Pattern morphology and dynamical scaling in the Cahn Hilliard model

Numerical simulations were carried out in two-dimensions of the dimensionless Cahn-Hilliard equation. Simulations were run for a factor of ten in time beyond previously reported results. The simulations also covered a broad range of values of the mean composition,$\left\langle \psi \right\rangle _0 $. To determine the dynamical scaling exponent, $b,$ an equation of the form $R_G (t)=at^b+c$ was fit to a measure of average domain size. In contrast to previous results, we found that $b$ varied substantially with$\left\langle \psi \right\rangle _0 $. The largest deviation from the Lifshitz-Slyozov value of 1/3 occurred at $\left\langle \psi \right\rangle _0 =0.15$, where $b=0.221\pm 0.04$. We used a measure of the non-circularity of minority domains to show that for $\left\langle \psi \right\rangle _0 \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} 0.20$ the domain shapes are not scale invariant for times exceeding our simulation times. We also point out the possible existence of a phase boundary $\left\langle \psi \right\rangle _{0,c} $that separates a phase with circular domains of minority component from a phase with non-circular minority component domains.