  In this paper, the mathematical properties and numerical discretizations of
multiphase models that simulate the phase separation of an $N$-component
mixture are studied. For the general choice of phase variables, the unisolvent
property of the coefficient matrix involved in the $N$-phase models based on
the pairwise surface tensions is established. Moreover, the symmetric
positive-definite property of the coefficient matrix on an $(N-1)$-dimensional
hyperplane --- which is of fundamental importance to the well-posedness of the
models --- can be proved equivalent to some physical condition for pairwise
surface tensions. The $N$-phase Allen-Cahn and $N$-phase Cahn-Hilliard
equations can then be derived from the free-energy functional. A natural
property is that the resulting dynamics of concentrations are independent of
phase variables chosen. Finite element discretizations for $N$-phase models can
be obtained as a natural extension of the existing discretizations for the
two-phase model. The discrete energy law of the numerical schemes can be proved
and numerically observed under some restrictions pertaining to time step size.
Numerical experiments including the spinodal decomposition and the evolution of
triple junctions are described in order to investigate the effect of pairwise
surface tensions.
