The Pennsylvania State University
The Graduate School
Department of Aerospace Engineering
Molecular Dynamics Simulation of Argon Droplet Evaporation with
Three Body Force Potentials
A Thesis in
Aerospace Engineering
by
Chad Jerey Ohlandt
Submitted in Partial Fulllment
of the Requirements
for the Degree of
Master of Science
August 1996
I grant The Pennsylvania State University the nonexclusive right to use this work
for the University's own purposes and to make single copies of the work available
to the public on a not-for-prot basis if copies are not otherwise available.
Chad Jerey Ohlandt
We approve the thesis of Chad Jerey Ohlandt.
Date of Signature
Michael M. Micci
Associate Professor of Aerospace Engineering
Thesis Adviser
Lyle N. Long
Associate Professor of Aerospace Engineering
Dennis K. McLaughlin
Professor of Aerospace Engineering
Head of the Department of Aerospace Engineering
iii
ABSTRACT
The importance of many-body forces in supercritical and subcritical argon droplet
evaporation is explored using molecular dynamics (MD) simulations with various potentials. The impact of the intermolecular potential on MD is established, and its
origins including many-body forces are reviewed. Three intermolecular potentials
were implemented to account for many-body forces | an eective Lennard-Jones
(LJ) 12-6 potential, the same Lennard-Jones 12-6 potential in combination with the
Axilrod-Teller (AT) triple dipole potential, and Barker-Fisher-Watts (BFW) potential
which includes AT.
The MD codes developed here used cubic, periodic boundary conditions. Force
calculations were limited using a cell link list and a cuto of 2.5 . Molecular displacements were conducted with a modied form of the velocity Verlet algorithm. A
multiple time step method was employed to reduce computational requirements of the
three body algorithms. The simulations were implemented on an IBM SP2 parallel
computer using up to 32 nodes. Parallelization of the MD code was accomplished
using a particle decomposition method.
The eective LJ and BFW potentials produce statistically similar simulations of
droplet evaporation but result in systems with dierent particle positions and velocities. An eective LJ potential has limited value for simulating multiphase systems
due to its attenuation to a certain density. However, the BFW potential involves
the calculation of three body forces which increase the computational demand by
an order of magnitude. An eective LJ with AT potential seems to be an arbitrary
combination of two potentials and exaggerates the eects of many-body forces which
both potentials independently attempt to produce.
iv
TABLE OF CONTENTS
LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
vi
LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii
ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : :
ix
1 Motivation and Background : : : : : : : : : : : : : : : : : : : : : : : :
1
1
3
3
5
6
8
9
10
1.1 Breakdown of Analytical Methods : : : : : : : : : : : : :
1.2 Breakdown of Simulation Methods : : : : : : : : : : : :
1.2.1 General Limitations of Continuum Methods : : :
1.2.2 Droplet Simulations using Continuum Methods :
1.3 Molecular Dynamic Simulations : : : : : : : : : : : : : :
1.3.1 Computational Demand and Parallel Computing :
1.3.2 Brief History of MD Simulations : : : : : : : : : :
1.4 Thesis Scope : : : : : : : : : : : : : : : : : : : : : : : : :
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2 Molecular Potentials : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11
2.1 Origins : : : : : : : : : : :
2.1.1 Long Range Forces
2.1.2 Short Range Forces
2.2 Common Potentials : : : :
2.3 Many-Body Forces : : : :
2.4 Choice of Potentials : : : :
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12
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23
3 Molecular Dynamics Algorithms with Three Body Forces : : : : : 27
3.1 Basic MD Code : : : : : : :
3.1.1 Boundary Conditions
3.1.2 Displacements : : : :
3.1.3 Initialization : : : : :
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27
27
29
29
v
3.1.4 Heating : : : : : : : : :
3.1.5 Parallelization : : : : : :
3.2 Three Body Force Calculations
3.2.1 Two Body Forces : : : :
3.2.2 Three Body Forces : : :
3.3 Multiple Time Step Method : :
3.4 Code Performance : : : : : : :
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30
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31
33
34
36
38
4 Results from Argon Droplet Evaporation with Dierent Potentials 45
General Overview : : : : : : :
Radial Distribution Functions
Energies : : : : : : : : : : : :
System Parameters : : : : : :
4.4.1 Temperature : : : : : :
4.4.2 Density : : : : : : : :
4.5 Regression Rates : : : : : : :
4.1
4.2
4.3
4.4
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66
5 Conclusions and Future Work : : : : : : : : : : : : : : : : : : : : : : : 69
5.1 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.1.1 Future Work : : : : : : : : : : : : : : : : : : : : : : : : : : : :
69
70
REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 73
vi
LIST OF FIGURES
2.1 Argon pair potential and force. : : : : : : : : : : : : : : : : : : : : :
2.2 Axilrod-Teller energy potential for a three body system. : : : : : : : :
2.3 Pairwise component of Barker-Fisher-Watts and the eective LennardJones potentials. : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
12
21
Periodic cubic boundaries. : : : : : : : : : : : : : : : : : : : : : : : :
Axilrod-Teller three body force in the X-direction. : : : : : : : : : : :
Axilrod-Teller three body force in the Y-direction. : : : : : : : : : : :
Code times for the rst 100 time steps. : : : : : : : : : : : : : : : : :
Fraction of computer time used by dierent elements of the codes. : :
Floating point operations and megaops ratings for the dierent codes.
The evolution of code speed over evaporation runs of 22,000 particle,
supercritical system. : : : : : : : : : : : : : : : : : : : : : : : : : : :
3.8 The evolution of code speed over evaporation runs of 27,000 particle,
subcritical system. : : : : : : : : : : : : : : : : : : : : : : : : : : : :
28
36
37
41
42
43
4.1 Argon Pv plot with ambient environments indicated. : : : : : : : : :
4.2 Lennard-Jones and Barker-Fisher-Watts energy potentials for a three
body system. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.3 Dierence between LJ and BFW potentials. : : : : : : : : : : : : : :
4.4 Radial distribution functions at 10 and 160 ps with a 200K, 7.5 MPa
environment. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.5 Radial distribution functions at 10 and 150 ps with a 300K, 3.0 MPa
environment. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.6 Kinetic, potential, and total energies for the supercritical case. : : : :
4.7 Temperature contours for all three axes at 150 ps for the supercritical
case, 200K, 7.5 MPa. : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.8 Temperature contours for all three axes at 150 ps for the subcritical
case, 300K, 3.0 MPa. : : : : : : : : : : : : : : : : : : : : : : : : : : :
45
3.1
3.2
3.3
3.4
3.5
3.6
3.7
26
44
44
46
47
49
50
52
54
55
vii
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
Temp. contours up to 150 ps for the supercritical case, 200K, 7.5 MPa
Temp. contours up to 350 ps for the supercritical case, 200K, 7.5 MPa
Temp. contours up to 150 ps for the supercritical case, 300K, 3.0 MPa
Temp. contours up to 600 ps for the supercritical case, 300K, 3.0 MPa
Density contours for all three axes at 150 ps for the supercritical case,
200K, 7.5 MPa : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Density contours for all three axes at 150 ps for the subcritical case,
300K, 3.0 MPa : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Density contours up to 150 ps for the supercritical case, 200K, 7.5 MPa
Density contours up to 350 ps for the supercritical case, 200K, 7.5 MPa
Density contours up to 150 ps for the subcritical case, 300K, 3.0 MPa
Density contours up to 600 ps for the subcritical case, 300K, 3.0 MPa
Regression rates for both environments and dierent potentials. : : :
56
57
58
59
60
61
62
63
64
65
68
viii
LIST OF TABLES
2.1 Energy contributions of various three body potentials : : : : : : : : :
2.2 Relative importance of three body contributions : : : : : : : : : : : :
2.3 Parameters for various potentials chosen. : : : : : : : : : : : : : : : :
22
22
26
3.1 Dierent size cases used for timing and performance measurements. :
38
4.1 The number of atoms found in the supercritical and subcritical droplet
simulations. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46
4.2 Regression rates for the subcritical case compared with d2 law prediction. 67
ix
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my advisors, Dr. Michael Micci
and Dr. Lyle Long. Additionally, I would like to extend my thanks to the other
members of my research group, Je Little, Terri Kaltz, Obika Nwobi, and Brian
Wong. Financial support for this work was provided by the Air Force Oce of
Scientic Research under Grant No. F49620-94-1-0133.
1
Chapter 1
Motivation and Background
What follows is an evaluation of available methods and models for simulating
multi-phase uid dynamic systems. The advantages and disadvantages of each are
discussed with a two fold purpose. First, it demonstrates the applicability of using
molecular dynamics (MD) to model droplet evaporation under supercritical ambient
conditions. Second, the importance of the atomic potential as the source of most of
MD's advantages and disadvantages is established. This leads to a presentation of
the scope of the thesis.
Numerous methods of modeling complex systems of matter have been developed
over the years. Analytical models and computational simulations are two broad categories in which most methods can be classied. However, as described below, many
of the methods have fundamental limiting assumptions or requirements which make
them unsuitable or of limited use in modeling particular complex systems. Molecular
dynamics, as the name indicates, is a method of modeling matter at the molecular
level. By returning to the molecular level with MD, systems too complex to model
using other methods can be simulated.
Although the following applies to any complex system that exceeds the assumptions and demands of the methods employed, this paper will focus on the evaporation
of droplets in near and supercritical ambient environments. This is due to the fact
that questions answered by this thesis were originally raised during attempts to model
such droplet evaporation 1].
1.1 Breakdown of Analytical Methods
The development of analytical methods revolves around taking a complex system and sifting out the most important or signicant features of the system. Then
with these fundamental features in mind, one continues to simplify the system with
2
appropriate assumptions until the system can be modeled with mathematical methods.
The limiting factors are the simplications. Simple geometries, axisymmetric,
inviscid, perfect gas, laminar, steady state, incompressible, and many others are examples of these assumptions. However, in today's world modeling is necessary for
many environments that cannot be accurately modeled with these assumptions.
In the case of droplet evaporation, analytical work conrmed by empirical evidence has led to the d2 evaporation law which is used as the foundation for most
models 2].
d2 = d20 + Kt
(1.1)
Above d, d0, K , t are the droplet diameter, initial droplet diameter, evaporation
coecient, and time, respectively. In order to predict droplet life time, the initial
droplet size and the evaporation coecient are needed. Since the droplet size can
be specied, the remaining need is a model that predicts K . Spalding developed
such a model in the late fties 2, 3]. The Spalding model assumes a quasi-steady
single spherical drop in a non-convective atmosphere with a certain ambient pressure
and temperature without gravity eects. The solution is derived by applying basic
mass and energy conservation at the droplet's surface and solving using boundary
conditions at the surface and at innity 4].
K = ; 8s Ds ln(1 + B )
l
(1.2)
Variables
Subscripts
B - Spalding transfer number s - surface
D - mass diusion coecient o - initial
- density
l - liquid, within the droplet
However, the Spalding transfer number, B , is a complex function of specic
heats, surface temperature, ambient temperature, mass fractions, and heat of vaporization. In terms of the equations, the model breaks down as the critical point is
approached and the heat of vaporization heads towards zero. Since B is inversely
3
proportional to the heat of vaporization, B becomes innite and K follows. Conceptually, the quasi-steady assumption on which Spalding's model is based breaks down
as the ambient environment approaches near critical. Energy from the environment
enters at a rate faster than the process of evaporation at the surface can consume.
Therefore, the interior begins to heat, and the system cannot be modeled as simply
a quasi-steady surface.
Beyond simple failure as described above, the model is also limited by our knowledge of parameters such as diusion coecients, specic heats, vapor pressure as a
function of temperature, and heat of vaporization as the ambient pressure increases
above even a few atmospheres. Additionally, it assumes perfectly spherical droplets
in a quiescent environment. Although other models exist that attempt to take into
account dierent elements that are neglected such as supercritical 5] or circulation
6] within the droplet, they explain little of what is actually occurring and simply
produce estimates for droplet lifetimes.
1.2 Breakdown of Simulation Methods
When analytical and mathematical approaches fail to produce sucient or accurate information, the next logical approach is to model the system numerically using
computational methods. While MD is a computational method, there is a signicant
dierence between it and other current nite element and nite dierencing methods.
As the names imply, these continuum methods divide the system into arbitrary nite
sections and then apply conservation principles between these divisions. The size of
the sections are dependent on the size of the system, computational power available,
and stability of the algorithms being used. In contrast, MD elements are dictated by
the molecular structure of matter. The nite methods require a clear understanding
of how neighboring elements or points interact within the continuum and how they
interact with the boundaries.
1.2.1 General Limitations of Continuum Methods
Most current computational uid dynamics (CFD) methods use some simplied
form of the continuity equation, Navier-Stokes equations, energy equation, and state
4
equations such as those listed below 7].
@ + @ (u ) = 0
@t @xk k @u
@p
@
@u
@
@u
@u
j
j
k
i @uj
@t + uk @x = ; @x + @x @x + @x @x + @x + fi
k
j
j
i
i
k
2 j
@e
@u
@
@T
@u
@u
@e
k
k
i @uj @uj
@t + uk @x = ;p @x + @x k @x + @x + @x + @x @x
k
k
j
j
k
j
i
i
p = RT
e = Cv T
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
However, to use these equations, dynamic viscosity, second viscosity coecient,
thermal diusion and specic heat, , , k and Cv , must either be constants or known
functions of pressure and temperature. Additionally, the state equation for a perfect
gas, p = RT , must be reliable within the pressure and temperature constraints of
the system being modeled. If not, then another more complicated state equation
will need to be substituted. The above equations also assume a single component
system. For simulations of multi-component systems, additional equations must be
included for the conservation of species concentrations. Such equations would require
coecients of mass diusion, Dab , which would add another function of temperature
and pressure that must be known for each component. However, in many complex
systems such as rareed ows, supercritical environments, shock regions, and highlyenergized plasmas, these coecients or relationships are not known accurately. As a
result, one lacks the information necessary to use continuum methods for simulating
such systems.
Finally, the geometry of continuum methods can also prove to be limiting. First,
the elements or units into which the continuum is broken into must be large enough
to statistically simulate macroscopic properties such as pressure and temperature.
However, some systems are physically too small, or certain characteristics of a system operate on a scale smaller than the grid. Again, the continuum methods fail to
accurately model certain realities. Since a continuum method by denition assumes
5
a continuous system, at some point there must be a dened boundary to the model
because simulation of innite systems is obviously impossible. The boundaries of the
simulation need to be dictated either as periodic or through some mathematical relation. Necessarily simplied boundary conditions, such as no-slip, perfectly smooth,
or sliding surfaces, often lead to inexplicable singularities where dierent boundary
conditions meet 8].
1.2.2 Droplet Simulations using Continuum Methods
Regardless of these diculties, recent attempts have been made to model droplet
evaporation in supercritical environments using CFD methods by Yang and Shuen
9, 10]. In the second paper, a model of liquid oxygen (LOX) droplet evaporation in
supercritical ambient hydrogen was implemented using full mass, momentum, energy,
and species conservation equations while neglecting body forces, viscous dissipation,
radiation, and chemical reactions. Mass and thermal diusion were represented by
Fick's and Fourier's laws respectively. The numerical scheme used was a fully implicit, dual time stepping integration routine developed previously by the authors.
Initially, a 90K LOX droplet of specied diameter (50-300 microns) is introduced
into an ambient hydrogen environment of specied pressure and temperature. The
code proceeds to employ conservation equations in the gas and liquid phases and
match them at the droplet surface. When the droplet surface temperature achieves
the critical mixing temperature, the code continues the simulation using conservation
equations for the single remaining phase and denes the droplet radius as the point
at which the critical mixing temperature is found.
Due to the importance that the mass diusion, the thermal conductivity, and
specic heat play in the model and their variation at near critical and supercritical
pressures and temperature, special attention was placed on their derivation. The
specic heat and thermal conductivity were determined by simply using the extended
corresponding-state model of Ely and Hanley 11, 12]. It was assumed that the critical divergence eects of LOX were canceled by the presence of hydrogen producing
a mixture with minimal divergence near critical states. The mass diusion was estimated using a corresponding-state approach suggested by Takahashi which uses
6
the Chapman-Enskog binary mass diusivity and corrects it based on reduced state
variables 13].
The state equation used was an ideal gas equation corrected using a compressibility factor. The compressibility factor was determined using a modied SoaveRedlich-Kwong equation 14] with empirical adjustments based on solid oxygen and
high pressure gaseous hydrogen experiments done by McKinley 15].
The fundamental diculty for most supercritical models is the determination of
the very important transport and thermodynamic parameters. The authors discuss
the accuracy of the corresponding-state model used to determine the mass diusivity. "This scheme appears to be the most complete to date, and has demonstrated
moderate success in the limited number of tests conducted 10]." Additionally, the
Soave-Redlich-Kwong equation used to determine the compressibility factor for the
state equation is primarily for hydrocarbons. Using modications, it can be adapted
to hydrogen, but this requires some empirical, vapor-pressure data. However, the
best empirical data the authors could nd was a single paper describing solid oxygen and gaseous hydrogen properties by McKinley. The unknown accuracy of these
very important coecients raise questions about the accuracy of the model in the
supercritical, single phase state.
Additionally, the CFD model imposes a number of geometric constraints on
the system. At subcritical environments two grids were used to separately model
the liquid and gas regions with conservation principles connecting the two regions.
Axisymmetric assumptions were used to limit the computational demand. While
these geometric constraints are reasonable, they do not necessarily reproduce droplet
evaporation in its truest form, particularly in supercritical environments.
1.3 Molecular Dynamic Simulations
For complex systems modeled poorly by analytical or continuum methods, one
possible solution is molecular dynamics. MD only requires knowledge of the interaction potential between each of the system's atomic components. With only this
information, a system can be modeled simply by initially positioning the atoms involved and watching what happens. Observation of the model will illustrate how the
7
system evolves in time without the simplifying assumptions found in the previous
models. Additionally, because of the fundamental nature of MD, a great deal more
information is available beyond what can be generally observed. Since the positions
and velocities for every particle are known throughout the simulation, mass and thermal diusion coecients, as well as temperature and density, can be derived from the
simulation for any denable region.
MD is the use of classical mechanics to model atoms and molecules as particles,
each with its own force eld that acts on all other particles within its sphere of
inuence. The force elds are determined from a potential equation. The gradient of
the potential equation gives the force that any two particular atoms act upon each
other. Given the total force on a particular particle and using Newton's second law of
motion, the acceleration of the particle can be determined from a = f=m. Knowing
its original velocity and position combined with the acceleration vector, the particle
can be moved to its new position and assigned a new velocity at a certain time in
the future. If this is done for every particle in the system and then the forces are
reevaluated for all the particles, then one time step has been completed. If the process
continues over a number of time steps, then we have successfully modeled that system
on the molecular level. For a more indepth study of molecular dynamics techniques
consult books by Haile 16] and Allen and Tildesley 17].
In its simplest form, one can imagine a rareed gas as if the atoms were billiard
balls moving through space. Whenever the distance between two particles is twice
the radius of the balls, they collide with one another. Upon collision, assuming
it's perfectly elastic, the momentum is redistributed equally through an impulse of
opposing force vectors on each ball. These force vectors accelerate the balls in opposite
directions, and they move o toward their next collision. The potential described
above is called a hard sphere potential which has been used in early simulations to
produce reasonable models of rareed gases.
It should be apparent that in concept, MD is an extremely powerful and extremely simple method of simulation. A time stepped, classical dynamics problem
becomes MD with the mere implementation of a particular atomic potential between
constituents of the system. The accuracy of the model depends only on two factors
8
| numerical error and the accuracy of the potential equation.
1.3.1 Computational Demand and Parallel Computing
The one great limitation of MD is the computational demand. For every time
step and for each particle in relation to every neighboring particle, the gradient of
the potential must be evaluated. Then, with the forces summed, the accelerations
must be calculated, and the velocities and positions updated. Additionally, for dense
systems such as liquids the time step has to be on the order of femtoseconds to
achieve stability. Even with the computational power available today, this limits one
to either short simulation times or very small systems. A liquid system of 108 atoms,
an unrealistically small system, will take hundreds of computer hours to run a few
seconds in simulation time. It has been this limitation that has restricted the use of
MD methods in the past.
However, a branch of high performance computing with an emphasis on parallel
computing has evolved over the last decade or so. Rather than spend millions developing a faster single main processor, it has become apparent that the economical way
to proceed is to combine the power of multiple mainstream, mass-produced processors
together. Although parallel processing has inherent ineciencies in comparison to serial supercomputing, the cost savings is undeniable. Additionally, as computer power
of processors found in desktop workstations continues to double every two years, the
development of economical teraop systems is only a few years away.
One of the most interesting features of MD algorithms is that they are merely an
immense collection of ordinary dierential equations. The total force on a particular
particle is only dependent on the positions of other nearby particles within a certain
sphere of inuence or cuto. As a result, the majority of particles in the simulation
have no signicant eect on any particular particle at any given point in time. This
means that the problem can easily be divided amongst independent processors. The
inherent ineciency in parallel computing is the necessity of dividing a problem up
between independent processors. However, MD algorithms by nature lend themselves
to this division of work.
In the near future with the development of more powerful high performance
9
parallel computers, the capability to conduct relatively large MD simulations will
become available. The MD technique will no longer be limited to microscopic systems
meant only to mimic certain molecular level phenomena, but will be applicable to real,
macroscopic systems.
1.3.2 Brief History of MD Simulations
With the possibility of modeling real sized systems using MD in the near future,
signicant changes in how the MD community will use MD algorithms will occur.
However, when the use of MD changes, one must reconsider some of the previous
assumptions and determine whether they still hold true. In order to do this, the
historical uses of MD must be reviewed.
The rst crude MD simulations were carried out the in the late fties by Alder
and Wainwright. In 1964, Rahman modeled the rst Lennard-Jones system. The
Lennard-Jones potential is the popular, realistic potential in use today which will be
discussed in detail later. In the seventies, numerous techniques were developed that
allowed simulations of complex molecules and transient systems. However, most work
was still limited to systems on the order of a few hundred particles 17]. Because of
these small systems sizes, the only real uses were the examination of certain atomic
and molecular level characteristics such as lattice structure energies and molecular
interactions between two complex molecules. Chemists began to apply the method
to modeling particular chemical phenomena. As the methods became more standardized, packages such as AMBER were developed for biochemists and other chemistry
related elds 18, 19]. Using these MD packages, chemists could watch complex
molecules interact with baths of simple molecules. Simulation results were also used
to corroborate chemical experimental work of all kinds. In the last decade, people
have begun simulating simple systems such as the diusion of gases into solids, interfaces between dierent liquids, or rareed gases 20, 21]. Nonetheless, most work has
been limited to chemical or molecular level phenomena.
The next step in the evolution of MD techniques is the application of MD to
macroscopic or "real" systems. While molecular level phenomena will always be of
interest, large scale use of microscopic methods will produce macroscopic eects. The
10
eld of gas dynamics has for years recognized the relationship between microscopic
interaction of atoms and macroscopic properties such as temperature, pressure, and
viscosity. As mentioned previously, the limitation has been the available computational power in order to simulate systems large enough to produce observable macroscopic eects. With this problem solved by the widespread development of parallel
computing, the application of MD to macroscopic modeling may continue. However,
this transition changes the demands on the potentials used in the MD simulations.
1.4 Thesis Scope
The purpose of this work was to explore how dierent molecular potentials
aect MD simulations of supercritical droplet evaporation. The importance of MD
is established above. Chapter 2 delves into the origins of intermolecular potentials.
The importance of dierent many-body forces is weighed. Finally, past and present
attempts to model these potentials are reviewed, and potentials which include three
body forces are chosen for more detailed evaluation.
Chapter 3 reviews the aspects of the code developed for this work. The various
MD techniques implemented in the code developed are listed, and the modications
necessary for the inclusion of three body forces are discussed. Finally, computational
performance and timings of the various codes are presented.
The results from the MD simulation of argon droplet evaporation under subcritical and supercritical ambient environments are found in Chapter 4. Radial distribution functions and system energies are followed by qualitative discussion of temperature, density, and surface tension contours. Droplet regression rates for the various
cases are then presented.
Finally, Chapter 5 presents the conclusions that can be drawn from the simulations and suggests directions for future work involving three body force potentials in
macroscopic MD simulations.
11
Chapter 2
Molecular Potentials
The cornerstone of all MD work is the molecular potential. In order to calculate
the accelerations, velocities, and positions of each molecule, it is necessary to determine the force acting on each molecule at that instant in time. The force acting upon
a particular molecule is the negative gradient of the energy potential.
dv ^i ; dv ^j ; dv k^
f (r) = ; 5 v(r) = ; dx
dy dz
(2.1)
All results and properties determined with MD simulations follow directly from the
energy potential.
It becomes necessary to determine the potential equation and constants that
produce the best simulation results for one's particular needs. In traditional MD
circles, if the goal was to reproduce diusion experiments with your MD simulation,
one would use potential parameters derived from diusion experiments or \tweak"
the parameters until results matched those from diusion experiments. However, as
mentioned in Chapter 1, MD is no longer limited to simple systems and fundamental
chemistry by computational demands. Currently, MD techniques are being applied to
the simulation of macroscopic or real systems. When attempting to reproduce reality,
the only appropriate criteria for choosing a potential is to nd the most realistic one
available.
This chapter rst explores the physical origins of molecular potentials and forces.
Historical and current attempts to model these potentials by science are then examined. A review of work concerning many body forces and their importance is
conducted. Finally, the choice of potential equations and constants for this work is
explained.
12
2.1 Origins
The intermolecular potential is the glue that holds our universe together |
producing matter in the forms we know, solid, liquid, and gas. Shown below in
Figure 2.1 is a sample energy potential function and its corresponding force function
between two argon atoms. As a matter of denition, negative force is considered an
attractive force. Upon examination, three distinct features are obvious. At larger, but
nite, distances there is an attractive force that acts on the atoms. At short distances
a strong repulsive force is evident. In between, where the long range attractive and
short range repulsive forces interact, there is a trough or well.
5
v(r) / ε
Argon Pair Potential and Force
interatomic pair potential
interatomic pair force
4
3
F(r) σ / ε = -∇ v(r) σ / ε
2
1
rm
1.0
1.5
2.0
2.5
rij /σ
3.0
0
-1
σ
-2
-3
repulsive
force
attractive force
-4
Figure 2.1. Argon pair potential and force.
This trough denes the dierent states of matter. Cold matter with little kinetic
energy sits comfortably in the bottom of the potential well. Molecules in liquids
have enough energy to move from one atom's potential well into another's, but the
13
repulsive, short range forces limit their compressibility. Finally, a gas has atoms ying
around outside of the inuence of other atoms. Potentials only come into play when
collisions occur, and the strong repulsive forces cause the atoms to bounce o each
other like pool balls.
The origin of intermolecular forces is electro-magnetic. Gravitational forces act
over long distances and require masses more signicant than atomic ones. Nuclear
forces are only signicant at much shorter distances than those that exist between
molecules. The electro-magnetic origins of intermolecular forces can be divided into
three parts, electrostatic, induction, and dispersion 22].
Electrostatic forces arise from molecules with inherent charge distributions such
as the permanent dipole moments found in H2O and HCl. The forces can be evaluated by dening a molecule's charge distribution with point charges and using a
Coulombic potential where i are the point charges in one molecule and j are those in
the other.
1 X Qi
= 4
0 i ri
vel (r) =
X
j
Qj (Qj )
(2.2)
(2.3)
The force is the negative gradient of the potential, ;5 vel (r). For charged molecules,
ions, the vel is proportional to 1r . For neutral molecules with dipole-dipole, dipolequadrupole, and quadrupole-quadrupole moments, the vel is proportional to r13 , r14 ,
and r15 , respectively. Electrostatic potential and force contributions are also extremely
orientationally dependent.
Induction occurs when a polarized or charged molecule encounters a non-polarized molecule and induces a polarization in the latter. The induction potential, vind(r)
is a function of the polarized molecule's electric eld and the polarizability of the nonpolarized molecule. For two net neutral molecules with one possessing a permanent
dipole moment, the induced potential, vind(r), is proportional to r16 .
14
Although electrostatic and induction potential contributions to the total molecular potential are important, the concerns of this research are limited to the dispersion
contribution. The argon \molecule", an argon atom, has no net charge or permanent moment. The extension of droplet evaporation under supercritical conditions
to include diatomic O2 and H2 molecules which also do not have electrostatic moments would also not require these contributions. However, should one wish to use
molecules with permanent charges or moments in their MD simulation, they would
need to include the above contributions.
The molecular potential of argon illustrated in Figure 2.1 is due solely to dispersion forces. The dispersion potential can therefore be subdivided in a similar manner
to the division of the overall potential. The long range, attractive forces arise from
the interaction of orbiting electrons in two neutral atoms via quantum mechanics.
The short range, repulsive forces occur when electron clouds from two neutral atoms
overlap. As a result of the overlap, the like charged electron clouds repel and push
back each other, exposing the positively charged nuclei to each other which then produce the repulsive force. The combination of the two forces creates the well in the
middle of the potential.
2.1.1 Long Range Forces
The long range forces can be derived from Rayleigh{Schrodinger perturbation
theory. Conceptually, the electron clouds mutually induce polar moments in each
other which then interact. The second order perturbation for a system of two atoms
produces a series of components based on the order of the induced polar moments.
The terms are proportional to r;2(l+1) where l is the sum of the moments halved |
dipole{dipole (l=2), dipole{quadrapole (l=3), and quadrapole{quadrapole (l=4).
E (2) (12) =
1
X
C2l+2
2l+2
l=2 r
vpairwise(r) = Cr66 + Cr88 + Cr1010 + :::
(2.4)
(2.5)
15
The same can be done to get the third order perturbation series. However, its leading
term is proportional to r111 which is considered insignicant 23].
It is important to realize that this analysis was done with two atoms in isolation.
The same can be done with three atoms in isolation. It turns out that the second
order perturbation of a three particle system is the summation of the second order
perturbation of two particle system between each of the three particles.
E (2) (123) = E (2) (12) + E (2) (23) + E (2) (31)
(2.6)
However, the third order perturbation of three particle has two contributions. Again,
the rst element is the pairwise additive combination of third order perturbation of
two particles, but the second factor is a new, non-additive component to the potential.
E (3) (123) = E2(3) (123) + E3(3) (123)
(2.7)
E2(3) (123) = E (3) (12) + E (3) (23) + E (3) (31)
(2.8)
E3(3) (123), the non-additive term of the third order perturbation of a three atom system, is also a series based on the moments of the three atoms. The leading term of
this series, the dipole{dipole{dipole term, is called the Axilrod-Teller potential and
will be discussed more in the following section on three body forces. This analysis
continues with additional atoms to produce 4-body, 5-body, ... n-body forces. However, these n-body potentials become less and less signicant with higher orders of n
due to increasing inverse power of r. Although it has not been proven, it is suspected
that the series of n-body potentials will converge for most atoms 24].
2.1.2 Short Range Forces
Our understanding of short range forces is less developed than that of long range
forces. In the case of long range forces, the wave functions of the individual atoms are
known, and their interaction can be calculated for cases with simple atoms. However,
when the atoms are close enough for the electron clouds to overlap, the wave functions
16
are disturbed and no longer known. Additionally, when the atoms are extremely close,
the nuclei interact directly repelling each other.
A number of attempts have been made to estimate these new, modied wave
functions and then complete the perturbation calculations. While not well understood, the resulting short range potential is of the form, Ae;br , where A and b are
constants 22, 25].
2.2 Common Potentials
The theoretical work described above produces a general potential equation due
to dispersion of the following form.
vdisp(r) = Ae;br + Cr66 + Cr88 + Cr1010 + non-additive contributions
(2.9)
However, our knowledge of and early attempts to model the intermolecular potential
occurred before our understanding of quantum mechanics was able to explain it. As a
result, many potential equations are simply equations that produce particularly good
curve ts to the available data. Additionally, due to the complexity of true potential
equations, there is a great deal of momentum to continue working with the simpler,
curve-tting equations.
The simplest potential is the hard-sphere potential with no attractive element
and an innite value at and less than some radius called . Although crude, this
potential is eective at reproducing the characteristics of molecular interaction in a
gas. In a gas the particles spend almost no time interacting with each others' potentials and merely collide occasionally at high velocities. The hard-sphere potential can
reproduce these collisions adequately.
The most broadly used potential, and still heavily used today, is the one due to
Lennard-Jones (LJ).
rm n
rm 6 n
6
vLJ (r) = n ; 6 r ; n ; 6 r
(2.10)
Above is the n-6 form of the LJ potential! below is the most commonly used form
17
with n=12, the Lennard-Jones 12-6 potential.
12 6 vLJ (r) = 4 r ; r
(2.11)
Comparing it with the model theoretical potential in Equation 2.9, the attractive
component of the LJ 12-6 matches the r16 relationship of the rst term, the dipole{
dipole interaction, from the long range perturbation theory. However, the repulsive,
1
r12 , component has no theoretical basis, but merely reproduces the exponential short
range forces well.
Although numerous other potentials have been created that model certain molecules better and more precisely than the LJ 12-6, the trade o in complexity with
the newer potentials is not usually considered worth the gain in accuracy. As a
matter of fact, it is that very question which this work seeks to answer in reference
to supercritical droplet evaporation. In general, two broad trends can be seen in the
development of newer potentials, those based in the theory which tend to be complex
and accurate and those created to remain simple but accurately model the potentials
nonetheless.
In the former case, a good example is the most accurate potential known for
argon by Barker and Pompe 26].
r" = rr
(2.12)
m
!
5
2
X
X
C
2
j
+6
vBP (r) = e(1;r) Ai("r ; 1)i ; (
+ r")2j+6
i=0
j =0
(2.13)
Barker and Pompe also used the Axilrod-Teller three body potential in conjunction
with the potential above. The exponential repulsive terms and the r16 , r18 , and r110
terms match the same found in the theoretical model potential in Equation 2.9.
However, this equation demands that 13 dierent constants be determined from a
curve t of experimental data.
In contrast, the n("r)-6 potential is an extension of the basic Lennard-Jones with
18
n as a function of separation 22].
n = 13 + ("r ; 1)
vn(r);6 (r) = 6 r";n ; n r";6
n;6
n;6
(2.14)
(2.15)
This potential reproduces the interaction of monatomic species extremely accurately
with only three constants, , , and rm.
Regardless, at this point in time, any potential is an approximate curve t based
on experimental data. The best potential for a particular simulation is one based on
experimental data found under the same conditions as those being simulated, and the
best equation is the one that produces the most accurate curve t of that data.
2.3 Many-Body Forces
All the potentials above are functions of the radial separation, r, between two
particular molecules. While the vast majority of potential energy in a system is
pairwise additive, the presence of third, fourth, or more molecules within a certain
sphere of inuence does modify the value of the potential energy in a non-additive
manner. In the pursuit of more accurate potentials, the next logical step is to consider
the impact of non-additive contributions to the systems.
In gases, these non-additive contributions would be negligible. As mentioned in
the previous section, gas molecules spend so little time within each others' spheres of
inuence that the hard core potential can be used to accurately simulate gas systems.
However, in dense systems of molecules such as solids, liquids, and supercritical environments the probability for tertiary molecules to inuence the interaction of two
primary molecules greatly increases. Doran and Zucker examined the signicance of
non-additive potential contributions in detail 24].
One could take the third order perturbation of a three particle system found in
Equation (2.7) and generalize it to produce the multi-body third order perturbation
19
energy. Doing the same for fourth order produces a similar equation.
E (3) =
E (4) =
X
X
e(3) (l1l2 ) +
e(4) (l1l2 ) +
X
X
e(3) (l1l2 l3)
e(4) (l1l2 l3) +
X
e(4) (l1 l2l3 l4)
(2.16)
(2.17)
The summations are over all possible combinations of moments with lx dening the
moment of the xth molecule. The values of lx often appear in both alphabetic and
numeric forms with dipoles (D/1), quadrupoles (Q/2), and octupoles (O/3). For the
sake of completeness, rst order perturbation, E (1) , is zero, and second order, E (2) ,
is the traditional pairwise contribution discussed above in Section 2.1.1.
The lowest order moments have the greatest contribution! each series listed
above diminishes as higher order moments correlate with greater inverse powers of r.
While convergence of these series has only been proven for a few rare examples, it
is generally held that most converge, albeit slowly. The rst term of the third order
perturbation pairwise component, e(3) (DD), is zero, and little is known about rest of
that component which is usually discarded. The triple-dipole term of the fourth order
perturbation, e(4) (DDD), accounts for 90% of the whole fourth order perturbation
energy, E (4) , and the rest of E (4) is generally ignored. For future reference, the
e(4) (DDD) is always a negative or attractive force. Other than e(4) (DDD), all that
P
remains is e(3) (l1l2 l3).
Considerable work has been done to explore the non-additive component of
the third order perturbation energy. Each term in the series of that component is
divided into a geometric function, w(3) (l1l2 l3), and an interaction function, z(3) (l1l2l3 ).
z(3) (l1 l2l3)) depends only on the atom variety of the molecules interacting.
e(3) (l1l2 l3) = z(3) (l1 l2l3 ) w(3) (l1l2l3 )
(2.18)
20
The geometric functions for dierent combinations of moments have been calculated.
1 cos 2 cos 3 )
w(3) (DDD) = 3(1 + 3 cos
3
r12r233 r313
9
cos
3
3 ; 25 cos 33 + 6 cos (1 ; 2 )
(3)
w (DDQ) = 16
r123 r234 r314
1
w(3) (QQD) = 15
5
64 r r4 r4 3(cos 3 + 5 cos 33) +
(2.19)
(2.20)
12 23 31
20 cos (1 ; 2 )(1 ; 3 cos 22 ) + 70 cos 2(1 ; 2 ) cos 3
15 1
w(3) (QQQ) = 128
r125 r235 r315 ;27 + 220 cos 1 cos 2 cos 3 +
490 cos 21 cos 22 cos 23 +
175 cos 2(1 ; 2 ) + cos 2(2 ; 3 ) + cos 2(3 ; 1 )
5 1
43 +
w(3)(DDO) = 32
r123 r235 r315 9 + 8 cos 23 ; 49 cos
6 cos (1 ; 2)(9 cos 3 + 7 cos 3)
(2.21)
(2.22)
(2.23)
The sources of these equations are listed in Doran and Zucker 24]. However, the
rst term in the series, w(3) (DDD), is the Axilrod-Teller triple dipole potential 27].
Notice that the higher order moments are proportional to greater powers of the inverse
of r making them less and less signicant. Additionally, it is readily apparent that
three body forces are extremely dependent upon the geometric position of the triplet
as evident in Figure 2.2.
The remaining multi-body potential of interest is w(4) (DDD) from above.
1 + cos 2 2 + 1 + cos 32 + 1 + cos 1 2
w (DDD) = 45
64
r126 r236
r236 r316
r316 r126
(4)
(2.24)
Doran and Zucker calculated the potential energy contributions of each of these
terms to the crystal structures of solid rare gases and compared it with the heat of
sublimation, H0, derived experimentally.
Table 2.1 demonstrates that three body forces are signicant under dense solid
21
2
Axilrod-Teller
Potential Energy
1
for I-J-K System
J
*
v =v/Nε
0.05
Y[σ] 0
I
0.04
0.03
0.02
-1
0.01
0.00
-0.01
-2
Cutoff - .9σ
-2
-1
0
1
2
X[σ]
Figure 2.2. The Axilrod-Teller energy potential for a three body system with I and J xed at (0,0)
and (0,1 ), respectively. The position of K determines the potential energy of the system.
and liquid conditions. Additionally, the energy from the fourth order perturbation
tends to cancel the contributions from the third order perturbation other than the
Axilrod-Teller potential. In Table 2.2, the relative importance of the Axilrod-Teller
potential is compared with the summation of other many-body potentials. Keeping
in mind that many-body energy contributions amount to only few percent of the total
potential energy, it becomes evident that three body forces other than the AxilrodTeller triple dipole are of small signicance and probably less than the error in most
available experimental data.
22
H0 e(3) (DDD) e(3) (DDQ) e(3) (QQD) e(3) (QQQ) e(4) (DDD)
Ne -1875.7
62.4
15.5
2.7
0.22
-4.8
A -7745.6
579.6
173.8
37.0
3.5
-126.7
Kr -12212.3
1004.0
220.1
34.3
2.4
-281.4
A -15817.7
1597.9
373.6
62.0
4.6
-636.6
Table 2.1. Energy contributions of various three body potentials to crystalline structures.
Jg;1 mol;1]24]
e(3) (DDD)
Ne
62.4
A
579.6
Kr
1004.0
A
1597.9
Pl6 DDD
ll e
1 2 3
=
l1l2 l3 ) + e(4) (DDD) Percent Dierence
(3) (
13.6
87.6
-24.6
-196.4
21.8%
15.1%
-2.5%
-12.3%
Table 2.2. Relative importance of three body contributionsJg;1mol;1 ]
There are two accepted methods to account for this additional three body potential in MD simulations. As most pairwise potentials are merely curve ts from
experimental data, selecting data for only dense systems will bias the potential to
compensate for three body eects. This method is referred to as an eective potential. The second method is to add a term to the potential in the form of the
Axilrod-Teller triple dipole equation.
The advantage of the rst method is its obvious simplicity. However, it has a
number of drawbacks. First, the Axilrod-Teller potential is extremely geometrically
dependent. So while the eective potential may produce a system's total potential
energy within a few percent of reality, it cannot simulate the real trajectories of
the molecules, which may aect studies of transport coecients or other interaction
dependent quantities. Additionally, an eective potential can only be considered
accurate for a simulation of uniform density which matches the data used to determine
it. As such, eective potentials would do a very poor job of simulating two phase
systems which by denition would have inherently dierent densities present.
23
The importance of adding Axilrod-Teller triple dipole terms to a pairwise potential equation was established by Barker, Fisher, and Watts 28]. Their work indicates
that the Axilrod-Teller potential can contribute as much as 5% of the potential energy of a system and aect the pressure by up to 50%. Additionally, their potential
performs well around the critical line and in the supercritical regime. Previous work
by Barker has established the importance of three body forces to the prediction of
transport properties 26]. Over the years, others have raised the issue of neglecting
the other many-body contributions under certain conditions, but a pairwise potential
with Axilrod-Teller has continued to prove the best model for argon over a large range
of densities and pressures 29].
2.4 Choice of Potentials
With a plethora of potentials for argon available, it was necessary to determine
which potentials would perform best for modeling supercritical droplet evaporation.
Traditionally, a potential would be chosen to best reproduce a specic system characteristic of interest under the required environment of T and p. However, our objective
is to simulate a complex, macroscopic system under supercritical conditions. Droplet
evaporation under supercritical conditions would require two phases of matter and
extremely dense ambient environments. Additionally, our interest is not limited to
one parameter of the system, but it includes a full spectrum of thermodynamic and
transport properties within localized regions of the simulation. The only solution is to
use the most accurate intermolecular potential available with the hopes of simulating
a \real" system.
As the purpose of this work is to explore the impact of these potentials to MD
droplet evaporation and weigh that with their computational demands, a number of
potentials were reviewed. The control is the established 12-6 Lennard-Jones potential
in common use today. Additionally, the importance of Axilrod-Teller contributions
to accuracy as described above indicate its necessary inclusion in combination with
a pairwise potential. As the form of the Axilrod-Teller three body forces is relatively
straightforward, the choice of a suitable companion pairwise potential proved to be
the more dicult one.
24
The computationally least demanding pairwise potential was the traditional
Lennard-Jones 12-6. The most accurate pair potential for argon to date was determined to be the Barker, Fisher, and Watts (BFW) potential which was dened to
include the Axilrod-Teller triple dipole term. Finally, a modern, more accurate potential with a simple form was sought as a balance of the two extremes above. The result
was the n("r)-6 potential which is a variation of the original Lennard-Jones potential,
but the rst exponent becomes a function of the separation distance, r 22]. While it
has only three adjustable parameters, the n("r)-6 has proven to be extremely accurate
for rare gases.
12 6
vLJ (r) =4 r ; r
6
n
;
n
;
6
vn6(r) = n ; 6 r" ; n ; 6 r"
r" = rr
n = 13 + ("r ; 1)
m
;
;
vBFW (r) = e(1;r) A0 + A1 ("r ; 1) + A2 ("r ; 1)2 + A3 ("r ; 1)3 +
C
C
C
8
10
6
4
5
A4("r ; 1) + A5("r ; 1) ; + r"6 + + r"8 + + r"10
(2.25)
(2.26)
(2.27)
Note the similarity between the BFW potential of the generic theoretical potential
listed in Equation 2.9. However, the form most used in MD work is the force interaction which is the negative gradient of the potential energy. For comparison, listed
below are the x-component derivatives of the gradient.
12 6
dv
LJ (r) 24
; dx = r2 2 r ; r x
5
5
m ; nrm
; dvndx6(r) = (n ; 6)62rr r"n + (n r
; 6)r7 (
n ; 6)2r7
m
6
; (n6;nr6)m r8 + 6 (nn ;+ 6) r"rrlogr""nr+1] x
m
(2.28)
(2.29)
25
(r) = e(1;r) ;A + A ("r ; 1) + A ("r ; 1)2 + A ("r ; 1)3+
; dvBFW
0
1
2
3
dx
rrm
(1;r)
A4("r ; 1)4 + A5 ("r ; 1)5 ; e rr (A1 + 2A2 ("r ; 1)+
m
2
3
3A3("r ; 1) + 4A4("r ; 1) + 5A5("r ; 1)4 ;
1 6C6r"4 + 8C8r"6 + 10C10r"8
rm2 (
+ r"6)2 (
+ r"8)2 (
+ r"10 )2
(2.30)
x
The force potentials above illustrate a number of things. Not only is the
Lennard-Jones 12-6 potential simple in presentation, but the fact that it has only
even powers of r means that it is not necessary to calculate r by taking the square
root of r2 which is a computationally expensive operation. Additionally, with computational demand in mind, the simplied, but accurate, n(r)-6 oers little advantage
over the BFW potential. Actually, it looks worse with non-integer exponents and a
log function which are all extremely computationally demanding.
In conclusion, three potentials were chosen to study the eects of incorporating more accurate potentials, mainly three body forces, into MD simulations of argon droplet evaporation under supercritical conditions | Lennard-Jones (LJ) 12-6,
Lennard-Jones 12-6 with three body Axilrod-Teller (AT), and Barker-Fisher-Watts
(BFW) which includes Axilrod-Teller. The parameters chosen were those for a liquid
argon eective LJ potential and appropriate ones for BFW and AT. Parameters are
listed in Table 2.3. For future reference, the pair potential component of BFW and
the eective LJ potentials are plotted in Figure 2.3.
26
=
=k=
Barker-Fisher-Watts22] rm =
=k=
3:41 10;10 m
119.8 K
3:761 10;10 m
142.1 K
C6=
C8=
C10 =
=
=
1.10727
0.16971
0.01361
12.5
0.01
Lennard-Jones17]
Axilrod-Teller28]
=
73:2 10;84 erg cm9
A0 = 0.27783
A1 = -4.50431
A2 = -8.33122
A3 = -25.2696
A4 = -102.0195
A5 = -113.25
Table 2.3. Parameters for various potentials chosen.
150
Argon Pair Potentials
100
Lennard-Jones 12-6 potential
Barker-Fisher-Watts pair potential component
50
v(r) / kB
[K]
1.0
1.5
2.0
2.5
3.0
0
rij /σ
-50
-100
-150
Figure 2.3. Pairwise component of Barker-Fisher-Watts and the eective Lennard-Jones potentials.
27
Chapter 3
Molecular Dynamics Algorithms with Three Body Forces
This chapter discusses some common elements of MD codes and how they were
implemented in the original code written by Little 30]. The following sections then
discuss how three body force calculations were added and how the resulting workload was decreased using the multiple time step (MTS) method. Finally, timings,
speedups, and parallel eciencies for the various codes developed are reviewed with
a discussion of scalability.
3.1 Basic MD Code
The original code used for this work was developed by Little 30]. Boundary
conditions were cubic and periodic with the droplet originally placed at the center
and recentered every 500 time steps. Force calculations were constrained by the use of
a cell based link list and with a cuto of 2.5 . Displacements were accomplished with
a modied form of the velocity Verlet algorithm. The system was initialized by fusing
a droplet equilibrated in a vacuum with an ambient environment of specied pressure
and temperature. Finally, the code was parallelized with the particle decomposition
method to achieve optimal load balancing.
The following is not intended to be a detailed explanation of how a MD code
works, but to catalog how the common elements of this MD code were implemented.
For more detailed explanations of the methods discussed below, the reader is referred
to the thesis work of Little 30] and books by Allen and Tildesley 17] and Haile 16].
3.1.1 Boundary Conditions
The system simulated was a cube of ambient environment with a liquid droplet
at the center of the cube. The boundary conditions on all sides were periodic. Periodic
boundary conditions necessitate both minimum imaging, the mirroring of twenty six
identical droplets and environments with the real droplet located at the center of
28
the cube, and physical conservation, an atom that exits the system through one side
is replaced by an identical atom entering from the opposite side. While the latter
requirement is not dicult, the minimum imaging can become extremely taxing as
every force calculation, the core of an MD program, would require a check to make
sure that position coordinates of an atom are those of the closest mirror image. For
example, two atoms on opposites sides of the simulation would not interact directly,
but each would interact with the mirror image of the other that is adjacent to it.
Figure 3.1 is a 2-D example of periodic boundaries from Little 30].
Figure 3.1. Periodic cubic boundaries.
The computational demand for this mirror imaging was minimized by making
the code cell aware. The system was subdivided into cells of box length 2.5 as
described in Section 3.2 for the force calculations. By taking advantage of this division
and making the code aware of which cells lay on the faces, edges, and corners of the
simulation, minimum image calculations would be limited to only those atoms in the
boundary cells.
29
3.1.2 Displacements
As mentioned previously, MD is merely a classical dynamics model of atomsized balls. At a point in time, the total force on each atom is evaluated. Following
Newton's 2nd Law, the acceleration due to that force is calculated, from which a
velocity is determined. Using these velocities and accelerations, the atom is moved a
nite amount based on the time step, and the process is repeated.
For computational stability reasons, this is dicult to do in a completely explicit
manner as described above. The preferred algorithm, which is partially implicit, is
the velocity Verlet algorithm listed below.
2
rxi+1 = rxi + %t vxi + %2t aix
1
vxi+ 2 = vxi + %2t aix
aix+1 = fx(ri+1)
1
vxi+1 = vxi+ 2 + %2t aix+1
(3.1a)
(3.1b)
(3.1c)
(3.1d)
However, the algorithm used in this work is a slightly modied form of the velocity
Verlet. Little 30] modied the algorithm by reordering it as follows.
aix+1 = fx(ri+1)
1
vxi+1 = vxi+ 2 + %2t aix+1
2
rxi+2 = rxi+1 + %t vxi+1 + %2t aix+1
3
vxi+ 2 = vxi+1 + %2t aix+1
(3.2a)
(3.2b)
(3.2c)
(3.2d)
This version is functionally the same as the velocity Verlet, but limits the memory
demand by removing the need to store the particle accelerations.
3.1.3 Initialization
A liquid argon droplet centered in various ambient environments were initialized
in a two step manner. First, two separate systems were created. A lattice of argon
30
atoms are heated to a liquid temperature and allowed to equilibrate in a vacuum
which due to surface tension eects produces a droplet. Ambient environments were
simply lattices of the appropriate density heated to the desired temperatures and
allowed to equilibrate. The size of the ambient environments was driven by the size
of the droplet. The environment had to be large enough to absorb the droplet's atoms
as it evaporated without resulting in a signicant change in the density, and therefore
pressure, of the environment.
The droplet was then fused into the ambient environment of choice by removing
all environment atoms that could see any droplet atom within a 90 angle of a vector
from the center of the system through the environment atom in question. In other
terms, any environment atom that could see any part of the droplet while facing
directly away from the center of the droplet was deleted. The complete system then
equilibrates in the rst 500-1000 time steps or 5-10 picoseconds of simulation time.
This procedure attempts to imitate a cold, cryogenic droplet suddenly injected into
a supercritical environment as one might nd in a typical liquid rocket engine.
3.1.4 Heating
Due to the temperature contrast between the droplet and the environments
studied in this work, as the droplet evaporates it absorbs kinetic energy from the
ambient environment cooling it. In order to maintain an ambient environment of
constant temperature, it is necessary to heat the outer edge of the environment.
Heating was limited to the outer layer of cells in the simulations. The heating
algorithm would sum the velocity magnitude of every atom within the outer cells to
determine their average temperature. The fraction of variation between the chosen
environment temperature and the actual temperature is determined. The fraction
of dierence would then be applied to the velocity magnitudes to produce a layer of
boundary cells at the dictated ambient temperature. This was done every time step.
3.1.5 Parallelization
As discussed in Chapter 1, MD is inherently parallel because the work is already
divided and each particle is oblivious to the majority of other particles in the system.
31
Three methods of dividing the work amongst multiple processors are particle decomposition, domain decomposition, and force decomposition. For detailed explanations
of each see Plimpton 31]. This MD code used particle decomposition.
The primary motivation behind choosing particle decomposition rests with the
density variation of our system and load balancing. In droplet evaporation the density
of the liquid is much greater than the surrounding environment. As a result, if
domain decomposition is used, a processor with a domain from the interior of the
droplet would take much longer to complete force calculations than a domain from
the environment which would have fewer particles due to the O(N 2) nature of the
work. Force decomposition trades a slight increase in computational overhead for a
reduction in communication time. However, the theoretical gains only outweigh the
losses when run on systems with at least 9 processors 31, 30]. In practice, there is
only a marginal decrease in communication time with under 25 processors. Since the
limiting factor for this code is already the computational demand, force decomposition
oers little benet.
Particle decomposition is done by dividing the particles up equally among the
dierent processors. Each processor calculates the forces and displacements for the
particles for which it is responsible. Then the new positions and velocities from each
processors particles are broadcast to all other processors with which the next time
steps begins. Not only does this method provide excellent load balancing, but its
implementation is simpler than the other methods.
3.2 Three Body Force Calculations
The core of any MD code is the force calculations. The displacement calculations
must be done for every atom in the system which is an order N computation, O(N ).
N being the number of particles in the system. The force calculations require the
pairwise summation of the forces from every other atom on the atom in question for
every atom in the system which is an O(N 2) operation. While tolerable in simulations
with a small number of particles, large simulations with large numbers of particles
quickly become too much for even today's supercomputers. Obviously to simulate
macroscopic systems such as droplet evaporation, something must be done to decrease
32
the order of magnitude of this operation. For future reference, to include three body
forces would require a summation of forces over every triplet in the system which
would be an O(N 3 ) operation.
Molecular potentials and corresponding forces have an extremely short range.
As a result, it is common practice to ignore the presence of any atoms outside a
cuto radius. Following usual practice, the cuto for this work was set at 2.5 . Even
though it is no longer necessary to calculate the full force potential between every
pair of atoms, a check is still required between every pair to see whether it is within
the cuto or not. This check alone can be expensive when done on an O(N 2 ) scale.
The solution is to divide the system into cells with box length equal to the cuto,
2.5 . Linked lists are maintained of the atoms that exist in each cell. Each atom
will be checked for interaction with every other atom in the same cell and with every
atom in the surrounding 26 cells. For every atom found within the cuto, full force
calculations are completed. The cell linked lists were updated every time step.
Additionally, most MD codes take advantage of Newton's 3rd Law | for every
action there is an equal and opposite reaction. Since the force of atom i on atom j is
the negative of the force of atom j on atom i, this force needs only to be calculated
once for every pair. However, due to parallelization issues, the cost of calculating
every interaction twice, i{j and j {i, was deemed less than the cost of communicating
which processors were to calculate which interactions. This is a result of the chosen
method of parallelization as described in Section 3.1.5.
The majority of computation time for this code, 90-95%, was devoted to three
body forces. As discussed in Section 3.1.2, displacement of molecules requires the
calculation of particle accelerations at each point in time. Accelerations are derived
from the net force on the particle, a=f /m. The two body and three body forces are
summed separately and added together to calculate the particle's acceleration.
fi = fi2b + fi3b =
X
i6=j
; 5 v2b (r) +
XX
j<k k6=i
j 6=i
; 5 v3b (r)
(3.3)
For each particle the negative gradients of the potential energy function must be
calculated and summed over all pairs for two body forces and all triplets for three
33
body forces.
3.2.1 Two Body Forces
For the sake of comparison, the gradient of the two body Lennard-Jones 12-6
potential as presented in Equation 2.28 takes the following form in Fortran code.
rsqinv = 1.d0/rsq
sqr2 = sigsq*rsqinv
sqr6 = sqr2**3
sqr12 = sqr6**2
fqr = (sqr12+sqr12-sqr6)*rsqinv
fx = fx + fqr*rxij
fy = fy + fqr*ryij
fz = fz + fqr*rzij
Given r2, it produces the three Cartesian components of the force of atom j on atom
i. This routine must be run on every other molecule found within the specied cuto
of 2.5 of the ith molecule. The routine above contains approximately 18 oating
point operations.
The other pairwise potential used in this work is the rst part of the Barker{
Fisher{Watts (BFW) energy potential. The x-component of its negative gradient can
be found in Equation 2.30. In contrast to the Lennard-Jones pair potential, the BFW
routine below requires an estimated 99 oating point operations or about 5 times as
much computational power to evaluate the pairwise force.
c rij and its forms
rij = sqrt(rsq)
rbar = rij*invrm
invrrm = (1.d0/rij)*invrm
c rbar to different powers
rbar2 = rbar*rbar
rbar4 = rbar2*rbar2
rbar6 = rbar4*rbar2
rbar8 = rbar6*rbar2
rbar10 = rbar8*rbar2
c Long range force geometric elements
invdelr6 = 1/(delta+rbar6)
invdelr8 = 1/(delta+rbar8)
invdelr10 = 1/(delta+rbar10)
34
c rminus to different powers
rminus = rbar - 1.d0
rminus2 = rminus*rminus
rminus3 = rminus*rminus2
rminus4 = rminus*rminus3
rminus5 = rminus*rminus4
c Reappearing scalars
expalpha = exp(-alpha*rminus)
aprime = expalpha*(A0+A1*rminus+A2*rminus2+
&
A3*rminus3+A4*rminus4+A5*rminus5)
c6prime = C6*invdelr6
c8prime = C8*invdelr8
c10prime = C10*invdelr10
c Calculate forces
fqr = alpha*invrrm*aprime &
expalpha*invrrm*(A1+2*A2*rminus+3*A3*rminus2+
&
4*A4*rminus3+5*A5*rminus4) &
invrm2*(6*rbar4*invdelr6*c6prime +
&
8*rbar6*invdelr8*c8prime +
&
10*rbar8*invdelr10*c10prime)
fx = fx + fqr*rxij
fy = fy + fqr*ryij
fz = fz + fqr*rzij
3.2.2 Three Body Forces
The Axilrod-Teller energy potential and its negative gradient take the following
form.
i cos j cos k )
vAT = (1 + 3 cos
3 3 3
r r r
(3.4)
ij jk ik
fAT = r5 r5 r5 5(rij2 rjk2 rik2 ; 3(rik rjk)(rik rij )(rij rjk ))( rrij2 + rrik2 )
ij jk ik
+ 3((rij rjk )(rik rij ) + (rik rij )(rik rjk ))rjk
+ 3((rij rjk )(rik rjk ))(rij + rik )
;2(rjk2 rik2 rij + rjk2 rij2 rik )
ij
ik
(3.5)
The bold letters represent vectors. The constant is a dependent only on the species
of atoms involved. For argon, is 73:2 10;84 erg cm9 28]. The code to calculate
the Axilrod-Teller force on the ith molecule due to the j and k molecules follows.
35
c Calculate dot products
rijrjk = (rxij*rxjk) + (ryij*ryjk) + (rzij*rzjk)
rijrik = (rxij*rxik) + (ryij*ryik) + (rzij*rzik)
rjkrik = (rxjk*rxik) + (ryjk*ryik) + (rzjk*rzik)
c Multiply dot products
rijrjk_rijrik = rijrjk*rijrik
rijrik_rjkrik = rijrik*rjkrik
rjkrik_rijrik_rijrjk = rjkrik*rijrik*rijrjk
rijrjk_rjkrik = rijrjk*rjkrik
c Calculate square roots and sr^5 place holder
rij = SQRT(rijsq)
rik = SQRT(riksq)
rjk = SQRT(rjksq)
rij5 = rij*rijsq*rijsq
rik5 = rik*riksq*riksq
rjk5 = rjk*rjksq*rjksq
c Repeated scalar quantities
nurrr = nu/(rij5*rik5*rjk5)
f1 = 5.d0*((rijsq*riksq*rjksq)-3.d0*rjkrik_rijrik_rijrjk)
fi2 = 3.d0*(rijrjk_rijrik + rijrik_rjkrik)
fi3 = 3.d0*rijrjk_rjkrik
rijsqinv=1.d0/rijsq
riksqinv=1.d0/riksq
c Calculate forces
fjkx3b =nurrr*(f1*(rxij*rijsqinv+rxik*riksqinv) +
*
fi2*rxjk +
*
fi3*(rxij+rxik) *
2.d0*(rjksq*riksq*rxij+rjksq*rijsq*rxik))
fjky3b =nurrr*(f1*(ryij*rijsqinv+ryik*riksqinv) +
*
fi2*ryjk +
*
fi3*(ryij+ryik) *
2.d0*(rjksq*riksq*ryij+rjksq*rijsq*ryik))
fjkz3b =nurrr*(f1*(rzij*rijsqinv+rzik*riksqinv) +
*
fi2*rzjk +
*
fi3*(rzij+rzik) *
2.d0*(rjksq*riksq*rzij+rjksq*rijsq*rzik))
This routine is applied to all triplets formed with the ith molecule where rij ,
rjk , and rik are all within the cuto of 2.5 . With about 120 oating point operations
the Axilrod-Teller component is comparable to the BFW force evaluation. However,
there are many more triplets than pairs that need to be evaluated. As such, the three
body force calculations still amount to the majority of the computational work.
The resulting forces are not simply repulsive and attractive along the relative
36
position vector. Figures 3.2 and 3.3 below indicate the unusual symmetry of three
body forces. In the example below, the X-Y plane is the plane dened by the triplet,
and the resulting Z-direction forces are then zero.
2
Axilrod-Teller
Force in X-direction
on Atom I
1
fx=fxσ/ε
*
J
0.30
Y[σ] 0
0.20
I
0.10
0.00
-1
-0.10
-0.20
-0.30
-2
Cutoff - .9σ
-2
-1
0
1
2
X[σ]
Figure 3.2. Axilrod-Teller three body force in the X-direction on I due to J and K. I and J are xed
at 0,0 and 0,1 , respectively. Position of K is determined from the axes.
3.3 Multiple Time Step Method
Even with a cell linked list and a cuto, the demands of including three body
force calculations are signicant. In order to further reduce the computational requirements, a multiple time step (MTS) method was implemented. In short, the
simulation used two dierent time steps, one for two body forces and one for three
body forces. MTS was originally developed for simulations constrained to small time
37
2
Axilrod-Teller
Force in Y-direction
on Atom I
1
fy=fyσ/ε
*
J
0.30
Y[σ] 0
0.20
I
0.10
0.00
-1
-0.10
-0.20
-0.30
-2
Cutoff - .9σ
-2
-1
0
1
2
X[σ]
Figure 3.3. Axilrod-Teller three body force in the Y-direction on I due to J and K. I and J are xed
at 0,0 and 0,1 , respectively. Position of K is determined from the axes.
steps by other factors by Streett and Tildesley 32].
Haile demonstrated that three body forces vary at a slower rate than two body
forces 33]. As a result, while pairwise forces are evaluated every time step, AxilrodTeller contributions need only be calculated once every few time steps. During the
intermediate steps, the three body forces for each particle are extrapolated linearly.
3b
3b
fm3b(t + t) = fm3b(t) + fm (t) ; fnm%(tt ; n%t) t
(3.6)
In the equation above, m is each Cartesian index of the force, x, y, or z, and t is
some multiple of %t less than n.
38
MTS method is dened by a pairwise time step, %t, and three body force evaluation interval, n. The equivalent time step for the three body forces is simply n%t.
Choosing these two parameters is primarily a function of code stability and system
energy conservation. While Haile concluded that %t=2:5 10;15 seconds and n=8
was optimal, this work used a value of n=2 and %t was simply the value required
for pairwise stability. For the supercritical environment with a temperature of 200K,
%t was 10:77 10;15 seconds. The subcritical environment had a higher ambient
temperature of 300K, and the value of %t was reduced to 7:5 10;15 seconds. The
value of the equivalent three body time step for both this work and Haile was on the
order of 20 femtoseconds. Most likely, for a given level of kinetic energy the equivalent
three body time step has a stability maximum just as the pairwise time step has a
maximum.
3.4 Code Performance
Two systems were used for the code timing and performance measurements.
Both systems were supercritical environments, 200K and 7.5 MPa, with liquid argon
droplets placed in the center. The systems sizes and their ratios are listed in Table
3.1. As the work really scales with the computation of three body force, the ratio
# of atoms Total Droplet Environment
Small Case 22,144 5,911
16,233
Large Case 96,480 30,859
65,621
Ratio L/S
4.36
5.22
4.04
Table 3.1. Dierent size cases used for timing and performance measurements.
of droplet sizes is more indicative of the ratio of real work than the ratio of system
size for codes involving the AT potential.
The timing results are sub-divided into three dierent parts. The computational
times represents times spent computing the forces, accelerations, and displacements.
Communication time is limited to the communication of the atoms' positions. Wait
time refers only to the time that passes from when one processor nishes the force
39
and displacement calculations and the last processor completes those calculations for
the time step. Link list time is devoted to rebuilding the cell link lists. Finally, miscellaneous time is all remaining time during each time steps and includes operations
such as calculating system properties, boundary heating, and system energies.
The times in Figure 3.4 clearly indicate the increased computational demand
of the three body codes. However, the communications times are identical for all
three codes. Communication requirements are directly proportional to the number of
particles in the simulation. This was a limiting factor in the original LJ code which
did not scale well with problem size. Since the communications were only a function
of the size of the system and were a signicant fraction of the total time as seen in
Figure 3.5, this implied a practical maximum problem size regardless of the number
of processors available. However, the three body code is primarily computationally
dependent and uses only a small fraction of total time for communication. Maximum
problem size should scale well with the maximum number of processors available.
While the three body codes do demonstrate good scalability for a xed problem
size, there is some loss of eciency due to a decrease in the megaop rates. This is
partially a result of the cell aware aspect of the code. Each processor goes through
every cell, every time step and calculates the forces for its molecules only. However,
with more processors and less particles per processor there is more overhead work
without an increase in the op rates. Additionally, the calculation of system characteristics and energies requires the communication of information from all processors
to the master processor which becomes more dicult with more processors.
A comparison of the LJ with AT and BFW potentials indicates little dierence
in computational demand. Even though the BFW pair potential is 5 times more
demanding than the LJ potential, the increase in computing time is slight. The three
body calculations dominate the simulation computationally because there are many
more triplets than pairs in the system.
Speed-ups and parallel eciencies indicate reasonable performance with an increase in the number of processors. However, megaop rates are somewhat disappointing for a machine with a theoretical peak of 250 megaops per processor. The
poor performance is a result of the large number of IF statements which must be
40
executed and do not count as oating point operations.
The timings below are from the rst 100 time steps. However, Figures 3.7 and
3.8 demonstrate that the work loads for the three body codes are not constant. As
the droplet evaporates the median value of density for the system drops. This both
decreases the number of triplets that must be checked and reduces the number of
three body calculations completed. Both curves tend toward an asymptote that is
dened by the average density of the system.
41
40
40
35
30
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
Time[min]
25
96k BFW
20
Computation Time Per Processor
for 100 Time Steps
30
22k LJ
25
Time[min]
35
Total Time Per Processor
for 100 Time Steps
15
15
10
10
5
5
0
96k LJ w/ AT
96k BFW
20
0
8
16
24
32
8
16
Processors
1.0
32
4.0
0.9
3.5
Communication Time Per Processor
for 100 Time Steps
0.7
22k LJ
0.6
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
0.5
96k BFW
Wait, Link List, and Misc. Times
Per Processor for 100 Time Steps
3.0
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
2.5
Time[min]
0.8
Time[min]
24
Processors
0.4
2.0
96k BFW
1.5
0.3
1.0
0.2
0.5
0.1
0.0
0.0
8
16
24
Processors
32
8
16
24
Processors
Figure 3.4. Code times for the rst 100 time steps.
32
1.0
0.9
0.9
0.8
0.8
0.7
0.7
Fraction of Total Time
1.0
0.6
0.5
0.4
Fraction of Total Time
Devoted to Computation
0.3
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
0.2
0.1
Fraction of Total Time
Devoted to Communication
0.6
22k LJ
0.5
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
0.4
96k BFW
0.3
0.2
0.1
96k LJ w/ AT
96k BFW
0.0
0.0
8
16
24
32
8
16
Processors
24
Processors
1.0
0.9
0.8
Fraction of Total Time Devoted
to Waits, Link List, and Misc.
0.7
Fraction of Total Time
Fraction of Total Time
42
22k LJ
22k LJ w/ AT
22k BFW
0.6
96k LJ
96k LJ w/ AT
96k BFW
0.5
0.4
0.3
0.2
0.1
0.0
8
16
24
32
Processors
Figure 3.5. Fraction of computer time used by dierent elements of the codes.
32
43
50
32
28
Floating Point Operations Per Second
(Mflops) Per Processor for 100 Time Steps
40
Algorithmic Speed-Up
Algorithmic Speed-Up
30
96k LJ w/ AT
96k BFW
20
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
20
96k BFW
16
12
8
10
4
0
0
8
16
24
32
8
16
24
Processors
Processors
100
90
80
70
Parallel Efficiency[%]
Megaflops/Processor
24
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
60
50
40
Parallel Efficiency
30
22k LJ
22k LJ w/ AT
22k BFW
96k LJ
96k LJ w/ AT
20
10
96k BFW
0
8
16
24
32
Processors
Figure 3.6. Floating point operations and megaops ratings for the dierent codes.
32
44
7
Time Required to
6
Complete 100 Time Steps
200K, 7.5MPA, LJ w/ 3b
200K, 7.5MPa, BFW
Time [min]
5
4
3
2
1
0
0
10000
20000
30000
40000
50000
60000
Time Steps
Figure 3.7. The evolution of code speed over evaporation runs of 22,000 particle, supercritical system.
7
Time Required to
6
Complete 100 Time Steps
300K, 3.0MPa, LJ w/ 3b
300K, 3.0MPa, BFW
Time [min]
5
4
3
2
1
0
0
10000
20000
30000
40000
50000
60000
Time Steps
Figure 3.8. The evolution of code speed over evaporation runs of 27,000 particle, subcritical system.
45
Chapter 4
Results from Argon Droplet Evaporation with Dierent Potentials
4.1 General Overview
Argon droplets were evaporated in supercritical and subcritical ambient environments using dierent potentials. One supercritical environment and one subcritical
environment were chosen for comparison and to match previous work 30]. The supercritical environment was 200K and 7.5 MPa! the subcritical environment was 300K
and 3.0 MPa. Both environments are indicated in the Pv diagram in Figure 4.1.
Each droplet was allowed to equilibrate to a temperature of 100K and the matching
8
Argon Properties
Supercritical Case
200K, 7.5MPa
7
Critical Property Zone
T > 150.8K
6
P > 4.9MPa
0K
0K
0K
Pressure (MPa)
30
20
16
5
Subcritical Case
4
300K, 3.0MPa
3
Critical
Temperature
Isotherm (151K)
2
Saturated
1
0
0.000
Saturated
Liquid
Vapor
0.005
0.010
0.015
0.020
3
Specific Volume (m /kg)
Figure 4.1. Argon Pv plot with ambient environments indicated.
0.025
46
vapor pressure before being fused with the environment. The resulting system sizes
are listed in Table 4.1. The physical size of both droplets is about 5 nanometers.
However, the total volume of the subcritical system is about 5.8 times larger than
the volume of the subcritical system because the ambient density is much lower. As
a result, the subcritical droplet appears smaller than the supercritical droplet when
in reality the subcritical environment is merely larger.
# of atoms
Total Droplet Environment
Supercritical Case 22,144 5,911
16,233
Subcritical Case 27,264 6,460
20,804
Table 4.1. The number of atoms found in the supercritical and subcritical droplet simulations.
As mentioned in Section 2.4, three dierent potentials were used in this work
| Lennard-Jones (LJ) pair potential, LJ with the Axilrod-Teller (AT) three body
potential added on, and Barker-Fisher-Watts (BFW) potential which includes AT. All
parameters and sources are listed in Table 2.3. Figure 4.2 is graphical representation
of LJ and BFW. They are obviously relatively similar in appearance as expected.
However, the calculated dierences are shown in Figure 4.3.
2
2
Barker-Fisher-Watts
Lennard-Jones
Potential Energy
1
J
Y[σ] 0
I
for I-J-K System
*
v =v/Nε
1
Potential Energy
for I-J-K System
J
*
v =v/Nε
2.00
2.00
1.50
Y[σ] 0
I
1.50
1.00
1.00
0.50
0.50
-1
-1
0.00
0.00
-0.50
-0.50
-1.00
-2
-1.00
-2
Cutoff - .9σ
Cutoff - .9σ
-2
-1
0
X[σ]
1
2
-2
-1
0
1
2
X[σ]
Figure 4.2. Lennard-Jones and Barker-Fisher-Watts energy potentials for a three body system with
I and J xed at 0,0 and 0,1 , respectively. The position of K determines the potential energy of the
system.
47
2
Difference in
Potential Energy
1
LJ-BFW
for I-J-K System
v*=v/Nε
J
0.60
Y[σ] 0
I
0.50
0.40
0.30
-1
0.20
0.10
0.00
-2
Cutoff - .9σ
-2
-1
0
1
2
X[σ]
Figure 4.3. Dierence between LJ and BFW potentials.
The major dierence between potentials appears as atom K approaches the
cuto of .9 . The cuto is a simply a line drawn beyond which K will never reach.
Its value comes from pair distribution functions of argon liquids as seen in Figure 4.4.
Within liquid argon an atom is most likely found approximately 1.1 distant from
another atom which dictates the choice of 0,1 for position of atom J. Additionally,
atoms are rarely found closer than 1 and never seen below .9 in a liquid which
was used to dene the cuto. However, it is interesting to note that the dierence
between these two potentials increases rapidly as .9 is crossed. This large dierence
is probably due to the dierence in repulsive elements of the two potentials | the
1
r12 compared to an exponential.
A test matrix of two environments by three potentials produces six dierent
48
cases. The pair distribution functions and system energies are presented for each
case. Then a qualitative examination of temperature, density, and surface tension
evolution is conducted. Finally, the regression rates of the droplets are plotted for
each case and discussed.
4.2 Radial Distribution Functions
The radial distribution functions were determined from the core of the simulation. Early in the simulation, the droplet is clearly a liquid. The subcritical case
retains the liquid form of the distribution function better than the supercritical case
which tends to atten out. The supercritical droplet is evolving slowly into a gas
even as it evaporates while the subcritical droplet remains a liquid. The radial distribution functions from around 150 picoseconds are truncated at 3 . Beyond which,
the distribution of the simulation core is more indicative of the environment than the
droplet which has decreased in size.
In general, the function is not greatly aected by the dierent potentials. Of the
dierences that do appear, the height of the initial peak for the LJ with AT is lower
just as Haile observed 33]. Since the true dierences between all three potentials are
most noticeable around .9 to 1.1 , it follows that the greatest aect on the radial
distribution functions would occur in the same region. However, the most interesting
characteristic is how closely the simple eective LJ and BFW match | well within
noise limits. In comparison, the LJ with AT is often lower than the other two which
indicates a higher potential energy and faster evaporation.
49
2.5
2.0
Radial Distribution Function
LJ, Timesteps 100-1100
LJ w/ 3b, Timesteps 100-1100
BFW, Timesteps 100-1100
1.5
g(r)
1.0
0.5
0.0
1
2
radius[σ]
3
4
Radial Distribution Function
2.5
LJ, Timesteps 15100-16100
LJ w/ 3b, Timesteps 15100-16100
BFW, Timesteps 20100-21100
2.0
1.5
g(r)
1.0
0.5
0.0
0
1
radius[σ]
2
3
Figure 4.4. Radial distribution functions at approximately 10 and 160 picoseconds with a 200K, 7.5
MPa environment.
50
2.5
2.0
Radial Distribution Function
LJ w/ 3b, Timesteps 100-1100
BFW, Timesteps 100-1100
LJ, Timesteps 100-1100
1.5
g(r)
1.0
0.5
0.0
1
2
radius[σ]
3
4
Radial Distribution Function
2.5
LJ w/ 3b, Timesteps 20100-21100
BFW, Timesteps 20100-21100
LJ, Timesteps 20100-21100
2.0
1.5
g(r)
1.0
0.5
0.0
0
1
radius[σ]
2
3
Figure 4.5. Radial distribution functions at approximately 10 and 150 picoseconds with a 300K, 3.0
MPa environment.
51
4.3 Energies
The plots for the system's kinetic, potential, and total energies do not present
any surprises. Between the dierent environments, the higher kinetic energy for the
subcritical case logically follows from its higher temperature just as the lower potential
energy of the supercritical environment results from its higher density.
While the variance between potentials is again minimal, there are two trends of
interest. First, the potential energies of all three potentials seem to approach some
asymptote as the evaporation continues. As the droplet evaporates, the density of the
system approaches something a little denser than the original ambient environment
density. However, this density is still signicantly lower than liquid density, and
the AT component of LJ with AT and BFW approaches zero. Without AT, all
three potentials are essentially identical, and the asymptote is dened by the pair
potentials only based on the nal density of the system. Second, the eective LJ
and BFW match extremely well with the LJ with AT having a consistently higher
potential energy.
The rising value of total energy is a direct result of the boundary heating conditions discussed in Section 3.1.4. The increase in energy tapers o in the supercritical
case simply because the droplet has already evaporated and the system is approaching
the ambient environment temperature.
2.0
1.5
1.5
1.0
1.0
Energy[joules*10-16]
2.0
0.5
Kinetic Energy
0.0
200K, 7.5MPA, LJ w/ 3b
200K, 7.5MPa, LJ
300K, 3.0MPa, LJ w/ 3b
300K, 3.0MPa, LJ
-0.5
Potential Energy
200K, 7.5MPA, LJ w/ 3b
200K, 7.5MPa, LJ
300K, 3.0MPa, LJ w/ 3b
0.5
300K, 3.0MPa, LJ
200K, 7.5MPa, BFW
300K, 3.0MPa, BFW
0.0
-0.5
200K, 7.5MPa, BFW
300K, 3.0MPa, BFW
-1.0
-1.0
0
100
200
300
400
500
600
0
700
100
200
300
400
500
600
700
Time[picoseconds]
Time[picoseconds]
2.0
1.5
Energy[joules*10-16]
Energy[joules*10-16]
52
1.0
0.5
Total Energy
0.0
200K, 7.5MPA, LJ w/ 3b
200K, 7.5MPa, LJ
300K, 3.0MPa, LJ w/ 3b
300K, 3.0MPa, LJ
-0.5
200K, 7.5MPa, BFW
300K, 3.0MPa, BFW
-1.0
0
100
200
300
400
500
600
700
Time[picoseconds]
Figure 4.6. Kinetic, potential, and total energy vs. time steps for the supercritical case.
53
4.4 System Parameters
The following provides a qualitative comparison based on temperature, density,
and surface tension. As all of these properties are functions of 3D space and time, it
becomes very dicult to display the results in a quantitative manner. The graphics
are cross sections cut through the center of the cubic simulation.
4.4.1 Temperature
Figures 4.7 and 4.8 are a compilation of cross sectional slices from all three axes
and each potential at 150 picoseconds. They demonstrate the immense dierence between supercritical and subcritical evaporation. Without the benet of surface tension
in the supercritical case, the temperature contours are clearly neither symmetric nor
the same for the dierent indices. However, even in the supercritical case, the slice
from each axis shows approximately the same amount of area for each level of the
contours visible. Although each axis is dierent, the general trends are still evident
in any single axis. The subcritical demonstrates good symmetry and consistency.
Now turning our attention the Figures 4.9 through 4.12 of temperature evolution,
the initial temperature for both environments of the LJ with AT seem colder than
the other two potentials. This is a result of the system re-equilibrating during the
rst 500 time steps or about 5 picoseconds of the simulation. Simulations for each
potential start with the exact same droplet and environment with identical positions
and velocities for every atom. However, the initial velocities and positions come from
a droplet and an environment equilibrated using a simple eective LJ potential. When
the simulation starts running with the LJ with AT potential, the droplet suddenly
has more potential energy and is no longer in equilibrium. The droplet quickly seeks
a new equilibrium which has a corresponding lower temperature. This does not occur
with the BFW, because it is so similar to the eective LJ potential.
Despite the slight decrease in droplet temperature, on the order of 2-3 degrees,
the LJ with AT clearly evaporates, in a temperature contour sense, faster than the
other two potentials. Both the eective LJ and BFW seem to parallel each other in
the rate of evaporation.
54
LJ w/ 3b
LJ
BFW
I Axis Slice
Temperature
T*=kT/ε
150 picoseconds
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
J Axis Slice
K Axis Slice
Figure 4.7. Temperature contours for all three axes at 150 picoseconds for the supercritical case,
200K, 7.5 MPa.
In the supercritical case, the eective LJ and BFW seem to indicate the same
general trends, but they clearly lead to dierent systems with dierent particle positions and velocities. This would indicate that when LJ and BFW are compared
statistically over the whole system they may produce the similar results. However,
on the microscopic level there is a clear dierence between the two. The subcritical
cases with LJ and BFW also follow the same general trends. While the systems do
appear to be dierent, the eects of surface tension makes the dierence much less
obvious.
When contrasting the temperature evolution of the subcritical and supercritical
cases, the core temperature in the supercritical increases towards the environment
55
LJ w/ 3b
LJ
BFW
I Axis Slice
Temperature
T*=kT/ε
150 picoseconds
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
J Axis Slice
K Axis Slice
Figure 4.8. Temperature contours for all three axes at 150 picoseconds for the subcritical case, 300K,
3.0 MPa.
temperature while the core in the subcritical case remains at a relatively constant
temperature while the surface evaporates away. This demonstrates why quasi-steady
models cannot predict supercritical lifetimes because the droplet core is not steady.
4.4.2 Density
Density contours are far better than temperature contours at dening a droplet's
surface. Nonetheless, they produce the same trends seen above. The LJ with AT cases
evaporate faster than the other two, and while the eective LJ and BFW produce the
same trends while developing into dierent systems with dierent particle positions
and velocities.
56
LJ w/ 3b
LJ
BFW
10 picoseconds
50 picoseconds
Temperature
T*=kT/ε
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
100 picoseconds
150 picoseconds
Figure 4.9. Temperature contours up to 150 picoseconds for the supercritical case, 200K, 7.5 MPa
57
LJ w/ 3b
LJ
BFW
200 picoseconds
250 picoseconds
Temperature
T*=kT/ε
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
300 picoseconds
350 picoseconds
Figure 4.10. Temperature contours up to 350 picoseconds for the supercritical case, 200K, 7.5 MPa
58
LJ w/ 3b
LJ
BFW
10 picoseconds
50 picoseconds
Temperature
T*=kT/ε
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
100 picoseconds
150 picoseconds
Figure 4.11. Temperature contours up to 150 picoseconds for the supercritical case, 300K, 3.0 MPa
59
LJ w/ 3b
LJ
BFW
300 picoseconds
400 picoseconds
Temperature
T*=kT/ε
0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0
500 picoseconds
600 picoseconds
Figure 4.12. Temperature contours up to 600 picoseconds for the supercritical case, 300K, 3.0 MPa
60
LJ w/ 3b
LJ
BFW
I Axis Slice
Density
ρ =ρσ
*
3
150 picoseconds
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
J Axis Slice
K Axis Slice
Figure 4.13. Density contours for all three axes at 150 picoseconds for the supercritical case, 200K,
7.5 MPa
61
LJ w/ 3b
LJ
BFW
I Axis Slice
Density
ρ =ρσ
*
3
150 picoseconds
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
J Axis Slice
K Axis Slice
Figure 4.14. Density contours for all three axes at 150 picoseconds for the subcritical case, 300K,
3.0 MPa
62
LJ w/ 3b
LJ
BFW
10 picoseconds
50 picoseconds
Density
*
3
ρ =ρσ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
100 picoseconds
150 picoseconds
Figure 4.15. Density contours up to 150 picoseconds for the supercritical case, 200K, 7.5 MPa
63
LJ w/ 3b
LJ
BFW
200 picoseconds
250 picoseconds
Density
*
3
ρ =ρσ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
300 picoseconds
350 picoseconds
Figure 4.16. Density contours up to 350 picoseconds for the supercritical case, 200K, 7.5 MPa
64
LJ w/ 3b
LJ
BFW
10 picoseconds
50 picoseconds
Density
*
3
ρ =ρσ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
100 picoseconds
150 picoseconds
Figure 4.17. Density contours up to 150 picoseconds for the subcritical case, 300K, 3.0 MPa
65
LJ w/ 3b
LJ
BFW
300 picoseconds
400 picoseconds
Density
*
3
ρ =ρσ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
500 picoseconds
600 picoseconds
Figure 4.18. Density contours up to 600 picoseconds for the subcritical case, 300K, 3.0 MPa
66
4.5 Regression Rates
The most denitive measurement of droplet evaporation is a density based regression rate. These regression rates were determined by counting the number of
atoms that are present in densities greater than the chosen droplet density throughout the system. The number, n 23 , is proportional to the radius of a spherical droplet
and is plotted in Figure 4.19.
In both supercritical and subcritical cases the LJ with AT potential evaporates
the droplet faster which is in line with previous observations. This is a direct result of
a shallow potential. As seen in Figure 2.3, the eective LJ potential has a decreased
or raised well depth to compensate for the presence of many body forces normally
found at liquid densities. When this is combined with AT potential, the well is raised
even more. As a result the LJ with AT more potential energy and needs less kinetic
energy to decrease the density of the droplet. In other words, the strength of the glue
that holds the droplet together is articially reduced when an eective LJ is combined
with AT potential.
Continuing previous trends, the BFW and eective LJ potentials are extremely
similar. The supercritical case is almost a perfect match, and the subcritical case is
o by less than 10% and often better. The greater divergence in the subcritical case
can be explained by the density of the environment. The eective LJ is attuned to
liquid densities. In the supercritical case both the droplet and the environment are
at typical liquid densities, but the subcritical environment has a signicantly lower
density. As such, the eective LJ fails to model the less dense subcritical environment
as well as the BFW potential. The AT component of BFW simply becomes negligible
at the lower densities and the forces are simply a result of the BFW pair potential
only.
Using the method for determining evaporation coecients from the regression
rate plots and a value for the evaporation coecient derived from the d2 law found in
Little 30], the regression rates for each potential can be compared with the analytical
techniques for the subcritical case. The regression slopes are a linear curve t of the
data in Figure 4.19. While both the LJ with AT and BFW potentials are closer
to the d2 law prediction than the LJ, this is not conclusive due to the error in the
67
Source
d2 Law 30]
LJ
LJ w/ AT
BFW
d 2=3
dt n
#2=3 /s]
;:54 1012
;:77 1012
;:63 1012
d 2
dt d
m2/s] Percent Dierence
;1:8 10;7
;1:4 10;7
-22%
;1:9 10;7
5.6%
;1:6 10;7
-11%
Table 4.2. Regression rates for the subcritical case compared with d2 law prediction.
determination of the d2 evaporation coecient as indicated by Little. Nonetheless,
it provides a method to gauge the signicance of dierent potentials on evaporation
rates.
68
350
Supercritical Environment
Regression Rates
300
200K, 7.5MPa, LJ w/ 3b
200K, 7.5MPa, LJ
200K, 7.5MPa, BFW
250
ρcut = 1.0e28 atoms/m3
200
n
2/3
150
100
50
0
0
100
200
300
400
500
600
Time[picoseconds]
350
Subcritical Environment
Regression Rates
300
300K, 3.0MPa, LJ w/ 3b
300K, 3.0MPa, LJ
300K, 3.0MPa, BFW
250
ρcut = 1.0e28 atoms/m
3
200
n
2/3
150
100
50
0
0
100
200
300
400
500
600
Time[picoseconds]
Figure 4.19. Regression rates for both environments and dierent potentials.
69
Chapter 5
Conclusions and Future Work
5.1 Conclusions
many-body forces are clearly signicant in argon at liquid densities. As such,
any MD simulations involving liquid densities or greater must account for these forces.
Two methods for doing such are to use an eective pairwise potential or to include
many-body potentials in the calculations. The results of using both methods on
supercritical and subcritical argon droplet evaporation has demonstrated that given
the right choice of potential parameters both methods produce statistically similar
results with dierent physical systems. However, eective potentials are accurate only
under conditions of similar density and temperature as the data from which they are
determined. While the accurate many-body potentials can function properly under a
broad range of densities and temperatures, their implementation requires an order of
magnitude increase in computing power.
The three potentials studied here were an eective Lennard-Jones (LJ) 12-6 pair
potential, the same Lennard-Jones potential used on combination with the AxilrodTeller (AT) triple dipole potential, and the Barker-Fisher-Watts (BFW) potential
which includes the Axilrod-Teller potential. Under supercritical conditions the eective LJ and the BFW produce very similar results. In the subcritical case they were
similar but not identical. This deviation is due to the fact that the eective LJ does
not model the less dense subcritical environment properly.
In both cases the LJ with AT potential originally used by Haile evaporated
the droplet faster than either of the other two. The author believes this is a direct
result of an articially shallow potential resulting from the arbitrary combination of
the two potentials. Looking at Figure 2.3, it is apparent that the well depth of the
eective LJ potential is less than the depth of the pairwise component of the BFW
potential. While it is hard to quantify the contributions of the AT potential to a
system, it has been shown that its net eect is to raise the potential energy of the
70
system. To compensate for net increase of potential energy from many-body forces,
the well depth of a pairwise potential would need to be raised as with the eective
LJ potential. However, the combination of an eective LJ with AT duplicates the
many-body contribution twice over. The result is a excessive decrease in the depth of
the energy potential. This means that less kinetic energy is required to overcome the
articially low potential that holds the droplet together, and the droplet evaporates
faster.
While the eective LJ and the BWF potentials produce the same system wide
characteristics and trends, they do not result in physically identical systems. The
eective potential does not reproduce true physical motion of molecules in the system
because it cannot duplicate the geometric dependence of three body forces as seen
in Figures 3.2 and 3.3. When averaged over numerous particles in a large system,
the eective potential is statistically accurate. Therefore, if the simulation goal is
to determine system wide characteristics such as regression rate, the eective LJ is
suitable. However, if one is interested in localized transport phenomena or physical
aspects such as droplet surface area, the eective LJ may not be suitable and the
BFW potential should be used. An additional limitation of the eective LJ potential
is its attenuation to a specic density. In simulations with large variations in density,
it would be prudent to use BFW.
5.1.1 Future Work
The evaporation of systems using the eective LJ and BFW potentials has been
observed to produce similar system characteristics but dierent particle positions and
velocities. However, the extent and importance of these dierences has not been quantied. Any future work that seeks to extract localized physical characteristics from
macroscopic MD simulations should consider this and try to evaluate the importance
of many-body forces.
Another direction of interest is the eect of potentials on simulations of much
higher temperatures. While higher temperatures should not aect the importance of
the long range, many-body forces focused on in this work, choosing the correct potential will still be an issue. Supercritical gases at liquid densities with high temperatures
71
would involve more collisions of much higher velocity atoms. The importance of the
poorly understood repulsive, short range force might increase. As the r112 term of
the LJ has no basis in theory, the LJ potential may fail to correctly model such an
environment. The complex repulsive term in the BFW potential or a similar potential
may be necessary for MD simulations of this quality.
In hindsight, there are a number of improvements that can be made to the
three body codes used in this study. These improvements could signicantly reduce
the computational demand and make the decision to incorporate three body eects
in MD simulations easier. Little 30] continued to develop the two body code by
introducing Verlet lists in addition to the cell linked lists. Rather than sample every
atom in the current and 26 surrounding cells to determine which neighbors are with
the cuto of 2:5 , only atoms on the Verlet list are checked. The Verlet list contains
all atoms within a sphere of 2:8 which should include every atom within the cuto.
The Verlet lists are updated every 20 time steps using the cell linked lists. For the
two body codes Little reported a decrease in computation time of 50%. However,
implementation of Verlet lists in three body algorithms would have an even greater
eect by reducing the number of particles over which all triplets need to be sampled
which would be a reduction of an O(N 3 ) operation. This could result in a three body
code that is only 2 to 3 times more demanding than two body codes.
Another possible improvement of less signicance would be to take advantage
of Newton's Third law. As mentioned, most MD codes do incorporate this concept
and calculate forces for each pair only once, but our codes do not because the added
communication was not worth the decrease in computation when using particle decomposition. However, three body codes are not limited by communication costs,
and it again may be worth using Newton's third law. The gains are not as great for
three body force calculations as for two body forces, because the three dierent force
vectors are not identical. The estimated gain might be a one third reduction in actually oating point operations. Additionally, there may be a reduction in sampling of
triplets. These gains would need to be weighed against the additional overhead and
communication costs.
Finally, an alternative approach to accounting for many-body forces might be
72
to modify the eective LJ potential. Its greatest weakness is its failure to accurately
model a wide range of densities. However, a simple solution would make the well
depth, , a function of local density. Summing the number of atoms within each
atom's cuto would be a trivial addition to the force algorithm. This local number
density from the previous time step could be used to determine the value of that is
appropriate for the force calculations. As no localized physical phenomena of interest
should be less then 5 to 10 , this may overcome the density limitations of eective
potentials.
73
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