Continuum modeling and simulation of a ferroelectric smectic-A liquid crystal

We construct a continuum model of a recently discovered ferroelectric smectic-A (SmA_{F}) liquid crystal that has a layered structure with orientational order and polarization along the layer normal. We take into account the layer order by a complex order parameter ψ, and the orientational order by a director n[over ⃗]. Screened electrostatic interaction of induced charges is also introduced. We carry out numerical calculations of the dynamics of phase transition from an isotropic phase directly to SmA_{F} for a simple case of small screening length. Then the electrostatic interaction is reduced to splay elastic energy and can be incorporated into the variation of the splay elastic constant (K_{1}). We show that the enhancement of the effective K_{1} significantly promotes the coarsening dynamics of the layered structure to a uniform one by suppressing the bending of the layers. We calculated three different characteristic lengths from the correlation of n[over ⃗], number density of the defects of n[over ⃗], and the structure factor of ψ. All the characteristic lengths for different effective K_{1} exhibit power-law time evolution whose exponent is close to 1/4, except for the case when the effective K_{1} is unrealistically small compared with the bend elastic constant K_{3}. Our model can be exploited to a wide class of problems concerning the structural properties of a ferroelectric SmA_{F} liquid crystal.