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 τ as R=[Bτ]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 k can be collapsed onto a scaling function Cov(δt,t̅ ), where δt=k^1/nB|τ₂-τ₁| and t̅ =k^1/nB(τ₁+τ₂)/2. Both analytically and numerically, the covariance is found to depend on δt only through δt/t̅ in the small-t̅ limit and δt/t̅ ^1-n in the large-t̅ 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 t̅ . In addition, the two-time, two-point order-parameter correlation function is found to scale as C(r/(Bⁿ√τ₁²ⁿ+τ₂²ⁿ),τ₁/τ₂), even for quite large distances r. The asymptotic power-law exponent for the autocorrelation function is found to be λ≈4.47, violating an upper bound conjectured by Fisher and Huse.