Interfaces of semi-infinite smectic liquid crystals and equations of state of infinite smectic stacks of semiflexible manifolds.

In this paper, we first elucidate the classical problem of the elastic free energy of a semi-infinite smectic-A liquid crystals, that fills the semispace above an interface (a boundary smectic layer) of a given shape. For the free energy of this interface, we obtain an effective interface Hamiltonian that takes into account the system discreteness introduced by the layered character of smectic-A liquid crystals. It is thus applicable to both short and long wavelength fluctuations of the interface shape. Next, we use our interface Hamiltonian to develop an efficient approach to the statistical mechanics of stacks of N semiflexible manifolds, such as two-dimensional smectic phases of long semiflexible polymers and three-dimensional lamellar fluid membrane phases. Within our approach, doing the practically interesting thermodynamic limit N--> infinity is reduced to considering a small stack, with just a few interacting manifolds, representing a subsystem of an infinite smectic. This dramatic reduction in the number of degrees of freedom is achieved by treating the first (the last) manifold of the small stack as an interface with the semi-infinite smectic medium below (above) the small stack. We illustrate our approach by considering in detail two-dimensional sterically stabilized smectic liquid crystals of long semiflexible polymers with hard-core repulsion. Smectic bulk (N= infinity ) equation of state and the universal constant characterizing entropic repulsion in these phases are obtained with a high accuracy from numerical simulations of small subsystems with just a few semiflexible polymers.