Solvation force in two-dimensional Ising strips

Finite-size contributions to the free energy of a fluid confined between two parallel walls, separated by a distance L, give rise to an excess pressure which is termed the solvation force ${\mathit{f}}_{\mathrm{solv}}$(L). Using exact transfer-matrix methods we calculate the analog of ${\mathit{f}}_{\mathrm{solv}}$ for a two-dimensional Ising strip of infinite length and finite width L≡na, where n is the number of layers and a is the lattice spacing, for bulk field h=0 and fixed (+/+) and (+/-) boundary conditions on the spins in the surface layers. + and - refer to up and down spins, respectively. ${\mathit{f}}_{\mathrm{solv}}^{\mathrm{}(++)\mathrm{}}$ is negative (attractive force) for all temperatures T and for a given L has its minimum above the critical temperature ${\mathit{T}}_{\mathit{c}}$. The amplitude of the force at the minimum is about 6.6 times the value at ${\mathit{T}}_{\mathit{c}}$, the ‘‘Casimir’’ amplitude. In the (+/-) strip a +- interface develops at all subcritical temperatures and entropic repulsion gives rise to a positive ${\mathit{f}}_{\mathrm{solv}}^{(+\mathrm{-})}$ which has its maximum slightly below ${\mathit{T}}_{\mathit{c}}$. Universal scaling forms are derived for both cases and accurate approximations, valid for low, near critical, and high temperatures, are obtained. In the scaling limit, L→∞,t[≡(T-${\mathit{T}}_{\mathit{c}}$)/${\mathit{T}}_{\mathit{c}}$]→0, the minimum of ${\mathit{f}}_{\mathrm{solv}}^{\mathrm{}(++)\mathrm{}}$ is given by ${\mathit{nt}}_{\min }$=1.2642 and the maximum of ${\mathit{f}}_{\mathrm{solv}}^{(+\mathrm{-})}$ by ${\mathit{nt}}_{\max }$=-0.2735. We compare and contrast our results with earlier predictions based on mean-field analyses and scaling arguments.