Thankfully, these regions
should only be encountered while the computer is performing line-minimisations, so
that we do not need any derivative information at these points. Because the real minima do not lie in these regions, we should never have to evaluate the gradient there,
because the line-minimisations (which do not use derivative information) will always
108
Technical Details and Algorithms
lead us away from the forbidden areas. However, they may sample the function in
these areas, which is the problem. The “shape” of the dummy function is not particularly important, as all it needs to do is provide a steep wall in the free energy so that
the minimiser turns around. The dummy function that we use looks like:
if (n.gt.1.0) then
func = func + (n-1.0)*HUGE
else if (n.lt.0.0) then
func = func + (-n)*HUGE
else if (s.gt.1.0) ...
if n is too high,
Ωdummy = Ωdummy + (n − 1) × 107
if n is too low,
Ωdummy = Ωdummy + (−n) × 107
do the next condition . . .
This is just a linear function of the amount by which the various boundary conditions
are violated. We need to choose lower bounds on the lattice spacings; generally,
as long as they don’t shrink by more than about 50% of their equilibrium values
everything works well. At the end of a minimisation, we must check that all of these
conditions are satisfied, so that Ωdummy = 0, and this procedure is not perturbing the
thermodynamics.
5.2.8 False s-Potential
In the solid phases the minimum value of the order parameter s, for reasonably low
temperatures, is almost exactly equal to 1 (or 0, for the symmetric solution). This
causes great difficulty with the minimisation routine, as there is a divergence in γ(n, s)
for s = 1 as mentioned above. We have observed that the free energy surface is fairly
flat in this region, and can solve this problem by introducing a false potential that
pushes s slightly away from s = 1, and is zero everywhere else. This is entirely for
calculational convenience, and does not correspond to any physical phenomena. The
potential we have used is
h
√
i
√
Ff alse (s) = A exp −B s + exp −B 1 − s ,
(5.25)
with A = 0.50 and B = 100.0. This is a convenient form; it has a value near 0.50 for
s near 1 or 0 and is extremely small everywhere else. The effect of this potential is to
move the minimum in s down to about 0.999 (or up to 0.001), at a free energy cost
of about 0.02 units. This constitutes a very small perturbation of the solid phase, and
does not affect any of the phase behaviour of the model. Its effect is to make the solid
melt at a very slightly lower temperature.
Chapter 6
BULK STUDIES
Before we can study a surface, we need the phase diagram of the corresponding
bulk material. This chapter contains the phase diagrams generated for many different
choices of potential, using both relaxed lattices and restrictions of the model where
the lattices were frozen at well-minimum spacings. We have done several studies on
the bulk phase diagram while varying the input parameters Vcore , r0 , the lattice spacings (if fixed) and the range of the potential, in order to determine how sensitive the
model is to these parameters. We compare our Lennard-Jones phase diagram to the
one obtained by Trayanov and Tosatti.
6.1 Isotropic Potentials
We calculated phase diagrams for the cut-and-shifted Lennard-Jones potential, both
with fully relaxed and frozen lattice spacings. The potential is cut-and-shifted at r cs =
3.0σ. These phase diagrams are shown in Figures 6.1 and 6.2 on pages 110 and 111.
The principal difference between the phase diagrams of the two models is that the
solid-liquid coexistence line is shifted to lower temperatures in the relaxed model; this
shift also accounts for the lower triple point pressure in this model, as the solid-vapour
and liquid-vapour coexistence lines are not much changed by the relaxed lattices.
The triple point temperature of argon in Lennard-Jones reduced units is Tt = 0.70,
and its triple point pressure is Pt = 1.6 × 10−3 [50]. The frozen-lattice model is actually closer to the correct temperature than the relaxed-lattice model. The critical point
of argon occurs at Tc = 1.25 and Pc = 0.116, so that the two models both underestimate the critical temperature by 10-15%, and find nearly double the correct critical
pressure. Table 6.1 (page 112) compares some of these data.
Neither model is outstanding, as both are in error by roughly the same magnitude in opposite directions. The relaxed-lattice model consistently overestimates the
density of the liquid phase, while the frozen-lattice model underestimates it. The two
109
110
Bulk Studies
0.8
Pressure
Solid
0.6
0.4
0.2
Liquid
Vapour
0.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.0
1.1
Temperature
Chemical Potential, µ
-2.5
Liquid
Solid
-3.0
Vapour
-3.5
-4.0
0.4
0.5
0.6
0.7
0.8
0.9
Temperature
Figure 6.1: (P, T ) and (µ, T ) phase diagrams of the Lennard-Jones system,
calculated with the model using fully relaxed lattices. The lattice spacings
are different for all three phases. The triple point occurs at Tt = 0.615446
and Pt = 3.8804 × 10−3 , at a chemical potential of µt = −3.557425.
6.1 Isotropic Potentials
111
Pressure
0.8
0.6
0.4
Solid
0.2
Liquid
Vapour
0.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Temperature
Chemical Potential, µ
-2.0
Solid
Liquid
-3.0
Vapour
-4.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Temperature
Figure 6.2: (P, T ) and (µ, T ) phase diagrams of the Lennard-Jones system, calculated with the model using lattices frozen at an inter-particle
separation of 21/6 σ. The triple point occurs at Tt = 0.688307 and Pt =
1.4798 × 10−2 , at a chemical potential of µt = −3.163117.
112
Bulk Studies
argon frozen-theory
Tt
0.70
0.6883
Pt
0.0016
0.0148
ρliq (Tt ) 0.8342
0.7691
Tc
1.25
1.06
Pc
0.116
0.232
ρliq (Tc ) 0.3156
0.2699
relaxed-theory
0.6154
0.0039
0.9386
1.10
0.252
0.3551
Table 6.1: Comparison of triple point and critical point data for argon and
both theories. Note that the two theories seem to fit the experimental data
equally well.
models nearly agree on the location of the critical point, and are in similar error on the
critical pressure. There are several reasons for these errors. Consider the nature of the
mean-field approximation. In a real (dense) liquid, each liquid particle is surrounded
by an icosahedral shell of other particles. This is surrounded by a less well-defined
second coordination shell, etc. This ordering behaviour is what gives rise to the peaks
in the radial distribution function of the liquid. The correlation of particles in the liquid causes a large energetic stabilisation of the liquid phase, due to energy-favourable
packing of the particles. This behaviour causes a drop in the free volume of each
particle, due to an increase in the depth of the local potential field. All of these effects
are absent in this model, in which the radial distribution function is effectively flat.
On this basis, we would expect that the melting temperature of the model would move
relative to that of the real material, and the direction that it moved would be given by
the relative weights of the entropic “bonus” afforded by the mean-field approximation
and the energetic penalty. Table 6.2 shows some data on the solid, liquid and vapour
phases at the triple point of both models.
In the relaxed-lattice model, the lattice spacings of the liquid phase are contracted
considerably, increasing its density to nearly that of the solid. This is an energeticallydriven effect; by reducing the lattice spacings and increasing the density per unit cell
n, the internal energy of the liquid phase is increased. In a real liquid, such behaviour
would correspond to a huge reduction in the entropy of the phase, but the mean-field
approximation precludes this because it depends on correlation between particles.
In the frozen-lattice model, the liquid is described better because we have fixed its
length scale to be approximately that of the real liquid, in which the first-neighbour
separation occurs near the potential well-minimum. In fact, the mean-field liquid allows for only six neighbours at this distance (at n = 1), and six more at the
next-nearest lattice-point distance, while in the true liquid, particles are on average
6.1 Isotropic Potentials
113
frozen-lattice
phase
ρ
n
s
solid
0.9990 0.9990 0.9986
liquid 0.7691 0.7691 0.500
vapour 0.0115 0.0115 0.500
a=b=c
1.587401
1.587401
1.587401
relaxed-lattice
phase
ρ
n
s
solid
0.9792 0.9993 0.9987
liquid 0.9386 0.8239 0.500
vapour 0.0032 0.0029 0.500
a=b=c
1.5982
1.5200
1.5360
Table 6.2: Properties of the frozen-lattice model (top) and the relaxedlattice model (bottom) at the triple point. a, b and c are the lattice spacings
in the fcc crystal equivalent
to unit cell measurements of the NaCl struc√
ture, with a, b, c = ri j 2 where ri j is the inter-particle separation.
twelve-coordinate.
Trayanov and Tosatti calculate in their model a triple point of Tt = 1.0662, Pt '
0.06, and a critical point of Tc = 1.229, Pc ' 0.125. This corresponds to a better fit
to the critical point than either of the models presented above, and a much worse fit
to the triple point. This is due to several different effects. First of all, as we shall
see, the use of a cut-and-shifted Lennard-Jones potential does have some effect on the
position of the coexistence lines, particularly those of the vapour phase, and the use
of a longer ranged potential would improve our critical point. The second difference
between these models is the choice of Vcore used in the determination of the liquid
free volume. Trayanov and Tosatti use a Vcore determined from hard-sphere studies
by Weeks, Chandler and Andersen [194,234] that is somewhat smaller that ours. This
causes an entropic stabilisation in the liquid phase, which can then become denser,
the effect of which is to increase the critical temperature and bring down the critical
pressure. Lastly, the different crystalline free volumes are responsible for the different
positions of the triple point. As already mentioned, it would be impossible to use
numerical integrations in a relaxed-lattice model, so that we turned to an approximate
solution. Because they use a fixed lattice, Trayanov and Tosatti do perform these
integrals, and obtain a better (and much larger) estimate of the crystalline free volume.
This causes the melting point to move to an artificially high temperature, due to a
poor description of the liquid phase. Essentially, their triple point is poor due to a
better description of the solid phase, while these models have a better triple point
due to a worse description! As we shall see in our analysis of the surfaces of the
114
Bulk Studies
Lennard-Jones crystal, the phase behaviour of the surface is strongly influenced by
the temperature at which it melts and the corresponding liquid density while it is
much less dependent on other factors.
6.2 Sensitivity of the Model
There are several parameters of this lattice model which are not derivable from a
first-principles approach, and must be chosen. These are:
1. Vcore , the volume “excluded” by a single particle.
2. r0 , the extension at which the harmonic potential in the approximation to the
crystalline free volume is fit.
3. a, b and c, the lattice spacings in the crystal. If the lattice is frozen, the phase
diagram will be dependent on these.
4. rcs , the radius at which the Lennard-Jones potential is cut-and-shifted.
Instead of recalculating the entire phase diagram for a range of combinations of
all of these parameters, we will instead consider them as independent variables and
look at their effect on the position of the triple point. Since we expect the position of
the triple point to strongly influence the surface behaviour of the model, this should
be a good test.
Study #1: Varying a, b and c in the frozen-lattice model
This study (see Table 6.3) shows that the relative stability of the solid and liquid
phases is strongly influenced by the choice of lattice cell dimensions, though this
is not at all surprising. The smallest cell dimensions shown correspond to a huge
compression (7% by volume) of the solid phase, so that it is destabilised and melts at a
much lower temperature. As we have explained, the liquid phase enjoys a compressed
lattice, so that this effect is accentuated. For any reasonably small perturbation of the
lattice spacings, though, the triple point only moves by a few percent. The pressure
is certainly the most sensitive variable shown. Considering the bulk phase diagrams
presented earlier, we expect that the critical point should be equally sensitive to the
choice of fixed lattice dimensions, since the energetics of the liquid phase are strongly
influenced by the lattice spacings, while the vapour phase should not be affected.
6.2 Sensitivity of the Model
a,b,c
1.550
1.575
1.585
24/6
1.590
1.595
1.600
1.625
115
Tt
0.5780
0.6566
0.6825
0.6883
0.6944
0.7057
0.7164
0.7637
µt
-3.6030
-3.2943
-3.1876
-3.1631
-3.1371
-3.0884
-3.0414
-2.8295
Pt
2.301 × 10−3
9.074 × 10−3
1.356 × 10−2
1.480 × 10−2
1.622 × 10−2
1.916 × 10−2
2.239 × 10−2
4.339 × 10−2
Table 6.3: Dependence of the triple point on the choice of lattice spacings
in the frozen-lattice model. These are the lattice spacings in the fcc crystal
equivalent
to unit cell measurements of the NaCl structure, with a, b, c =
√
ri j 2 where ri j is the inter-particle separation.
Study #2: Variation of the triple point with r0
The value of r0 used in the studies already presented is 0.5rnn (rnn is the nearestneighbour distance in the lattice); in this study we multiply that by various values of
r0 /0.5rnn , and recalculate the triple point.
The triple point varies over a considerable range in Table 6.4 (page 116), but is
only really perturbed by choosing the fitting point very close to the lattice site (small
r0 ). This illustrates the fundamental difficulty with this fitting procedure, that the potential well has very steep walls that vary as (ri j − r)−6 for very small displacements,
and (ri j − r)−12 for larger ones, and we are trying to fit it with a quadratic function of
r. Furthermore, very small choices of r0 , which may describe the “visited” part of the
potential better, cause unfortunate difficulties with the fitting procedure described earlier so that the lattice spacings in the relaxed model grow to very large values and the
solid evaporates. For larger choices of r0 the triple point moves only slightly, so that
we expect the shape of the fit is not changing much in this region. For the last point
shown, the trend of the previous points is reversed; for this extension, the particle has
actually moved “over” a repulsive wall, and is in a more attractive region again. This
is an unrealistic choice of extension, but is included for completeness.
In the relaxed-lattice model, (see Table 6.5, page 116) the triple point is more
sensitive to smaller choices of r0 that in the frozen-lattice model, and is equally insensitive to larger choices. Also, for smaller choices of r0 the density of the solid
phase at the triple point is strongly perturbed in this model, while it remains very
close to 1.0 in the frozen-lattice model.
116
Bulk Studies
r0 /0.5rnn
0.600
0.700
0.800
0.900
0.950
1.000
1.050
1.100
1.200
1.300
Tt
0.8740
0.8114
0.7588
0.7176
0.7015
0.6883
0.6781
0.6708
0.6642
0.6649
µt
-2.8787
-2.9574
-3.0366
-3.1076
-3.1376
-3.1631
-3.1834
-3.1983
-3.2121
-3.2106
Pt
7.903 × 10−2
4.895 × 10−2
3.082 × 10−2
2.044 × 10−2
1.717 × 10−2
1.480 × 10−2
1.314 × 10−2
1.204 × 10−2
1.109 × 10−2
1.119 × 10−2
Table 6.4: Position of the triple point as a function of the r0 parameter
in the frozen-lattice model. The “1.000” value is that used for the bulk
studies presented earlier.
r0 /0.5rnn
0.800
0.900
1.000
1.050
1.100
1.200
Tt
0.7093
0.6523
0.6154
0.6033
0.5869
0.5868
µt
-3.3267
-3.4609
-3.5574
-3.5911
-3.6378
-3.6378
Pt
1.3765 × 10−2
6.6824 × 10−3
3.8804 × 10−3
3.1916 × 10−3
2.4195 × 10−3
2.4200 × 10−3
ρsolid
0.9376
0.9593
0.9791
0.9827
0.9878
0.9878
a,b,c
1.6211
1.6091
1.5983
1.5963
1.5936
1.5936
Table 6.5: Position of the triple point as a function of the r0 parameter in
the relaxed-lattice model. The a,b and c values are for the solid phase.
The “1.000” value is that used for the bulk studies presented earlier.
6.2 Sensitivity of the Model
Vcore
0.5181
0.5921
0.6661
0.7401
0.8141
0.8881
117
Tt
0.6095
0.6341
0.6607
0.6883
0.7153
0.7502
µt
-3.5840
-3.4519
-3.3099
-3.1632
-3.0192
-2.8860
Pt
3.470 × 10−3
5.640 × 10−3
9.201 × 10−3
1.478 × 10−2
2.298 × 10−2
3.389 × 10−2
Table 6.6: Position of the triple point as a function of the Vcore parameter
in the frozen-lattice model. Vcore = 0.7401 is the volume enclosed by the
well-minimum radius of the Lennard-Jones potential.
Vcore
0.5921
0.6661
0.7401
0.8141
0.8881
Tt
0.5049
0.5582
0.6154
0.6693
0.7149
µt
-4.1579
-3.8623
-3.5574
-3.2833
-3.0598
Pt
2.680 × 10−4
1.110 × 10−3
3.880 × 10−3
1.039 × 10−2
2.152 × 10−2
a,b,c
1.5855
1.5913
1.5983
1.6103
1.6177
Table 6.7: Position of the triple point as a function of the Vcore parameter
in the relaxed-lattice model. The a,b and c values are for the solid phase.
Study #3: Variation of the triple point with Vcore
Table 6.6 shows that the triple point parameters are not extremely sensitive to this
quantity for small perturbations of Vcore . Larger values of Vcore lead to an entropic
de-stabilisation of the liquid phase so that the triple point moves to higher temperatures and the solid phase is globally stable over a larger range of temperatures. The
chemical potential at the triple point also varies with Vcore ; because the vapour phase
is not perturbed much by different choices of the core volume, what we are seeing
here is the melting line moving to different temperatures while the sublimation line
stays in much the same place.
The data in Table 6.7 shows that in the relaxed-lattice model we get a larger sensitivity on Vcore due to the peculiar contraction of the liquid phase lattice spacings
discussed earlier. For smaller choices of Vcore the liquid lattice contracts even more
(see below) and the triple point moves to even lower temperatures, since the free
energy of the solid phase is effectively independent of this quantity.
118
Bulk Studies
rcs
3.0σ
3.5σ
4.0σ
5.0σ
6.0σ
7.0σ
8.0σ
Tt
0.6883
0.7079
0.7164
0.7258
0.7298
0.7318
0.7328
µt
-3.1631
-3.2994
-3.3821
-3.4633
-3.4997
-3.5183
-3.5283
Pt
1.478 × 10−2
1.420 × 10−2
1.350 × 10−2
1.296 × 10−2
1.271 × 10−2
1.258 × 10−2
1.251 × 10−2
Table 6.8: Dependence of the triple point on the range of the potential, in
the frozen-lattice model. rcs is the range at which the potential is cut-andshifted.
Study #4: Range of the potential in the frozen-lattice model
The data in Table 6.8 shows that the choice to cut-and-shift the potential at 3.0σ constitutes a small perturbation of this system, which is much smaller than those due
to arbitrary choices of the other parameters. For longer-ranged potentials, the triple
point moves to slightly higher temperatures and lower pressures. However, performing calculations with very long cutoffs is quite expensive; there are approximately
2150 particles inside an 8σ radius in the solid phase, whereas there are only 113 inside a 3σ radius. Although this is not a problem in calculations using fixed lattices
(since the summations need be done only once), in our relaxed-lattice model it would
be prohibitively expensive.
f
Study #5: Variation of Vc with r0 in the frozen-lattice model
This study was performed at T = 0.60; the chemical potential does not affect the
f
crystalline free volume. No actual solutions are necessary for this study, since Vc
is a function of temperature, r0 , and the lattice spacings only. Table 6.9 shows that
the crystalline free volume can be made to vary over a range of about ±25% if the
r0 parameter is adjusted over a range of about ±40%. Although the total variation is
small, since the entropy goes as the logarithm of the free volume this can cause quite
a perturbation of the phase diagram, as we have shown.
Trayanov and Tosatti expect a crystalline free volume near the triple point of approximately 8.0 × 10−3 , which is considerably higher than any determined by these
fitting procedures. This is roughly in keeping with estimates of the entropy of the solid
phase which can be determined from its heat capacity [235, 236]. However, the entropy of the solid phase is increased by collective motions of atoms (phonons), so that
we would expect a very accurate mean-field treatment to underestimate this quantity.
6.2 Sensitivity of the Model
119
r0 /0.5rnn
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
f
Vc
2.966 × 10−3
2.318 × 10−3
1.828 × 10−3
1.474 × 10−3
1.230 × 10−3
1.073 × 10−3
9.860 × 10−4
9.540 × 10−4
Table 6.9: Dependence of the crystalline free volume on the r0 parameter,
in the frozen-lattice model. These runs were performed at T = 0.60; the
chemical potential does not influence the crystalline free volume.
frozen-lattice
Vcore
ρliquid Pressure
0.4441 0.8573 0.7879
0.5181 0.8446 0.5627
0.5921 0.8280 0.5240
0.6661 0.8059 0.3806
0.7401 0.7764 0.2331
0.8141 0.7375 0.0875
0.8881 0.6883 -0.0478
Vcore
0.5181
0.5921
0.6661
0.7401
0.8141
0.8881
1.0362
relaxed-lattice
ρliquid a, b, c
1.0340 1.5025
1.0062 1.5067
0.9651 1.5118
0.9123 1.5176
0.8493 1.5236
0.7795 1.5293
0.6281 1.5390
Pressure
1.4226
1.1712
0.9047
0.6345
0.3782
0.1531
-0.1702
Table 6.10: Results for the sensitivity of both models to the Vcore parameter
at T = 0.70, µ = −3.00.
Our initial choice of r0 = 0.5rnn was in part prompted by severe difficulties with the
sampling recipe described in Chapter 4. When the sampled radius is too small, large
extensions of the lattice spacings can actually lead to situations where the lattice point
has a higher energy than points on the ellipse, so that the routine calculates negative
values for A, B and C. This problem can be avoided if r0 is taken large enough that
the sampled points are in a highly repulsive region. Of course, this leads to a poorer
fit at the bottom of the potential well, and so to a smaller calculated free volume.
Study #6: The dependence of ρliquid on Vcore at fixed T and µ
This study was performed at T = 0.70, µ = −3.00, with both fixed lattice spacings
and relaxed spacings.
Table 6.10 (page 119) shows that the density of the liquid at a given state point
is very dependent on the choice of core volume, and also that this dependence is
120
Bulk Studies
roughly the same for the frozen-lattice and relaxed-lattice models. Again, this model
is expected to dramatically overestimate the free volume of the liquid phase, so that
the choice of a larger (than otherwise) core volume may help to reduce this error. As
we see from the tables, larger core volumes cause a drop in the liquid density. In a
real liquid this would be compensated somewhat by increased attractive forces due to
correlation between particles, which favour a denser liquid.
Conclusions about sensitivity
In general, all of the parameters which are not determined from the analysis have
some effect on the phase diagrams and characteristics of the phases in the model. The
position of the triple point temperature is not extremely sensitive to any of them, so
that small perturbations in these values have only small effects on the phase diagram.
For large perturbations of some of the parameters, especially the lattice spacings in the
frozen-lattice model, the phase diagram is strongly perturbed, but this is reasonable.
We see that the good agreement of the triple point in the frozen-lattice model with
simulation results (and experiments) is somewhat fortuitous, but that had we chosen
other reasonable values for these parameters, the agreement would not have been
dramatically worse.
6.3 Anisotropic Potentials
It is not difficult to introduce anisotropy into the system at the mean-field level if
we neglect rotational-translational coupling. In this study, we work with a potential
function of the form
!
!
1
1
1
Ui j ri j , pi , p j = 4ε 12 − 6 + 4ε 12 × δ2 P2 (cos θ) pi + p j
ri j
ri j
ri j
!
1
+ 4ε 12 × δ02 pi p j + δ002 pi p j P2 (cos θ) .
(6.1)
ri j
In this expression, P2 (cos θ) is a function of the angle θ that the vector from one
particle to the other makes with the nematic director n̂. The three coefficients δ 2 , δ02 ,
and δ002 are what determine the “shape” of the molecule. This is a repulsive shape
potential, in which the attractive forces between molecules are isotropic and given by
the Lennard-Jones potential. In reduced units, the ε factors in all of the terms may be
eliminated.
In simple theories of liquid-crystals, no attempt is made to fit the (single) adjustable parameter in the potential (which looks like δ02 ) to any well-defined property
6.3 Anisotropic Potentials
Drtarget
0.50
0.50
0.40
0.40
0.40
0.30
1.20
2.00
3.00
3.50
121
Rrtarget
1.10
1.20
1.10
1.20
1.30
1.30
0.70
0.70
0.70
0.65
δ2
δ02
0.2919 -0.1042
0.2412 -0.1091
0.3160 0.01467
0.3040 -0.1231
0.3416 -0.1315
0.4141 -0.1476
-0.063 -0.0695
-0.182 -0.0976
-0.254 -0.1120
-0.276 -0.1169
error
0.000443
0.005232
0.003463
0.000984
0.014213
0.003931
0.071670
0.03624
0.01757
0.02606
Table 6.11: Attempts to fit different “molecules” with the potential form
6.1. In this table, Drtarget and Rrtarget are the anisotropy ratios for the welldepth and radius of the target “molecule”. The data is obtained by min2
2
imising the function F = Dr (δ2 , δ02 ) − Drtarget + Rr (δ2 , δ02 ) − Rrtarget .
The quoted error is the value of F. The last four rows of data correspond
to oblate molecules.
of the molecules being studied. The potential function in these theories is effectively
Ui (pi , hpi) = Api hpi, where A is independent of the temperature and pressure but not
the volume. This parameter describes an “effective” orientational coupling between
the particles, and replaces the ε of the Lennard-Jones potential as a scalable parameter. In this study, we expect that our choice of potential form will provide a very poor
fit to the potential of any molecule but a homonuclear diatomic, and even the fit in
that case will be unsatisfactory. The reason, of course, is that we are truncating the
spherical harmonic expansion of the potential at very low order which will only allow
for a poor approximation to most intermolecular potentials.
As a demonstration, we can try to fit this potential, using only the δ 2 and δ02 parameters, to the actual potentials of diatomic Lennard-Jones molecules. The fit criteria
will be that the length ratio Rr (defined as the long axis of the molecule divided by the
short axis of the molecule) and the well-depth ratio Dr (the end-to-end well-depth divided by the side-to-side well-depth) are as close as possible to those of the LennardJones diatomics. The results of this test are shown in Table 6.11. This potential is
not too bad at describing nearly spherical particles (Rr nearly 1), but fails quickly for
larger shape anisotropies. In fact, by varying these parameters we can cover quite a
range of the Dr quantity; the model falls short on the ratio of axial lengths.
A good Lennard-Jones diatomic fit to the oxygen or nitrogen diatomic molecule
will have an inter-nuclear separation of 0.32σ [237]; this potential will have D r ' 0.49
and Rr ' 1.127, which puts it between the first two entries of Table 6.11. Molecules
122
Bulk Studies
like chlorine, bromine and iodine are fit well by an inter-nuclear separation of
0.52σ [238, 239]. These ratios are not very well fit by our procedure, so that even
these very simple molecules will not be particularly well described by this theory.
This potential could definitely be improved, principally by adding attractive terms
into the anisotropic part. As we shall see, the failure of the relaxed-lattice surface calculations will preclude the inclusion of better potentials into this work, since they
can only be used in bulk calculations at this time. Although dramatically better
anisotropic pair potentials are available that do not rely on using multiple interaction
sites [240, 241], in the mean-field approximation these would be nearly equivalent to
the ones we are using, since it is the solution method that dictates the truncation at
low order (l = 2), rather than the potential.
6.3.1 Relaxed-Lattice Anisotropic Phase Diagrams
As a demonstration that the bulk phase diagrams can be obtained for these potentials
we have chosen several arbitrary sets of the three shape parameters and determined
the phase diagrams of the corresponding materials. Two of these phase diagrams are
presented here.
In Figure 6.3 are the phase diagrams of the first of these choices of potential, with
δ2 = 0.30, δ02 = −0.10, and δ002 = 0.10. These parameters correspond to molecules
with short:long axis length ratios of 1:1.20, and a well-depth ratio (D r , as above) of
2.90. This is an artificially high ratio for such a short molecule. The phase diagram
contains two more phases than that of the simple Lennard-Jones potential. The first is
an orientationally ordered solid, which exists at low temperatures. For certain values
of the pressure or chemical potential, this phase melts directly into the liquid phase;
for other values, it first undergoes a disordering transition to an orientationally disordered solid, which then melts at a higher temperature. This transition is accompanied
by a decrease in the density of this phase, due to changes in the lattice parameters.
Both solid phases have densities per lattice cell (n) nearly equal to 1; it is the relaxation of the lattice spacings that makes the denser orientationally ordered solid
possible. For lattice spacings held fixed at their isotropic-potential values, the ordered
solid is destabilised and does not exist.
The second new phase is an orientationally ordered liquid, a nematic liquid crystal
[223,224]. In fact, due to the lattice geometry and the use of n and s order parameters,
this is the only type of liquid crystal that can exist in this system; there is no possibility
for smectic order within the liquid phase. This liquid is denser than the disordered
liquid that appears at higher temperatures, as we would expect.
All three of the melting lines in this phase diagram curve backwards, which indi-
123
Pressure
10.00
5.00
Ordered Solid
Nematic
Liquid
6.3 Anisotropic Potentials
Disordered Liquid
Disordered Solid
Vapour
0.00
0.4
0.6
0.8
1.0
Temperature
1.0
-1.0
-3.0
Ordered Solid
Chemical Potential, µ
Nematic
Liquid
Disordered Liquid
Disordered Solid
-5.0
Vapour
-7.0
0.4
0.6
0.8
1.0
Temperature
Figure 6.3: (P, T ) and (µ, T ) phase diagrams of the anisotropic–potential
model, with relaxed lattices and δ2 = 0.30, δ02 = −0.10 and δ002 = 0.10.
The summations are performed along the (111) planes of the bulk crystal,
with the nematic director chosen perpendicular to these planes.
124
Bulk Studies
cates that the liquid phases, both ordered and disordered, are too dense and too compressible. This appeared as a less dramatic effect in the relaxed-lattice phase diagram
of the Lennard-Jones system. There are now two triple points along the solid-vapour
coexistence line, so that in a study of the surface behaviour of this system one would
look for both surface-induced disordering and surface-induced melting behaviour.
In the mean-field theory, once a phase has disordered so that it has p eq ' 0, the
average potential between two particles looks just like the Lennard-Jones potential.
We might expect the phase behaviour to be the same as well, were it not for the
fact that the free volume calculations are always performed at p ≡ 1. In addition,
although the particles may be orientationally disordered, their core volume (Vcore ) is
not the same as that of spherical particles.
We repeated these calculations for a different choice of δ parameters, with δ 2 =
0.50, δ02 = −0.10, and δ002 = 0.10. For these parameters, we calculate Rr = 1.34 and
Dr = 5.71, which indicates a slightly thinner particle than before, with a dramatically deeper side-by-side well depth. We expect that this should further stabilise the
orientationally ordered phases, which turns out to be the case, as shown in Figure 6.4.
In this phase diagram, the orientationally ordered solid has been sufficiently stabilised so that the disordered solid (though it exists as a metastable state) is no longer
anywhere the global minimum of the free energy. The disordered liquid remains, of
course. The four phases nearly meet at a “quadruple point”, and tuning of the δ parameters would undoubtably allow this to happen. As before, the melting line has a
negative slope, indicating that the liquid phase is unphysically dense; we would expect that an ordered solid would be the densest phase possible for a simple potential
such as this, but that is not the behaviour shown in this diagram.
Note that both of the triple points for this potential occur at significantly higher
temperatures than any of the ones previously seen. The melting triple point is at
T = 0.8337, and the nematic/isotropic triple point is at T = 0.8390; in the previous
anisotropic model, the melting transition occurred at T = 0.6060. This reflects the fact
that increasing the anisotropy parameters stabilises the ordered phases more than the
disordered ones, so that the disordering transitions move to higher temperatures. Furthermore, the dramatic stabilisation of the ordered phases is due to the behaviour of
the potential function, which for these anisotropy parameters has a unphysically deep
side-by-side potential well. A more realistic potential would have an R r that scaled
roughly as Dr (consider, for instance, these ratios for linear diatomic and triatomic
molecules).
6.3 Anisotropic Potentials
125
0.0
nematic
liquid
Chemical Potential, µ
nematic
liquid
-4.8
-2.0
solid
-5.2
-4.0
liquid
ordered
solid
disordered
liquid
-5.8
0.80
vapour
0.84
-6.0
vapour phase
-8.0
0.5
0.7
0.9
1.1
1.3
1.5
Temperature
Figure 6.4: (µ, T ) phase diagram of the anisotropic–potential model, with
relaxed lattices and δ2 = 0.50, δ02 = −0.10 and δ002 = 0.10. The summations
were performed as before. The inset shows a magnified view of the two
triple points of the system.
126
Bulk Studies
6.3.2 Frozen-Lattice Anisotropic Phase Diagrams
We cannot expect particles interacting through an anisotropic potential to have the
same minimum-energy lattice spacings (or geometry) as those that interact through
an isotropic one. This was the principal reason that in this study we have included the
lattice spacings as order parameters in the free energy minimisation process. In fact,
our attempts to find the phase diagram of anisotropic potentials using frozen lattice
spacings have almost always found ordered phases that are quite unstable and do not
appear in the phase diagrams. This is principally due to the “shape” nature of the
potential function that we use; the inclusion of longer-ranged anisotropic attractive
terms might change this behaviour.
For certain choices of the potential δ parameters, however, the interaction between
particles is independent of the direction vector between their lattice points. That is, if
we choose δ2 = 0 and δ002 = 0, with δ02 6= 0, the interaction between cells is independent
of their relative orientation, which implies that the free-energy minimum of the lattice
spacings will be that of the isotropic crystal, perhaps scaled by some constant factor.
Therefore, for these choices of the potential parameters, we can perform frozen-lattice
studies of the surface.
This choice of the potential parameters corresponds to a potential of the form U =
Ui + Api p j , with Ui the isotropic part. This is the potential studied in the traditional
formulation of the Maier-Saupe theory of liquid crystals [225, 242]. In these theories,
even the lattice structure is usually averaged out, so that the problem becomes one
of solving for a self-consistent value of p, where all of the terms arising from the
nature of the orientational energy and the geometry of the system have been subsumed
into A. This is viewed as an ‘effective” orientational potential, and can be used to
describe the nematic–isotropic transition. In order to describe the smectic phases of
a liquid crystal, in which molecules are ordered into layers, some way of describing
the “smectic density wave” must be introduced into the theory.
We have calculated the phase diagrams using these δ02 6= 0 potentials for several
choices of δ02 , using the same fixed lattice spacings that we used when studying the
Lennard-Jones crystal. The first of these phase diagrams is shown in Figure 6.5. This
is calculated with δ02 = −0.30, and exhibits four phases, the “extra” one being an
ordered solid phase. These phases meet in two triple points. This potential does not
anywhere exhibit a liquid crystal phase. The reason is that the effective orientational
potential is short-ranged, so much so that in the frozen-lattice liquid this potential is
insufficient to induce ordering. In this Figure it appears as though the disordering line
in the solid phase is perfectly vertical; in fact, it has a very large and positive slope,
due to the fact that the two phases have practically the same density. In the limit of
6.3 Anisotropic Potentials
127
-2.0
Chemical Potential, µ
Disordered Solid
-3.0
Ordered Solid
-4.0
Liquid
-5.0
Vapour
-6.0
-7.0
0.5
0.7
0.9
1.1
1.3
Temperature
Figure 6.5: Bulk phase diagram for the anisotropic potential with δ 2 = 0,
δ02 = −0.30 and δ002 = 0, calculated with lattice spacings frozen at the
Lennard-Jones well-minimum distances.
Chemical Potential, µ
Disordered
Solid
-3.0
Ordered Solid
Liquid
-5.0
Vapour
-7.0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Temperature
Figure 6.6: Bulk phase diagram for the anisotropic potential with δ 2 = 0,
δ02 = −0.40 and δ002 = 0, calculated with lattice spacings frozen at the
Lennard-Jones well-minimum distances.
128
Bulk Studies
high pressure, this line becomes vertical.
In the second phase diagram (Figure 6.6, page 127), the anisotropic coupling is
stronger (δ02 = −0.40), so that the disordering line in the solid phase moves past the
solid-liquid-vapour triple point to a much higher temperature. As before, the disordering transition line is nearly vertical. In these phase diagrams, the triple point has
again moved to higher temperatures than in the isotropic system, so that we expect,
among other things, that the triple point pressure will be correspondingly higher and
that the density of the liquid phase at Tt will be correspondingly lower. In all of these
models, the density of the liquid phase at liquid-vapour coexistence drops smoothly
with increasing temperature from values of approximately 0.80 at T ' 0.70 to its
critical point density (ρ ' 0.35) at T ' 1.10.
Although these frozen-lattice anisotropic models are not very realistic, they do
provide a route to studying such things as the orientational order-disorder transition
of the solid surface (in the δ02 = −0.30 model), and also the onset of orientational
order at the solid-liquid interface in the δ02 = −0.40 model. Extensive studies of the
interplay between the nematic, smectic, and isotropic phases of the free surface of a
liquid crystal have been performed using just such a model [202,212], but one without
the possibility of a solid phase, and without the free volume terms necessary for the
realistic description of condensed phases in the lattice-model method.
Chapter 7
SURFACES OF THE
LENNARD-JONES CRYSTAL I.
BEHAVIOUR OF THE (100), (110)
AND (111) FACES
This chapter contains the results of studies on the (100), (110) and (111) faces of
the Lennard-Jones crystal, using the frozen-lattice model described in Chapter 6. We
have determined the structure of the solid-vapour interface for each exposed face over
a wide range of temperatures by performing many numerical minimisations of the
surface tension. By looking at the surface excess heat capacity we can locate the
phase transitions (both continuous and first-order), and identify the physical nature of
the transitions by looking at the equilibrium structure of the surface at temperatures
slightly above and slightly below the transition temperatures.
7.1 Overview
Figure 7.1 on page 131 shows the surface tension as a function of temperature for all
three crystal faces. The (100) and (110) curves are smooth, while the kinks in the
(111) curve indicate first-order transitions in the structure of the surface. The data for
the two “open” faces are roughly in line with the simulation results of Broughton and Gilmer
[132], while the (111) data show quite different behaviour. As we shall see, the behaviour of the three different faces can be explained by considering several competing
effects, the most important of which is the way that the interspersed-sublattice model
is applied to study the (111) face of the crystal.
129
130
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
this page intentionally left blank
7.1 Overview
131
2.5
Surface Tension, γ
2.0
(100)
(110)
(111)
1.5
1.0
0.30
0.40
0.50
Temperature
0.60
Figure 7.1: Surface tension versus temperature for the (100), (110), and
(111) faces of the Lennard-Jones crystal, using the frozen-lattice model.
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
132
7.2 Results for the (100) Surface
Side View, two planes
Top View, one plane
|
l
|
l
|
l
|
l
|
l
|
|
l
|
l
|
l
l
|
l
|
l
|
l
Figure 7.2: Geometry of the (100) lattice plane. The dashed box indicates
one possible choice of the orientation of a single lattice cell which contains
one point from each sublattice. Solid circles are from one sublattice, open
circles are from the other. Note that each plane parallel to the surface
contains points from both sublattices, and each lattice cell lies entirely
within one plane.
The (100) face of the crystal is “open”, so that there are holes in the top layer. The
top two layers of a (100) face of the lattice are shown in Figure 7.2. These holes are
actually the octahedral holes in the occupied sublattice, so that if we were considering
NaCl, both sodium and chloride ions would be present in the exposed lattice plane,
in a square lattice. In the lattice-model this means that the lattice cells lie “in” the
lattice plane, so that the disordering described by the s order parameter occurs in
a particular lattice plane. We could choose the lattice cells to contain particles in
two lattice planes, but there is no physical reason for doing so. We expect that this
would lead to a poorer description of the solid because, although the energy of a
given configuration is independent of this choice, the free volume term is not. The
free volume interpolation is based on the s parameter. If we choose the lattice cells to
include points from adjacent planes, then a given configuration will have a different
s-profile than if we choose lattice cells “in-plane”.
Figure 7.3 shows the density profiles through the solid-vapour interface for the
entire temperature range studied. At low temperatures these profiles are very sharp,
which means that the crystal face is un-relaxed, so that there are no vacancies in
the top layer of the crystal and no adatoms attached to it. As the temperature is
raised, the density of the top layer drops slowly and the density of the first adlayer
increases. Note that the density of the adlayer increases faster than the density of
the top crystalline layer decreases, so that the number of particles in the condensed
7.2 Results for the (100) Surface
133
δT=0.0015, T=0.6676 to 0.6883
1.0
n(layer)
0.8
0.6
0.4
T=0.6676
T=0.6883
0.2
0.0
5.0
15.0
25.0
δT=0.02, T=0.30 to 0.660
1.0
n(layer)
0.8
0.6
T=0.660
T=0.30
0.4
0.2
0.0
5.0
15.0
Layer Number
25.0
Figure 7.3: Density profiles of the (100) face solid-vapour interface. The
lower of the two graphs contains overlaid profiles from low temperatures,
where the crystal surface is “clean”, up to about 96% of the melting temperature. The upper graph contains density profiles in the surface melting
regime, from about 0.98Tt to Tt .
134
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
phase increases. In the corresponding physical system we would say that material is
condensing from the vapour phase onto the crystal surface. This behaviour would not
be observed in any of the molecular dynamics studies performed on this system to
date, as all such studies have been performed in the (N,V, T ) or (N, P, T ) ensembles.
Furthermore, in these studies, the vapour phase in the simulation box is approximated
by a hard vacuum. While this constitutes only a small perturbation away from the
phase coexistence curve (one that is generally smaller than the fluctuations of pressure in the simulation box), the absence of vapour-phase particles prohibits any such
adsorption being observed.
When the temperature has been raised to within a few percent of the melting temperature, the interface has broadened to four or five lattice-planes in width, depending
on how this width is defined. This is in keeping with predictions of several simulators [130, 243], and also indicates agreement with the prediction that liquid-like
behaviour of the top few layers of the crystal does not occur except very close to the
triple point.1
For temperatures within about 2% of the melting temperature, a liquid layer (in
this study characterised by its density and crystallinity, rather than dynamical data)
intrudes between the solid and vapour phases. This is surface melting. The thickness
of the liquid layer does not become appreciable (more than two layers thick) until
within about 0.2% of the melting temperature (0.14 K, for argon) after which the
thickness of the layer diverges quickly. The density plots of the thinner liquid-layer
configurations indicate that the liquid layer grows in a layer-wise fashion for the first
few layers.
In Figure 7.4 are the surface tension, surface excess entropy, and surface excess
heat capacity as functions of temperature. As expected, the heat capacity diverges at
the melting temperature. There are several interesting features present in this curve.
At temperatures between about 0.35 and 0.45, there is a broad peak which corresponds to the onset of disordering of the interface. Zhu and Dash [87] find in their
calorimetry experiments a broad feature in the surface excess heat capacity of argon
slightly below its roughening temperature, which they determine to be about 0.8Tt , or
T = 0.56 for this model. As already noted, this model cannot be used to study surface
roughening or reconstructions as they happen in the directions parallel to the interface, but since the width of the local interface should also change at the roughening
transition we may be observing a remnant of that here.
1 These studies and characterisations of the “liquid-ness” of the surface layers rely principally on the
comparison of calculated diffusion constants with the diffusion constant of bulk liquid. Unfortunately,
in a mean-field model, all phases have the diffusivity and compressibility of an ideal gas, so that we
cannot directly compare these data.
7.2 Results for the (100) Surface
135
10000
Energy, Energy/T
1000
surface tension, γ
Sex
Cex
100
10
1
0.35
0.45
0.55
Temperature
0.65
Figure 7.4: Thermodynamic functions of the (100) solid-vapour interface.
These are the surface tension γ, the surface excess entropy Sex , and the surface excess heat capacity (at constant volume) Cex . The entropy and heat
capacity were determined by numerically differentiating the γ(T ) data.
The odd feature in the heat capacity data at T = 0.54 is an artifact caused
by a bad point in the coexistence line data used to generate the chemical
potential values used in this study.
136
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
At higher temperatures, in the surface melting regime, there are several peaks
in the heat capacity curve which are shown magnified in Figure 7.5. These peaks
correspond to the liquid layer growing by one lattice spacing in thickness in a continuous fashion, and are reminiscent of the layer critical points observed in adsorption/calorimetry experiments. The last peak is a weak first-order transition, wherein
the thickness of the liquid layer grows by one lattice spacing discontinuously. It is
hard to obtain good solutions to the model at temperatures much closer to the triple
point, because the liquid layer begins to grow very quickly, and it becomes very difficult to guess a starting point for the minimisation that is within the radius of convergence of the global minimum. Also, as soon as the liquid-vapour interface is depinned from the solid-liquid interface, the free energy hypersurface develops many
local minima corresponding to different metastable thicknesses of the liquid layer,
which makes location of the global minimum much more difficult.
7.2 Results for the (100) Surface
137
Cex (Surface Excess Heat Capacity)
10000
1000
100
10
0.678
0.680
0.682
0.684
0.686
0.688
Temperature
Figure 7.5: Thermodynamics of the (100) solid-vapour interface at high
magnification, using a temperature discretisation of δT = 6.2 × 10 −5 . Two
curves are superimposed; surface tension from a heating run and from
a cooling run. The noise and negative dips at T = 0.6878 indicate the
presence of a weak first-order transition.
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
138
7.3 Results for the (110) Surface
Top View, two layers
Side View, two layers
w
g
w
g
w
g
l
|
l
|
l
|
w
g
w
g
w
g
w
g
w
g
w
g
l
|
l
|
l
|
l
|
l
|
l
|
Figure 7.6: Geometry of the (110) lattice plane. In the “Top View”, two
lattice planes are shown, the smaller points being in the second plane.
The dashed box indicates the orientation of a lattice cell. As in the (100)
geometry, each lattice cell lies entirely within one plane. In the side view
we show the same two lattice planes.
Much like the (100) face, the (110) crystal face is also “open”, so that there are
octahedral holes in the occupied sublattice present on the surface. One difference
between the two faces is that in the (100) face, of the six Oh holes around a lattice site
four are “in-plane”, while in the (110) face only two are “in-plane”. Also, as we can
see from Figure 7.6, the lattice sites are arranged in a rectangular array on the (110)
face, while the array is square on the (100) face. As before, a single lattice cell will
lie in a single lattice plane, so that s describes “in-plane” order.
The (110) face exhibits smoother behaviour than the (100) face does, which is
shown clearly by the curves in Figure 7.7. As in the (100) system, there is a broad feature in approximately the right place to be identified with the roughening/broadening
of the interface. In this system the peak is centred at approximately T = 0.52, which
is a somewhat higher temperature than we saw in the (100) data. This may be due
to the fact that the (110) surface excess energy is higher than that of the (100) face
and it broadens somewhat at low temperatures, so that this “roughening” occurs at
higher temperatures because the “un-roughened” surface is more stable. At higher
temperatures, Cex diverges smoothly and without trace of layer-wise behaviour. The
absence of layer-wise transitions in the (110) system compared with the (100) system
might be explained by the different surface energetics of the two faces. The (100) face
has a lower surface tension at low temperatures than the (110) face so the potential
well-depth in the top layer of the crystal is greater for the (100) face. Therefore, we
would predict a stronger “ordering field” for liquid adsorbed onto the (100) face than
the (110) face, so that liquid adsorbed onto the (100) face would be more likely to
7.3 Results for the (110) Surface
139
10000
surface tension, γ
Sex
Cex
Energy, Energy/T
1000
100
10
1
0.30
0.40
0.50
0.60
Temperature
Figure 7.7: Thermodynamic functions of the (110) solid-vapour interface.
These are the surface tension γ, the surface excess entropy Sex , and the surface excess heat capacity (at constant volume) Cex . The entropy and heat
capacity were determined by differentiating the γ(T ) data numerically.
140
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
be partially ordered (and thus display layering phenomena) than liquid adsorbed onto
the (110) face. We expect that if the crystal melted at a lower temperature, the (110)
face would also exhibit layer-wise transitions.
The density profiles in Figure 7.8 are already slightly rounded at T = 0.30, and
broaden at lower temperatures than do the density profiles of the (100) system. In the
surface melting regime the liquid layer appears to be thicker than that of the (100)
face, but because the lattice planes are closer together in the (110) system these layers
are of the same thickness. This layer grows much more smoothly, so that it is possible
to observe slightly thicker surface melts. As before, as soon as the two interfaces
become disconnected it becomes difficult to obtain further data.
7.3 Results for the (110) Surface
141
δT=0.0015, T=0.6583 to 0.6883
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
5.0
15.0
25.0
δT=0.02, T=0.30 to 0.660
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
5.0
15.0
25.0
Layer Number
Figure 7.8: Density profiles of the (110) plane solid-vapour interface. The
lower graph contains the density profiles from a very low temperature,
where the crystal face is “clean”, up through about 96% of the melting
temperature. The upper graph contains profiles from the surface melting
regime.
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
142
7.4 Results for the (111) Surface
Side View, three layers
Top View, three layers
}
d
}
d
m
}
m
}
d
m
d
}
d
m
m
d
c
d
c
d
z
}
z
}
z
m
j
m
j
m
Figure 7.9: Geometry of the (111) plane. In the “Top View” are three sublattice layers, with the open circles being two layers of the Na sublattice
and the solid circles the Oh sites between them. In the “Side View” the
same three lattice planes are shown. Note that a lattice cell must contain
sublattice sites with different z–coordinates.
The (111) face of the fcc lattice is close-packed, so that there are no O h (or even
tetrahedral!) holes in the exposed plane. In NaCl, this means that the exposed face
consists entirely of either sodium or chloride ions. In the lattice-model, this means
that a single lattice cell contains two sublattices sites with different z-coordinates, so
that the s-parameter describes disordering shared between two lattice planes. We take
the s-parameter to describe a Na-plane and the Cl-plane above it; this choice does
influence our results. As before, all of the parts of the model are independent of this
orientation of the lattice cell except the free volume approximation and the combinatorial γ function2 . Alternatively, we could choose the s-parameter to connect a Naplane and the Cl-plane below it; this was the choice made by Trayanov and Tosatti
in their lattice model. We have performed some calculations with this choice, and
found that it has a large effect. With the second definition, the (111) crystal surface
remains ordered up to Tt , and does not surface melt. At all temperatures above the
surface-layer disordering transition, this choice of offset gives a dramatically higher
surface tension than the same surface in calculations using the first definition.
The (111) face (with the “up” lattice cells) undergoes a series of layer-wise transitions as the temperature is raised. The thermodynamics curves shown in Figure 7.10
cover six separate transitions. The first two transitions that occur are at T = 0.396 and
T = 0.547, and are first-order transitions corresponding to disordering of the top and
2
If we refer back to Equation 4.61, the mathematical expression for this difference is that in the
“down” case we choose z 1 = 3, z0 = 3 and z−1 = 0, and in the “up” case we choose z 1 = 0, z0 = 3 and
z−1 = 3. In our studies, we have used the second choice. Furthermore, this peculiarity of the (111) face
breaks the symmetry between the two sublattices. The “up” choice for the Na sublattice is the same
as the “down” choice for the Cl sublattice, so that the behaviour of the surface is dependent on which
sublattice is populated in the solid phase. This was not the case for the (100) or (110) faces.
7.4 Results for the (111) Surface
450.0
143
surface tension, γ
Sex
Cex
Energy, Energy/T
350.0
250.0
150.0
50.0
-50.0
0.35
0.45
0.55
0.65
Temperature
Figure 7.10: Thermodynamic functions of the (111) solid-vapour interface. These are the surface tension γ, the surface excess entropy S ex , and
the surface excess heat capacity (at constant volume) Cex . The entropy
and heat capacity were determined by differentiating the γ(T ) data numerically. The two “fin–like” features at lower temperatures are caused
by differentiating a discontinuity in the surface excess entropy; e.g., are
first-order transitions.
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
144
sublattice density
low-T branch
1.0
0.8
0.6
0.4
0.2
0.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
12.0
13.0
14.0
15.0
sublattice density
high-T branch
1.0
0.8
0.6
0.4
0.2
0.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
z
Figure 7.11: Sublattice-densities for the (111)-face transition at T =
0.396. The densities per layer of each sublattice are shown separately
because the two sublattices are offset in the z-direction. z is in units of σ.
The top “layer” of the crystal is composed of the two sublattice planes on
either side of z = 13σ.
second-to-top layers of the solid surface. The fin-like features seen in the heat capacity are the result of differentiating the discontinuity in Sex to obtain the heat capacity;
in a first order phase transition only the thermodynamic potential is a continuous
function of the temperature, and all of its derivatives are discontinuous. These two
transitions are shown explicitly in Figures 7.11 and 7.12. Figure 7.11 shows the two
solutions to the equations at that temperature. For T < 0.396, the thermodynamically
stable density profile is the “low-T branch”, while for T > 0.396 the stable profile is
the “high-T branch”. At T = 0.396 these two profiles have the same surface tension
and may coexist.
The transition at T = 0.396 corresponds to a slight disordering of the top layer
of the crystal. As the temperature is raised, this layer continues to disorder until
T = 0.547, when the second layer from the top undergoes a similar transition, shown
in Figure 7.12. At this temperature, the width of the interface is still only about three
molecular diameters, which is slightly smaller than the widths of either of the “open”
faces at the same temperature. The (111) face has the lowest surface tension of the
three un-relaxed crystal faces at low temperature, so that the energetic penalty for
disordering it is the greatest.
The effects of the choice of lattice cell are clearly indicated in Figure 7.11. In
the surface disordering transition, some particles are promoted from the top layer to
7.4 Results for the (111) Surface
145
sublattice density
low-T branch
1.0
0.8
0.6
0.4
0.2
0.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
12.0
13.0
14.0
15.0
sublattice density
high-T branch
1.0
0.8
0.6
0.4
0.2
0.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
z
Figure 7.12: Sublattice-densities for the (111)-face transition at T =
0.547. This transition corresponds to a partial disordering of the secondto-top crystal plane.
sublattice density
T=0.6317 (above the transition)
1.0
0.8
0.6
0.4
0.2
0.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
15.0
16.0
17.0
sublattice density
T=0.6257 (below the transition)
1.0
0.8
0.6
0.4
0.2
0.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
z
Figure 7.13: Sublattice-densities for the (111)-face transition for the continuous transition near T = 0.6286. The two configurations shown are
taken at δT = ±0.003 above and below the transition temperature.
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
146
sublattice density
T=0.6589 (above the transition)
1.0
0.8
0.6
0.4
0.2
0.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
15.0
16.0
17.0
sublattice density
T=0.6649 (below the transition)
1.0
0.8
0.6
0.4
0.2
0.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
z
Figure 7.14: Sublattice-densities for the (111)-face transition for the continuous transition near T = 0.6620. The two configurations shown are
taken at δT = ±0.003 above and below the transition temperature.
sublattice density
T=0.6801 (above the transition)
1.0
0.8
0.6
0.4
0.2
0.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
11.0
12.0
13.0
sublattice density
T=0.6761 (below the transition)
1.0
0.8
0.6
0.4
0.2
0.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
z
Figure 7.15: Sublattice-densities for the (111)-face transition for the continuous transition near T = 0.6780. The two configurations shown are
taken at δT = ±0.002 above and below the transition temperature.
7.5 Surface Melting
147
disordered sites slightly above that layer. They are not really adatoms, but occupy O h
holes above vacancies in the top crystal surface. If we had chosen the other definition
of the lattice cell, then the same {n, s} profile would correspond to those particles
occupying Oh holes between the top and second layers of the surface. Since the
solid layers have very few vacancies, this violates some of the packing conditions
(equations 5.21–5.24) and corresponds to a highly unfavourable situation.
As the temperature is increased towards the triple point, three critical points very
reminiscent of adsorption critical points are crossed, with the interfacial width growing by one disordered layer each time. These transitions (located by their heat capacity peaks) occur at T = 0.6286, 0.6620 and 0.6780. They are shown in Figures 7.13–
7.15. These thicker interfaces are liquid-like, indicating that in this model the (111)
face begins to melt at a much lower temperature than do the open faces. However, a
more precise statement would be that liquid condenses onto the (111) face more than
onto the other two faces. This is to some extent an artifact of the lattice-geometry
of the model and the combinatorial function used to describe the layered system,
γ ({ni }, {si }). In the (110) and (100) faces, any material adsorbed onto the surface is
restricted by the lattice geometry so that the lowest layer of adsorbed material must
have s ' 1, so that NNa−Cl = 0. In the (111) system, this restriction is not really
present; a given lattice cell is only connected by γ ({ni }, {si }) to cells in its own layer,
and cells in the layer immediately above it. This allows material adsorbed onto the
surface to disorder freely, allowing for more adsorbed material, smoother interfaces,
and lower surface tension.
7.5 Surface Melting
As we have already explained in Chapter 2, for short-ranged potentials or thin liquid
layers the growth of a surface melted layer is expected to proceed logarithmically. In
these studies, we have avoided the need to arbitrarily define a measure of the thickness of the liquid-layer. We are already calculating the position of the Gibbs dividing
surface in the slab for other reasons, and observed that for a partially-melted solid
surface the Gibbs dividing surface falls in the centre of the quasi-liquid layer. In fact,
we can use the position of this surface (zd ) as a measure of liquid layer growth, without needing to calculate “10-90” widths or other such things, which are not as well
defined. There is one difficulty with using zd as a measure of liquid layer thickness,
which is that in a large simulation slab there are many different solutions with the
same surface tension, but different values of zd ; this is obvious, since we can just
displace any solution by one lattice spacing in either direction, and nothing will be
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
148
Position of Gibbs Dividing Surface (σ)
11.0
9.0
7.0
5.0
1.5
(100)
(110)
(111), cooling curve
(111), heating curve
regression-fit
2.5
3.5
-log10(Tt - T)
Figure 7.16: Position of the Gibbs dividing surface plotted against
log10 (Tt − T ) for all three faces. The heavy lines are linear regression
fits to the various data sets. The vertical offset is immaterial, and is determined by the number of solid layers in the slab; the Gibbs dividing surface
is measured relative to the first layer in the calculation. Provided that the
solid-liquid interface doesn’t move from one temperature to another, this
is a good measure of the liquid layer thickness.
7.6 Comparison With Previous Lattice Models
149
changed. In practice, for small increments δT between successive data points these
displacements never occur, and the position of zd as a function of temperature is a
reliable measure of liquid-layer growth.
Figure 7.16 shows the position of the Gibbs dividing surface plotted against
log10 (Tt − T ) for all three faces. All of the faces exhibit the predicted logarithmic
growth of the liquid layer thickness. Furthermore, the slopes of each data set are the
same, indicating that the growth rate of the liquid layer is unperturbed by the underlying lattice structure. The wavy aspect of the (100) curve is another manifestation of
the series of critical transitions crossed when this face surface melts. The (110) curve
is very smooth, as expected. The (111) curve exhibits two hysteresis loops, which
correspond to first-order transitions where the thickness of the liquid layer increases
by one lattice spacing. These transitions occur at higher temperatures than those described in our analysis of the (111) face data; the first of them is present in Figure 7.10.
In general, whenever hysteresis loops are present in one or more of the order parameters a first-order transition is present, and many such transitions were found in this
manner in the previous analysis of the Blume-Emery-Griffiths model. The presence
of hysteresis loops does not yield any information about the stability or persistence
of metastable states. This would require calculation (by some other method) of the
widths of the free energy minima, as well as the height of the transition-state barrier
between them.
7.6 Comparison With Previous Lattice Models
Trayanov and Tosatti performed a similar set of calculations to these using a model
which is very similar to this one excepting the way in which we handle the free volume. Our solution technique differs in that we use a minimisation-based procedure,
while they performed iterative root-findings to find the minimum, but this should not
make any difference. However, we obtained markedly different results using what
appears to be a similar model.
First of all, Trayanov and Tosatti observed surface non-melting for the (111) face
crystal, and did not see any of the layer-wise adsorption transitions that we have
found. The reason for this is almost certainly that they have chosen the “other” definition of the lattice cell, one that connects (via the combinatorial function) a lattice
cell with its neighbours in its own layer and those below it. This causes a great restriction on both disordering of the top layer of the crystal and the adsorption of additional
material, and raises the solid-liquid interfacial tension enough that no surface melting
occurs. Trayanov and Tosatti attributed this fact to a lack of “in-plane” disordering
150
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
in their model, which is definitely present, but it is more precisely due to the peculiarities of the combinatorial function and free volume approximations employed. As
these calculations have shown, using the other definition of the lattice cell removes
this difficulty. That is, although using the other choice of cell does not remove the artificiality of these approximations from the model, it does minimise their perturbation
of the surface tension.
Secondly, Trayanov and Tosatti see, for both the (100) and (110) faces, entirely
smooth surface melting. We will show in the next chapter that the absence of “wiggles” in their surface-melting curve (the absence of critical layer transitions) is due
to their surfaces melting at too high a temperature, so that they are well past the endpoints of these transition lines. We have shown that for different values of the input
parameters (Vcore , r0 , etc.) that result in significantly higher melting temperatures,
all three surfaces melt entirely smoothly. Evidently, the position of the “roughening”
temperature of this model is not particularly dependent on these quantities, while the
melting temperature is. Since the behaviour of the surface has been shown to depend
strongly on the relative positions of these two temperatures, it is important to get the
melting temperature correct in order to see the correct behaviour at the surface.
7.7 Comparison with Simulations and Experiments
As we have already mentioned, an extensive simulation study of the surface of the
Lennard-Jones crystal over a wide range of temperatures was performed by Broughton
and Gilmer. Rosato, Ciccotti and Pontikis (RCP) did a second study of the premelting
behaviour of the (110) face of the Lennard-Jones solid, and interpreted their results
in a somewhat different way. Broughton and Gilmer felt that they were observing
the onset of a surface melting transition, signalled by considerable increases in the
diffusivities in the surface layer and predictions from free-energy curves that the topmost layers of the system would melt completely at a temperature slightly below the
melting temperature, which they predict to be T = 0.997 × TM . This is a higher temperature than any they simulated, but they argue that attempts to simulate it would be
impractical because the equilibration time will become very long as the free energy
driving force to equilibrate the thickness of the liquid layer will be very small.
In the RCP study, the roughening transition is successfully identified, and occurs
at the temperature measured by Zhu and Dash. However, RCP find no evidence of
surface melting, even though the range of temperatures at which they simulate extends
much nearer to the melting point. They feel that this indicates that the surface remains
disordered but crystalline all the way to the bulk melting transition.
7.7 Comparison with Simulations and Experiments
151
Zhu and Dash have observed that the experimental heat capacities of multilayer
argon and neon films on graphite are explained well using the Landau free energy
models of surface melting, and that their data can be fit to the predicted functional
forms that come from these theories. This indicates that even if the surface of these
films is not “melted”, it certainly behaves as mean-field level theories predict a melted
surface should. Obviously, the predictions of RCP must be reconciled with these data.
In fact, the highest temperature investigated by RCP is T = 0.643, with a predicted
melting temperature of T = 0.68, so that they have approached to within about 5.5%
of the melting temperature. The study presented here predicts that the width of the
interface remains stable at approximately four molecular diameters until considerably
closer to the melting temperature, and that substantially “liquid” behaviour will not
occur at the surface until within 1-2% of the triple point.
In all of the molecular dynamics simulation studies done to date on this kind of
system, one of the implicit constraints of the simulation was that the number of particles in the simulation box was constant and equal to the number of particles in the
crystalline slab with perfectly clean crystal faces. Although this is practically unavoidable in simulation work, it can be a considerable problem. Consider a system
where the global minimum free energy configuration of the system consists of surfaces where, on average, 10% of the sites in the top layer of the crystal are vacant and
25% of the sites available to an adatom are occupied. In a constant-N simulation, this
would be an unreachable configuration, since the number of particles in the system
has to be equal to that of a system where 0% of surface sites are empty, and 0% of
adlayer sites are occupied. In a physical system, this configuration could be reached
by evaporation of some other part of the crystal, so that the extra 15% of adsorbed
material could come from the vapour phase. This kind of equilibration is not possible
in a typical simulation box.
In order to determine the effects of this constraint, we have performed several
additional calculations. In each calculation, the system was allowed to equilibrate
fully at T = 0.40, where the crystal surface is effectively clean. The total number of
particles in the calculation (per unit surface area) was then fixed, and the temperature
increased, repeating the minimisation at each new temperature, with the constraint
that the number of particles per unit area not change. The results of this study are
presented in Figure 7.17 (page 152). In each face, although the perturbation is small,
the effects of the constraint are definitely noticeable. The (100) face is the most perturbed, but only at fairly low temperatures, before the interface has broadened to its
“roughened” width. The greatest deviation occurs near the pseudo-roughening transition identified earlier, where the fully-relaxed equilibrium structure of the surface is
Surfaces of the Lennard-Jones Crystal I. Behaviour of
The (100), (110) and (111) Faces
152
(100)
(110)
(111)
2.6
2.4
2.4
Surface Tension
2.2
2.2
2.0
2.2
1.8
2.0
2.0
1.6
1.4
1.8
1.8
constrained
unconstrained
1.6
0.3 0.4 0.5 0.6
T
1.2
1.6
0.3 0.4 0.5 0.6
T
1.0
0.3 0.4 0.5 0.6
T
Figure 7.17: Surface studies done at constant ∑i ni , compared to fully relaxed calculations. For all three faces, the perturbation caused by the constraint is small, with the largest effect in the (100) system around T = 0.55.
forbidden by the constraint. The (110) face hardly perturbed at all by this restriction.
The (111) face is noticeably influenced by the constant–N condition, but the effect
is only to move all of the transitions to slightly higher temperatures. In general, the
perturbation to the surface tension between the corresponding constrained and unconstrained systems is never more than a few percent. All three of these constrained
systems also appear to surface melt. We expect, based on this data, that simulation
studies in the (N,V, T ) ensemble induce only a small error in the thermodynamics
and phase behaviour because of the constant–N condition. Although there are systems where this error could dramatically effect other results, such as in studies of
crystal morphology, these problems are not often studied in this way.
Chapter 8
SURFACES OF THE
LENNARD-JONES CRYSTAL II.
SENSITIVITY TESTS AND
CONSTRAINED MELTING
BEHAVIOUR
This chapter contains the results of a series of tests to determine the effects of liquid
density and triple point temperature on the surface behaviour measured in Chapter 7.
We determine that the degree of layering behaviour is strongly influenced by the melting temperature and the density of the liquid phase near the triple point, with higher
densities and lower temperatures leading to more pronounced layering transitions. We
have also performed a set of calculations measuring the surface tension of the interface as a function of the disordered layer thickness. These results help to explain why
the behaviour of the surface is so dependent on the liquid density, as well as why the
three crystal faces we have studied behave so differently.
8.1 Sensitivity of the Surface Behaviour to Input Parameters
In the study of the bulk system, we found that by varying the different input parameters of the model, principally r0 and Vcore , we could move the triple point about over a
reasonably wide range of temperatures. In this section, we will perform a similar set
of calculations but will focus on the behaviour of the surface of the (100) face of the
crystal just below the triple point. That is, we will tune the input parameters so that
the triple point moves over a range of temperatures, and see how this affects the sur153
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
154
study
LJ
G
A
D
C
H
F
B
aniso
ρliq (Tt )
0.769
0.800
0.820
0.6703
0.755
0.795
0.682
0.599
0.696
Tt
0.6883
0.6883
0.6885
0.8056
0.8088
0.8052
0.7173
0.8740
0.8278
behaviour
2 continuous, 1 first-order
1 continuous, 1 first-order
2 continuous, 3 first-order
completely smooth
faint trace of transition
2 continuous
completely smooth
completely smooth
completely smooth
Table 8.1: Results of the surface-sensitivity studies. The observed behaviour is shown as a function of the triple point temperature and liquid
density. The study names are not important, and are included for reference.
face melting behaviour. In addition, we can adjust these parameters so that the density
of the liquid phase at the triple point varies as well, so that we can correlate surface
behaviour with this quantity. The density of the liquid phase at the triple point has
a strong influence on both the solid-liquid surface tension γsl and the liquid-vapour
tension γlv .
Each calculation performed in this manner requires several preliminary calculations. First, we must accurately locate the new triple point of the “tuned” model.
Second, we must obtain the solid-vapour coexistence line as (T, µ) data over the range
of temperatures that will be studied, so that at each new temperature we can insure
that the system is at phase coexistence. At this point, calculations can be run. Because some of the systems exhibit first-order transitions, calculations must be run
both “heating” and “cooling” the surface over this range of temperature, so that any
metastable solutions are identified.
Table 8.1 summarises the results of these studies. We have measured the nearmelting behaviour of many systems, and classified them according to the type of
transitions observed. This data exhibits two trends. The first is that increasing the
liquid density increases the number of first-order transitions present in this temperature range. The second is that increasing the temperature decreases the number
of first-order transitions. Of all of the studies presented, first-order transitions are
only observed for ρliq > 0.769, so that lower densities always show either completely
smooth growth of the liquid layer, or the beginnings of continuous layer transitions.
In Figure 8.1 we plot the progression of the Gibbs dividing surface (z d ) into the
slab system against log10 (Tt − T ), as in Figure 7.16. This is a reasonably good mea-
8.1 Sensitivity of the Surface Behaviour to Input Parameters
155
Position of Gibbs Dividing Surface (σ)
17.0
16.0
15.0
‘A’ - cooling
‘A’ - heating
‘G’ - cooling
‘G’ - heating
‘LJ’
14.0
13.0
1.0
2.0
3.0
4.0
5.0
-log10(Tt - T)
Figure 8.1: Position of the Gibbs dividing surface versus the logarithm of
the reduced temperature for three different systems which all melt very
near to T = 0.6883. These systems have significantly different liquid densities at this point. The liquid density in the ‘A’ study is ρA = 0.820, the
density in the ‘G’ study is ρG = 0.800 and the density in the ‘LJ’ study is
ρLJ = 0.769.
156
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
sure of the thickness of the liquid layer above the slab, and in any case surface phase
transitions are prominent in this kind of plot. The three different sets of data are for
three different choices of r0 and Vcore , chosen such that all of these systems have a
triple point temperature of Tt = 0.6883 ± 0.0002 but have different liquid densities at
this temperature. In general, choosing Vcore smaller increases the density of the liquid phase, and then the triple point can be moved back to the correct temperature by
tuning r0 . The lowest-density data in this plot is that corresponding to the choice of
parameters used in the frozen-lattice Lennard-Jones study. The next two sets of data
correspond to higher densities of the liquid phase. Figure 8.1 indicates that increasing
the liquid density increases the layer-wise “character” of the onset of surface melting.
Higher densities show sharper transitions, and have better defined hysteresis cycles.
For lower densities, the transitions are more likely to be continuous, and are not as
well defined. Note that the rate of liquid-layer growth (indicated by the average slope
of the curves) is effectively the same for all three systems.
This might indicate that denser liquids show more structure at the solid-liquid
interface, so that the system favours an integer number of liquid layers at the onset of
surface melting rather than fractional coverages. This kind of free energy dependence
would lead to first-order transitions between states with different numbers of layers,
as each would be a local minimum of the free energy. In a system with a weaker
dependence these metastable states might disappear, or the barriers between them
could be smaller, so that continuous transitions would replace first-order ones and
fewer features would appear in the zd plots.
Figure 8.2 shows similar zd plots for a different group of systems, where again the
three curves correspond to different liquid densities and all three systems have a triple
point near T = 0.806. In this case, the densities are slightly lower than those of the
systems in Figure 8.1; although we can tune the densities and triple point temperatures
over a certain range by varying the r0 and Vcore parameters, there are many mutually
inaccessible values of ρliq and Tt . In this Figure, the two curves corresponding to
lower densities show a smooth progression of the Gibbs dividing surface across the
slab, with faint traces of layer-wise behaviour visible in the ‘C’ data. The densest
liquid shown undergoes two continuous transitions over this temperature range. This
data is taken at a liquid density of 0.795, which is higher than that of the lowest
density shown in Figure 8.1 which exhibits two continuous transitions and one firstorder transition. Increasing the temperature smooths out the free energy surface, so
that at higher temperatures layer-wise behaviour is suppressed.
8.1 Sensitivity of the Surface Behaviour to Input Parameters
157
Position of Gibbs Dividing Surface (σ)
18.0
16.0
14.0
‘H’
‘D’
‘C’
12.0
1.0
2.0
3.0
4.0
5.0
-log10(Tt - T)
Figure 8.2: Position of the Gibbs dividing surface versus the logarithm
of the reduced temperature for three systems which all melt very near to
T = 0.806, but have significantly different liquid densities at this point.
The densities of the triple point liquid for these systems are ρH = 0.795,
ρC = 0.755 and ρD = 0.670. The odd feature in the ‘H’ curve at low
temperature is due to a motion of the solid-liquid interface within the slab.
158
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
this page intentionally left blank
Cex (Surface Excess Heat Capacity)
8.2 Calculation of the Surface Tension at Fixed Liquid Layer Thickness
159
10000
1000
100
‘H’
‘D’
‘C’
10
0.780
0.790
0.800
0.810
Temperature
Figure 8.3: Surface excess heat capacity for three different systems which
all melt very near to T = 0.806, but have significantly different liquid
densities at this point. The densities of the triple point liquid for these
systems are ρH = 0.795, ρC = 0.755, and ρD = 0.670.
In order to see the progression of layer-wise behaviour with increasing liquid
density more clearly, we plot the surface excess heat capacities for the systems that
melt near T = 0.806. These are shown in Figure 8.3. The onset of layer-wise behaviour is very apparent here; the low-density curve shows no features and is entirely
smooth, the middle-density curve shows broad oscillations, and the higher-density
curve shows oscillations and well-defined peaks at the transition temperatures. This
supports the hypothesis that the liquid density at the triple point is closely linked to
the way that the surface melts.
8.2 Calculation of the Surface Tension at Fixed Liquid
Layer Thickness
One way to help understand what happens in a system undergoing surface melting is
to determine the surface tension as a function of the liquid layer thickness, at different
temperatures. We expect that these curves will show a minimum at the equilibrium
structure which will move about as a function of temperature. For temperatures ap-
160
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
proaching the triple point, this curve should flatten out so that the minimum becomes
much shallower, and its position will quickly diverge in a positive direction [198].
Henderson [55] suggests that oscillations in this curve are responsible for layering
transitions, and can cause pseudo-surface melting to replace complete surface melting.
8.2.1 Calculation Technique
In studies of the wetting of a fixed substrate wall by a vapour, this kind of study
is somewhat simpler. Rather than plot the surface tension against some arbitrary
definition of the liquid layer width, it is easier and more precise to just plot γ against
nex , the excess adsorbed material. For thick layers, the number of adsorbed liquid
layers will be nex /ρliq . It is not trivial to get these data. In a simulation study, this
would require accurate measurement of the free energy of the system at many different
coverages of adsorbed material, using a difficult technique such as Grand Canonical
Monte Carlo. In a free minimisation, in either a lattice model or density functional
theory, the amount of mass in the system is allowed to vary; this must be constrained
at each new coverage in order to construct γ(nex ).
There are many ways to construct “artificial” density profiles at different coverages or liquid layer thicknesses, without performing expensive minimisations. However, using un-relaxed density profiles can lead to spurious results, and it is necessary
to allow these profiles to relax via some constrained minimisation, or by several Picard iterations [55].
In a substrate-wetting system, the best way to perform this calculation would
seem to be to constrain the total mass in the system to some fixed value ntotal =
nex +Vtot ρvap . which, over a small temperature range, is effectively the same as constraining nex to a fixed value. In such a system, we could then find the equilibrium
structure of the system subject to this constraint, increment ntotal by a small amount,
and find the equilibrium structure again, eventually obtaining the entire γ(n ex ) curve
at a given temperature.
If the system is close to solid-liquid coexistence, this method has a potential failing. Because of the constraint, crystalline material might appear in the calculational
slab. This would not affect the correctness of the calculation, but it would make interpretation of the data much more difficult.
In the surface melting system presented here, matters are somewhat more complicated. We are interested in obtaining γ as a function of liquid layer thickness for
non-equilibrium values of this thickness. There are several problems. First of all,
it is not easy to define a liquid layer thickness at all, especially in a lattice system.
8.2 Calculation of the Surface Tension at Fixed Liquid Layer Thickness
161
In a continuous order parameter system, such as a Landau free energy model, it is
easy to define a liquid-layer thickness that varies continuously with perturbations of
the density profile. For instance, we could define it as the distance between the two
inflection points of the density profile, or the distance between the intersection of the
profile with n = 0.90 (say) and n = 0.10 [243]. In a lattice model, we would have to
fit a continuous curve to the discrete ρi data in order to use this method. In this work,
we use position of the Gibbs dividing surface (zd ) as a measure of the liquid layer
thickness, because it does not involve a fitting procedure which could distort the data.
In order to do these measurements, then, the parameter to constrain is the position
(in σ) of zd . Recall that for fixed lattice spacings, zd is defined by the relation (see
Equation 5.2)
zd
∑i ci − z d
(8.1)
∑ ni − cα nc nα − cβ nc nβ = 0.
i
Since nα , cα = cβ , nβ , nc and the ci are all fixed, constraining zd to a fixed value is
equivalent to constraining ∑i ni to a fixed value. This suggests a recipe where we
solve for the surface structure at constant ∑i ni , increment ∑i ni by a small amount,
and re-solve.
There are technical problems with this recipe. Consider a situation where we
increment the total mass to a value higher than its equilibrium value. There are several
possibilities. The first is that the minimum found by our program corresponds to a
metastable solution with a thicker-than-equilibrium liquid layer. Another possibility
is that the solid-liquid interface moves one or more layers into the slab, to “take up”
the extra density. This is likely to happen since making the liquid layer thicker is
thermodynamically unfavourable, while moving the whole solid-vapour interface one
lattice spacings to the left or right is not thermodynamically unfavourable, because
those two phases are in coexistence. In order to do these calculations without the
position of the interface fluctuating within the slab, we must “pin” it. This is easily
accomplished. We observe that the width of the interface as measured by the s-profile
is virtually unchanging, even through surface melting. This is because the liquid and
vapour have the same value of s. The way that we fix the position of the solid is by
requiring that all layers in the slab above some s j have si≥ j ≡ 0.50. This has the effect
of fixing the solid-liquid (or solid-vapour) interface between the 1 rst and jth layers of
the slab. This additional constraint allows for the stable calculation of meaningful
γ(zd ) curves. If the position of the solid-liquid interface is fluctuating, then z d is not
a meaningful measure of the state of the system; if it is fixed, then the motion of z d
is a useful measure of surface melting. We expect that this recipe will only work for
temperatures very near to the triple point; for lower temperatures (and sharper free
162
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
energy minima) it is unlikely that these measures will have the desired effect.
8.2.2 Results
We have performed these calculations to obtain γ(zd ) for all three crystal faces of the
frozen-lattice Lennard-Jones system, at several different temperatures.
Figure 8.4 shows the γ(zd ) data for three different temperatures very near to
melting for the (110) face system. Note that the differences between the first two
(∆T = 0.0011) and second two (∆T = 0.000203) temperatures approach the triple
point by an order of magnitude difference, so that we expect a roughly linear progression of the minimum of the curves from one graph to the next. This is essentially what
we observe. In the topmost graph, the minimum appears to be shallower than that of
the middle graph, but this is due to the great discrepancy in vertical scale between the
two. There is a small kink in the lowest temperature graph, due to a one-layer motion
of the solid-liquid interface, probably because it was not constrained tightly enough.
In the third graph, shown at high magnification (the scale is over 250 times finer than
in the top graph) we can see a faint oscillation in the curve. This oscillation does not
decay with increasing liquid-layer thickness, which means that it is not due to long
range correlations with the solid surface. In fact, it is due to the underlying lattice
structure perturbing the liquid-vapour interface, which is the only thing moving in
this calculation. Because these oscillations are (approximately) only 3 × 10 −6 energy
units in magnitude, they do not appear in any of our other data on the (110) surface.
In Figure 8.4 the minimum of the free energy curve moves by roughly 2σ between
the first and second temperatures and by 3σ between the second and third temperatures. With each temperature step the “depth” of the minimum decreases by an order of magnitude, so that the well depth goes roughly like the reduced temperature
(Tt − T ), while the liquid layer thickness grows as − log(Tt − T ). (We have checked
numerically that this is true; a “well depth” can be defined as the difference in γ between the minimum and a point a short distance away in the positive direction, say
0.5σ.)
Figure 8.5 (page 164) and Figure 8.6 (page 165) show the generated density profiles and γ(zd ) curves for the (111) and (100) faces, respectively. These graphs show
much more clearly the oscillation in the γ data due to the effect of the lattice on
the liquid-vapour interface. Except for the magnitude of the oscillations, the surface
tension curves look very much like those of the (110) face. The periodicity of the lattice is actually visible in the density profiles themselves; a repeating pattern of surface
structures is visible in the density data for both the (111) and (100) surfaces. Although
it appears that the two interfaces are entirely disconnected, under high magnification
Surface Tension
8.2 Calculation of the Surface Tension at Fixed Liquid Layer Thickness
163
1.62000
1.61000
1.60000
(T=0.687)
1.59000
5.0
10.0
15.0
10.0
15.0
10.0
15.0
Surface Tension
1.58150
1.58100
1.58050
1.58000
(T=0.68810)
1.57950
5.0
Surface Tension
1.57630
1.57625
1.57620
(T=0.688303)
1.57615
5.0
Gibbs Dividing Surface (σ)
Figure 8.4: Surface tension as a function of liquid layer thickness, in the
(110) system. The three different plots correspond to three different temperatures; the triple point of this system is Tt = 0.68830538. Note that
the three different plots have widely different scales; in the first, the vertical scale covers ∆γ = 0.04, in the second, ∆γ = 0.002, and in the third,
∆γ = 0.00015.
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
164
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
5.0
15.0
25.0
Layer Number
Surface Tension
0.8380
0.8370
0.8360
0.8350
0.8340
8.0
10.0
12.0
14.0
16.0
18.0
Gibbs Dividing Surface (σ)
Figure 8.5: Surface tension and density profiles in the (111) surface melting system, at T = 0.688300. The top graph shows the density profiles
generated using the constraint method; between profiles, the total density in the box was increased by ∆ntotal = 0.30. The bottom graph shows
surface tension as a function of liquid layer thickness, with a step of
∆ntotal = 0.15.
8.2 Calculation of the Surface Tension at Fixed Liquid Layer Thickness
165
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
5.0
15.0
25.0
Layer Number
Surface Tension
1.6880
1.6875
1.6870
1.6865
1.6860
7.0
9.0
11.0
13.0
15.0
17.0
19.0
Gibbs Dividing Surface (σ)
Figure 8.6: Surface tension and density profiles in the (100) surface melting system, at T = 0.688303. The top graph shows the density profiles
generated using the constraint method; between profiles, the total density in the box was increased by ∆ntotal = 0.30. The bottom graph shows
surface tension as a function of liquid layer thickness, with a step of
∆ntotal = 0.15.
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
Surface Tension (offset)
166
1.6871
(110)
(100)
(111)
1.6869
1.6867
1.6865
1.6863
12.0
13.0
14.0
15.0
Gibbs Dividing Surface (σ)
Figure 8.7: Oscillations of γ(zd ) for all faces. The data are shifted on the
vertical axis so that they all lie in the same region. The smooth curves are
splines fit to the data points.
the density of the liquid layer slopes smoothly down across the layer, although only
by a small amount. An arbitrary recipe for constructing these profiles without doing
constrained minimisations would not reveal this feature.
We have already considered in Chapter 1 how a solid surface can induce structure
in a liquid. In fact, a liquid-vapour interface can also induce structure in a liquid,
and this theory has been borne out by density functional calculations on simple liquids [47]. In a surface melting situation, the oscillatory parts of h(z) caused by both
interfaces should interfere, and the equilibrium thickness of the liquid film will be influenced by the constructive or destructive nature of this interference. However, these
effects should all decay exponentially with increasing liquid layer thickness. Furthermore, Evans et al showed [47] that these effects should all decay with the same decay
lengths and periodicities, so that the oscillations we are observing here must be due
to something else.
Figure 8.7 shows magnified sections of the γ(zd ) curves for all three faces, vertically shifted (but not rescaled) so that they fit in the same plot. We see that the (111)
oscillations are roughly one order of magnitude greater than those of the (100) data,
8.3 Conclusions
167
which are in turn one order of magnitude greater than those of the (110) data. Figure 8.7 also shows that the three different curves do not have the same period; this
makes sense because, while they all have a period of one lattice spacing, the distance
between lattice planes is different in the three models. It is interesting to note that the
strongest oscillations due to the lattice are for the system with the largest inter-layer
spacing, and the weakest oscillations are for the system with the smallest spacing.
This might indicate that for a finer-discretisation lattice model these effects will be
less important and we will recover more of the physical behaviour of the system.
8.3 Conclusions
We can now explain all of the different surface melting behaviour observed in the
previous chapters. The piece of information that allows us to do this is knowing the
relative magnitudes of the oscillations induced in the γ(zd ) curve by the underlying
lattice structure. In the frozen-lattice Lennard-Jones study, we found that the (110)
face of the crystal melted entirely smoothly, while the (100) face melted via several
continuous layering transitions and a single (observable) first-order transition. The
(111) face of the crystal melted via several continuous transitions, followed by several
first-order transitions. The reason for all of this is simple. In the case of the (110) face,
the oscillations induced by the lattice structure are so small as to be negligible (but still
measurable). Therefore, the mean-field γ(zd ) curve is smooth, and the progression of
zd across the slab with increasing temperature is also smooth. This face completely
surface melts, so that at coexistence, the crystal-vapour interface is wet by an infinitely
thick layer of the liquid phase.
The (100) face exhibits very different behaviour. The mean-field γ(z d ) curve for
this face is similar to that of the (110) face, but the oscillations in the curve due to
the lattice structure are larger and no longer negligible. At lower temperatures, near
the onset of surface melting, the minimum in γ(zd ) is narrow and well defined. As
the system is heated, this minimum moves across oscillations caused by the lattice,
which causes smooth and continuous layer-wise transitions to appear. It is because the
minimum is narrow that these transitions are continuous; only for comparatively large
oscillations would metastable states appear. As the temperature is increased further,
the minimum in the (smoothed) γ(zd ) curve broadens and becomes much shallower,
so that metastable states appear due to the oscillations and at higher temperatures we
see first-order layering. This is an unusual progression; in most systems first-order
behaviour occurs at lower temperatures than continuous behaviour.
In the (111) system, the oscillations in γ(zd ) are of greater magnitude than in the
168
Surfaces of the Lennard-Jones Crystal II. Sensitivity Tests
and Constrained Melting Behaviour
(100) system, so we would expect to see this crossover to first-order behaviour at
lower temperatures, which is precisely what happens. The low-temperature phase
transitions in the (111) system may also be dependent on these large-magnitude oscillations, though they are more likely caused by other peculiarities of the lattice model.
In the previous section, we demonstrated that in the (100) face system, the appearance of layer-wise behaviour was controlled by the triple point temperature and
the triple point liquid density. These two parameters influence the structure of the
liquid-vapour interface, so that for higher liquid densities this interface is more sensitive to the lattice structure, while for higher temperatures it becomes less sensitive.
These trends make intuitive sense; for a more pronounced interface (lower temperatures, higher liquid densities) we would expect more dependence on the offset of the
interface relative to the lattice.
Therefore, we conclude that the different behaviour observed for the different
crystal faces and the temperature and ρliq dependence of this behaviour is controlled
by the liquid-vapour interface, which is perturbed by the use of a reference lattice
in our calculations. A better mean-field theory would not show this non-decaying
oscillation in γ(zd ). In fact, a better mean-field lattice theory would show entirely
smooth γ(zd ) curves, due to a lack of correlations in the liquid phase.
Some DFT calculations [55] indicate that in wetting systems (of which this is one)
there should be an oscillation in γ(zd ), but an exponentially decaying one. This has
many ramifications. These oscillations are caused by oscillations in the liquid density
profile as a function of z. As we saw in Chapter 1, Figure 1.14, the h(z) oscillations
are much larger near the wall than the oscillations measured in this theory, at least
over the layer thicknesses studied. (Recall that even the oscillations in the (111) data
only amount to just 0.029% of the total surface tension, with an amplitude of around
0.0005 energy units.) This would mean that real systems should exhibit more firstorder layering behaviour than this model predicts, not less. Whether real systems
exhibit first-order layering, continuous layering, or a combination of the two depends
on the relative size of the variation in the smoothed γ(zd ) and the magnitude of the
oscillations.
In this system, there is a crossover from smooth behaviour to oscillatory/firstorder behaviour when the well depth becomes smaller that the amplitude of the oscillations. In a real system, the oscillations will decay exponentially. However, the
minimum in γ(zd ) becomes shallower as (Tt − T ), and broadens quickly as well. In
fact, the position of the minimum moves away proportional to − log(Tt − T ), so that
the exponential decay of the oscillation means that the amplitude of the oscillations
around the minimum of γ(zd ) decay linearly with (Tt − T ). We have observed that
8.3 Conclusions
169
the depth of the potential well also varies linearly with this quantity. This means that
there can still be a crossover at some temperature, because the minimum is becoming
shallower as the oscillations near it are decaying. Whether this transition happens
or not depends on the slopes of the two linear decay functions. Even in a system
where the oscillations are small near the surface, there can still be a crossover to a
first-order behaviour so that the system will only exhibit incomplete surface melting.
If the oscillations near the surface are sufficiently strong, then first-order behaviour
will dominate over this entire temperature range, and the crystal will not completely
surface melt. If the oscillation always dies off before the minimum in γ(z d ) is reached,
complete melting should occur.
170
Chapter 9
RELAXED-LATTICE STUDIES OF
THE SURFACE
In this chapter we explain how our attempts to study surfaces using the relaxed-lattice
model failed. The crystalline free volume approximation that we described in Chapter 4 is badly behaved under strongly anisotropic conditions, which results in bizarre
and unphysical results for the lattice spacings near the surface. We have attempted to
make the lattice “stiff”, which should alleviate these problems, but this solution was
not very successful.
9.1 How to Do the Calculation
There are two problems which must be dealt with in the application of the relaxedlattice model to study surface phase transitions. The first is that the choice of lattice
spacings in the model is not trivial. They are different in the bulk phases, but cannot
be different (except in the z direction normal to the surface) in the slab system. The
second problem is that our recipe for the crystalline free volume is badly behaved in
very anisotropic situations, such as at surfaces.
We handle the lattice geometry difficulties in the following way. In the slab system, the a and b lattice spacings lie parallel to the surface. These spacings must be
the same in every layer of the slab system regardless of the phase occupying it; this is
necessary to avoid lattice mismatches, which cannot be dealt with at this level of theory. The ci which describe the distances between successive layers of the system are
allowed to vary. Our modification of the recipe for frozen-lattice surfaces is simple.
First we solve for the fully relaxed bulk solid phase, to put at the bottom of the slab.
Then we solve for the relaxed vapour phase (to put at the top of the slab), with a and
b values constrained to those determined in the bulk solid calculation. This slightly
perturbs the vapour phase, but not very much. Since the vapour phase is entropically
171
172
Relaxed-Lattice Studies of the Surface
stabilised all that really matters is the total volume of the lattice cell, so that if a and
b are fixed then c moves to compensate, recovering an equilibrium value of v 0 . This
process leads to asymmetric cells in the vapour and liquid phases. In the slab calculation, the top and bottom few layers are fixed, and a and b are fixed at their bulk
solid values for all layers of the slab. We must apply the constraint that M ex = 0: the
surface excess number of lattice cells is equal to zero. The recipe for this is explained
in Chapter 5.
9.2 Tests of the Method
We have attempted to perform these calculations for the test case of the (100) surface solid-vapour interface. We have found that the crystalline free volume is badly
behaved at the surface layer; the ci that contains the distance between the top crystalline layer and the first vapour layer grows to a value on the order of 9σ, which is
much larger than its (relaxed) bulk crystalline value of 1.60σ. This is unphysical, and
clearly an artifact. It is due to the recipe by which we fit the harmonic potential to the
f
actual one in the calculation of Vc . Our recipe requires taking several points at a fixed
distance in lattice spacings, and using the average potential at that average distance
(in σ, not lattice spacings) to calculate the spring constants. It is probably the use of
averages that is the difficulty. By moving one of the sampling points very far away
(by increasing the ci at the surface), the averaged depth of the potential is reduced,
so that the spring constants are reduced and the crystalline free volume is increased.
Were the points sampled at a fixed distance in σ, rather than lattice spacings, this
might not occur. Unfortunately, that causes other problems. The major one is that as
the lattice relaxes in the minimisation, particles move into and out of radial cutoffs
of other particles, so that terrible convergence difficulties appear. Even using r 0 in
σ rather than lattice spacings may not entirely solve the problem, because increasing
the ci at the surface still increases the crystalline free volume of the top layer of the
slab, with no energetic penalty.
We attempted to fix this problem by introducing a false potential function based on
the lattice spacings; in other words, we made the lattice itself stiff, with an equilibrium
structure at the spacings used in the frozen-lattice experiments. The potential used
was
f (c) = Cb exp [Ca (c − cmax )]
(9.1)
in the bulk phase calculations, where Cb and Ca are proportionality constants with
typical values of 10 and 20, and cmax is the maximum “allowed” extension of c; 2.3811
in these experiments. (2.3811 = 24/6 × 1.5). In the surface part of the calculations,
9.2 Tests of the Method
173
2.00
relaxed model
frozen model
1.90
ci
1.80
1.70
1.60
1.50
1.40
0.0
10.0
20.0
Layer Number
Figure 9.1: These are the c-profiles for frozen-lattice and relaxed-lattice
calculations for the (100) solid-vapour interface at T = 0.60. In this calculation, the bulk phases are kept at the typical frozen-lattice lattice spacings for both runs. This graph demonstrates the poor performance of the
relaxed-lattice model on the solid surface even with false c-potentials in
place.
the function was just evaluated in each layer:
f ({ci }) = ∑ Cb exp [Ca (ci − cmax )] .
(9.2)
i
This potential is very positive for c near or above cmax , and near zero for c near
its energy-minimum value. We have done some test calculations with this potential.
The ci profile for a typical solution is shown in Figure 9.1. In this calculation, the
Mex condition has been enforced by requiring ∑i ci to be fixed at the isotropic value,
and the bulk calculations were performed with frozen lattices. This is not entirely
correct; we should be constraining it, but to a slightly different value which takes
into account the different equilibrium lattice spacings in the solid and vapour phases.
(This was explained in Chapter 5.) The problem that arises is evident in this graph.
The ci at the surface expands up to some value which is determined by our choice
of false c-potential. The lattice spacings just below that expand slightly as well, and
all of this expansion is made up for by considerable contraction of the vapour phase
lattice spacings. In a system constrained to a different ∑i ci , the solid behaves in
the same way and the vapour phase compensates as much as is necessary. Note that
for this choice of false potential the surface relaxation is unphysically large. We
174
Relaxed-Lattice Studies of the Surface
could tune the false potential until the c-profile looked more reasonable, but this is not
very statisfying and the results we obtained with such a model would be completely
dependent on (and very sensitive to) the false potential.
9.3 Conclusions
Although we have derived the necessary thermodynamics to use relaxed lattices in this
system, the approximations that we have made are not robust enough to be used in the
surface studies with relaxed lattices. A more accurate calculation of the crystalline
free volume would doubtless solve the problem, but we have not come up with one
at this time. Numerical integrations should work but are too expensive to use in these
calculations. Because we expect relaxed-lattice surface calculation to be about two
orders of magnitude more expensive than frozen-lattice calculations anyway, this is
unacceptable.
Our false-potential method would allow us to proceed, but there would be very
little point. The surface tension is quite sensitive to the rigidity of the lattice; full
relaxation allows for a 30% decrease in the solid-vapour surface tension! Use of a
restricted (but still flexible) lattice would give spurious results; among other things,
we expect that the surface melting condition γsv > γsl + γlv will never be fulfilled.
These problems could not be anticipated from preliminary bulk calculations, since
they only consider a single lattice cell repeated periodically. In that system, expansion of a single lattice length corresponds to an expansion of the entire crystal in the
corresponding direction, and was never observed in our calculations. Only once the
computer program to do the surface calculations was completed could we determine
whether or not it would work.
Unfortunately, this failure means that we cannot perform surface calculations on
the relaxed-lattice anisotropic systems studied in Chapter 6, so that these and other
systems will have to wait for an improved model.
Chapter 10
STUDIES OF SURFACES WITH
ANISOTROPIC POTENTIALS
In Chapter 6 we used the frozen-lattice model to determine phase diagrams for two
different model anisotropic potentials, with δ02 = −0.30, and δ02 = −0.40. The first
of these has an order-disorder transition in the solid phase that occurs at a lower
temperature than the triple point, so that the sublimation line contains a first-order
orientational transition. The second potential also has an order-disorder transition in
the solid phase, but at a temperature above the triple point so that the melting line
contains a disordering transition. In this chapter we calculate the surface behaviour
for the low-index faces for both of these models. Rather than go into equal detail in
the consideration of each crystal face as in the study of the Lennard-Jones crystal, we
will focus attention on the (100) face in these studies of anisotropic potentials.
10.1 Surfaces of the δ02 = −0.30 Crystal
10.1.1 (100) Surface
In Figure 10.1 (page 176) are the n and p profiles of the (100) solid–vapour interface
for the δ02 = −0.30 potential, over a range of temperatures extending from the fairly
low value of T = 0.51 to just below the bulk orientational transition at T = 0.7985.
That is, we are plotting pi versus i, where i is the layer index and pi is the equilibrium
value of the orientational order parameter p in the ith layer of the system. The density
profiles are plotted in the same way. At low temperatures the density profile is quite
sharp, indicating a clean crystal surface with very little adsorbed material present.
As before, at higher temperatures the profile “rounds off”, broadening to a width of
five or six layers at the temperature of the orientational transition. This is slightly
wider than the (100) solid-vapour interface seen in the Lennard-Jones system, and the
175
176
Studies of Surfaces with Anisotropic Potentials
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
7.0
12.0
17.0
22.0
12.0
17.0
22.0
1.0
p(layer)
0.8
0.6
0.4
0.2
0.0
7.0
Layer Number
Figure 10.1: n and p profiles for the (100) face solid-vapour interface, near
the solid disordering transition, over the temperature range T = 0.510 to
T = 0.79. This data is for the δ02 = −0.30 potential function.
10.1 Surfaces of the δ02 = −0.30 Crystal
177
additional width is probably due to the higher temperatures involved.
The p profiles show that at low temperatures the first adsorbed layer (layer #17
in the plot) has some orientational order, which drops off quickly with increasing
temperature. This order in the dilute monolayer is an energetic effect due to its contact
with the solid surface. Recall that the positional order in the first adsorbed layer is
also quite high, due to the constraints imposed by the γ(n, s) function. Even at the
lowest temperature shown, orientational disorder at the surface extends one or two
layers into the bulk crystal.
As the temperature is raised towards the bulk disordering transition, two things
happen. The first is that the value of p in the bulk crystal drops smoothly to a value
of about 0.5 at the transition temperature. At this temperature the ordered system
is in thermodynamic equilibrium with the disordered system, which is characterised
by p ' 0 and slightly different values of the n and s parameters. The second effect
is that the “orientational width” of the interface increases, so that layers further and
further away from the surface of the crystal begin to disorder. As the temperature gets
very near to the transition, the width of this interface grows quickly, so that the solidvapour interface is “wet” by a thick layer of disordered solid. At these temperatures,
the width of the order-disorder interface in the solid phase is very large and it is
difficult to define the width of the disordered layer.
Above the bulk orientational transition, the entire solid phase is orientationally
disordered. Figure 10.2 (page 178) shows density profiles of the (100) solid-vapour
interface as the temperature is raised from just above the disordering transition to just
below the solid-liquid-vapour triple point. These profiles indicate that the (100) face
of the disordered crystal surface melts, as did the (100) face of the Lennard-Jones
crystal. This melting is much smoother than in the isotropic case, so that there are no
layer-wise transitions apparent from this plot (compare with Figure 7.3). In fact, the
progression of zd across the slab is entirely smooth, as shown in Figure 10.6. It is not
surprising that the disordered crystal surface melts because it is nearly the same as the
Lennard-Jones crystal; once orientational order has been lost the two are very similar.
10.1.2 (110) and (111) Surfaces
Rather than show plots for the density profiles of all three faces, we will plot some of
their thermodynamic functions, and compare these. The surface tensions of all three
crystal faces as functions of temperature are shown in Figure 10.3 (page 179). In each
case the line is divided into two segments; this is because this data must be obtained
in two separate runs, as there is a first-order transition on the line! As in the Lennard-
178
Studies of Surfaces with Anisotropic Potentials
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
12.0
17.0
22.0
27.0
Layer Number
Figure 10.2: n profiles for the (100) face solid-vapour interface, at the
onset of surface melting. The temperature range shown is T = 0.798 to
T = 0.827. This data is for the δ02 = −0.30 potential function.
Jones crystal, the (111) face has the lowest surface tension for low temperatures,
with the (100) next lowest and the (110) face highest. There is a crossing between the
(100) and (110) lines, though this time at a higher temperature. The (111) curve in this
picture is much smoother than that of the Lennard-Jones study (Figure 7.1), indicating
that there are no low-temperature first-order transitions for this crystal face. Adsorbed
material builds up on this face as before, but in a smooth fashion.
At temperatures below the orientational transition, all three of the crystal faces of
the δ02 = −0.30 potential have higher surface tensions than the corresponding faces
of the Lennard-Jones crystal. This is an energetic effect; at very low temperatures
the surface tension is nearly equal to the surface excess energy, which will be higher
for potentials with deeper minima. At higher temperatures the surface tensions in the
anisotropic systems drop well below those of the Lennard-Jones crystal faces. This
is because the anisotropic systems can support a broader interface than the isotropic
ones, since they melt at a much higher temperature. A broader interface is entropically
favourable, stabilising the surfaces at higher temperatures.
Figure 10.4 shows the surface excess heat capacities for all three faces over the
entire temperature range studied. There are many interesting features to be found in
this graph. In the (111) data, there is a broad peak at T ' 0.67 (remember that this is
a logarithmic scale!) at the formation of the first adlayer on the close-packed surface
10.1 Surfaces of the δ02 = −0.30 Crystal
179
3.0
(100)
(111)
(110)
Surface Tension
2.5
2.0
1.5
1.0
0.5
0.0
0.60
0.70
0.80
Temperature
Cex (Surface Excess Heat Capacity)
Figure 10.3: Surface tension as a function of temperature for all three
faces, using the δ02 = −0.30 potential function.
10000
1000
(100)
(111)
(110)
100
10
1
0.60
0.70
0.80
Temperature
Figure 10.4: Surface excess heat capacity for all three faces over the entire
temperature range studied, for the δ02 = −0.30 potential. This data was
obtained by numerical differentiation of the surface tension as a function
of temperature.
180
Studies of Surfaces with Anisotropic Potentials
of the crystal. A much weaker feature is centred at T ' 0.76, corresponding to the
growth of a second adlayer. In the (110) data there is a faint peak at T ' 0.69 which
corresponds to the broadening of the interface, and is reminiscent of the feature in the
Lennard-Jones (110) crystal face data at T ' 0.55. All three of the faces show a rise
in the heat capacity data just below the bulk orientational transition. This is due to the
continuous wetting of the interface by the disordered phase of the solid.
For temperatures higher than the disordering point, all three faces behave the same
and all surface melt very smoothly. This is reflected in the heat capacity data, which
is practically the same for all three faces above T = 0.80. Recall that in the LennardJones crystal this was not the case, so that the onset of surface melting was qualitatively different for the three faces.
The excess entropies for these surfaces are shown in Figure 10.5. This graph
shows that the change in surface excess entropy of the orientational transition is negadisordering
tive (∆Sex
< 0), which makes intuitive sense. If the solid phase is ordered then
disordering the top few layers will greatly increase the surface excess entropy, while
if the solid phase is disordered no such gain is possible. There are broad features
in the entropy curves for the (111) and (110) faces, which correspond to the “extra”
entropy added by the adsorption of an adlayer in the first case, and the broadening
of the interface in the second case. In this data the three faces do not have the same
appearance for all temperatures above T = 0.80; only much closer to the triple point
do they have nearly identical excess entropies.
Figure 10.6 shows the position of the Gibbs dividing surface as a function of
the logarithm of the reduced temperature for all three faces. This data confirms the
conclusion based on the heat-capacity data, that the three faces all behave the same
near surface melting in this model. In Figure 10.6 we see from the linear regression
data that the lines are not quite perfectly straight, and appear to curve very slightly
downwards at higher temperatures. This is probably due to the error in rounding
off the triple point temperature to T = 0.8278. For temperatures within 0.001 or so
of this, the error in the logarithmic function due to the truncation at four significant
figures will be appreciable.
10.1 Surfaces of the δ02 = −0.30 Crystal
181
Sex (Surface Excess Entropy)
20.0
(100)
(111)
(110)
15.0
10.0
5.0
0.0
0.60
0.70
0.80
Temperature
Figure 10.5: Surface excess entropy as a function of temperature for all
three faces, for the δ02 = −0.30 potential. This data was obtained by numerical differentiation of the surface tension as a function of temperature.
Gibbs Dividing Surface (σ)
19.0
18.0
17.0
16.0
15.0
14.0
(100)
(111)
(110)
linear regression
13.0
12.0
1.5
2.5
3.5
-log10(0.8278 - T)
Figure 10.6: Position of the Gibbs dividing surface as a function of reduced temperature, for the δ02 = −0.30 potential function. All three faces
are shown. Note that T = 0.8278 is the solid-liquid-vapour triple point for
this potential.
182
Studies of Surfaces with Anisotropic Potentials
10.2 Surfaces of the δ02 = −0.40 Crystal
In this study we examine the surface behaviour of the frozen-lattice model with the
potential given by δ2 = 0, δ02 = −0.40 and δ002 = 0. The corresponding phase diagram
is shown in Figure 6.6. This potential has a larger degree of anisotropy than the
δ02 = −0.30 potential just described, which should have several effects. Because the ε
parameter has not been rescaled, the ordered solid in this model will have significantly
greater lattice energies than either the Lennard-Jones solid or the δ 02 = −0.30 solid,
which implies a higher melting point. Also, the core volume of the particles will be
somewhat smaller, so that the liquid phase at a given temperature will be slightly more
dense.
10.2.1 (100) Surface
Figure 10.7 shows the n and p profiles of the (100) interface of the δ 02 = −0.40
anisotropic potential over a wide range of temperatures. At the lowest temperature
shown, the crystal surface is completely clean and the density profile is very sharp.
At this temperature (T = 0.452) the very small amount of material in the first adlayer has some orientational order, with p equal to about half of its bulk value. As
the temperature is raised towards the triple point, the bulk value of p drops smoothly,
as it did for the less anisotropic δ02 = −0.30 potential. As before, the surface of the
crystal disorders as it broadens so that by about T = 0.75 the interface is four or five
layers in width (narrower than that of the δ02 = −0.30 potential!) and completely disordered. This narrower width is probably due to energetic factors, which favour a
sharper interface over a broader one.
The last few configurations shown in Figure 10.7 indicate that this system also
surface melts, and over a much larger temperature range than the other systems we
have studied. The last two or three configurations shown appear to be melted, so
that a liquid layer is visible roughly 0.06 temperature units below the triple point, at
about 0.97Tt . In the Lennard-Jones crystal, we did not see any appreciable melting
behaviour until well within this range. In this system the density of the vapour phase
at the triple point is considerably higher than that in either the Lennard-Jones system
or the δ02 = −0.30 system, and the density of the liquid shoulder at surface melting is
considerably lower. These two effects combine to increase the equilibrium width of
the liquid layer at any given temperature, so that we should expect to see the formation
of a liquid layer at lower temperatures.
For this potential, the disordering transition of the solid phase always occurs at
temperatures higher than the triple point, so that there is a second triple point where
10.2 Surfaces of the δ02 = −0.40 Crystal
183
1.0
p(layer)
0.8
0.6
0.4
0.2
0.0
10.0
15.0
20.0
25.0
15.0
20.0
25.0
1.0
n(layer)
0.8
0.6
0.4
0.2
0.0
10.0
Layer Number
Figure 10.7: n and p profiles for the (100) face solid-vapour interface,
using the δ02 = −0.40 potential function. The temperature range is T =
0.452 to T = 0.952, with δT = 0.02.
184
Studies of Surfaces with Anisotropic Potentials
1.0
n(layer)
0.9
0.8
0.7
0.6
0.5
10.0
15.0
20.0
25.0
30.0
15.0
20.0
25.0
30.0
1.0
p(layer)
0.8
0.6
0.4
0.2
0.0
10.0
Layer Number
Figure 10.8: n and p profiles for the (100) face solid-liquid interface, near
the disordering point. The range of temperature covered is T = 0.9520 to
T = 1.0631.
10.2 Surfaces of the δ02 = −0.40 Crystal
185
the ordered solid, disordered solid, and liquid are all in coexistence. As shown in
Chapter 6, this occurs at T = 1.0633, µ = −4.474, at a pressure of P = 1.534. This is
almost directly above the critical point on the (T, µ) phase diagram.
Figure 10.8 shows the n and p profiles of the solid-liquid interface as the system
is moved along the solid-liquid coexistence line from the conventional triple point up
towards this second one. The density profiles through the interface change very little
over this range; the density of the liquid changes by about 13%, but the shape and
width of the interface is essentially constant. The p profiles show, as in the case of
disordering at the solid-vapour interface of Figure 10.1, a smooth progression of the
order/disorder interface into the bulk of the crystal accompanied by a broadening of
that interface. The p profiles indicate a small degree of orientational order through
most of the solid-liquid interface.
Two other lines terminate at this triple point: the order–disorder line of the solid
phase, and the disordered-solid melting line. We have calculated the structure of the
interfaces along these lines over a range of temperatures. The interface between orientationally ordered and disordered parts of a solid crystal consists of a smooth p
profile that drops from its ordered value to zero over a width of four or five lattice
spacings. This interface has the same structure over a very large part of the coexistence line; for extremely high pressures it becomes a bit sharper. Furthermore, this
interface is metastable well below the melting point, meaning that nothing at all happens to the interface as it cools through the second triple point. (Melting of the crystal
would, of course, ensue from some other place at that temperature.) Likewise, the interface between the disordered solid and the liquid exhibits no interesting behaviour
over a wide range of temperature. The structure of the interface looks much like the
n-profiles of Figure 10.8, with the p-profiles being flat lines. This interface is also
metastable through the second triple point.
10.2.2 (110) and (111) Surfaces
The trend in surface tensions followed in the Lennard-Jones study and in the δ 02 =
−0.30 study is still evident for this potential, as shown in Figure 10.9 (page 186).
The (111) face is the global minimum, and there is a crossing between the (100) and
(110) faces. A small kink in the (111) surface tension at T ' 0.64 is visible here,
indicating a first-order transition of some sort. In all other respects, these curves are
unremarkable.
The surface excess heat capacities for all three faces over the same range of temperature are shown in Figure 10.10 (page 186). The suspected transition in the structure of the (111) interface is apparent as a noisy peak centred at T ' 0.64. This is a
186
Studies of Surfaces with Anisotropic Potentials
Surface Tension
3.0
2.0
1.0
0.0
0.55
(100)
(111)
(110)
0.65
0.75
0.85
0.95
Temperature
Cex (Surface Excess Heat Capacity)
Figure 10.9: Surface tension as a function of temperature for all three
faces, for the δ02 = −0.40 potential.
(100)
(111)
(110)
10000
100
1
0.55
0.65
0.75
0.85
0.95
Temperature
Figure 10.10: Surface excess heat capacity for all three faces over the entire temperature range studied, for the δ02 = −0.40 potential. This data was
obtained by numerical differentiation of the surface tension as a function
of temperature.
10.2 Surfaces of the δ02 = −0.40 Crystal
187
Sex (Surface Excess Entropy)
20.0
(100)
(111)
(110)
15.0
10.0
5.0
0.0
0.55
0.65
0.75
0.85
0.95
Temperature
Figure 10.11: Surface excess entropy as a function of temperature for all
three faces, for the δ02 = −0.40 potential. This data was obtained by numerical differentiation of the surface tension as a function of temperature.
first-order transition, but a very weak one; a strong transition would not have a peak
and would just show up as huge noise in the derivatives, while a continuous transition
would be just a peak in the heat capacity. This feature seems to do both. It corresponds, as before, to the formation of a dense adlayer on the solid surface. There is
a broad feature at T = 0.80 which corresponds to the formation of a second adlayer.
The (110) and (100) curves are very similar, with the (110) curve being the smoother
of the two. As before, all three faces appear to surface melt in the same way in this
system. The (110) curve being “smoother” than the (100) curve is consistent with the
results from the Lennard-Jones study, but not the results from the δ 02 = −0.30 study
where the trend was different. Evidently, the trends in the behaviour of the anisotropic
surfaces are not necessarily the same as those of the isotropic system, and crossovers
between different kinds of behaviour must occur at different degrees of anisotropy.
The surface excess entropies of the three crystal faces are shown in Figure 10.11.
Since there is no disordering transition along this coexistence line in this model, the
discontinuity in all three lines observed in Figure 10.5 is not present. The transition
in the (111) face at T = 0.64 is very apparent in this data. The sharp change in
Sex at T = 0.64 probably indicates a weak first-order transition. Over most of the
temperature range shown, the (111) face has a higher surface excess entropy than the
(100) or (110) faces. This is due to the layers of disordered material adsorbed onto
this face.
188
Studies of Surfaces with Anisotropic Potentials
Gibbs Dividing Surface (σ)
19.0
18.0
(110)
(111)
(100)
17.0
16.0
15.0
1.5
2.0
2.5
3.0
-log10(0.9518-T)
Figure 10.12: Gibbs dividing surface plot for the δ02 = −0.40 potential.
T = 0.9518 is the temperature of the solid-liquid-vapour triple point in
this model. The vertical scale and offset are immaterial, and depend on
the size of the slab used in the calculation.
Figure 10.12 shows the motion of the Gibbs dividing surface for all three faces
as a function of the logarithm of the reduced temperature. As was the case for the
δ02 = −0.30 potential, all three lines are perfectly straight with exactly the same slope,
indicating isotropic liquid layer growth without any effects from the periodicity of
either the lattice or the potential field at the surface. This is hardly surprising as
this system has the highest triple point temperature that we have studied, and we
expect that higher temperatures should have the effect of smoothing out transitions.
On the other hand, the solid phase is still orientationally ordered at these temperatures
(though the liquid phase is not), which might somehow influence the liquid layer
growth. The short-ranged nature of the potential seems to preclude this, resulting in
entirely smooth melting behaviour. This surface melting behaviour correlates well
with the data presented in Chapter 8; both of the anisotropic systems we have studied
have high melting points and do not exhibit layering transitions in the surface melting
regime.
10.3 Conclusions
Although the models studied in this chapter are not very realistic, they do exhibit
many interesting features and these results ought to be useful in the interpretation of
10.3 Conclusions
189
more realistic models. We have determined that if the orientational transition of the
solid phase occurs at a temperature below the triple point, then solid disordering can
be initiated at the surface and the surface of the crystal, regardless of exposed face,
will be wet by a layer of disordered crystal. Furthermore, this transition does not
significantly perturb the structure of the surface, though it does dramatically affect
the excess thermodynamic properties. The requirements for this type of behaviour to
be observed are that the anisotropic part of the potential be short-ranged (surface disordering may occurs for long-ranged anisotropic potentials, but we have not studied
them here!), and that the lattice geometry (space group) of the crystal doesn’t change
through the disordering transition.
If the disordering temperature occurs above the triple point, then the surface of
the crystal in contact with the liquid undergoes a very similar transformation as it is
heated through the disordering transition. Since the liquid in this model is entirely
isotropic under these conditions, it is not surprising that the interface behaves this
way. Also, for these materials the surface of the crystal does surface melt, so that
near the triple point the ordered crystal face is wet by a layer of isotropic liquid. This
happens for all three of the crystal faces we have studied.
These calculations support many experimental observations that weakly
anisotropic molecules such as methane and diatomic oxygen behave very much like
the inert gases, at least as far as the structure of disordered crystal surfaces is concerned. Since these materials all undergo orientational transitions at temperatures
much lower than their melting points, their crystals are orientationally disordered
near melting, so that their surface behaviour is not influenced by their anisotropy.
Presumably, this would be the case for any kind of molecule where the orientational
correlation in the liquid phase is only weak and short-ranged. For molecules where the
liquid has a greater degree of orientational order, the presence of the surface should
significantly perturb the structure of the liquid at the solid-liquid interface, which
should affect γsl and the surface phase behaviour.
190
Chapter 11
CONCLUSIONS
We have derived a model for the interfacial behaviour of a simple material using a
lattice-cell approximation to the grand partition function. This model can be solved
within the mean-field approximation using standard numerical techniques. We can
use this approach to study the surface behaviour of molecules which interact through
a simple potential which can include low-order anisotropic terms. In order to make
this approach more realistic, we have included a free volume approximation based on
that used by Trayanov and Tosatti in their approach to a somewhat simpler problem.
This new approximation is simpler and less expensive to evaluate than previous ones,
with no degradation in performance.
We have applied this method to study the behaviour of the (100), (110) and (111)
surfaces of the Lennard-Jones crystal all along the sublimation line. We have characterised the low-temperature “roughening” transitions of these surfaces, some lowtemperature layer-wise transitions, and the surface melting behaviour for all three
faces. These studies showed a higher degree of layer-wise behaviour than was observed before in approximate models of these systems. We have determined that the
reason for this is that the triple point temperature of this model is closer to the true
value then those obtained from other lattice models.
We have done all of these calculations to greater accuracy and precision than those
presented by Trayanov and Tosatti. In addition, we used much finer temperature grids
and performed studies of the sensitivity of the surface behaviour to our approximations. These are the most complete studies of this type to date, and this precision
is necessary in order to determine what is really happening in the model. Without
extremely precise calculations and small temperature increments, we would not have
found the results presented in Chapter 8 and would not have been able to explain all
of the behaviour observed in the surface melting in Chapter 7. These calculations are
also much cheaper than density functional studies performed on the same systems.
Although high-quality DFT calculations are more reliable, very extensive studies are
191
192
Conclusions
still prohibitively expensive and have not really been done.
One of the successes of this study is that we have discovered that the difficulty with
the (111) surface is not due to the “lack of in-plane motion” [184] but really arises
from the use of particular definitions of the lattice cell in the combinatorial γ function.
In our calculations we have made measurements using both possible definitions of the
lattice cell, and have found that the behaviour of the surface is strongly dependent on
this choice. It turns out that the choice of offset which we have studied does not
constrain the surface structure as much, and we have recovered the surface melting
behaviour expected for this face.
Although we have all of the necessary tools for using relaxed-lattice models to
study surface phase transitions, we have not been able to do these studies because of
problems with the calculation of the crystalline free volume. Our own approximation
should work well for small displacements of the lattice spacings away from the bulk
values, but it is badly behaved at longer extensions and has unfortunate problems at
surfaces. Although this might have been anticipated, it was not obvious and could
not be tested until all of the computer code was completed and working, which took
some time. In order to improve the calculation of the crystalline free volume, we
need a recipe for fitting the potential that does not involve the lattice spacings in the
same way. As already pointed out, if we fit the ellipsoidal potential form with more
points taken at fixed extensions (in σ) we might solve the problem, but there seem to
be difficulties with convergence caused by this method. Also, the more points that are
used to fit the form, the more expensive the calculation becomes.
Even without the use of relaxed lattice spacings, we can introduce a degree of
anisotropy into the calculation, which considerably complicates both the bulk phase
diagram and the surface phase behaviour. We have demonstrated that for short-ranged
anisotropic potentials the solid surface disorders at lower temperatures than does the
bulk solid, and near to the orientational transition this disordered layer completely
wets the solid-vapour interface. This is also true when the transition occurs above the
triple point so that the disordered solid wets the solid-liquid interface. Both of the
systems studied completely surface melt, indicating that the state of the solid surface
in these models is not particularly important in describing surface melting.
In order to explain the dependence of the surface behaviour on the triple point
temperature, we did another set of calculations with several slightly different models.
We also measured the surface tension of the solid-vapour interface as a function of
“liquid layer thickness” in order to determine the shape of the free energy surface
controlling the phase behaviour. These revealed that the reference lattice perturbs
the surface tension in a well-defined way. The strength of this perturbation is face-
193
dependent, which helps explain the dramatic differences we found in the behaviour
of the different crystal faces. In Chapter 8 we explicitly determined the effects of
the reference lattice on the liquid-vapour interface, and thus on the surface melting
behaviour of the solid. Although this is trivial in the analysis of some lattice models,
it has not been done in any more complex models, or in any model in the context
of surface melting. The underlying periodicity of the lattice is reflected in the nondecaying oscillation present in the γ(zd ) data for all three crystal faces. In each case,
the magnitude of the oscillation induced by the lattice controls the kind of growth at
the surface. Very small oscillations in the (110) curve correlate well with an entirely
smooth surface melting, while larger oscillations in the (100) and (111) faces cause
stepwise behaviour, and continuous and first-order phase transitions. This behaviour
is all mediated by the temperature and liquid density at the triple point. These two
factors control the structure of the liquid-vapour interface, so that they influence the
amplitude of the oscillation in the surface tension. For higher temperatures and lower
densities, the interface is broader and less sensitive to lattice offsets, while for lower
temperatures and higher densities the interface is sharper and more sensitive, leading
to larger amplitude oscillations.
The non-decaying oscillations in γ(zd ) lead to a remarkable variety of phase behaviour. For very thin liquid layers (at the onset of surface melting), the oscillations
are only a perturbation and do not control the system, so that the layer-wise transitions
are either nonexistent or very weak continuous transitions. At temperatures closer to
the triple point, the smoothed γ(zd ) broadens considerably and its minimum becomes
shallower. This has the effect of increasing the importance of the oscillations, so that
they control the system and we see first-order layering.
In Chapter 8 we explained how the decay of the oscillations in a real system and
the broadening of the minimum in γ(zd ) are competing effects, the winner of which
determines whether or not the system completely surface melts or only incompletely
surface melts. These effects are not measurable using this model because oscillations
in liquid structure are not present, so that we cannot make any prediction about the
surface melting extremely near to the triple point. However, we note that the scales
involved seem to indicate that any reasonably strong oscillation will cause the system
to exhibit only thin-layer pseudo-melting. In our calculations, even an oscillation less
that 1% of the value of the surface tension within about 5σ of the solid surface is
sufficient to cause first-order layering behaviour, and thus pseudo-melting. Since the
large amplitude of oscillations near the wall in the h(z) curves presented in Chapter 1
indicate that the oscillation in free energy should be significant out to at least 5σ, we
expect that the complete surface melting exhibited by this model is an artifact of the
194
Conclusions
lack of correlation in the liquid phase, and that all three of the faces of a rare-gas
crystal should only incompletely surface melt.
We proposed that to settle this question would require extensive simulations of
surface systems using methods (GC-MC) that could determine the surface tension as
a function of some constrained liquid layer thickness. The Grand Canonical Monte
Carlo method is difficult to use in systems such as this; the simulation box contains
both fully ordered regions and low-density disordered regions, so that particle insertion/elimination techniques are likely to have great difficulty sampling the different
phases “evenly”. A more appealing route to obtaining γ(l) data would be to perform
an umbrella-sampling simulation on the surface system, where the umbrella potential
is a function of the liquid layer thickness. We have already determined that z d (relative to some fixed solid layer) can be used as a measure of the liquid layer thickness;
an umbrella potential could be used to force the system to explore a range of values
of zd at some fixed T and P. From this data we could determine a Landau free energy
for the system, and thus the Gibbs free energy [244]. This simulation method would
not require particle insertions or other difficult Monte Carlo moves. As we explained
in Chapter 7, using a fixed number of particles does perturb the surface tension, but
in the surface melting regime this effect is very small.
We have done the most complete study of a complex lattice model of surface melting to date. We have determined that the techniques that we have used to make this
model tractable are not sufficiently correct to avoid unphysical perturbations due to
the lattice-cell approximation. These perturbations are well-defined, and we can account for their contributions to the behaviour of the model. We have made progress
towards a more elaborate model where the reference lattice is flexible, but this ran
into difficulty and could not be used for these studies. This improvement is necessary
for our original goal of modelling the surface behaviour of crystals of anisotropic
molecules. However, we have made some progress towards this goal by using restricted anisotropic potentials and frozen lattices, which have produced interesting
results.
Appendix A
A PATHOLOGICAL MINIMISATION
In general, minimisation, optimisation and root-finding algorithms all work by iterating a procedure until some calculated quantity is less than an input tolerance, at
which point the routines claim that they have successfully converged and return. For
instance, in the Numerical Recipes minimisations the convergence test looks like
IF (2.0*ABS(FP-FRET).LE.FTOL*(ABS(FP)+ABS(FRET))) RETURN
where FP and FRET are the two most recent function values obtained, and FTOL is a
functional tolerance. That is, when the length of a single step downhill is smaller
(relative to the function value) than the tolerance, the routine finishes. Some routines (usually root-finders) use absolute tolerances instead of relative ones. Because
multidimensional minimisation routines work by performing many successive onedimensional minimisations, other tolerances enter into the program. This kind of
convergence test is suitable provided that the tolerances are all chosen correctly and
that the function to be minimised doesn’t have too rocky a surface. In general, it
is a very good idea to check that the point returned by the minimiser is actually a
minimum by testing the length of the gradient vector, which introduces yet another
tolerance.
In the development of a program to study the model discussed in Chapter 4, we
experienced terrible problems with convergence, many of which were due to inappropriate choices of the many tolerances needed by the minimisation routines. In order to
illustrate some of the difficulties involved with performing these types of calculations
to the limits of machine precision, we can study much simpler functions that have
minima of a similar shape.
As an example of a function with these difficulties, consider
f (x) = −A (x − 1)2 − log x
195
(A.1)
196
A Pathological Minimisation
with derivative equal to
1
f 0 (x) = 2A (1 − x) − .
x
(A.2)
√
2A ± 2 A2 − 2A
.
x0 =
4A
(A.3)
This derivative has zeros at
The (−) zero is a minimum. This minimum is similar to the “solid” solution of our
free energy function with respect to n, where the quadratic term is like the energy and
the logarithmic term is contained in γ(n, s). For different values of A, the minimum of
f (x) occurs in different places; for large A it approaches zero as roughly 1/A. Some
representative values of A and x0 are shown in Table A.1. First consider the A =
100 case. If we perturb the function away from the minimum by −0.0000253 =
−2.53 × 10−5 , so that x = 0.005, then ∆ f = ( f (x) − f (x0 )) = 1.261 × 10−5 , while
∆ f 0 = ( f 0 (x) − f 0 (x0 )) = f 0 (x) = −1.00, which is five orders of magnitude larger.
If we look at the A = 1000 case, then we find that a perturbation of only −2.5 ×
−7
10 causes a functional perturbation of ∆ f = −2.0 × 10−7 , and a perturbation in the
gradient of ∆ f 0 = −1.00 again which is six orders of magnitude larger.
If we perform a Taylor expansion of f (x) around this minimum, we find for A =
1000 that
f (x − x0 ) − f (x0 ) = 1.998 × 106 (x − x0 )2
+ 2.663 × 109 (x − x0 )3
+ 3.992 × 1012 (x − x0 )4
+ ...,
(A.4)
so that for (x − x0 ) > 10−3 , all of the higher terms are very significant.
The point of this analysis is that in a numerical minimisation of this function, if we
want to use both the function value and the magnitude of the gradient vector as “success” criteria, then the tolerance chosen for the gradient vector must be chosen many
orders of magnitude larger than that for the function value for the two be satisfied at
A
10
20
100
1000
x0
0.05278
0.02567
0.0050253
0.00050025
Table A.1: Values of A and the corresponding position of the minimum x 0 .
197
roughly the same step in the minimisation. On a computer using double precision
arithmetic, we expect that errors of order 10−16 will be introduced in every operation.
If many thousands of operations are necessary for a single function evaluation (for
instance, sums over a lattice), then the precision we expect for the function (and an
analytically evaluated gradient) might be of order 10−12 or 10−13 , or even worse if we
are ever dividing large numbers by small ones or doing other imprecise operations.
A function-based minimisation would have to be set to quit before this tolerance is
reached, since noise will disrupt the calculation at this point. If the behaviour of the
function is as poor as the one just described, then a high-precision choice for the
function tolerance might be ftol = 10−10 , with a corresponding gradient tolerance
of gtol = 10−4 . However, because the gradient and function are evaluated to the
same precision, there is “extra information” still available at this point. This could
be extracted with a gradient-based optimisation algorithm, such as the “polishing”
procedure described in Chapter 5.
198
Appendix B
PROGRAMS
All of the studies in this thesis were done on the DAP 600 computer at the Department
of Applied Mathematics and Theoretical Physics or on Silicon Graphics workstations
in the University Chemical Laboratory. We have written all of the programs from
scratch except for some of the numerical routines, which came either from NETLIB
([email protected]) or Numerical Recipes [42]. All of the code was written in FORTRAN 77. These programs are all in varying states of disrepair, and will be freely
given, without guarantee, to anyone who is interested.
B.0.1 AFM - Simulations
As we have already explained, the molecular dynamics simulations presented in Chapter 1 were done on a DAP 600, an SIMD-architecture parallel machine. The DAP
speaks its own highly stylised version of FORTRAN, so that these programs are not
portable to any other kind of machine. The architecture of this machine requires a
server workstation so that two programs are required for any study; one for the server,
which interprets input data into a form recognisable by the DAP, and the DAP program itself. DAP codes are generally very specialised in order to make the most
of the machine’s peculiar architecture; the algorithms that we used for the neighbors
lists and other parts of simulations would not be at all efficient on a more conventional
computer.
B.0.2 AFM - Integral Equations
These calculations were all performed on Indigo R3000 Silicon Graphics workstations in the Chemical Laboratory. The necessary programs are fairly short and simple
to write. The only difficulty that must be overcome is that the FFT routines need to
be correctly scaled; most library routines do not do this as default.
199
200
Programs
B.0.3 BEG Calculations
These calculations were also all performed on Indigo R3000 Silicon Graphics workstations. The programs are fairly short and straightforward. We have two separate
programs, one that uses minimisations and one that uses root-findings. As a warm-up
to these programs we have also written a program which calculates the mean-field
surface behaviour of the simple Ising model.
B.0.4 “A Better Theory” Calculations
The program that implements the theory described in Chapters 4 and 5 is called SURF,
and is reasonably well-maintained. SURF has a long history; it started with a program
that only did bulk calculations (BULK), with which we tried several other γ functions
and free volume approximations before implementing the slab geometry calculation
in SURF. This program can do all of the necessary tasks for the studies presented
here, including finding triple points, coexistence lines, and performing the constrained
studies in Chapter 8. SURF compiles on SGI machines and DEC machines; it has not
been tested on any others at this time. Most of the calculations were done on Indigo2
R8000 workstations. The code used to do these studies is approximately 8000 lines
long, which makes it the longest program used in this work by a large margin. SURF
is resonably well documented and has a manual that explains its workings in detail.
B.0.5 This Thesis
This thesis was written in LATEX [245], using a number of non-standard packages,
including epsf, mathptm, fancyheadings and cite [246]. The plots were all prepared
using the ACE/gr program by P. J. Turner. The format of the thesis itself is all contained in my own thesis.cls class file, which is based on the standard book.cls.
Surface plots were prepared with Mathematica [247], and configuration snapshots
were done with the IGP program by K. D. Hammonds.
References
[1] S. Lem (1975) The Cyberiad: Fables for the Cybernetic Age, Mandarin, London.
[2] S. J. O’Shea, M. E. Welland and T. Rayment (1992) Appl. Phys. Lett., 60(19),
2356.
[3] S. J. O’Shea, M. E. Welland and J. B. Pethica (1994) Chem. Phys. Lett., 223,
336.
[4] J. N. Israelachvili and G. E. Adams (1978) J. Chem. Soc. Faraday Trans. I, 74,
975.
[5] R. G. Horn and J. N. Israelachvili (1981) J. Chem. Phys., 75(3), 1400.
[6] J. N. Israelachvili (1985) Chemica Scripta, 25, 7.
[7] F. F. Abraham (1978) J. Chem. Phys., 68(8), 3713.
[8] J. J. Magda, M. Tirrell and H. T. Davis (1985) J. Chem. Phys., 83(4), 1888.
[9] I. K. Snook and W. van Megen (1980) J. Chem. Phys., 72(5), 2907.
[10] L. J. Douglas, M. Lupkowski, T. L. Dodd and F. van Swol (1993) Langmuir,
6(9), 1442.
[11] L. D. Gelb and R. M. Lynden-Bell (1993) Chem. Phys. Lett., 211(4,5), 328.
[12] L. D. Gelb and R. M. Lynden-Bell (1994) Phys. Rev. B, 49(3), 2058.
[13] M. P. Allen and D. J. Tildsley (1987) Computer Simulation of Liquids, Clarendon Press, Oxford.
[14] J. E. Lennard-Jones (1931) Proc. Phys. Soc., 43(240), 461.
[15] J. E. Lennard-Jones (1924) Proc. Camb. Phil. Soc., 22, 105.
201
202
References
[16] W. A. Steele (1973) Surf. Sci., 36, 317.
[17] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNiola and J. R.
Haak (1984) J. Chem. Phys., 81(8), 3684.
[18] J. N. Israelachvili (1992) Intermolecular and Surface Forces, Academic Press,
London, 2nd edn.
[19] J. P. Hansen and I. R. McDonald (1976) Theory of Simple Liquids, Academic
Press, London.
[20] L. S. Ornstein and F. Zernike (1914) Proc. Akad. Sci. (Amsterdam), 17, 793.
[21] J. K. Percus and G. J. Yevick (1958) Phys. Rev., 110(1), 1.
[22] G. Stell (1963) Physica, 29, 517.
[23] J. M. J. van Leeuwen, J. Groeneveld and J. De Boer (1959) Physica, 25, 792.
[24] E. Meeron (1960) J. Math. Phys., 1(3), 192.
[25] T. Morita (1960) Prog. Theor. Phys., 23(5), 829.
[26] G. S. Rushbrooke (1960) Physica, 26, 259.
[27] L. Verlet (1960) Nuovo Cim., 18(1), 77.
[28] J. Chihara (1973) Prog. Theor. Phys., 50(2), 409.
[29] W. G. Madden and S. A. Rice (1980) J. Chem. Phys., 72(7), 4208.
[30] F. J. Rogers and D. A. Young (1984) Phys. Rev. A, 30(2), 999.
[31] G. Zerah and J. P. Hansen (1986) J. Chem. Phys., 84(4), 2336.
[32] S. M. Foiles, N. W. Ashcroft and L. Reatto (1984) J. Chem. Phys., 80(9), 4441.
[33] A. A. Broyles (1960) J. Chem. Phys., 33(2), 456.
[34] A. A. Broyles, S. U. Chung and H. L. Stalin (1962) J. Chem. Phys., 37(10),
2462.
[35] T. K. Vanderlick, L. E. Scriven and H. T. Davis (1989) J. Chem. Phys., 90(4),
2422.
[36] R. Kjellander and S. Sarman (1990) Mol. Phys., 70(2), 215.
References
203
[37] R. Kjellander and S. Sarman (1991) Mol. Phys., 74(3), 665.
[38] P. Attard, D. R. Bérard, C. P. Ursenbach and G. N. Patey (1991) Phys. Rev. A,
44(12), 8224.
[39] D. Henderson and M. Plischke (1992) J. Chem. Phys., 97(10), 7822.
[40] M. J. Gillan (1979) Mol. Phys., 38(6), 1781.
[41] K.-C. Ng (1974) J. Chem. Phys., 61(7), 2680.
[42] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery (1988) Numerical Recipes in C, Cambridge University Press, Cambridge, 2nd edn.
[43] G. Zerah (1985) J. Computat. Phys., 61, 280.
[44] F. S. Beckman (1960) in Mathematical Methods for Digital Computers (Edited
by A. Ralson and H. S. Wilf), vol. 1, 62–72, John Wiley and Sons, Inc., New
York.
[45] D. Henderson, F. F. Abraham and J. A. Barker (1976) Mol. Phys., 31(4), 1291.
[46] H. Margenau and G. M. Murphy (1943) The Mathematics of Chemistry and
Physics, D. Van Nostrand Co., Inc., New York.
[47] R. Evans, R. J. F. Leote de Carvalho, J. R. Henderson and D. C. Hoyle (1994)
J. Chem. Phys., 100(1), 591.
[48] R. Evans, J. R. Henderson, D. C. Hoyle, A. O. Parry and Z. A. Sabeur (1993)
Mol. Phys., 80(4), 755.
[49] P. Attard, C. P. Ursenbach and G. N. Patey (1992) Phys. Rev. A, 45(10), 7621.
[50] R. C. Weast and M. J. Astle (1981) Handbook of Chemistry and Physics, CRC
Press, Cleveland, 62nd edn.
[51] E. R. Dobbs and G. O. Jones (1957) Rep. Prog. Phys., 20, 516.
[52] J. M. Yeomans (1992) Statistical Mechanics of Phase Transitions, Clarendon
Press, Oxford.
[53] P. Ehrenfest (1933) Proc. Roy. Akad. Sci. (Amsterdam), 36, 153.
[54] H. Nakanishi and M. E. Fisher (1982) Phys. Rev. Lett., 49(21), 1565.
204
References
[55] J. R. Henderson (1994) Phys. Rev. E, 50(6), 4836.
[56] S. Dietrich and M. Schick (1985) Phys. Rev. B, 31(7), 4718.
[57] J. W. Cahn (1977) J. Chem. Phys., 66(8), 3667.
[58] G. Jacucci and N. Quirke (1983) Phys. Lett. A, 98(5,6), 269.
[59] R. H. Swendsen (1978) Phys. Rev. B, 18(1), 492.
[60] J. D. Weeks and G. H. Gilmer (1979) Adv. Chem. Phys., 40, 157.
[61] R. Pandit, M. Schick and M. Wortis (1982) Phys. Rev. B, 26, 5112.
[62] J. Kahng and C. Ebner (1989) Phys. Rev. B, 40(16), 11269.
[63] C. Ebner (1983) Phys. Rev. B, 28(5), 2890.
[64] M. Faraday (1860) Proc. Roy. Soc. (London), 10, 440.
[65] A. Lied, H. Dosch and J. H. Bilgram (1994) Phys. Rev. Lett., 72(22), 3554.
[66] E. Mazzega, U. del Pennino, A. Loria and S. Mantovani (1976) J. Chem. Phys.,
64(3), 1028.
[67] S. Valeri and S. Mantovani (1978) J. Chem. Phys., 69(11), 5207.
[68] M. Maruyama, M. Bienfait, J. G. Dash and G. Coddens (1992) J. Crystal Growth, 118, 33.
[69] L. A. Wilsen and J. G. Dash (1995) Phys. Rev. Lett., 74(25), 5076.
[70] M. Elbaum, S. G. Lipson and J. G. Dash (1993) J. Crystal Growth, 129, 491.
[71] M. W. Orem and A. W. Adamson (1969) J. Coll. Int. Sci., 31(2), 278.
[72] J. Ocampo and J. Klinger (1983) J. Phys. Chem., 87, 4167.
[73] J. Ocampo and J. Klinger (1983) J. Phys. Chem., 87, 4325.
[74] V. I. Kvlividze, V. F. Kiselev, A. B. Kurzaev and L. A. Ushakova (1974)
Surf. Sci., 44, 60.
[75] Y. Mizuno and N. Hanafusa (1987) J. Physique C, 48(3), 511.
[76] D. Nason and N. H. Fletcher (1975) J. Chem. Phys., 62(11), 4444.
References
205
[77] I. Golecki and C. Jaccard (1978) J. Phys. C: Solid State Phys., 11, 4229.
[78] D. Beaglehole and D. Nason (1980) Surf. Sci., 96, 357.
[79] Y. Furukawa, M. Yamamoto and T. Kuroda (1987) J. Physique C, 48(3), 495.
[80] R. Gangwar and R. M. Suter (1990) Phys. Rev. B, 42(4), 2711.
[81] J. P. Coulomb and O. E. Vilches (1984) J. Physique, 45, 1381.
[82] J. P. Coulomb, T. S. Sullivan and O. E. Vilches (1984) Phys. Rev. B, 30(8),
4753.
[83] J. Z. Larese (1993) Acc. Chem. Res., 26(7), 353.
[84] J. Z. Larese and Q. M. Zhang (1990) Phys. Rev. Lett., 64(8), 922.
[85] J. Z. Larese and Q. M. Zhang (1995) Phys. Rev. B, 51(23), 17023.
[86] D.-M. Zhu and J. G. Dash (1986) Phys. Rev. Lett., 57(23), 2959.
[87] D.-M. Zhu and J. G. Dash (1988) Phys. Rev. B, 38(16), 11673.
[88] J. L. Seguin, J. Suzanne, M. Bienfait, J. G. Dash and J. A. Venables (1983)
Phys. Rev. Lett., 51(2), 122.
[89] M. S. Pettersen, M. J. Lysek and D. L. Goodstein (1989) Phys. Rev. B, 40(7),
4938.
[90] H. S. Youn and G. B. Hess (1990) Phys. Rev. Lett., 64(8), 918.
[91] P. Day, M. LaMadrid, M. Lysek and D. Goodstein (1993) Phys. Rev. B, 47(12),
7501.
[92] J. M. Phillips, Q. M. Zhang and J. Z. Larese (1993) Phys. Rev. Lett., 71(18),
2971.
[93] J. Krim, J. G. Dash and J. Suzanne (1984) Phys. Rev. Lett., 52(8), 640.
[94] P. Zeppenfeld, J. Goerge, M. Büchel, R. David and G. Comsa (1994) Surf. Sci.,
318, L1187.
[95] M. Maruyama (1989) J. Crystal Growth, 94, 757.
[96] M. Bienfait and J. M. Gay (1991) in Phase Transitions in Surface Films 2
(Edited by H. Taub et al), 307–325, Plenum Press, New York.
206
References
[97] M. Bienfait, J. P. Coulomb and J. P. Palmari (1987) Surf. Sci., 182, 557.
[98] M. Bienfait (1987) Europhys. Lett., 4(1), 79.
[99] M. Bienfait, J. M. Gay and H. Blank (1988) Surf. Sci., 204, 331.
[100] J. M. Gay, J. Suzanne and J. P. Coulomb (1990) Phys. Rev. B, 41(16), 11346.
[101] J. P. Coulomb, K. Madih, B. Croset and H. J. Lauter (1985) Phys. Rev. Lett.,
54(14), 1536.
[102] M. Bienfait, P. Zeppenfeld, J. M. Gay and J. P. Palmari (1990) Surf. Sci., 226,
327.
[103] J. Z. Larese, M. Harada, L. Passell, J. Krim and S. Satija (1988) Phys. Rev. B,
37(9), 4735.
[104] S. Chandavarkar, R. M. Geertman and W. H. de Jeu (1992) Phys. Rev. Lett.,
69(16), 2384.
[105] X.-Y. Liu (1993) Phys. Rev. B, 48(3), 1825.
[106] D. Beaglehole, E. Z. Radlinska, B. W. Ninham and H. K. Christenson (1991)
Langmuir, 7(9), 1843.
[107] J. M. Gay, J. Suzanne, G. Pepe and T. Meichel (1988) Surf. Sci., 204, 69.
[108] H. K. Kim, Y. P. Feng, Q. M. Zhang and M. H. W. Chan (1988) Phys. Rev. B,
37(4), 1745.
[109] J. Z. Larese, L. Passell, A. D. Heidemann, D. Richter and J. P. Wicksted (1988)
Phys. Rev. Lett., 61(4), 432.
[110] J. Z. Larese, L. Passell and B. Ravel (1988) Can. J. Chem., 66, 633.
[111] H. S. Nham and G. B. Hess (1988) Phys. Rev. B, 38(7), 5166.
[112] M. Maruyama, M. Bienfait, F. C. Liu, Y. M. Liu, O. E. Vilches and F. Rieutord
(1993) Surf. Sci., 283, 333.
[113] P. Zeppenfeld, M. Bienfait, F. C. Liu, O. E. Vilches and G. Coddens (1990)
J. Physique, 51(17), 1929.
[114] R. Chiarello, J. P. Coulomb, J. Krim and C. L. Wang (1988) Phys. Rev. B,
38(13), 8967.
References
207
[115] J. Krim, J. P. Coulomb and J. Bouzidi (1987) Phys. Rev. Lett., 58(6), 583.
[116] H. S. Youn and G. B. Hess (1990) Phys. Rev. Lett., 64(4), 443.
[117] B. Pluis, J. M. Gay, J. W. M. Frenken, S. Gierlotka, J. F. van der Veen, J. E.
MacDonald, A. A. Williams, N. Piggins and J. Als-Nielsen (1989) Surf. Sci.,
222, L845.
[118] B. Pluis, J. W. M. Frenken and J. F. van der Veen (1987) Physica Scripta, T19,
382.
[119] J. W. M. Frenken, B. J. Hinch, J. P. Toennies and Ch. Wöll (1990) Phys. Rev. B,
41(2), 938.
[120] J. W. M. Frenken, F. Huussen and J. F. van der Veen (1987) Phys. Rev. Lett.,
58(4), 401.
[121] J. K. Kristensen and R. M. J. Cotterill (1977) Phil. Mag., 36(2), 437.
[122] B. Pluis, A. W. Denier van der Gon, J. F. van der Veen and A. J. Riemersma
(1990) Surf. Sci., 239, 265.
[123] J. W. M. Frenken, R. J. Hamers and J. E. Demuth (1990) J. Vac. Sci. Technol. A,
8(1), 293.
[124] E. Holub-Krappe, K. Horn, J. W. M. Frenken, R. L. Krans and J. F. van der
Veen (1987) Surf. Sci., 188, 335.
[125] J. Villain, D. R. Grempel and J. Lapujoulade (1985) J. Phys. F: Met. Phys., 15,
809.
[126] J. F. van der Veen, B. Pluis and A. W. Denier van der Gon (1980) in Chemistry
and Physics of Solid Surfaces VII (Edited by R. Vaneslow and R. F. Howe),
455–489, Springer-Verlag, Berlin.
[127] J. F. van der Veen and J. W. M. Frenken (1986) Surf. Sci., 178, 382.
[128] J. G. Dash (1991) in Phase Transitions in Surface Films 2 (Edited by
H. Taub et al), 339–356, Plenum Press, New York.
[129] J. Q. Broughton and G. H. Gilmer (1983) J. Chem. Phys., 79(10), 5095.
[130] J. Q. Broughton and G. H. Gilmer (1983) J. Chem. Phys., 79(10), 5105.
208
References
[131] J. Q. Broughton and G. H. Gilmer (1983) J. Chem. Phys., 79(10), 5119.
[132] J. Q. Broughton and G. H. Gilmer (1986) J. Chem. Phys., 84(10), 5741.
[133] J. Q. Broughton and G. H. Gilmer (1986) J. Chem. Phys., 84(10), 5749.
[134] J. Q. Broughton and G. H. Gilmer (1986) J. Chem. Phys., 84(10), 5759.
[135] V. Rosato, G. Ciccotti and V. Pontikis (1986) Phys. Rev. B, 33(3), 1860.
[136] R. M. Lynden-Bell (1990) Surf. Sci., 230, 311.
[137] D. J. Adams (1975) Mol. Phys., 29(1), 307.
[138] J. E. Lane and T. H. Spurling (1976) Aust. J. Chem., 29, 2103.
[139] J. P. R. B. Walton and N. Quirke (1989) Mol. Sim., 2, 361.
[140] J. M. Phillips (1990) Phys. Lett. A, 147(1), 54.
[141] J. M. Phillips and C. D. Hruska (1989) Phys. Rev. B, 39(8), 5425.
[142] C. D. Hruska and J. M. Phillips (1988) Phys. Rev. B, 37(7), 3801.
[143] J. M. Phillips and N. Shrimpton (1992) Phys. Rev. B, 45(7), 3730.
[144] J. M. Phillips and T. R. Story (1990) Phys. Rev. B, 42(11), 6944.
[145] J. M. Phillips (1995) Phys. Rev. B, 51(11), 7186.
[146] A. Alavi and S. Chandavarkar (1994) Surf. Sci., 302(3), L331.
[147] A. Boutin, B. Rousseau and A. H. Fuchs (1993) Surf. Sci., 287, 866.
[148] A. Boutin and A. H. Fuchs (1993) J. Chem. Phys., 98(4), 3290.
[149] R. Lipowsky and W. Speth (1983) Phys. Rev. B, 28(7), 3983.
[150] G.-J. Kroes (1992) Surf. Sci., 275, 365.
[151] J. P. Rose and R. S. Berry (1993) J. Chem. Phys., 98(4), 3246.
[152] J. W. Cahn, J. G. Dash and H.-Y. Fu (1992) J. Crystal Growth, 123(1-2), 101.
[153] N. Cabrera, J. M. Soler, J. J. Sáenz, N. Garcı́a and R. Miranda (1984) Physica B, 127, 175.
References
209
[154] M. J. de Oliveira and R. B. Griffiths (1978) Surf. Sci., 71, 687.
[155] W. F. Saam (1983) Surf. Sci., 125, 253.
[156] A. A. Migdal (1975) Zh. Eksp. Teor. Fiz., 69(4), 1457.
[157] L. P. Kadanoff (1976) Ann. Phys. (NY), 100, 359.
[158] C. Ebner (1981) Phys. Rev. A, 23(4), 1925.
[159] J. A. Niemen and K. Kaski (1990) Phys. Rev. B, 41(4), 2321.
[160] D. Stauffer, J. S. Ho and M. Sahimi (1991) J. Chem. Phys., 94(2), 1385.
[161] P. Wagner and K. Binder (1986) Surf. Sci., 175, 421.
[162] R. Moss and P. Harrowell (1994) J. Chem. Phys., 100(10), 7630.
[163] A. Patrykiejew and K. Binder (1992) Surf. Sci., 273, 413.
[164] K. Binder and D. P. Landau (1992) Phys. Rev. B, 46(8), 4844.
[165] K. Binder and D. P. Landau (1988) Phys. Rev. B, 37(4), 1745.
[166] K. Binder and D. P. Landau (1984) Phys. Rev. Lett., 52(5), 318.
[167] K. Binder and D. P. Landau (1992) J. Chem. Phys., 96(2), 1444.
[168] M. den Nijs (1991) in Phase Transitions in Surface Films 2 (Edited by
H. Taub et al), 247–267, Plenum Press, New York.
[169] M. den Nijs, E. K. Riedel, E. H. Conrad and T. Engel (1985) Phys. Rev. Lett.,
55(16), 1689.
[170] M. den Nijs (1992) Phys. Rev. B, 46(16), 10386.
[171] M. den Nijs (1991) Phys. Rev. Lett., 66(7), 907.
[172] H. van Beijeren (1977) Phys. Rev. Lett., 38(18), 993.
[173] R. H. Swendsen (1977) Phys. Rev. Lett., 38(11), 615.
[174] M. den Nijs (1990) Phys. Rev. Lett., 64(4), 435.
[175] M. P. Nightingale, W. F. Saam and M. Schick (1984) Phys. Rev. B, 30(7), 3830.
[176] R. H. Swendsen (1976) Phys. Rev. Lett., 37(22), 1478.
210
References
[177] H. Asada (1990) Surf. Sci., 230, 323.
[178] C. S. Jayanthi (1991) Phys. Rev. B, 44(1), 427.
[179] Y. Teraoka and T. Seto (1992) Surf. Sci., 261, 275.
[180] Y. Teraoka and T. Seto (1993) Surf. Sci., 283, 371.
[181] Y. Teraoka (1993) Surf. Sci., 294, 273.
[182] C. Ebner (1980) Phys. Rev. A, 22(6), 2776.
[183] S. Yashonath and D. D. Sarma (1984) Chem. Phys. Lett., 110(3), 265.
[184] A. Trayanov and E. Tosatti (1988) Phys. Rev. B, 38(10), 6961.
[185] A. Trayanov and E. Tosatti (1992) Solid State Comm., 84(1/2), 177.
[186] T. V. Ramakrishnan and M. Yussouff (1979) Phys. Rev. B, 19(15), 2775.
[187] T. V. Ramakrishnan (1982) Phys. Rev. Lett., 48(8), 541.
[188] P. Tarazona (1985) Phys. Rev. A, 31(4), 2672.
[189] W. A. Curtin and N. W. Ashcroft (1985) Phys. Rev. A, 32(5), 2909.
[190] M. S. Wertheim (1963) Phys. Rev. Lett., 10(8), 321.
[191] R. J. Baxter (1968) Aust. J. Phys., 21, 563.
[192] E. Thiele (1963) J. Chem. Phys., 39(2), 474.
[193] N. F. Carnahan and K. E. Starling (1969) J. Chem. Phys., 51(2), 635.
[194] J. D. Weeks, D. Chandler and H. C. Andersen (1971) J. Chem. Phys., 54(12),
5237.
[195] H. C. Andersen, J. D. Weeks and D. Chandler (1971) Phys. Rev. A, 4(1), 1597.
[196] H. Löwen, R. Ohnesorge and H. Wagner (1994) Ber. Bunsenges. Phys. Chem.,
98(3), 303.
[197] R. Ohnesorge, H. Löwen and H. Wagner (1994) Phys. Rev. E, 50(6), 4801.
[198] B. Pluis, D. Frenkel and J. F. van der Veen (1989) Surf. Sci., 239, 282.
References
211
[199] J. E. Lennard-Jones and A. F. Devonshire (1939) Proc. Roy. Soc. (London),
A169, 317.
[200] J. E. Lennard-Jones and A. F. Devonshire (1939) Proc. Roy. Soc. (London),
A170, 464.
[201] H. Mori, H. Okamoto and S. Isa (1972) Prog. Theor. Phys., 47(4), 1087.
[202] Z. Pawlowska, G. F. Kventsel and T. J. Sluckin (1988) Phys. Rev. A, 38(10),
5342.
[203] W. L. Bragg and E. J. Williams (1934) Proc. Roy. Soc. (London), A145, 699.
[204] T. Muto and Y. Takagi (1955) in Solid State Physics (Edited by F. Seitz and
D. Turnbull), vol. 1, Academic Press, Inc., New York.
[205] J. A. Pople and F. E. Karasz (1961) J. Phys. Chem. Solids, 18(1), 28.
[206] F. E. Karasz and J. A. Pople (1961) J. Phys. Chem. Solids, 21(3/4), 294.
[207] T. Sakka, M. Iwasaki and Y. Ogata (1991) J. Chem. Phys., 95(4), 2688.
[208] S. Chandrasekhar, R. Shoshidhar and N. Tara (1970) Mol. Crys. Liq. Crys., 10,
337.
[209] S. Chandrasekhar, R. Shoshidhar and N. Tara (1971) Mol. Crys. Liq. Crys., 12,
245.
[210] L. M. Amzel and L. N. Becka (1969) J. Phys. Chem. Solids, 30, 521.
[211] D. Ronis and C. Rosenblatt (1980) Phys. Rev. A, 21(5), 1687.
[212] Z. Pawlowska, G. F. Kventsel and T. J. Sluckin (1987) Phys. Rev. A, 36(2), 992.
[213] A. N. Berker, S. Ostlund and F. A. Putnam (1978) Phys. Rev. B, 17(9), 3650.
[214] L. D. Gelb (1994) Phys. Rev. B, 50(15), 11146.
[215] M. Blume, V. J. Emery and R. B. Griffiths (1971) Phys. Rev. A, 4(3), 1071.
[216] W. Hoston and A. N. Berker (1991) Phys. Rev. Lett., 67(8), 1027.
[217] K. Huang (1963) Statistical Mechanics, John Wiley & Sons, New York, 2nd
edn.
[218] J. G. Kirkwood (1950) J. Chem. Phys., 18(3), 380.
212
References
[219] T. L. Hill (1956) Statistical Mechanics, Principles and Selected Applications,
McGraw-Hill Book Co., Inc., New York.
[220] D. J. Cleaver and M. P. Allen (1993) Mol. Phys., 80(2), 253.
[221] P. A. Lebwohl and G. Lasher (1972) Phys. Rev. A, 6(1), 426.
[222] W. L. McMillan (1971) Phys. Rev. A, 4(3), 1238.
[223] W. L. McMillan (1972) Phys. Rev. A, 6(3), 936.
[224] S. Chandrasekhar (1992) Liquid Crystals, Cambridge University Press, Cambridge, 2nd edn.
[225] W. Maier and A. Saupe (1959) Z. Naturforsch., 14A, 882.
[226] C. G. Gray and K. E. Gubbins (1984) Theory of Molecular Fluids, vol. 1,
Clarendon Press, Oxford.
[227] R. J. Buehler, R. H. Wentorf, Jr., J. O. Hirschfelder and C. F. Curtiss (1951)
J. Chem. Phys., 19(1), 61.
[228] D. Chandler (1987) Introduction to Modern Statistical Mechanics, Oxford University Press, New York.
[229] J. W. Gibbs (1876) Trans. Conn. Acad., III, 108.
[230] J. W. Gibbs (1928) Collected Works of J. Willard Gibbs, vol. 1, Longmans,
Green and Co., New York, Pages 219-231.
[231] L. D. Landau and E. M. Lifshitz (1959) Statistical Physics, vol. 1, Pergamon
Press, Oxford, 3rd edn., Translated by J. B. Sykes and M. J. Kearsley.
[232] H. Asada and H. Sekito (1991) Surf. Sci., 258, L697.
[233] J. E. Lennard-Jones and A. F. Devonshire (1937) Proc. Roy. Soc. (London),
A163, 53.
[234] L. Verlet and J.-J. Weis (1972) Phys. Rev. A, 5(2), 939.
[235] F. A. Lindemann (1910) Physik. Z., 11(14), 609.
[236] G. N. Lewis and M. Randall (1923) Thermodynamics and the Free Energy of
Chemical Substances, McGraw-Hill Book Co., Inc., New York.
References
213
[237] J. Barojas, D. Levesque and B. Quentrec (1973) Phys. Rev. A, 7(3), 1092.
[238] K. Singer, A. Taylor and J. V. L. Singer (1977) Mol. Phys., 33(6), 1757.
[239] P. S. Y. Cheung and J. G. Powles (1975) Mol. Phys., 30(3), 921.
[240] B. J. Berne and P. Pechukas (1972) J. Chem. Phys., 56(8), 4213.
[241] J. G. Gay and B. J. Berne (1981) J. Chem. Phys., 74(6), 3316.
[242] W. Maier and A. Saupe (1960) Z. Naturforsch., 15A, 287.
[243] B. B. Laird and A. D. J. Haymet (1989) J. Chem. Phys., 91(6), 3638.
[244] R. M. Lynden-Bell, J. S. van Duijneveldt and D. Frenkel (1993) Mol. Phys.,
80(4), 801.
[245] L. Lamport (1985) LATEX: A Document Preparation System, Addison-Wesley
Publishing Co., Inc., Reading, Massachusetts, 2nd edn.
[246] M. Goosens, F. Mittelbach and A. Samarin (1994) The LATEX Companion,
Addison-Wesley Publishing Co., Inc., Reading, Massachusetts.
[247] S. Wolfram (1988) Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley Publishing Co., Inc., Redwood City, California, 2nd
edn.
[248] B. Breathed (1986) Bloom County Babylon, Little, Brown and Co., Boston.
214
Berkeley Breathed, Bloom County Babylon [248]