Universality classes for the critical wetting transition in two dimensions

Abstract

Universality classes and critical exponents for the wetting transition in two dimensions are determined with the use of a continuum planar solid-on-solid model. Effective substrate potentials $\mathrm{BU}\left(f\right)<0\mathrm{}$ falling off no slower than $f^{-2\mathrm{}}$, where $f$ is the distance from the substrate, are shown to lead to wetting transitions at finite potential strength $B$. For potentials which go to zero at least exponentially fast with $f$, the interface free energy $F_s$ is analytic in the thermal scaling field $t$. In the case of longer-range substrate potentials with a finite first moment, $F_s$ remains proportional to $t^2$ to leading order, but higher-order nonanalytic terms in $t$ appear, while in the borderline case $\mathrm{}\left|U\right(f\left)\right|\sim f^{-2\mathrm{}}$, $F_s$ has an essential singularity at the wetting transition. For potentials going to zero more slowly than $f^{-2\mathrm{}}$, there is no transition at finite $B$. It is shown that $F_s\sim B^{\frac{2}{\mathrm{}\left|a\right|\mathrm{}}}$ for $B\to 0^{+}$ for potentials asymptotically proportional to $f^{-2-a}$, $a<\mathrm{}0\mathrm{}$.