Evolution of speckle during spinodal decomposition

Abstract

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 $τ$ as $R = [B τ]^n$ 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 ${\bf k}$ can be collapsed onto a scaling function $Cov(δt,\bar{t})$, where $δt = k^{1/n} B |τ_2-τ_1|$ and $\bar{t} = k^{1/n} B (τ_1+τ_2)/2$. Both analytically and numerically, the covariance is found to depend on $δt$ only through $δt/\bar{t}$ in the small-$\bar{t}$ limit and $δt/\bar{t} ^{1-n}$ in the large-$\bar{t}$ 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 $\bar{t}.$ In addition, the two-time, two-point order-parameter correlation function is found to scale as $C(r/(B^n\sqrt{τ_1^{2n}+τ_2^{2n}}),τ_1/τ_2)$, even for quite large distances $r$. The asymptotic power-law exponent for the autocorrelation function is found to be $λ\approx 4.47$, violating an upper bound conjectured by Fisher and Huse.