Length scales for wetting transitions: Beyond the continuum Landau approximation for the interfacial binding potential

Abstract

We show that the length scale ${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$, which appears in the interfacial binding potential of the effective Hamiltonian description of wetting transitions, should be identified with the ``true'' or exponential range of the pair-correlation function of the bulk (wetting) phase, rather than the Ornstein-Zernike (second-moment) bulk correlation length ${\ensuremath{\xi}}_{\mathit{b}}$. Since ${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$>${\ensuremath{\xi}}_{\mathit{b}}$, this implies that the parameter \ensuremath{\omega}=${\mathit{k}}_{\mathit{B}}$T${\mathrm{\ensuremath{\alpha}}}^{2}$/4\ensuremath{\pi}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}, which determines the values of critical exponents for critical wetting in three dimensions, is significantly smaller than initial estimates of this quantity. For the simple-cubic Ising model (\ensuremath{\alpha}${\ensuremath{\xi}}_{\mathit{b}}$${)}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\approxeq}1.46 for T\ensuremath{\gtrsim}${\mathit{T}}_{\mathit{R}}$, the roughening temperature, and depending on the magnitude of the interfacial stiffness \ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}, we conclude \ensuremath{\omega}\ensuremath{\lesssim}0.5 is appropriate for such temperatures. We discuss the implications for recent Monte Carlo studies of critical wetting.