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In this paper, the mathematical properties and numerical discretizations ofmultiphase models that simulate the phase separation of an $N$-componentmixture are studied. For the general choice of phase variables, the unisolventproperty of the coefficient matrix involved in the $N$-phase models based onthe pairwise surface tensions is established. Moreover, the symmetricpositive-definite property of the coefficient matrix on an $(N-1)$-dimensionalhyperplane --- which is of fundamental importance to the well-posedness of themodels --- can be proved equivalent to some physical condition for pairwisesurface tensions. The $N$-phase Allen-Cahn and $N$-phase Cahn-Hilliardequations can then be derived from the free-energy functional. A naturalproperty is that the resulting dynamics of concentrations are independent ofphase variables chosen. Finite element discretizations for $N$-phase models canbe obtained as a natural extension of the existing discretizations for thetwo-phase model. The discrete energy law of the numerical schemes can be provedand numerically observed under some restrictions pertaining to time step size.Numerical experiments including the spinodal decomposition and the evolution oftriple junctions are described in order to investigate the effect of pairwisesurface tensions.

Multiphase Allen-Cahn and Cahn-Hilliard Models and Their Discretizations with the Effect of Pairwise Surface Tensions