  We investigate the phase-field approximation of the Willmore flow. This is a
fourth-order diffusion equation with a parameter $\epsilon>0$ that is
proportional to the thickness of the diffuse interface. We show rigorously that
for well-prepared initial data, as $\epsilon$ trends to zero the level-set of
solution will converge to motion by Willmore flow before the singularity of the
later occurs. This is done by constructing an approximate solution from the
limiting flow via matched asymptotic expansions, and then estimating its
difference with the real solution. The crucial step and also the major
contribution of this work is to show a spectrum condition of the linearized
operator at the optimal profile. This is a fourth-order operator written as the
sum of the squared Allen-Cahn operator and a singular linear perturbation.
