  In this paper, we construct and analyze a uniquely solvable, positivity
preserving and unconditionally energy stable finite-difference scheme for the
periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels
system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free
energy potential. The proposed scheme is based on a convex-concave
decomposition of the given energy functional with two variables, and the
centered difference method is adopted in space. We provide a theoretical
justification that this numerical scheme has a pair of unique solutions, such
that the positivity is always preserved for all the singular terms, i.e., not
only two phase variables are always between $0$ and $1$, but also the sum of
two phase variables is between $0$ and $1$, at a point-wise level. In addition,
we use the local Newton approximation and multigrid method to solve this
nonlinear numerical scheme, and various numerical results are presented,
including the numerical convergence test, positivity-preserving property test,
energy dissipation and mass conservation properties.
