Phase-field model with finite interface dissipation Ingo Steinbach a,⇑ , Lijun Zhang a , Mathis Plapp b a Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, D-44801 Bochum, Germany b Physique de la Matie `re Condense ´e, E ´ cole Polytechnique, CNRS, 91128 Palaiseau, France Received 16 January 2012; accepted 23 January 2012 Available online 7 March 2012 Abstract In rapid phase transformations, interfaces are often driven far from equilibrium, and the chemical potential may exhibit a jump across the interface. We develop a model for the description of such situations in the framework of the phase-field formalism, with separate concentration fields in each phase. The key novel feature of this model is that the two concentration fields are linked by a kinetic equation which describes the exchange of components between the phases, instead of an equilibrium partitioning condition. The associated rate constant influences the interface dissipation. For rapid exchange between the phases, the chemical potentials are equal in both coexisting phases at the interface as in previous models, whereas in the opposite limit strong non-equilibrium behavior can be modeled. This is illustrated by simulations of a diffusion couple and of solute trapping during rapid solidification. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase transformations; Phase field models; Diffusion 1. Introduction If two (or more) materials in an arbitrary thermody- namic state are brought into contact, the whole system starts to evolve towards thermodynamic equilibrium. Although this picture of a thermodynamic process is quite commonly used as a Gedankenexperiment, little is actually known about its initial stage when the two materials are far from common equilibrium. To be more precise, most ther- modynamic models fail to describe the condition at the interfaces in the first moment of the contact. The question then remains as to how to determine the composition of the contact plane between the two materials as a function of time. 1 The situation is schematically described in Fig. 1. Two materials a and b of initial compositions c 0 a and c 0 b are brought into contact. The compositions are, in general, out of chemical equilibrium, i.e. the chemical potentials (or diffusion potentials, which are the tangents to the free energy curves at the compositions c 0 a and c 0 b ) do not coin- cide. As soon as a common interface is formed, its compo- sition can change. The entire region between c min and c max has a positive free energy balance with respect to the two- phase state. It is a plausible assumption that the interface composition should lie somewhere between c 0 a and c 0 b . It is not possible, however, to determine the interface compo- sition based on thermodynamics alone, since we are not in a thermodynamic equilibrium and the problem is a kinetic one. A kinetic model which takes into account finite inter- face dissipation is needed, such as the ones discussed in Refs. [1,2]. Here, we develop as an alternative a “phase- field” type (diffuse-interface) model within the framework of the phase-field theory. A direct comparison between both types of models will be given in a forthcoming publication. A second example for strong non-equilibrium effects at interfaces, from a general point of view closely related to the previous one, is solute trapping or the transition to par- titionless transformation in rapid solidification [3,4]. Here, the driving force for transformation is strong throughout 1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2012.01.035 ⇑ Corresponding author. E-mail address: ingo.steinbach@rub.de (I. Steinbach). 1 Throughout this paper, we will treat only the case of an alloy with completely mixing solutes. The model, however, can easily be generalized to different thermodynamic variables in problems with potential jumps across the interface. www.elsevier.com/locate/actamat Available online at www.sciencedirect.com Acta Materialia 60 (2012) 2689–2701