Phase-field model for front propagation in a temperature gradient: selection and competition between the correlation...

A phase-field model is presented to study the propagation and the selection of a front in directional growth. The phase transition can be first or second order and is described by a nonconserved order parameter. In general, the thermal length l(u) (inversely proportional to the temperature gradient) is much larger than the correlation length l(phi), which gives the width of the front, and there is no direct competition between them (epsilon =l(phi)/l(u)<<1). In this paper, we consider a situation where these two lengths can be of the same order of magnitude (epsilon =l(phi)/l(u) close to 1). This happens in liquid crystals at the nematic-cholesteric phase transition. The problem of the front selection is solved theoretically by first performing an asymptotic analysis of the governing equations in the limit epsilon-->0, and then by solving the equations numerically. The main result is that the front is selected in a single way (no continuum of solutions) as long as epsilon not equal 0, whatever the velocity and the order of the phase transition. Finally, we show that the order parameter profile and the front temperature can change significantly when epsilon approaches 1.