ADSORPTION ISOTHERMS

discontinuous jumps: layering

transitions

some layering transitions

coexistence pressure

monolayer condensation

bilayer condensation

= 0.450.60

two-phase region

liquid-vapour transition of monolayer

two-phase region

two-phase region

two-phase region

two-phase region

at two-phase coexistence

LVSLSV

LVSLSV

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

if there exists such that there is a wetting transition, this is of 2nd order

Y(s)

s

COMPLETE WETTING

T=TW

COMPLETE WETTING

T>TW

PARTIAL WETTING

T<TW

PARTIAL WETTING

T<TW

area under curve )()(2

)( 0022

VsVsVs

s

V

s

V

dYd

contribution from hard interaction

contribution from attractive interaction

(with correlations = step function)

aba<s

a(TW )

abs<

Adsorption isotherms: Langmuir's model

Kr adsorbed on exfoliated graphite at T=77.3K

Vapour sector

Ns adatoms

s binding energy

N adsorption sites (N > Ns)

Distinguishable, non-interacting particles

The partition function is:

i

N

ss

EN

si eNNN

NeZ

)!(!

!

Using Stirling's approx., the free energy is:

)1log()1(loglog kTNNZkTF ssN

NN s / coverage

Chemical potential of the film:

1

log,,

kTdN

dF

N

Fs

TNTNf

ss

At low coverage

Film and bulk vapour are in equilibrium:

3

log1

logp

kTkTkTs

1

3*

se

kT

pp

ss eep ...1* linear for low (Henry's law)

This allows for an estimation of adsorption energies s by measuring the p-slope

Langmuir considers no mobility

Fowler and Guggenheim neglect xy localisation, consider full mobility (localisation only in z) and again no adatom interaction

N

i

is m

pNH

1

2

2

A = surface area

N

N

sAe

NZ

2!

1

The free energy is:

Again, calculating f and equating to of the (ideal) bulk gas:

Fowler and Guggenheim's model

2

logN

AekTNF s

snep 2*

ANn / (two-dimensional density)

Linear regime: has to do with absence of interactions

Es

Binder and Landau

Monte Carlo simulation of lattice-gas model with parameters for adsorption of H on Pd(100)

Limiting isotherm for

Corrections from 2D virial coefficients

T

Multilayer condensation in the liquid regimeellipsometric adsorption measurements of pentane on graphiteKruchten et al. (2005)

two-phase regions

2D critical points

Full phase diagram of a monolayerPeriodic quasi-2D solid

Commensurate or incommensurate?

Ar/graphite (Migone et al. (1984)

incommensurate solid

commensurate monolayer incommensurate monolayer

two length scales:• lattice parameter of graphite• adatom diameter

three energy scales:• adsorption energy• adatom interaction• kT (entropy)

(also called floating phase)º3033

Kr/graphite

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

sksk

kTn

,,

population of phonons with frequency

sk ,

),( sk

mode with force constant2

,, skskmf

kTkT

nxfsk

sksksksksk

,

,,,

2

,,2

1

22

,

,

2

skm

kTx

sk

The total mean displacement is

2

1

22 )(

g

dm

kTx

1a

1L

Using the Debye approximation for the density of states:

D

Dg

2,

3,)(

2

The mean square displacement when L goes to infinity is

Therefore, the periodic crystal structure vanishes in the thermodynamic limit

However, the divergence in <x2> is weak: in order to have , L has to be astronomical!

Da

LDLag

dm

kTx

2log

3,const)(11

22

2

1

22 ax

This is for the harmonic solid; there are more general proofs though

XY model and Kosterlitz-Thouless (KT)

jiji JssJ cos

Freely-rotating 2D spins

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:

DLL

DLL

DLL

3in )/2(

2in )/2(

1in )/2(

3

2

grows without limit: ordered state robust w.r.t. T

goes to a constant: spin wave stable and no ordered state

limiting case (in fact NO)

Even though there is no long-range order, there may exist quasi-long-range 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

r

Jr

J 22

2,1

2

2

rr

The total energy is

a

LJ

r

JdrE

L

a

v log lattice

parameterThe free energy is

a

LkTJ

a

LkT

a

LJTSEF vvv log2loglog

2

the vortex centre can be located at (L/a)2 different sites

When Fv = 0 vortex will proliferate: ...571.12

J

kTc

Vortices interact as

a

rvKv ij

ji logVortices 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>Tc there is a disordered phase, with free vortices and free bound vortex pairs

• For T<Tc there is QLRO (bound vortex pairs)

• For T=Tc there is a continuous phase transition

K renormalises to a universal limiting value and then drops to zero

/ijrji ess cTTfor

)(Tji rss cTTfor

4

1

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

• Bond-orientational order: correlations between directions of relative vectors between neighbouring atoms w.r.t. fixed crystallographic axis:

Two-dimensional melting

'rrg

)'()(66 ' rrierrg

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

Burgers vector

Dislocations

incr

easi

ng T