Surface ordering and finite-size effects in liquid-crystal films

Motivated by recent experiments on liquid-crystal films, we study the development of heat-capacity singularities with the number of layers, n. With enhanced surface couplings, there are distinct peaks due to surface ordering and bulk ordering, at temperatures ${\mathit{T}}_{\mathit{S}}$(n) and ${\mathit{T}}_{\mathit{B}}$(n), respectively. Comparing exact results from the two-dimensional Ising model with a mean-field theory, we find in both cases that the evolution of the peaks is controlled by a length scale ${\xi }_{\mathit{S}}$ for decay of surface order into the bulk. The bulk transition temperature is governed by a scaling function ${\mathit{F}}_{\mathit{S}}$(n/${\xi }_{\mathit{S}}$), which decreases with n for n${\xi }_{\mathit{S}}$, but increases with n for n>${\xi }_{\mathit{S}}$. Qualitative trends for thin films agree well with experimental observations.