  We present and analyze a new second-order finite difference scheme for the
Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau
(MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes
energy potential. This numerical scheme with unconditional energy stability is
based on the Backward Differentiation Formula (BDF) method time derivation
combining with Douglas-Dupont regularization term. In addition, we present a
point-wise bound of the numerical solution for the proposed scheme in the
theoretical level. For the convergent analysis, we treat three nonlinear
logarithmic terms as a whole and deal with all logarithmic terms directly by
using the property that the nonlinear error inner product is always
non-negative. Moreover, we present the detailed convergent analysis in
$\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use
the local Newton approximation and multigrid method to solve the nonlinear
numerical scheme, and various numerical results are presented, including the
numerical convergence test, positivity-preserving property test, spinodal
decomposition, energy dissipation and mass conservation properties.
