Eur. Phys. J. B (2013) 86: 422
https://doi.org/10.1140/epjb/e2013-40674-1

Regular Article

Pattern formation in the models with coupling between order parameter and its gradient

¹ Bogolyubov Institute for Theoretical Physics, NAS Ukraine, Metrologichna 14-b, 03680 Kyiv, Ukraine
² Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, 305-8568 Tsukuba, Japan

^a e-mail: bohdan.lev@gmail.com

Received: 16 July 2013
Received in final form: 4 September 2013
Published online: 9 October 2013

Abstract

A possibility of pattern formation in the models of the first order phase transitions with coupling between the order parameter and its gradient is discussed. We use the standard model of phase transitions extended to the higher derivatives of the order parameter that makes possible to describe the formation of various spatial distributions of the order parameter after phase transition. An example of the simple model with coupling between the order parameter and its gradient is considered. The proposed model is analogical to the mechanical nonlinear oscillator with the coordinate-dependent mass or velocity-dependent elastic module. The exact solution of this model is obtained that can be used to predict the order parameter distribution in the case of a spinodal decomposition.