Elsevier

Physica D: Nonlinear Phenomena

Volume 134, Issue 4, 10 December 1999, Pages 385-393
Physica D: Nonlinear Phenomena

A generalized field method for multiphase transformations using interface fields

https://doi.org/10.1016/S0167-2789(99)00129-3Get rights and content

Abstract

The recently developed multiphase field method, describing the interaction between an arbitrary number of individual phase fields with individual characteristics, is reformulated by the use of interface fields. This reformulation allows for the decomposition of the nonlinear multiphase field interactions into pairwise interaction of interface fields. This removes some difficulties in the treatment of triple points or higher order interactions that occurred in the original model. The interface fields being defined in a (2Ñ) dimensional space, where Ñ is the order of the multiple point, can be interpreted being the generalized coordinates for this variational problem. The considered example of a multiphase change problem indicates clearly that a relaxation ansatz for the evolution of the field variables towards the minimum of the free energy is warranted only for generalized coordinates, while a relaxation ansatz using functionally dependent variables and the Lagrange formalism in general mixes time and energy scales.

Introduction

The multiphase field theory [1] was originally developed for the description of first order phase transformations in a system containing N>2 different phases φα,φβ,…,φN and their gradients ∇φα,∇φβ,…,∇φN. The equations of motion of the φα towards the minimum of the free energy F are derived using a relaxation ansatzτφ̇α=∂∇φα∂φαf({φα}),where f({φα}) denotes the Gibbs free energy density of the N phase system as defined in [1], τ is a relaxation constant.

Since in a multiphase problem the {φα} are connected by the constraintNα=1φα=1orNα=1φ̇α=0,the φα are not independent field variables.

In the original model [1] the nonlinearities arising from this fact were attributed to triple point energies and energies of multiple interactions of higher order. According to the physical assumption that these energies have negligible influence on the total energy of the system, the corresponding nonlinearities were neglected.

Garcke et al. [2] have shown that this approximation violates conservation of interfacial stresses at multiple points (Young’s law). As an explanation for this violation one may consider that the phase boundaries in equilibrium are straight lines (2D) or planes (3D) and the angles between the boundaries are independent of the length scale. On the scale of the phase boundary thickness, the multiple phase point then fills the whole volume under consideration. Thus the multiple phase energies will influence the local physics significantly, though they are negligible in the system altogether.

By use of a Lagrange multiplier λ and treating the N phases α=1,…,N independent, the equations of motion of the φα are found:τφ̇α=∂∇φα∂φαf({φα})+λ1Nφα−1=∂∇φα∂φαf({φα})+λ.The Lagrange multiplier λ accounts for the constraint (2) or (3).

It was shown by Garcke et al. [2] that the ansatz (4) conserves the interfacial stress balance in the sharp interface limit with isotropic interfaces and by Nestler and Wheeler [3] that it holds also for arbitrary interface anisotropy.

There arise however two severe problems. The first is the definition of the relaxation constant τ in (4). As it is well known, the relaxation rate of an interface into equilibrium strongly depends on the type of interface, e.g. solid–liquid or solid–solid. The right-hand side of (4), however – besides the pairwise contributions related to one type of interface, that were used in the original model – contains higher order contributions, related to triple points. These contributions can hardly be attributed with an individual timescale. Therefore a decomposition of these terms related to specific boundaries αβ is necessary.

The second problem is the coupling of the phase field equations (4) to outer fields like temperature. The phase change αβ results in a local energy change δE related to the latent heat of that specific phase change Lαβ, while there is no evidence that a multiple phase change can be related to a triple energy L̃αβγ. Again therefore decomposing (4) into pairwise contributions is necessary.

In fact, these problems are two facets of the same difficulty: how to fix the time and energy scale of a multiple phase change in a multiphase system. In this paper, a formal transformation of the phase field variables φα onto a set of (2N) interfacial field variables ψαβ is described, that allows for the decomposition of (4) in the desired way. Moreover, this transformation leads to a definition of the multiphase change problem in a (2N) dimensional space being more general than the original definition on the N−1 manifold of the N phase system, connected by the constraint (2).

Section snippets

The phase field variables φα of a multiphase system and the free energy functional F

In the classical phase field theory [4], [5], the phase field variable φ(x,t) is defined as a continuous function in space and time on an Euclidean point space Ω. φ may be identified with the solid density that varies continuously from 1 (solid) to 0 (liquid) over the interface region with a thickness δ. The liquid density is then given by 1−φ(x,t), and φ(1−φ) may be interpreted as the interface density. Values of φ(x,t)<0 and φ(x,t)>1 are formally allowed but considered to be non-physical.

The

The interface fields ψαβ and their equations of motion

As can be seen from the constraint (2) resp. (7) the {φα} do not form an independent set of functional variables of the multiphase system. They are defined on a Ñ−1 dimensional manifold.

For Ñ phases (2Ñ) interfaces between two phases α and β can be formed. We define a set of (2Ñ) antisymmetric interface fields {ψαβ},α<β and its complement {ψ̃βα},α<β, ψ̃βα=−ψαβ. In the following we skip the tilde on the complement. For Ñ=2 a unique interface field ψαβ can be defined on the basis of φα and φβ:

Interpretation of the interface fields ψαβ

The interface fields ψαβ are by their definition restricted to the finite interface regions ΩαΩβ.

At dual phase boundaries αβ only one interface field ψαβ is defined locally. At triple points αβγ three interface fields overlap ψαβ, ψαγ, ψβγ, and at multiple points of order Ñ, (2Ñ) interface fields overlap.

The most remarkable feature of these fields is the fact that a variation δψαβ at a multiple point, where e.g. 0<φγ,φα,φβ<1, only affects the phases φα and φβ, but is influenced by φγ. φγ

Conclusion

A generalized field method for multiphase transformations is derived using interface fields. The interface fields ψαβ may either be treated as linear transformations of the phase fields φα or as linear independent fields defined in a (2Ñ) dimensional space, in general being of higher dimension than the dimension Ñ−1 of the manifold of the phase fields φα. In both cases, the interface fields are seen to be generalized coordinates of the variational problem.

Not using these generalized

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