ADSORPTION ISOTHERMS
discontinuous
jumps: layering
transitions
some layering
transitions
bilayer
condensation
monolayer
condensation
coexistence
pressure
t = 0.45
0.60
two-phase
region
two-phase
region
two-phase
region
two-phase
region
two-phase
region
liquid-vapour
transition of
monolayer
at two-phase coexistence
 SV   SL   LV
q>0
 SV   SL =  LV
q=0
Y(rs)
Y(rs) = Q(rs)
rs
Y(rs)
if there exists e
such that there
is a wetting
transition, this
is of 2nd order
PARTIAL
WETTING
T<TW
COMPLETE
WETTING
T=TW
PARTIAL
WETTING
T<TW
COMPLETE
WETTING
T>TW
rs
rs

r drY ( r ) = r dr r  e  = 2 r
V
V
2
s
 rV2   r s  rV  =  0 ( r s )   0 ( rV )
area under curve
contribution from
hard interaction
contribution from
attractive
(with correlations =
interaction
step function)
a0, b|a|, l<ls
a0, b>0, ls<l
a(TW )=0
Adsorption isotherms: Langmuir's model
Ns adatoms
Kr adsorbed on exfoliated
graphite at T=77.3K
Vapour sector
es binding energy
N adsorption sites (N > Ns)
Distinguishable, non-interacting
particles
The partition function is:
Z N =  e   Ei =
i
Using Stirling's approx., the free energy is:
N!
e  Ne s
N s !( N  N s )!
q = Ns / N
coverage
F = kT log Z N =  Ne s  N s kTq log q  (1  q ) log( 1  q )
Chemical potential of the film:
dq
q
 F 
 F 
=
=

e

kT
log



s

N

q
dN
1q

 N s ,T 
 N s ,T
f = 
Film and bulk vapour are in equilibrium:
q
 kT 
 e s  kT log
= kT log  3 
1q
 p 
p3 qe  e s
p 
=
kT
1q
*
At low coverage
p = qe
*
 e s
1  q  ...  qe
 e s
linear for low q (Henry's law)
This allows for an estimation of adsorption energies es by
measuring the p-q slope
Fowler and Guggenheim's model
Langmuir considers no mobility
Fowler and Guggenheim neglect xy localisation, consider full
mobility (localisation only in z) and again no adatom interaction
Es
N
2
i
p
H =  Ne s  
i =1 2m
The free energy is:

 Ae 
F =  N e s  kT log 
2 
 N 

n = N / A q
e s
1  Ae 
ZN =


N !  2 
N
Linear regime: has to do
A = surface area
with absence of interactions
Again, calculating f and equating
to  of the (ideal) bulk gas:
p =  ne
*
2
 e s
(two-dimensional density)
Binder and Landau
Monte Carlo simulation of lattice-gas model with parameters for
adsorption of H on Pd(100)
Limiting isotherm for T = 
Corrections from 2D virial
coefficients
two-phase regions
2D critical points
Multilayer condensation in the liquid regime
ellipsometric adsorption measurements of pentane on graphite
Kruchten et al. (2005)
Full phase diagram of a monolayer
Periodic quasi-2D solid
Commensurate or
incommensurate?
Ar/graphite (Migone et al. (1984)
incommensurate
solid
Kr/graphite
two length scales:
• lattice parameter of graphite
• adatom diameter
commensurate monolayer
3  3  30º
three energy scales:
• adsorption energy
• adatom interaction
• kT (entropy)
incommensurate monolayer
(also called floating phase)
Kr/graphite
Specht et al. (1984)
Two-dimensional crystals
Absence of long-range order in 2D (Peierls, '30)
There is no true long-range order in 2D at T>0 due to excitation
of long wave-length phonons with   kT
nk , s
kT
=
k , s
population of phonons with frequency k , s

( k , s ) mode with force constant
f k ,s = mk2,s
1 
kT
2



f k , s xk , s = nk , s k , s =
k , s = kT
2
k , s
The total mean displacement is
xk2 , s =
2
x
2
 a 1
2kT
m k2 ,s
kT
g ( )
=
d 2

m 1

 L1
Using the Debye approximation for the density of states:
 2 , 3D
g ( )  
 , 2D
The mean square displacement when L goes to infinity is
1
1

a

L
 const , 3D
kT
g ( ) 
=
d 2  
L

log  
2D
m 1


a

2
x2
Therefore, the periodic crystal structure vanishes in the
thermodynamic limit
However, the divergence in <x2> is weak: in order to have
x 2  a 2 , L has to be astronomical!
This is for the harmonic solid; there are more general proofs though
XY model and Kosterlitz-Thouless (KT)
Freely-rotating 2D spins
 
 Jsi  s j =  J cosi   j 
The ground state is a perfectly ordered arrangement of spins
But: there is no ordered state (long-range order) for T>0
Consider a spin-wave excitation:
The energy is:
L(2 / L) in 1D
L2 (2 / L) in 2 D
L3 (2 / L) in 3D
goes to a constant: spin wave
stable and no ordered state
limiting case (in fact NO)
grows without limit: ordered
state robust w.r.t. T
Even though there is no long-range order, there may exist quasi-longrange order
No true long-range order: exponentially decaying correlations
• True long-range order: correlation function goes to a constant
• Quasi-long-range order(QLRO): algebraically decaying correlations
QLRO corresponds to a critical phase
Not all 2D models have QLRO:
• 2D Ising model has true long-range order (order parameter n=1)
• XY model superfluid films, thin superconductors, 2D crystals (order
parameter n=2) only have QLRO
Spin excitations in the XY model can be discussed in terms of
vortices (elementary excitations), which destroy long-range order
vortex
antivortex
topological charge = +1
topological charge = -1
We calculate the free energy of a vortex
The contribution from a ring a spins situated a distance r from the
vortex centre is
J
J
2
q 2r  =
2
r
2 1
q =
= ,
2r r
The total energy is
L
J
L
Ev =  dr
= J log
r
a
a
The free energy is
2
lattice
parameter
L
L
L
Fv = Ev  TS v = J log  kT log   = J  2kT  log
a
a
a
the vortex centre can be located
at (L/a)2 different sites
When Fv = 0 vortex will proliferate:
kTc 
= = 1.571...
J
2
Vortices interact as
 = Kvi v j log
rij
Vortices of same vorticity attract each other
a
Vortices of different vorticity repel each other
But one has to also consider bound vortex pairs
They do not
disrupt order at
long distances
-1
+1
Easy to excite
Screen vortex
interactions
KT theory: renormalisation-group treatment of screening effects
Confirmed experimentally for 2D supefluids and superconductor
films. Also for XY model (by computer simulation)
Predictions:
• For T>Tc there is a disordered phase, with free vortices and free
bound vortex pairs
 
r /
si  s j  e ij
   for T  Tc
• For T<Tc there is QLRO (bound vortex pairs)
 
si  s j  r  (T )
1

4
for T  Tc
• For T=Tc there is a continuous phase transition
K renormalises to a universal limiting value and then drops to zero
Two-dimensional melting
The KT theory can be generalised for solids: KTHNY theory
There is a substrate. Also, there are two types of order:
• Positional order: correlations between atomic positions
 
Characterised e.g. by g  r  r ' 
• Bond-orientational order: correlations between directions of
relative vectors between neighbouring atoms w.r.t. fixed
crystallographic axis:


 
6 i q ( r ) q ( r ') 
g6  r  r '  = e
The analogue of a vortex is a a disclination
A disclination disrupts long-range positional order, but not the
bond-orientational order
In a crystal disclinations are bound in pairs, which are dislocations,
and which restore (quasi-) long-range positional order
increasing T
Dislocations
Burgers vector