Linear splay elasticity in surface-induced films of tilted smectic liquid crystals

The prefrozen films that may be observed at the surface of isotropic liquid crystal droplets, close to the isotropic-smectic phase transition, or surface-induced films, are essentially asymmetric. If moreover, the molecules are tilted inside the smectic layers, as in the smectic-C (SmC) or smectic-C(A) (SmC(A)) phases, the c director that we may define as the order parameter of the film, is a real vector. Thus, the surface-induced films of MHTAC exhibit vectorial or polar properties, though the molecules are not chiral. The film free energy therefore contains a surface-elasticity term, K(1)c(2) c, that is a linear function of the splay distortion, and that may be negative enough to promote a mechanical instability. A spontaneous c distortion, or c modulation, then invades the whole film and produces an array of parallel stripes, with a typical four-fringe periodicity when observed between crossed polarisers. Here, we present optical measurements of the distortion for different film thicknesses, and we propose a linear analysis of the data to test our model. Due to the limitations inherent to the Fourier expansion that we use, the calculations are valid only between two limits: a large q limit where splay domains collapse into disclination lines, and a small q borderline below which the distortion evolves towards a system of independent solitons. We find that the sign of the spontaneous splay elastic constant K1 alternates as a function of the number of layers, N, a property that is reminiscent of the alternate structure of the SmC(A) phase. We argue that the two-dimension elastic constant, K1, originates from the interactions between the molecules in contact to the isotropic phase, and we deduce K1 approximately -(-1)(N)x10(-11) N and the elastic anisotropy, with a ratio of the splay over bend elastic constants K(s)/K(b) approximately 4.5. Similar properties could be observed also in other types of ultrathin films, e.g., in free-standing, ferroelectric (SmC*) or antiferroelectric (SmC(*)(A)), films, in Langmuir films, and even in particular biological films. In some cases, a second, electric instability may occur and superimpose onto the elastic one.