Critical behavior of surface-layer magnetization at bulk Tc: Extraordinary transition

Critical adsorption and critical Casimir forces in the canonical ensemble

Critical properties of a liquid film between two planar walls are investigated in the canonical ensemble, within which the total number of fluid particles, rather than their chemical potential, is kept constant. The effect of this constraint is analyzed within mean-field theory (MFT) based on a Ginzburg-Landau free-energy functional as well as via Monte Carlo simulations of the three-dimensional Ising model with fixed total magnetization. Within MFT and for finite adsorption strengths at the walls, the thermodynamic properties of the film in the canonical ensemble can be mapped exactly onto a grand canonical ensemble in which the corresponding chemical potential plays the role of the Lagrange multiplier associated with the constraint. However, due to a nonintegrable divergence of the mean-field order parameter profile near a wall, the limit of infinitely strong adsorption turns out to be not well-defined within MFT, because it would necessarily violate the constraint. The critical Casimir force (CCF) acting on the two planar walls of the film is generally found to behave differently in the canonical and grand canonical ensembles. For instance, the canonical CCF in the presence of equal preferential adsorption at the two walls is found to have the opposite sign and a slower decay behavior as a function of the film thickness compared to its grand canonical counterpart. We derive the stress tensor in the canonical ensemble and find that it has the same expression as in the grand canonical case, but with the chemical potential playing the role of the Lagrange multiplier associated with the constraint. The different behavior of the CCF in the two ensembles is rationalized within MFT by showing that, for a prescribed value of the thermodynamic control parameter of the film, i.e., density or chemical potential, the film pressures are identical in the two ensembles, while the corresponding bulk pressures are not.

Calculation of the shift exponent for the two-layer three-state Potts model using the transfer matrix method

A finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature and critical exponent of the symmetric and the asymmetric two-layer three-state Potts models. For similar intralayer interactions our calculation of the shift exponent phi confirm some scaling arguments which predict phi = gamma, where gamma is the susceptibility exponent. For unequal intralayer interactions we have obtained phi = 0.5 which differ from the prediction phi = gamma/2 of a generalized mean-field theory.

Critical adsorption and Casimir torque in wedges and at ridges

Geometrical structures of confining surfaces profoundly influence the adsorption of fluids upon approaching a critical point T(c) in their bulk phase diagram, i.e., for t= (T- T(c) ) / T(c) -->+/-0 . Guided by general scaling considerations, we calculate, within the mean-field theory, the temperature dependence of the order parameter profile in a wedge with opening angle gamma<pi and close to a ridge (gamma>pi) for T>< T(c) and in the presence of surface fields. For a suitably defined reduced excess adsorption Gamma (+/-) (gamma,t-->+/-0) approximately Gamma (+/-) (gamma) mid R:tmid R:(beta-2nu) we compute the universal amplitudes Gamma (+/-) (gamma) , which diverge as Gamma (+/-) (gamma-->0) approximately 1/gamma for small opening angles, vary linearly close to gamma=pi for gamma<pi , and increase exponentially for gamma-->2pi . There is evidence that, within the mean-field theory, the ratio Gamma (+) (gamma) / Gamma (-) (gamma) is independent of gamma . We also discuss the critical Casimir torque acting on the sides of the wedge as a function of the opening angle and temperature.

Exact phase diagrams for an Ising model on a two-layer Bethe lattice

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two-layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system and find that when /J3/-->0, DeltaT(c) identical with[T(c)(J3)-T(c)(0)]/T(c)(0) approximately [J3]/J(1)/(1/psi), where T(c)(J3) is the phase-transition temperature for the system with interlayer coupling constant J3 and the shift exponent psi is 1 for J(1)=J(2) and is 0.5 for J1 not equal to J2. Such results are consistent with predictions of a scaling theory. We also derive equations for DeltaT(c) when /J3/ approaches infinity.