Ice crystals and snowflakes are out-of-equilibrium growth shapes which are a result of a nonlinear growth dynamics as a consequence of the extremal property of the associated thermodynamic potential. A special role during the pattern formation play kink solutions that represent the different state of order at the phase boundaries. The mechanisms of the kink formation give an insight into the dynamics of phase transitions in particular the formation and growth of ice nuclei. In this paper is described a relationship between the classical nucleation theory and Kobayashi’s phase field theory for ice crystal growth. The critical length of the nuclei is derived from the linear stability analysis for the phase field model and is identified with the result of the classical nucleation theory. We modify original Kobayashi’s phase field model by including freezing point depression due to salt in order to describe the phase boundary of the fine network and cavities filled with brine which are formed during the freezing process in sea ice.