Published February 17, 2021 | Version 1
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Lecture "Cahn-Hilliard-type phase-transition dynamics"

  • 1. WWU Münster, Institut für Theoretische Physik

Description

The data deposit contains an introductory  video lecture by Uwe Thiele, WWU Münster, Institut für Theoretische Physik on "Cahn-Hilliard-type phase-transition dynamics" together with the slides in pdf format and this info file.

The lecture gives a basic introduction to the archetypical case of a conserved gradient dynamics, i.e., Cahn-Hilliard-type models as obtained in phenomenological nonequilibrium thermodynamics with the help of Onsager’s variational principle.

The lecture should be suitable for advanced Bachelor students, Master students and beginning PhD students of the natural sciences and other interested people. It is a stand-alone lecture on the behaviour described by the Cahn-Hilliard equation for a single order parameter field, but was given in the context of a lecture course "Introduction to the Theory of Phase Transitions" (ITPT). Background information on gradient dynamics may be obtained in the lecture "Introduction to Nonequilibrium Thermodynamics - Onsager’s variational principle"  (see https://dx.doi.org/10.5281/zenodo.4545320).

We start with a brief recapitulation of the concept of gradient dynamics on an underlying energy functional, show that Cahn-Hilliard-type dynamics indeed always decreases the underlying energy before introducing the proper Cahn-Hilliard model by specifying a particular energy functional (square-gradient terms and local double-well potential). Then the linear stability of homogeneous steady states is analysed (dispersion relations are calculated and discussed), and fully nonlinear (heterogeneous) steady are discussed (where reference is made to the continuation tutorial ACCH, see https://dx.doi.org/10.5281/zenodo.4545409) in the context of the phase diagram for phase decomposition and the transition between supercritical and subcritical primary bifurcations. Then the  fully nonlinear behaviour is illustrated by time simulations. This part is finalised by brief discussions of the bifurcation behaviour in 2d and domain coarsening.

Subsequently, two possible extensions are introduced, namely, the consideration of mixed conserved and non-conserved dynamics for one field and the usage of more complicated energy functionals (e.g., for crystallisation). For the former, the linear stability of homogeneous steady states is discussed. For the latter, as an example the phase-field-crystal (PFC) model and its relation to Dynamical Density Functional Theory (DDTF) is explained  before the PFC model is used to model an advancing crystallisation front. The lecture concludes with a summary and outlook.

The second archetypical case of nonconserved gradient dynamics, i.e., Allen-Cahn-type dynamics, is considered in the accompanying lecture "Allen-Cahn-type phase-transition dynamics" (see https://dx.doi.org/10.5281/zenodo.4545328).

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