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arXiv:cond-mat/9803193 · Physical review. B, Condensed matter

Anomalous corrections to the Kelvin equation at complete wetting

Enrico Carlon, A. Drzewiński, Jos Rogiers

Published Aug 15, 1998• 32 citations
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Abstract

We consider an Ising model confined in an $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ geometry with identical surface fields at the boundaries. According to the Kelvin equation the bulk coexistence field scales as $1/L$ for large $L;$ thermodynamics and scaling arguments predict higher-order corrections of the type ${1/L}^{2}$ and ${1/L}^{5/3}$ at partial and complete wetting, respectively. Our numerical results, obtained by density-matrix renormalization techniques for systems of widths up to $L=144,$ are in agreement with a ${1/L}^{2}$ correction in the partially wet regime. However, at complete wetting we find a large range of surface fields and temperatures with a correction to scaling of type ${1/L}^{4/3}.$ We show that this term is generated by a thin wetting layer whose free energy is dominated by the contacts with the wall. For $L$ sufficiently large we expect a crossover to a ${1/L}^{5/3}$ correction as predicted by the theory for a thick wetting layer. This crossover is indeed found in a solid-on-solid model, which provides a simplified description of the wetting layer and allows the study of much larger systems than those available in a density-matrix renormalization calculation.

DOI
10.1103/physrevb.58.5070

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Physical review. E•Feb 1, 2017•0 citations
We consider an Ising model confined in an L\ifmmode×\else\texttimes\fi\ensuremath∞L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}L\ifmmode×\else\texttimes\fi\ensuremath∞ geometry with identical surface fields at the boundaries. According to the Kelvin equation the bulk coexistence field scales as 1/L1/L1/L for large L;L;L; thermodynamics and scaling arguments predict higher-order corrections of the type 1/L2{1/L}^{2}1/L2 and 1/L5/3{1/L}^{5/3}1/L5/3 at partial and complete wetting, respectively. Our numerical results, obtained by density-matrix renormalization techniques for systems of widths up to L=144,L=144,L=144, are in agreement with a 1/L2{1/L}^{2}1/L2 correction in the partially wet regime. However, at complete wetting we find a large range of surface fields and temperatures with a correction to scaling of type 1/L4/3.{1/L}^{4/3}.1/L4/3. We show that this term is generated by a thin wetting layer whose free energy is dominated by the contacts with the wall. For LLL sufficiently large we expect a crossover to a 1/L5/3{1/L}^{5/3}1/L5/3 correction as predicted by the theory for a thick wetting layer. This crossover is indeed found in a solid-on-solid model, which provides a simplified description of the wetting layer and allows the study of much larger systems than those available in a density-matrix renormalization calculation.
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​.
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.
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.
The global phase diagram of wetting in the two-dimensional Ising model is obtained through the exact calculation of the surface excess free energy. In addition to a surface field for inducing wetting, a surface-coupling enhancement is also included. The wetting transition (of second order) is critical for any finite ratio of surface coupling J_{s} to bulk coupling J, and becomes of first order in the limit J_{s}/J→∞. However, for J_{s}/J≫1, the critical region is exponentially small and is practically invisible to numerical studies. A distinct preasymptotic regime exists in which the transition displays first-order character. In this regime, surprisingly, the surface susceptibility and surface specific heat develop a divergence and show anomalous scaling with an exponent equal to 3/2.
We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent $\ensuremath{\beta}$, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by $\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}$ requires $\ensuremath{\beta}=1/2$ ($\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3$, and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=2$), the thermodynamic scaling establishes that ${\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}$, which in contrast requires $\ensuremath{\beta}=0$ (${\mathrm{\ensuremath{\Delta}}}_{s}=4$), where $\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and ${\mathrm{\ensuremath{\Delta}}}_{s}$ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., $\ensuremath{\beta}=0$) and a direct measurement of the susceptibility critical exponent $\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2$, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.