In this paper we present a second order accurate (in time) energy stable numerical
scheme for the Cahn-Hilliard (CH) equation, with a mixed finite element approximation
in space. Instead of the standard second order Crank-Nicolson methodology,
we apply the implicit backward differentiation formula (BDF) concept to derive
second order temporal accuracy, but modified so that the concave diffusion term is
treated explicitly. This explicit treatment for the concave part of the chemical potential
ensures the unique solvability of the scheme without sacrificing energy stability. An
additional term $A$τ∆($u^{k+1}$−$u^k$) is added, which represents a second order Douglas-Dupont-type
regularization, and a careful calculation shows that energy stability is
guaranteed, provided the mild condition $A$≥$\frac{1}{16}$ is enforced. In turn, a uniform in time $H^1$ bound of the numerical solution becomes available. As a result, we are able to
establish an $ℓ^∞$(0,$T$;$L^2$) convergence analysis for the proposed fully discrete scheme,
with full $\mathcal{O}$ (τ^{2}+$h^2$) accuracy. This convergence turns out to be unconditional; no scaling
law is needed between the time step size $τ$ and the spatial grid size $h$. A few
numerical experiments are presented to conclude the article.