Exploiting the mapping between a binary mixture and the Ising model we have analyzed the critical fluctuations by means of the density-matrix renormalization group technique. The calculations have been carried out for a two-dimensional Ising strip subject to equal strong surface fields. It was found that the critical Casimir force displays significantly different behavior on opposite sides of the capillary condensation line, especially below the critical temperature. It can be concluded that in a real binary mixtures the most attractive force appears at temperatures near T_C and at reservoir compositions slightly away from the critical composition.
Exploiting the mapping between a binary mixture and the Ising model we have analyzed the critical fluctuations by means of the density-matrix renormalization group technique. The calculations have been carried out for a two-dimensional Ising strip subject to equal strong surface fields. It was found that the critical Casimir force displays significantly different behavior on opposite sides of the capillary condensation line, especially below the critical temperature. It can be concluded that in a real binary mixtures the most attractive force appears at temperatures near T_C and at reservoir compositions slightly away from the critical composition.
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.
Using the quasi-exact density-matrix renormalization-group method we calculate the solvation forces in two-dimensional Ising films of thickness L subject to identical algebraically decaying boundary fields with various decay exponents p. At the bulk critical point the solvation force acquires a universal contribution which is long-ranged in L due to the critical fluctuations, a phenomenon known as the critical Casimir effect. For p = 2, 3 and 50, we study the scaling behaviour of the solvation force along the pseudo-phase coexistence and along the critical and sub-critical isotherms.
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 films $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ 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) ${H}_{1}{=H}_{L}=\ensuremath{\infty}$ (normal transition) by the density-matrix renormalization-group method. Universal scaling functions for magnetization profiles, the excess magnetization $\ensuremath{\Gamma},$ the longitudinal correlation length ${\ensuremath{\xi}}_{\ensuremath{\Vert}},$ 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 $y=\mathrm{sgn}(H)L|H{|}^{\ensuremath{\nu}/\ensuremath{\Delta}}\ensuremath{\approx}\ifmmode\pm\else\textpm\fi{}1$ 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.
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.
A scaling theory and a related conjecture presented recently by de Gennes and Fisher to predict the effects of walls inserted in two near-critical binary fluid mixtures, are checked theoretically by exact analytic calculations for $n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, two-dimensional Ising model strips with a surface field, h1, imposed on the first layer; exact calculations with a second surface field, hn, imposed on the nth layer are also reported. It is verified, in particular, that the effects on properties observed close to one wall of a second wall at distance D decay, at the critical point, as Dd1, where d is the spatial dimensionality. In addition to the scaling limits, the leading corrections are calculated explicitly and presented graphically.