Critical Casimir forces for films with bulk ordering fields

The confinement of long-ranged critical fluctuations in the vicinity of second-order phase transitions in fluids generates critical Casimir forces acting on confining surfaces or among particles immersed in a critical solvent. This is realized in binary liquid mixtures close to their consolute point Tc which belong to the universality class of the Ising model. The deviation of the difference of the chemical potentials of the two species of the mixture from its value at criticality corresponds to the bulk magnetic filed of the Ising model. By using Monte Carlo simulations for this latter representative of the corresponding universality class we compute the critical Casimir force as a function of the bulk ordering field at the critical temperature $T=T_{c}$ . We use a coupling parameter scheme for the computation of the underlying free energy differences and an energy-magnetization integration method for computing the bulk free energy density which is a necessary ingredient. By taking into account finite-size corrections, for various types of boundary conditions we determine the universal Casimir force scaling function as a function of the scaling variable associated with the bulk field. Our numerical data are compared with analytic results obtained from mean-field theory