Nonlinear development of morphological instabilities in explosive crystallization

Under appropriate conditions, a straight, steadily advancing explosive crystallization front undergoes a morphological instability against modes that are wavy in space and oscillatory in time. We study the effect of nonlinearity on the shape of such a front near the onset of instability, in a limit in which the wavelengths of the unstable modes are long and their oscillation frequencies are low. For driven crystallization we derive an amplitude equation describing the slow evolution of the Fourier coefficients of the front shape. Numerical integration of this equation shows that initially localized perturbations always develop into spatial patterns having a definite wavelength that depends on the growth parameters but not on the initial conditions. This observation then makes it possible to predict the selected wavelength analytically. For self-sustained crystallization we find that, in the analogous limit, the front generally comes to a halt as a result of the instability.