On the behavior of the Casimir force in an exactly solvable model of a liquid film with an ordering field: The case of Dirichlet boundary conditions

We study a model of a critical fluid described by the standard φ4 Ginzburg-Landau Hamiltonian and determine the Casimir force in the case of Dirichlet-Dirichlet boundary conditions imposed along the finite dimension of the film. Physically, the fluid can be a simple fluid near its liquid-vapor critical point, or a binary liquid mixture close to the corresponding demixing point. Mathematically, the stable states of the system are determined by the minimizers of the functional which leads to finding the solution of the corresponding Euler-Lagrange nonlinear differential equations under the considered boundary conditions. We solve these equations in terms of Weierstrass and Jacobi elliptic functions and give analytic representation of the Casimir force as a function of both the temperature and ordering field. The derived expressions are then evaluated numerically in the plane of the temperature and the ordering external field.