Wall effects in critical systems: Scaling in Ising model strips

Abstract

A scaling theory and a related conjecture presented recently by de Gennes and Fisher to predict the effects of walls inserted in two near-critical binary fluid mixtures, are checked theoretically by exact analytic calculations for $n\times \infty$, two-dimensional Ising model strips with a surface field, $h_1$, imposed on the first layer; exact calculations with a second surface field, $h_n$, imposed on the nth layer are also reported. It is verified, in particular, that the effects on properties observed close to one wall of a second wall at distance $D$ decay, at the critical point, as $\frac{1}{D^d}$, where $d$ is the spatial dimensionality. In addition to the scaling limits, the leading corrections are calculated explicitly and presented graphically.