© 2017

Interfacial Wave Theory of Pattern Formation in Solidification

Dendrites, Fingers, Cells and Free Boundaries


  • Offers the most comprehensive fundamental treatment of the subject matter available

  • Considers many applications in a very broad range of fields: condensed matter physics, materials sciences, fluid dynamics and chemical engineering

  • Compares in-depth theory and available experimental data


Part of the Springer Series in Synergetics book series (SSSYN)

About this book


This comprehensive work explores interfacial instability and pattern formation in dynamic systems away from the equilibrium state in solidification and crystal growth. Further, this significantly expanded 2nd edition introduces and reviews the progress made during the last two decades. In particular, it describes the most prominent pattern formation phenomena commonly observed in material processing and crystal growth in the framework of the previously established interfacial wave theory, including free dendritic growth from undercooled melt, cellular growth and eutectic growth in directional solidification, as well as viscous fingering in Hele-Shaw flow. It elucidates the key problems, systematically derives their mathematical solutions by pursuing a unified, asymptotic approach, and finally carefully examines these results by comparing them with the available experimental results.

The asymptotic approach described here will be useful for the investigation of pattern formation phenomena occurring in a much broader class of inhomogeneous dynamical systems. In addition, the results on global stability and selection mechanisms of pattern formation will be of particular interest to researchers working on material processing and crystal growth.

The stability mechanisms of a curved front and the pattern formation have been fundamental subjects in the areas of condensed-matter physics, materials science, crystal growth, and fluid mechanics for some time now. This book offers a stimulating and insightful introduction for all physicists, engineers and applied mathematicians working in the fields of soft condensed-matter physics, materials science, mechanical and chemical engineering, fluid dynamics, and nonlinear sciences.


interfacial instability viscous fingering Hele-Shaw cell Solidification of melts CFD with free and moving boundaries Directional solidification of binary mixtures Dendritic growth

Authors and affiliations

  1. 1.Dept. of Mathematics and StatisticsMcGill UniversityMontrealCanada

About the authors

Dr Jian-Jun Xu is a professor in the Department of Mathematics and Statistics at McGill University, Canada. He is an outstanding applied mathematician, working in the interdisciplinary area of applied mathematics, condensed matter physics, material science and fluid dynamics. He has published four monographs and about one hundred research papers. His expertise includes but is not limited to: asymptotics and numerical analysis, dynamical systems of non-Linear PDE with particular emphasis in the areas of solidification physics,  interfacial wave theory, pattern formation and crystal growth.


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“I was particularly pleased with Xu’s coverage of material that is often missing from conventional texts on pattern growth and nonequilibrium phenomena in condensed matter. … such an ambitious, advanced text will surely be of great interest to mathematical physicists and researchers in chemical physics, engineering, and materials science.” (Domenico Truzzolillo, Physics Today, March, 2018)