Post on 19-Dec-2015
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