Boundary conditions influence on the behavior of the Casimir force: A case study via exact results on the Ginzburg-Landau type fluid system with a film geometry

Abstract

The thermodynamic Casimir force in a fluid system depends on the conditions the bounding surfaces exert on either the confined fluid, or binary liquid mixture. This influence is described by imposing proper boundary conditions on the order parameter describing the fluid. We present a brief review and some new results on the force dependence on the example of the Ginzburg-Landau Φ4 model via exact results and series expansions of the corresponding order-parameter profiles. We consider the (+, +), (+, −), Dirichlet-Dirichlet and Neumann-Neumann boundary conditions and a combination of them. We focus on a system with a film geometry close to its bulk critical point in which the order parameter satisfies one of the above mentioned boundary conditions. Solving the corresponding boundary-value problem of one nonlinear differential equation in terms of Weierstrass and Jacobi elliptic functions we report analytic representation of the Casimir force. We study the behavior of the force depending on both the temperature and an external ordering field acting on the system. Our results are in a full agreement with the general arguments of the finite-size scaling theory. We confirm the expectation that the force is attractive if the boundary conditions are the same on each boundary and repulsive otherwise.