Nanoparticles can be beautiful, as in stained glass windows, or they can be ugly as in wear
and corrosion debris from implants. We estimate that there will be about 70 000 papers in …

This book discusses classical results, as well as recent developments, related to the Cahn–
Hilliard equation. It is based on the lectures that I gave at the CBMS-NSF Conference on the …

In this paper, we review the recent development of phase-field models and their numerical
methods for multi-component fluid flows with interfacial phenomena. The models consist of a …

Despite major scientific, medical and technological advances over the last few decades, a
cure for cancer remains elusive. The disease initiation is complex, and including initiation …

This is the first paper in a two-part series in which we develop, analyze, and simulate a
diffuse interface continuum model of multispecies tumor growth and tumor-induced …

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–
Hilliard equation, with second order accuracy in time and the fourth order finite difference …

In this paper we present and compare two unconditionally energy stable finite-difference
schemes for the phase field crystal equation. The first is a one-step scheme based on a …

This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

We develop a thermodynamically consistent mixture model for avascular solid tumor growth
which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and …

The phase-field method has recently emerged as the most powerful computational tool for
simulating complicated dendrite growth. However, these simulations are still limited to two …