Stability of a supercooled liquid to periodic density waves and dynamics of freezing

We present a theoretical analysis of the dynamic stability of a supercooled simple liquid to finite amplitude, periodic, density waves. The analysis is based on a nonlinear diffusion equation (NLDE) which is known to provide an accurate hydrodynamic description of the density fluctuations occurring on a molecular length scale. The NLDE leads to a system of coupled, nonlinear equations for the temporal evolution of the time-dependent order parameters. These equations are then cast in a form appropriate for perturbative evaluation of the generalized relaxation rates. Numerical values of the relaxation rates are obtained for the bcc and the fcc density waves and also for the 2d hcp lattice. A stability limit is reached when the relaxation rates of all the order parameters go to zero simultaneously. The values of the amplitudes of the density waves at the stability limit are found to remain finite at large supercooling, indicating the absence of a classical spinodal point for a simple liquid. The coupling between various density modes at the reciprocal lattice vectors of the solid is found to play an important role in the dynamics of ordering in a supercooled liquid. We find that because of these mode coupling effects, there is a dynamical constraint on the formation of the bcc phase. Such a constraint is absent for simple fcc forming systems. The relation of the present work with nucleation is discussed