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

6.  = 0.45 0.60

7. two-phase region liquid-vapour transition of monolayer two-phase region

8. at two-phase coexistence

11.  

14. Y(s)Y(s) Y (  s ) = Q (  s )

15. if there exists  such that there is a wetting transition, this is of 2 nd order Y(s)Y(s)

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

17. area under curve

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

27. a  b  a  < s a(T W )  a  b  s <

28. Adsorption isotherms: Langmuir's model Kr adsorbed on exfoliated graphite at T=77.3K Vapour sector N s adatoms  s binding energy N adsorption sites (N > N s ) Distinguishable, non-interacting particles The partition function is: Using Stirling's approx., the free energy is: coverage

29. Chemical potential of the film: At low coverage Film and bulk vapour are in equilibrium: linear for low  (Henry's law) This allows for an estimation of adsorption energies  s by measuring the p-  slope

30. Langmuir considers no mobility Fowler and Guggenheim neglect xy localisation, consider full mobility (localisation only in z) and again no adatom interaction A = surface area The free energy is: Again, calculating  f and equating to  of the (ideal) bulk gas: Fowler and Guggenheim's model (two-dimensional density) Linear regime: has to do with absence of interactions EsEs

31. 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

32. Multilayer condensation in the liquid regime ellipsometric adsorption measurements of pentane on graphite Kruchten et al. (2005) two-phase regions 2D critical points

33. Full phase diagram of a monolayer Periodic quasi-2D solid Commensurate or incommensurate?

34. Ar/graphite (Migone et al. (1984) incommensurate solid

35. 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) Kr/graphite

36. Specht et al. (1984)

37. 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 population of phonons with frequency mode with force constant The total mean displacement is

38. 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 is weak: in order to have, L has to be astronomical! This is for the harmonic solid; there are more general proofs though

39. 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 T>0 Consider a spin-wave excitation: The energy is: 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)

40. 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

41. vortex topological charge = +1 antivortex topological charge = -1

42. 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

43. When F v = 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 +1 They do not disrupt order at long distances Easy to excite Screen vortex interactions

44. 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 For T<T c there is QLRO (bound vortex pairs) For T=T c there is a continuous phase transition K renormalises to a universal limiting value and then drops to zero

45. 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

46. 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

47. Burgers vector Dislocations increasing T