  In confined systems near a continuous phase transition the long-ranged
fluctuations of the corresponding order parameter are subject to boundary
conditions. These constraints result in so-called critical Casimir forces
acting as effective forces on the confining surfaces. For systems belonging to
the Ising bulk universality class corresponding to a scalar order parameter the
critical Casimir force is studied for the film geometry in the crossover regime
characterized by different surface fields at the two surfaces. The scaling
function of the critical Casimir force is calculated within mean field theory.
Within our approach, the scaling functions of the critical Casimir force and of
the order parameter profile for finite surface fields can be mapped by
rescaling, except for a narrow crossover regime, onto the corresponding scaling
function of the so-called normal fixed point of strong surface fields. In the
crossover regime, the critical Casimir force as function of temperature
exhibits more than one extremum and for certain ranges of surface field
strengths it changes sign twice upon varying temperature. Monte Carlo
simulation data obtained for a three-dimensional Ising film show similar
trends. The sign of the critical Casimir force can be inferred from the
comparison of the order parameter profiles in the film and in the semi-infinite
geometry.
