  The paper presents a model of lateral phase separation in a two component
material surface. The resulting fourth order nonlinear PDE can be seen as a
Cahn-Hilliard equation posed on a time-dependent surface. Only elementary
tangential calculus and the embedding of the surface in $\mathbb{R}^3$ are used
to formulate the model, thereby facilitating the development of a fully
Eulerian discretization method to solve the problem numerically. A hybrid
method, finite difference in time and trace finite element in space, is
introduced and stability of its semi-discrete version is proved. The method
avoids any triangulation of the surface and uses a surface-independent
background mesh to discretize the equation. Thus, the method is capable of
solving the Cahn-Hilliard equation numerically on implicitly defined surfaces
and surfaces undergoing strong deformations and topological transitions. We
assess the approach on a set of test problems and apply it to model spinodal
decomposition and pattern formation on colliding surfaces. Finally, we consider
the phase separation on a sphere splitting into two droplets.
