Diapositiva 1

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ADSORPTION ISOTHERMS
discontinuous jumps layering transitions
some layering transitions
bilayer condensation
monolayer condensation
coexistence pressure
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t 0.45
0.60
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two-phase region
two-phase region
two-phase region
two-phase region
two-phase region
liquid-vapour transition of monolayer
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at two-phase coexistence
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q gt 0
q 0
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Y(rs) Q(rs)
Y(rs)
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Y(rs)
if there exists e such that there is a wetting
transition, this is of 2nd order
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PARTIAL WETTING TltTW
PARTIAL WETTING TltTW
COMPLETE WETTING TTW
COMPLETE WETTING TgtTW
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area under curve
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contribution from hard interaction
contribution from attractive interaction
(with correlations step function)
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alt0, blta, lltls
a(TW )0
alt0, bgt0, lsltl
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Adsorption isotherms Langmuir's model
Ns adatoms -es binding energy N adsorption sites
(N gt Ns) Distinguishable, non-interacting
particles The partition function is
Kr adsorbed on exfoliated graphite at T77.3K
Vapour sector
Using Stirling's approx., the free energy is
coverage
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Chemical potential of the film
Film and bulk vapour are in equilibrium
At low coverage
linear for low q (Henry's law)
This allows for an estimation of adsorption
energies es by measuring the p-q slope
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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
Linear regime has to do with absence of
interactions
A surface area
The free energy is
Again, calculating mf and equating to m of the
(ideal) bulk gas
(two-dimensional density)
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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
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two-phase regions
2D critical points
Multilayer condensation in the liquid
regime ellipsometric adsorption measurements of
pentane on graphite Kruchten et al. (2005)
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Full phase diagram of a monolayer
Periodic quasi-2D solid Commensurate or
incommensurate?
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Ar/graphite (Migone et al. (1984)
incommensurate solid
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Kr/graphite

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

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

commensurate monolayer
incommensurate monolayer
(also called floating phase)
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Kr/graphite
Specht et al. (1984)
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Two-dimensional crystals
Absence of long-range order in 2D (Peierls,
'30) There is no true long-range order in 2D at
Tgt0 due to excitation of long wave-length phonons
with
population of phonons with frequency
mode with force constant
The total mean displacement is
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Using the Debye approximation for the density of
states
The mean square displacement when L goes to
infinity is
Therefore, the periodic crystal structure
vanishes in the thermodynamic limit However, the
divergence in ltx2gt is weak in order to have
, L has to be astronomical!
This is for the harmonic solid there are more
general proofs though
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XY model and Kosterlitz-Thouless (KT)
Freely-rotating 2D spins The ground state is a
perfectly ordered arrangement of spins But there
is no ordered state (long-range order) for
Tgt0 Consider a spin-wave excitation
The energy is
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
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• 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 n1)
• XY model superfluid films, thin superconductors,
  2D crystals (order parameter n2) only have QLRO
• Spin excitations in the XY model can be discussed
  in terms of vortices (elementary excitations),
  which destroy long-range order

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vortex topological charge 1
antivortex topological charge -1
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We calculate the free energy of a vortex The
contribution from a ring a spins situated a
distance r from the vortex centre is
The total energy is
lattice parameter
The free energy is
the vortex centre can be located at (L/a)2
different sites
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When Fv 0 vortex will proliferate
Vortices interact as
Vortices of same vorticity attract each
other Vortices of different vorticity repel each
other
But one has to also consider bound vortex pairs
They do not disrupt order at long distances Easy
to excite Screen vortex interactions
-1
1
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• 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 TgtTc there is a disordered phase, with free
  vortices and free bound vortex pairs
• For TltTc there is QLRO (bound vortex pairs)
• For TTc there is a continuous phase transition
• K renormalises to a universal limiting value
  and then drops to zero

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

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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
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increasing T
Dislocations
Burgers vector