Marginal stability analysis of the phase field crystal model in one spatial dimension

The problem of wavenumber k and velocity V selection for a solid front invading an unstable homogeneous phase is considered. A marginal stability analysis is used to predict k and V for the parabolic and hyperbolic (or modiefied) phase field crystal models in one dimension. It is shown that the marginally selected wave number of the periodic crystal monotonically increases with increasing undercooling and relaxation times. At high undercooling and relaxation times it is found that the system can select a k that is unstable to an Eckhaus instability in the bulk phase. This may imply a transition to highly defected or glassy states in higher dimensions.