Evolution of speckle during spinodal decomposition

Time-dependent properties of the speckled intensity patterns created by scattering coherent radiation from materials undergoing spinodal decomposition are investigated by numerical integration of the Cahn-Hilliard-Cook equation. For binary systems which obey a local conservation law, the characteristic domain size is known to grow in time $\tau$ as $R=[B\tau {]}^n$ with $n=1/3,$ where B is a constant. The intensities of individual speckles are found to be nonstationary, persistent time series. The two-time intensity covariance at wave vector $\mathbf{k}$ can be collapsed onto a scaling function $\mathrm{Cov}(\delta t,\overline{t}),$ where $\delta {t=k}^{1/n}B|{\tau }_2-{\tau }_1|$ and $\overline{t}{=k}^{1/n}B({\tau }_1+{\tau }_2)/2.\mathrm{}$ Both analytically and numerically, the covariance is found to depend on $\delta t$ only through $\delta t/\overline{t}$ in the small-$\overline{t}$ limit and $\delta t/\overline{t}{}^{1-n}$ in the large-$\overline{t}$ limit, consistent with a simple theory of moving interfaces that applies to any universality class described by a scalar order parameter. The speckle-intensity covariance is numerically demonstrated to be equal to the square of the two-time structure factor of the scattering material, for which an analytic scaling function is obtained for large $\overline{t}.$ In addition, the two-time, two-point order-parameter correlation function is found to scale as $C\left({r/(B}^n\sqrt{{\tau }_1^{2n}+{\tau }_2^{2n}}\right),{\tau }_1/{\tau }_2),$ even for quite large distances r. The asymptotic power-law exponent for the autocorrelation function is found to be $\lambda \approx 4.47,$ violating an upper bound conjectured by Fisher and Huse.