Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Systems described by an O(n) symmetrical ϕ⁴ Hamiltonian are considered in a d-dimensional film geometry at their bulk critical points. The critical Casimir forces between the film’s boundary planes B_j, j=1,2, are investigated as functions of film thickness L for generic symmetry-preserving boundary conditions ∂_nϕ=c∘_jϕ. The L-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form f_res≈D(c₁L^Φ/ν,c₂L^Φ/ν)/L^d-1 when d<4, where c_i are scaling fields associated with the variables c∘_i and Φ is a surface crossover exponent. Explicit two-loop renormalization group results for the function D(c₁,c₂) at d=4-ϵ dimensions are presented. These show that (i) the Casimir force can have either sign, depending on c₁ and c₂, and (ii) for appropriate choices of the enhancements c∘_j, crossovers from attraction to repulsion and vice versa occur as L increases.