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Fig. 1: (a) Sharp interface. (b) Diffuse interface.where M is a mobility. The term g describes how the free energy variesas a function of the order parameter; at constant T and P , this takesthe typical form (Appendix 1)†:∫g =V[g 0 {φ, T } + ɛ(∇φ) 2 ]dV (1)where V and T represent the volume and temperature respectively. Thesecond term in this equation depends only on the gradient of φ and henceis non–zero only in the interfacial region; it is a description therefore ofthe interfacial energy. The first term is the sum of the free energies ofthe precipitate and matrix, and may also contain a term describing theactivation barrier across the interface. For the case of solidification,g 0 = hg S + (1 − h)g L + Qfwhere g S and g L refer to the free energies of the solid and liquid phases† If the temperature varies then the functional is expressed in termsof entropy rather than free energy.
espectively, Q is the height of the activation barrier at the interface,h = φ 2 (3 − 2φ) and f = φ 2 (1 − φ) 2Notice that the term hg S + Qf vanishes when φ =0(i.e. only liquid ispresent), and similarly, (1 − h)g L + Qf vanishes when φ =1(i.e. onlysolid present). As expected, it is only when both solid and liquid arepresent that Qf becomes non–zero. The time–dependence of the phasefield then becomes:∂φ∂t = M[ɛ∇2 φ + h ′ {g L − g S }−Qf ′ ]The parameters Q, ɛ and M have to be derived assuming some mechanismof transformation.Two examples of phase–field modelling are as follows: the first iswhere the order parameter is conserved. With the evolution of compositionfluctuations into precipitates, it is the average chemical compositionwhich is conserved. On the other hand, the order parameter is not conservedduring grain growth since the amount of grain surface per unitvolume decreases with grain coarsening.Cahn–Hilliard Treatment of Spinodal DecompositionIn this example, the order parameter is the chemical composition.In solutions that tend to exhibit clustering (positive ∆H M ), it is possiblefor a homogeneous phase to become unstable to infinitesimal perturbationsof chemical composition. The free energy of a solid solution whichis chemically heterogeneous can be factorised into three components.
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