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

Approximation of surface diffusion flow: a second order variational Cahn--Hilliard model with degenerate mobilities