Small-Angle X-ray Scattering Size Parameters and Higher Moments of the Particle-Size Distribution Function in the Asymptotic Stage of Ostwald Ripening

Small-angle X-ray scattering (SAXS) is a powerful tool to study the kinetics of phase separation in materials. A simple procedure is presented that allows one to prove if the particle-size distribution established in a system in the late stages of phase separation corresponds to the predictions of the classical Lifshitz-Slyozov-Wagner (LSW) theory for the asymptotic stage of Ostwald ripening. The method is based on the correlations between certain SAXS size parameters and the higher moments of the LSW size distribution functions for diffusion-limited or reaction-limited ripening. It is suggested that the use of these size parameters, which can be obtained with high accuracy from the scattering curve, is frequently more advantageous than a direct comparison of the experimentally obtained size distributions with the asymptotic size-distribution functions predicted by the LSW theory. The method is applicable if the suppositions made in the LSW theory that the precipitated particles should be homogeneous spheres with volume fraction tending to zero are fulfilled. The method is applied to a photochromic glass; although the silver-halide precipitates contained in the glass develop according to the power law of diffusion-limited Ostwald ripening, their size distribution is shown not to correspond to the features of the LSW size distribution. Consequently, in this case the LSW theory cannot describe quantitatively the kinetics of ripening.