  This paper tackles the approximation of surface diffusion flow using a
Cahn--Hilliard-type model. We introduce and analyze a new second order
variational phase field model which associates the classical Cahn--Hilliard
energy with two degenerate mobilities. This association allows to gain an order
of approximation of the sharp limit. In a second part, we propose some simple
and efficient numerical schemes to approximate the solutions, and we provide
numerical 2D and 3D experiments that illustrate the interest of our model in
comparison with other Cahn--Hilliard models.
