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Journal of Physics A Mathematical and General

Magnetization profiles for a Ising strip with opposite surface fields

Anna Maciołek

Published Jul 21, 1996• 25 citations
Publisher

Abstract

We obtain an exact solution of the d = 2 infinite Ising strip with surface fields of opposite signs , . Earlier predictions by Parry et al are confirmed of the crucial importance of wetting for a system confined between two parallel walls that exert competitive surface fields. The transition of the magnetization profile m(z) from the partial wetting regime below the critical wetting temperature at a single wall with a surface field , to the soft-mode single phase in the temperature range , where is the critical temperature of the bulk system, is described in detail. A scaling ansatz is verified for the singular part of the surface excess free energy, and for the profile, . The modification of the magnetization profile near one wall, due to wetting, is studied. Near the (-) wall at the profile is found to be a linear function of the scaled distance but its slope is different than for the case of perfect asymmetry system hence these two cases belong to the different universality regime.

DOI
10.1088/0305-4470/29/14/011

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We obtain an exact solution of the d = 2 infinite Ising strip with surface fields of opposite signs , . Earlier predictions by Parry et al are confirmed of the crucial importance of wetting for a system confined between two parallel walls that exert competitive surface fields. The transition of the magnetization profile m(z) from the partial wetting regime below the critical wetting temperature at a single wall with a surface field , to the soft-mode single phase in the temperature range , where is the critical temperature of the bulk system, is described in detail. A scaling ansatz is verified for the singular part of the surface excess free energy, and for the profile, . The modification of the magnetization profile near one wall, due to wetting, is studied. Near the (-) wall at the profile is found to be a linear function of the scaled distance but its slope is different than for the case of perfect asymmetry system hence these two cases belong to the different universality regime.
For the title system the column-column transfer matrix is set up and diagonalized by exact calculation. The free energy and correlation length follow exactly from the eigenvalues of the auxiliary problem, whereas other quantities such as magnetization profiles, heat capacities, etc., are calculable numerically with arbitrary accuracy. The case of surface fields with the opposite signs is studied in some detail. The transition from exponentially decaying to oscillatory eigenvectors is discussed and related to wetting transitions. For the important case of the perfect asymmetry system the finite-size scaling near the wetting temperature Tw{\mathit{T}}_{\mathit{w}}Tw​ of surface tension and correlation length parallel to the walls, is derived. Also the scaling of magnetization profiles is found. \textcopyright{} 1996 The American Physical Society.
The properties of a fluid, or Ising magnet, confined in an $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ geometry with opposing surface fields at the walls, is studied by density matrix renormalization techniques. In particular we focus on the effect of gravity on the system, which is modeled by a bulk field whose strength varies linearly with the distance from the walls. It is well known that, in the absence of gravity, phase coexistence is restricted to temperatures below the wetting temperature. We find that gravity restores phase coexistence up to the bulk critical temperature, in agreement with previous mean field results. A detailed study of the scaling to the critical point, as $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty}$, is performed. The temperature shift scales as 1/LyT{1/L}^{{y}_{T}}1/LyT​, while the gravitational constant scales as 1/L1+yH{1/L}^{{1+y}_{H}}1/L1+yH​, with yT{y}_{T}yT​ and yH{y}_{H}yH​ the bulk thermal and magnetic exponents, respectively. For weak surface fields and LLL not too large, we also observe a regime where the gravitational constant scales as ${1/L}^{{1+y}_{H}\ensuremath{-}{\ensuremath{\Delta}}_{1}{y}_{T}}$ $({\ensuremath{\Delta}}_{1}$ is the surface gap exponent), with a crossover, for sufficiently large LLL, to a scaling of type 1/L1+yH{1/L}^{{1+y}_{H}}1/L1+yH​.
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.
Abstract. Effective Hamiltonian models predict non-universal critical singularities for two--dimensional wetting transitions with long-ranged forces. We verify these predictions by studying delocalization transitions in an infinitely long Ising strip, of width L (lattice spacings), with long-ranged surface fields that have opposite sign at each surface. The extrapolated asymptotic value for the exponent βs does not confirm to the predicted non-universality but instead approaches the same universal value representative of systems with short-ranged forces. The crossover of the scaling behaviour of the transition lines is presented. Moreover, contrary to the existing predictions, the critical wetting transition for p = 2 has been found.