Volume-fraction dependence of the scaling function for phase-separating systems

In the last decades, the kinetics of phase separation following a quench into the miscibility gap have been studied in a variety of systems by small-angle scattering techniques. A scaling behavior in the late stages of coarsening, first observed in computer simulations, was found to be a `universal' feature of the small-angle scattering intensity S(k,t) with S(k,t) = S[k_m(t),t]F[k/k_m(t)], where k_m(t) is the value of the modulus k of the scattering vector at which S(k,t) has its maximum. Furthermore, it was observed recently that, for many systems ranging from liquid and polymer mixtures to solid alloys and computer simulation models, the scaling function F(x) does not change appreciably from one system to another when the volume fraction is kept constant. This paper first reviews work carried out jointly with J. L. Lebowitz and O. Penrose, discussing various features of the scaling function F leading to a simple analytical expression suitable to fit experimental data. New small-angle X-ray scattering results on the shape of the scaling function for dilute Al-Ag and Cu-Fe systems are then presented and compared with the predictions of the model.