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arXiv:0808.2966 · Physical Review E

Interplay of complete wetting, critical adsorption, and capillary condensation

A. Drzewiński, Anna Maciołek, Artur Barasiński, S. Dietrich

Published Apr 28, 2009• 28 citations
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Abstract

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.

DOI
10.1103/physreve.79.041145

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Bulk critical exponents in a presence of the capillary condensation

A. Drzewiński, Katarzyna Szota

The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found. Introduction In the thermodynamic limit of the two-dimensional system the correlation length and some other quantities diverge at the critical point (T = Tc, H = 0) [1]. One of them is magnetization that vanishes above Tc in zero bulk field (the paramagnetic regime). Phase coexistence occurs for temperatures T T). It results in a jump of the magnetization m = –∂f/∂H at H = 0, whereas the A. Drzewinski, K. Szota 30 limit value mb = m (T = const, H → 0) is finite and named the spontaneous magnetization. In the other words, when the thermodynamic path for fixed but very small H (the dashed arrow at Fig. 1) is closer and closer to the H = 0 axis, then the magnetization resembles more and more the spontaneous magnetization (full dots in Fig. 2). Fig. 1. Numerically determined coexistence curves for finite systems in the (T, H) phase diagram. The thick solid line indicates the bulk coexistence line. The dashed arrow denotes the thermodynamic path for the infinite systems but others for finite systems Fig. 2. Spontaneous magnetization mb as a function of temperature along the thermodynamic paths presented in the Fig. 1 with a constant shift –10 away from the coexistence lines There is an interesting question. What happens when the system is confined in the presence of the boundary fields. Is it possible to apply the same procedure and, exploiting the magnetizations along the (curvilinear) coexistence lines, reconstruct the behaviour of the spontaneous magnetization? Bulk critical exponents in a presence of the capillary condensation 31 1. The model and the method Our model Hamiltonian for the infinitely long strip has the form:         − − ∑ ∑ ∑ ∑ l k, l k, k L k, 1 k k l' , k' l k, σ H + σ h + σ h σ σ J = H ,1 1 where the first sum is over all nearest-neighbour pairs and we have measured the surface field h1 and the bulk field H in units of J. In order to find out the free energy f and the longitudinal correlation length ξ|| we have used the density-matrix renormalization group (DMRG) [3]. It was originally proposed by White as a new tool for diagonalization of quantum spin chains [4] and was later adapted to classical two-dimensional equilibrium statistical mechanics by Nishino [5]. The DMRG allows one to study much larger systems (up to L = 650 in the present paper) than is possible with standard exact diagonalization methods (typically L ~50) and provides data with remarkable accuracy. We estimate that the errors in the plots are smaller than the symbol size.

Journal of Applied Mathematics and Computational Mechanics•Jan 1, 2008•0 citations

The wetting layers For long-range wall-particle potentials

Artur Barasiński, A. Drzewiński

The complete wetting in 2D Ising strips subject to identical surface fields decaying as h1z −p is studied by means of the density-matrix renormalization-group technique. Using different criteria the thickness of a layer is determined along some isotherms above the wetting temperature. It is found that magnetization profiles are characterized by wide interfacial regime.

Journal of Applied Mathematics and Computational Mechanics•Jan 1, 2007•0 citations
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.
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.
We examine critical adsorption for semi-infinite thermodynamic systems of the Ising universality class when they are in contact with a wall of the so-called normal surface universality class in spatial dimension d=3 and in the mean-field limit. We apply local-functional theory and Monte Carlo simulations in order to quantitatively determine the properties of the energy density as the primary scaling density characterizing the critical behaviors of Ising systems besides the order parameter. Our results apply to the critical isochore, near two-phase coexistence, and along the critical isotherm if the surface and the weak bulk magnetic fields are either collinear or anticollinear. In the latter case, we also consider the order parameter, which so far has yet to be examined along these lines. We find the interface between the surface and the bulk phases at macroscopic distances from the surface, i.e., the surface is "wet." It turns out that in this case the usual property of monotonicity of primary scaling densities with respect to the temperature or magnetic field scaling variable does not hold for the energy density due to the presence of this interface.
Effective Hamiltonian models predict nonuniversal critical singularities for two-dimensional wetting transitions with marginal long-ranged forces. We test these predictions by studying interfacial delocalization transitions in an infinitely long Ising strip, of width L (lattice spacings), with external fields that are long ranged and have opposite signs at each surface. Finite-size scaling suggests that the shift of the delocalization temperature T(c)(L) below the (semi-infinite) wetting temperature T(w) scales as L(-1/beta(s)) with beta(s) the adsorption critical exponent. Density-matrix renormalization-group methods allow us to study the behavior of T(c)(L) for L up to several hundred lattice spacings. For short-ranged forces the method recovers the universal value of beta(s)=1 known from the exact solution. While marginal long-ranged forces strongly influence the finite-size scaling of T(c)(L) , the extrapolated asymptotic value for the exponent beta(s) does not appear to confirm the predicted nonuniversality, but instead approaches the same universal value representative of systems with short-ranged forces.
We investigate two dimensional critical Ising films of width L with surface fields H(1)=H(L) in the crossover between ordinary (H(1)=0) and normal (H(1)=infinity) transitions. Using exact transfer-matrix diagonalization and density matrix renormalization-group (DMRG) methods, we calculate magnetization profiles m(z), the excess magnetization Gamma, and the analog of the solvation force f(solv) as functions of H1 for several L. Scaling functions of the above quantities deviate substantially from their asymptotic forms at fixed points for a broad region of the scaling variable LH21 approximately L/l(1), where l(1) is the length induced by the surface field H1. The scaling function for /f(solv)/ has a deep minimum near LH(2)(1)=1, which is about one order of magnitude smaller than its value at both fixed points (the "Casimir" amplitude). For weak H1 (l(1)>L) the magnetization profile has a maximum at the center of the film, and f(solv) decays much faster than L-2. For stronger H1 (1<l(1)<L), the magnetization has two maxima at a distance approximately l(1) from the walls, and the solvation force decays much slower than L-2. For L>>l(1) the solvation force decays according to the universal power law f(solv) approximately L(-2). The results of the approximate DMRG method show remarkable agreement with the exact ones.
The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found. Introduction In the thermodynamic limit of the two-dimensional system the correlation length and some other quantities diverge at the critical point (T = Tc, H = 0) [1]. One of them is magnetization that vanishes above Tc in zero bulk field (the paramagnetic regime). Phase coexistence occurs for temperatures T T). It results in a jump of the magnetization m = –∂f/∂H at H = 0, whereas the A. Drzewinski, K. Szota 30 limit value mb = m (T = const, H → 0) is finite and named the spontaneous magnetization. In the other words, when the thermodynamic path for fixed but very small H (the dashed arrow at Fig. 1) is closer and closer to the H = 0 axis, then the magnetization resembles more and more the spontaneous magnetization (full dots in Fig. 2). Fig. 1. Numerically determined coexistence curves for finite systems in the (T, H) phase diagram. The thick solid line indicates the bulk coexistence line. The dashed arrow denotes the thermodynamic path for the infinite systems but others for finite systems Fig. 2. Spontaneous magnetization mb as a function of temperature along the thermodynamic paths presented in the Fig. 1 with a constant shift –10 away from the coexistence lines There is an interesting question. What happens when the system is confined in the presence of the boundary fields. Is it possible to apply the same procedure and, exploiting the magnetizations along the (curvilinear) coexistence lines, reconstruct the behaviour of the spontaneous magnetization? Bulk critical exponents in a presence of the capillary condensation 31 1. The model and the method Our model Hamiltonian for the infinitely long strip has the form:         − − ∑ ∑ ∑ ∑ l k, l k, k L k, 1 k k l' , k' l k, σ H + σ h + σ h σ σ J = H ,1 1 where the first sum is over all nearest-neighbour pairs and we have measured the surface field h1 and the bulk field H in units of J. In order to find out the free energy f and the longitudinal correlation length ξ|| we have used the density-matrix renormalization group (DMRG) [3]. It was originally proposed by White as a new tool for diagonalization of quantum spin chains [4] and was later adapted to classical two-dimensional equilibrium statistical mechanics by Nishino [5]. The DMRG allows one to study much larger systems (up to L = 650 in the present paper) than is possible with standard exact diagonalization methods (typically L ~50) and provides data with remarkable accuracy. We estimate that the errors in the plots are smaller than the symbol size.