  We present both exact and numerical results for the behavior of the Casimir
force in $O(n)$ systems with a finite extension in one direction when the
system is subjected to surface fields that induce helicity in the order
parameter. We show that for such systems the Casimir force in certain
temperature ranges is of the order of $L^{-2}$, both above and below the
critical temperature, $T_c$, of the bulk system. An example of such a system
would be one with chemically modulated bounding surfaces, in which the
modulation couples directly to the system's order parameter. We demonstrate
that, depending on the parameters of the system, the Casimir force can be
either attractive or repulsive. The exact calculations presented are for the
one dimensional $XY$ and Heisenberg models under twisted boundary conditions
resulting from finite surface fields that differ in direction by a specified
angle and the three dimensional Gaussian model with surface fields in the form
of plane waves that are shifted in phase with respect to each other.
Additionally, we present exact and numerical results for the mean field version
of the three dimensional $O(2)$ model with finite surface fields on the
bounding surfaces. We find that all significant results are consistent with the
expectations of finite size scaling.
