  In this paper, we aim to study the motions of interfaces and coarsening rates
governed by the time-fractional Cahn--Hilliard equation (TFCHE). It is observed
by many numerical experiments that the microstructure evolution described by
the TFCHE displays quite different dynamical processes comparing with the
classical Cahn--Hilliard equation, in particular, regarding motions of
interfaces and coarsening rates. By using the method of matched asymptotic
expansions, we first derive the sharp interface limit models. Then we can
theoretically analyze the motions of interfaces with respect to different
timescales. For instance, for the TFCHE with the constant diffusion mobility,
the sharp interface limit model is a fractional Stefan problem at the time
scale $t = O(1)$. However, on the time scale $t = O({\varepsilon}^{1/{\alpha}}
)$ the sharp interface limit model is a fractional Mullins--Sekerka model.
Similar asymptotic regime results are also obtained for the case with one-sided
degenerated mobility. Moreover, scaling invariant property of the sharp
interface models suggests that the TFCHE with constant mobility preserves an
${\alpha}/3$ coarsening rate and a crossover of the coarsening rates from
${\alpha}/3$ to ${\alpha}/4$ is obtained for the case with one-sided
degenerated mobility, which are in good agreement with the numerical
experiments.
