Critical depletion of fluids in pores: Competing bulk and surface fields

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.

Multilayer approximation for a confined fluid in a slit pore

A simple Lennard-Jones fluid confined in a slit nanopore with hard walls is studied on the basis of a multilayer structured model. Each layer is homogeneous and parallel to the walls of the pore. The Helmholtz energy of this system is constructed following van der Waals-like approximations, with the advantage that the model geometry permits to obtain analytical expressions for the integrals involved. Being the multilayer system in thermodynamic equilibrium, a system of non-linear equations is obtained for the densities and widths of the layers. A numerical solution of the equations gives the density profile and the longitudinal pressures. The results are compared with Monte Carlo simulations and with experimental data for Nitrogen, showing very good agreement.Received: 23 December 2009, Accepted: 24 February 2010; Edited by: D. A. Stariolo; DOI: 10.4279/PIP.020002

Interplay of complete wetting, critical adsorption, and capillary condensation

The excess adsorption Gamma in two-dimensional Ising strips (infinityxL), subject to identical boundary fields at both one-dimensional surfaces decaying in the orthogonal direction j as -h1j(-p), is studied for various values of p and along various thermodynamic paths below the bulk critical point by means of the density-matrix renormalization-group method. The crossover behavior between the complete-wetting and critical-adsorption regimes, occurring in semi-infinite systems, is strongly influenced by confinement effects. Along isotherms T=const the asymptotic power-law dependences on the external bulk field, which characterize these two regimes, are pre-empted by capillary condensation. Along the pseudo-first-order phase-coexistence line of the strips, which varies with temperature, we find a broad crossover regime in which both the thickness of the wetting film and Gamma increase as functions of the reduced temperature tau but do not follow any power law. Above the wetting temperature the order-parameter profiles are not slablike but exhibit wide interfacial variations and pronounced tails.

Nonmonotonic crossover from adsorption to desorption in supercritical fluid near a weakly attractive surface

The density profiles of a supercritical Lennard-Jones fluid near a weakly attractive surface are used to study the excess adsorption Gamma of supercritical fluid in a wide density range. We report the observation of two extrema of Gamma along the isotherm as a function of density. The attractive fluid-wall interaction potential tends to make Gamma positive, whereas the missing neighbor effect tends to make Gamma negative. The latter effect enhances with increasing fluid density due to the growth of the fluid-fluid attraction that results in the crossover from adsorption to depletion (Gamma=0) at some particular fluid density. With approaching the critical point, Gamma decreases as the bulk correlation length and passes through a minimum when the average density of confined fluid is close to the bulk critical density. Variation of Gamma with temperature and density is determined by the monotonic trend from adsorption to desorption upon increasing fluid density and by the increase of the absolute value of Gamma when approaching the critical point. Interplay of these trends results in two extrema of Gamma in the temperature-density plane. This effect may be responsible for the crossover from adsorption to desorption with approaching the critical temperature observed experimentally along the isotherms and along the critical isochore. The density profiles of a supercritical fluid are found to be exponential and the power law behavior predicted theoretically was not detected even when the bulk correlation length xi achieves 11 molecular diameters.

Off-critical Casimir effect in Ising slabs with symmetric boundary conditions in d=3

Extended de Gennes-Fisher (EdGF) local-functional method has been applied to the thermodynamic Casimir effect away from the critical point for systems in the Ising universality class confined between parallel plane plates with symmetric boundary conditions [denoted (ab)=(++)]. Results on the universal scaling functions of the Casimir force W++(y) (y is a temperature-dependent scaling variable) and Gibbs adsorption G[over ](y) are presented in spatial dimension d=3. Also, the mean-field form of the universal scaling function of the Gibbs adsorption G[over ](y) is derived within the local functional theory. Asymptotic behavior of W++(y) for large values of the scaling variable y is analyzed in general dimension d.

Simplified crossover droplet model for adsorption of pure fluids in slit pores

We present a generalized crossover (GC) model for the excess adsorption of pure fluids at a flat solid-liquid interface, which reproduces scaling behavior of the excess adsorption in the critical region and is reduced to the classical, van der Waals-type analytical model far away from the bulk critical point. In developing this model, we used the density-functional theory (DFT) approach for the order parameter profile calculations with a generalized corresponding states model for the local free-energy density. The GC DFT model well represents the available experimental adsorption data for Kr/graphite, C2H4/graphite, C3H8/graphite, CO2/silica, and SF6/graphite systems in the entire density range 0 < rho < or = 3rhoc and temperatures up to 1.7Tc. In the critical region 0.5 rhoc < r < or = 1.5rhoc and T < or = 1.15Tc, the GC DFT model is consistent with the predictions of the asymptotic renormalization-group crossover model for the critical adsorption in a semi-infinite system developed earlier. For the excess adsorption on the critical isochore, both theories predict a scaling-law behavior Gamma proportional tau(-nu+beta), but fail to reproduce a "critical depletion" of the excess adsorption along the critical isochore of the SF6/graphite system near Tc. We show that an anomalous decrease of adsorption observed in this system at tau = T/Tc - 1 < 10(-2) can be explained by finite-size effect and develop a simplified crossover droplet (SCD) model for the excess adsorption in a slit pore. With the effective size of the pore of L = 50 nm, the SCD model reproduces all available experimental data for SF6/graphite, including the critical isochore data where tau-->0, within experimental accuracy. At L >> xib (where xib is a bulk correlation length) the SCD model is transformed into the GC DFT model for semi-infinite systems. Application of the SCD model to the excess adsorption of carbon dioxide on the silica gel is also discussed.

Critical adsorption in a well-defined geometry

A fluid's density profile near a wall is predicted to assume a universal shape near the liquid-vapor critical point, a phenomenon termed critical adsorption. This universal shape is predicted to depend on the boundary conditions of the fluid at the walls and is predicted to be a function of the ratio z/xi, where z is the distance from the wall and xi is the bulk correlation length. A body of evidence confirms the analogous phenomenon of critical adsorption in binary fluids near the critical demixing point, but in the simple liquid-vapor system the experimental situation is not as clear. For example, critical adsorption of SF6 was observed in porous glass for reduced temperature t=T/T(c)-1>10(-3). However, for t<10(-3) a desorption behavior is seen. This desorption has so far resisted rigorous theoretical explanation. We report measurements of the critical adsorption of nitrogen inside a capacitor gap with a simple parallel plate geometry and open gap of 3 microm. Unlike the previous experiments with SF6, the data show a monotonic increase in the adsorption between t=5 x 10(-4) and t=10(-6), consistent with theoretical prediction and without any indication of desorption.

Near-critical confined fluids and Ising films: density-matrix renormalization-group study

Two-dimensional Ising strips subject to identical surface fields h(1)=h(2) > or =0 are studied for temperatures above and below the bulk critical temperature T(c) and a range of bulk fields h by means of the density-matrix renormalization-group method. In the case of nonvanishing surface fields, the near-critical behavior of the solvation force f(solv), total adsorption Gamma, inverse longitudinal correlation length xi(parallel)( -1) and specific heat C(H) is strongly influenced by the (pseudo) capillary condensation that occurs below T(c). We obtain scaling functions of f(solv), Gamma, and xi(parallel)(-1). C(H) exhibits a weakly rounded singularity on crossing the pseudocoexistence line. We contrast these results with those for the case of free boundaries where, for temperatures slightly below T(c), f(solv) and C(H) exhibit a sharp extremum away from h=0. Our results have direct repercussions for the properties of near-critical Ising films in three dimensions and we argue that the long-ranged solvation (Casimir) force in confined fluids should be more attractive in the neighborhood of the capillary critical point than exactly at the bulk critical point.

Crossover between ordinary and normal transitions in the presence of a bulk field

We investigate two-dimensional Ising films at the critical temperature T(c) and nonzero bulk magnetic field h using the density-matrix renormalization-group method. The crossover between ordinary (h(1)=0) and normal (h(1)=infinity) transitions corresponding to finite values of the surface fields h(1)=h(2), is studied. The structure and the solvation force f(solv) as a function of h, crucially depend on the value of h(1). Scaling functions for f(solv) and the longitudinal correlation length are given and discussed.

Effect of bulk magnetic field on critical Ising films

Two-dimensional Ising films L x infinity in a nonvanishing bulk magnetic field H are studied at the bulk critical temperature Tc for two choices of surface fields (a) H1 = HL = 0 (ordinary transition), and (b) H1 = HL = infinity (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization gamma, the longitudinal correlation length xi parallel, and for the analog of the solvation force fsolv are found and discussed. When H1 = 0 the scaling function for fsolv has two symmetric minima at y = sgn(H)L magnitude of H nu/delta approximately +/- 1 with an amplitude at the minimum about 3.8 times the value at H = 0, the Casimir amplitude. For the normal transition the scaling function for fsolv has a single minimum near the continuation of the pseudocoexistence (capillary condensation) line, with an amplitude about 100 times the Casimir amplitude.

Effects of confinement on critical adsorption: absence of critical depletion for fluids in slit pores

The adsorption of a near-critical fluid confined in a slit pore is investigated by means of density functional theory and by Monte Carlo simulation for a Lennard-Jones fluid. Our work was stimulated by recent experiments for SF6 adsorbed in a mesoporous glass, which showed the striking phenomenon of critical depletion, i.e., the adsorption excess Gamma first increases but then decreases very rapidly to negative values as the bulk critical temperature T(c) is approached from above along near-critical isochores. By contrast, our density functional and simulation results, for a range of strongly attractive wall-fluid potentials, show Gamma monotonically increasing and eventually saturating as the temperature is lowered toward T(c) along both the critical (rho=rho(c)) and subcritical isochores (rho<rho(c)). Such behavior results from the increasingly slow decay of the density profile away from the walls, into the middle of the slit, as T-->T(+)(c). For rho<rho(c) we find that in the fluid the effective bulk field, which is negative and which favors desorption, is insufficient to dominate the effects of the surface fields which favor adsorption. We compare this situation with earlier results for the lattice gas model with a constant (negative) bulk field where critical depletion was found. A qualitatively different behavior of the density profiles and adsorption is found in simulations for intermediate and weakly attractive wall-fluid potentials, but in no case do we observe the critical depletion found in experiments. We conclude that the latter cannot be accounted for by a single pore model.