Surface ordering and destruction of the first order phase transition smectic A-smectic C$^*$ in free standing smectic films

The thermodynamic behaviour of the first order phase transition smectic A-smectic C$^*$ in free standing films is studied numerically. Profiles of the tilt angle across the film, the free energy and average structural parameters $\langle \Theta \rangle$ and $\langle \cos\Theta \rangle$ are calculated. By diminishing the number of smectic layers the phase transition disappears at a critical film thickness $d_{\rm c}$, which is approximately one order of magnitude larger than the penetration length of the surface induced order. Computer simulation results describe well recent experimental data