  We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard
model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E, 55:R3844-R3846,
1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes,
and F. Nabet. Arch. Ration. Mech. Anal., 233(2):837-866, 2019]. The scheme is
shown to preserve the key properties of the continuous model, namely mass
conservation, positivity of the concentrations, the decay of the energy and the
control of the entropy dissipation rate. This allows to establish the existence
of a solution to the nonlinear algebraic system corresponding to the scheme.
Further, we show thanks to compactness arguments that the approximate solution
converges towards a weak solution of the continuous problems as the
discretization parameters tend to $0$. Numerical results illustrate the
behavior of the numerical model.
