ADSORPTION ISOTHERMS discontinuous jumps: layering transitions bilayer condensation monolayer condensation coexistence pressure some layering transitions

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

SV
 
SL
= 
LV q > 0 q = 0

Y ( r s
)
Y ( r s
) = Q ( r s
)

if there exists e such that there is a wetting transition, this is of 2 nd order r s
Y ( r s
)

PARTIAL
WETTING
T<T
W
PARTIAL
WETTING
T<T
W
COMPLETE
WETTING
T=T
W
COMPLETE
WETTING
T>T
W

r
V r s  d r
Y ( r
)
= r
V r s  d r

r  e

=

2
 r s
2  r
V
2


 r s
 r
V

= 
0
( r s
)
 
0
( r
V
) area under curve

contribution from hard interaction contribution from attractive interaction
(with correlations = step function)

a
0, b
| a
|, l
< l s a
0, b
>0, l s
< l a(T
W
)
=0

Adsorption isotherms: Langmuir's model
Kr adsorbed on exfoliated graphite at T=77.3K
N s adatoms
e s binding energy
N adsorption sites ( N > N s
)
Distinguishable, non-interacting particles
Vapour sector
The partition function is:
Z
N
=  i e
 
E i
=
N s
!
(
N !
N

N s
)!
e
 
N e s
Using Stirling's approx., the free energy is:
F
=  kT log Z
N
= 
N e s

N s kT
 q log q  q =
N s
/ N coverage
( 1
 q
) log( 1
 q
)


Chemical potential of the film:
 f
=

F

N
N s
, T
=

F
 q d q
N s
, T dN
=  e s
 kT log q
1
 q
Film and bulk vapour are in equilibrium:
 e s
 kT log q
1
 q
=  kT log

 kT p
 3

 p
*  p
 3 kT
= q e
 e s
1
 q
At low coverage p
* = q e
 e s

1
 q 
...

 q e
 e s linear for low q
(Henry's law)
This allows for an estimation of adsorption energies e s measuring the p q slope by

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
E s
H
= 
N e s
 i
N 
=
1
The free energy is: p i
2
2 m
F
= 
N


 e s
 kT log



Ae
N
 2 





Z
N
=
1
N !


Ae
e s
 2


N
Linear regime: has to do
A = surface area with absence of interactions
Again, calculating to
 f and equating
 of the (ideal) bulk gas: p
* =  2 ne
 e s n
=
N / A
 q
(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 three energy scales :
• adsorption energy
• adatom interaction
• kT (entropy) commensurate monolayer
3

3

30 º incommensurate monolayer
(also called floating phase)

Specht et al. (1984)
Kr/graphite

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 n k

, s
=
( k

, s ) kT

 k

, s population of phonons with frequency mode with force constant f k

, s
= m
 k , s

 k , s
1
2 f k

, s x k
2
, s
= n  k , s

 k

, s
= kT

 k

, s

 k

, s
The total mean displacement is
= kT x
2 = kT m

2 

1
, s
 a

1 d x k


L

1

= g (

)
2
2 kT m

2
 k , s

Using the Debye approximation for the density of states: g (

)






2
,
, 3 D
2 D
The mean square displacement when L goes to infinity is x
2 = kT m


1
2  d
 g (

)

2


 a

1 
L

1
L log a

 const

, 3 D
2 D
Therefore, the periodic crystal structure vanishes in the thermodynamic limit
However, the divergence in < x 2 > 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

Freely-rotating 2D spins 
J
 s i
 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 1 D
L
2
( 2

/ L ) in 2 D
L
3
( 2

/ L ) in 3 D 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 topological charge = +1 antivortex 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
 q =
2

2
 r
=
1
, r
J
2
 q
2
 
=

J r
The total energy is
E v
= 
L a dr

J r
= 
J log
L a
The free energy is
F v
=
E v

TS v
= 
J log
L
 kT log a lattice parameter
L a
2
=
 
J

2 kT
 log
L a the vortex centre can be located at ( L/a ) 2 different sites

When F v
= 0 vortex will proliferate:
Vortices interact as
 =  
Kv i v j log r ij a kT c
J
=

2
=
1 .
571 ...
Vortices of same vorticity attract each other
Vortices of different vorticity repel each other
But one has to also consider bound vortex pairs
-1 +1
They do not disrupt order at long distances
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>T c there is a disordered phase, with free vortices and free bound vortex pairs s
 i
 s
 j
 e
 r ij
/
    for T

T c

• For
T<T c
 s i
• For
T=T c there is QLRO (bound vortex pairs)
 s
 j
 r
 
( T )  
1
4 for T there is a continuous phase transition

T c

K renormalises to a universal limiting value and then drops to zero

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: g
6
 r

 r

'

= e
6 i
 q
( r

)
 q
( r

' )


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

Dislocations
Burgers vector