Critical Casimir effect in the Ising strips with standard normal and ordinary boundary conditions and the grain boundary

•Critical Casimir effect in Ising strips with internal grain boundary is considered.•Casimir amplitudes are derived as functions of the grain boundary strength.•Structure of short-distance expansions near the grain boundary is studied in detail.

We consider critical Casimir force in the Ising strips with boundary conditions defined by standard normal and ordinary   surface universality classes containing also the internal grain boundary. Using exact variational approach of Mikheev and Fisher we have elaborated on behaviors of Casimir amplitudes Δ++(g)Δ++(g), ΔOO(g)ΔOO(g) and Δ+O(g)Δ+O(g), corresponding to normal–normal, ordinary–ordinary and mixed normal–ordinary boundary conditions, respectively, with gg as a strength of the grain boundary. Closed analytic results describe Casimir amplitudes Δ++(g)Δ++(g) and ΔOO(g)ΔOO(g) as continuous functions of the grain boundary’s strength gg, changing the character of the Casimir force from repulsive to attractive and vice versa for certain domains of gg. Present results reveal a new type of symmetry between Casimir amplitudes Δ++(g)Δ++(g) and ΔOO(g)ΔOO(g). Unexpectedly simple constant result for the Casimir amplitude Δ+O(g)=π12 we have comprehensively interpreted in terms of equilibrium states of the present Ising strip as a complex interacting system comprising two sub-systems. Short-distance expansions of energy density profiles in the vicinity of the grain boundary reveal new distant-wall correction amplitudes that we examined in detail. Analogy of present considerations with earlier more usual short-distance expansions near one of the (N), (O) and (SB) boundaries, as well as close to surfaces with variable boundary conditions refers to the set of scaling dimensions appearing in the present calculations but also to the discovery of the de Gennes–Fisher distant wall correction amplitudes.