Smectic-A edge dislocations in a very thin cell

We study stable "bookshelf" smectic-A structures within a very thin plane-parallel cell of thickness L in which the mismatch between surface preferred (d(s)) and intrinsic (d(0)) smectic layer thicknesses occurs. The Landau-Ginzburg approach based on a complex smectic order parameter is used. For a weak enough smectic positional anchoring strength W smectic layers adopt the modified bookshelf profile. In a thick enough cell with increasing W a lattice of edge dislocations is continuously formed at the confining surfaces and then depinned from them. The structure with dislocations is formed when the condition d(0)/(zeta(s)/d(0)/d(s)-1/) approximately 2 is fulfilled, where zeta(s) is the positional surface anchoring extrapolation length. If the cell is thin enough the dislocations formed at opposite cell plates annihilate and consequently the smectic layers adopt a locked bookshelf structure. This transition is discontinuous and takes place when d(0)/(L/d(0)/d(s)-1/) approximately 5 is realized. To observe these transitions in a cell of thickness L approximately 1microm the conditions W approximately 10(-6) J/ m(2) and /d(0)/d(s) - 1/ approximately 5 x 10(-4) have to be fulfilled. All the three qualitatively different structures coexist at the triple point.