Spinodal decomposition kinetics: the initial and intermediate stages

摘要

This text is a treatment of spinodal decomposition kinetics ofcolloidal systems in the initial and intermediate stages. When a stable, homogeneous system is quenched into an unstable or meta-stable state, density inhomogeneities will develop, which ultimately lead to complete phase separation, where two phases are in coexistence. The kinetics of the phase separation process can be described by analyzing equations of motion for the relevant order parameter. In the present text, the gas-liquid phase separation of colloids is considered, where the relevant quantity is the macroscopic density. Here, the "gas phase" is a colloidal fluid of low concentration, while the "liquid phase" is a more concentrated colloidal fluid. The goal of this text is to develop a microscopic approach for spinodal decomposition kinetics, where the starting point is the Smoluchowski equation. This is an equation of motion for the probability density function of the phase space coordinates of the colloidal particles (the colloidal analogue of the Liouville equation for molecular systems). In section 1 the various stages that can be distinguished during spinodal decomposition, starting from a homogeneous system, are introduced. Spinodal decomposition kinetics in the initial stage, where density inhomogeneities have just started to develop, is discussed in section 2. First, the classic Cahn-Hilliard thermodynamic approach is considered, after which a microscopic rederivation of these results is given, based on the Smoluchowski equation. Section 2 concludes with a microscopic interpretation for the origin of the spinodal instability. In section 3 an alternative definition of the spinodal and binodal from a kinetic point of view is discussed, and the experimental relevance of the spinodal is considered. It turns out that the location of the theoretically well defined spinodal cannot be determined experimentally with arbitrary precision. Decomposition kinetics in the intermediate stage is treated in section 4. In the intermediate stage, density inhomogeneities are not small anymore, so that non-linear equations of motion must be considered. Due to the existence of a dominant length scale in the intermediate stage, dynamic scaling is expected. A dynamic scaling relation for the structure factor is derived, after which the Smoluchowski equation is analysed in the intermediate stage. The solution of the non-linear equation of motion for the density confirms the dynamic scaling relation for the structure factor, and leads to an explicit expression for the dynamic scaling function. Contraryto the thermodynamic type of approach, the effect of externally imposed shearing motion can be analysed in a quite straightforward manner, using the kinetic approach discussed in section 2. This is the subject of section 5. Theoretical predictions are compared to experimental findings in section 6. Finally, a few exercises are added, together with an overview of relevant literature (although this is by no means a complete overview).