  We study the crossover from the ordinary to the normal surface universality
class in the three-dimensional Ising universality class. This crossover is
relevant for the behavior of films of binary mixtures near the demixing point
and a weak adsorption at one or both surfaces. We perform Monte Carlo
simulations of the improved Blume-Capel model on the simple cubic lattice. We
consider systems with film geometry, where various boundary conditions are
applied. We discuss corrections to scaling that are caused by the surfaces and
their relation with the so called extrapolation length. To this end we analyze
the behavior of the magnetization profile near the surfaces of films. We obtain
an accurate estimate of the renormalization group exponent y_{h_1}=0.7249(6)
for the ordinary surface universality class. Next we study the thermodynamic
Casimir force in the crossover region from the ordinary to the normal surface
universality class. To this end, we compute the Taylor-expansion of the
crossover finite size scaling function up to the second order in h_1 around
h_1=0, where h_1 is the external field at one of the surfaces. We check the
range of applicability of the Taylor-expansion by simulating at finite values
of h_1. Finally we study the approach to the strong adsorption limit h_1
\rightarrow \infty. Our results confirm the qualitative picture that emerges
from exact calculations for stripes of the two-dimensional Ising model, [D. B.
Abraham and A. Maciolek, Phys. Rev. Lett. 105, 055701 (2010)], mean-field
calculations and preliminary Monte Carlo simulations of the 3D Ising model, [T.
F. Mohry et al, Phys. Rev. E 81, 061117 (2010)]: For certain choices of h_1 and
the thickness of the film, the thermodynamic Casimir force changes sign as a
function of the temperature and for certain choices of the temperature and h_1,
it also changes sign as a function of the thickness of the film.
