Travelling-wave amplitudes as solutions of the phase-field crystal equation

Abstract

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E 70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B 75, 064107 (doi:10.1103/PhysRevB.75. 064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen–Cahn type (Galenko & Jou 2005 Phys. Rev. E 71, 046125 (doi:10.1103/ PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the tanh method (Malfliet & Hereman 1996 Phys. Scr. 15, 563–568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput. 154, 713–723 (doi:10.1016/ S0096-3003(03)00745-8)). The general tanh solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general tanh solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D 308, 1–10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.