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.
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.
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 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.
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.
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.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
The solvation force for two‐dimensional Ising stripes in the presence of boundary fields is calculated via exact diagonalization of the transfer matrix in two cases: the symmetric case corresponds to identical boundary fields and the antisymmetric case to exactly opposite boundary fields. In the symmetric case the solvation force is always negative (attractive), while in the antisymmetric case it is positive (repulsive) at high temperatures and negative at low temperatures. It changes sign close to the critical wetting temperature characterizing the semi‐infinite system. The properties of the solvation force are discussed, and the scaling function describing its dependence on temperature, boundary field, and stripe’s width is proposed.