Critical adsorption and critical Casimir forces in the canonical ensemble

Abstract

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 threedimensional 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 non-integrable divergence of the mean field order parameter (OP) 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.

Dynamics of the critical Casimir force for a conserved order parameter after a critical quench

“…Recently, progress has been made, e.g., within molecular dynamics [93,94] and within the lattice Boltzmann method [95][96][97] towards simulation of critical dynamics and determining CCFs [98]. Moreover, these simulation methods typically realize the canonical ensemble and thus allow one to test the ensemble differences of the CCF [52,53].…”

“…Both in and out of equilibrium, the bulk pressure can be obtained from a scaling limit: 12) which is taken by keeping the relevant thermodynamic control parameters fixed (see Refs. [41,52,53] for further details concerning ensemble differences). The thermally averaged dynamical CCF K , in units of k B T , is expected to exhibit the following scaling behavior:…”

Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions

“…Generally, the bulk term scales ∝ L and the surface term ∝ L 0 , while the residual terms vanish exponentially for L → ∞ (see, e.g., Refs. [15,32]). We remark that for realistic fluid films, longranged van der Waals forces provide algebraically decaying non-universal contributions to the residual free energy [35,37,38,56].…”

“…The CCF (per transverse area A and k B T ) in the grand canonical and the canonical ensemble takes the following scaling form [2,5,15]:…”

“…In fact, CCFs are typically studied for fluid films which can exchange particles with their environment-a situation which realizes the grand canonical ensemble. However, global OP conservation, which is applicable for the canonical ensemble, can induce drastic changes of the CCF [15,16]. Hitherto, only a few studies have focused on the effect of a global OP constraint on the critical behavior [17][18][19][20][21].…”