Multiscale, hybrid mixture theory for swelling systems with interfaces

A three-scale problem involving swelling media is considered. The structure of the system (e.g., clay, polymers, etc.) is assumed to be characterized by particles composed of more than one phase (usually solid and liquid) which may swell or shrink. The swelling (or shrinkage) is the result of mass transfer from (to) a multicomponent solution which is in contact with the particles. Mesoscopic equations for the particles themselves are reviewed. The macroscopic equations for the combination of swelling particles and bulk solution as well as their interfaces, considered as a mixture, are obtained here. Constitutive theory for the system is derived. We choose the independent variables and derive constitutive restrictions for four cases of a dual-porosity multiple-component swelling media: a mesoscopic model which assumes no interfacial effects, a mesoscopic model which includes interfacial effects, a macroscopic model which assumes no interfacial effects, and a macroscopic model which includes interfacial effects. For each case, the entropy inequality is fully expressed using a Lagrange multiplier technique. Novel definitions for macroscopic pressures and chemical potentials are given, and generalized Darcy\u27s and Fick\u27s laws are presented. Although these models are developed with a clay soil in mind, the results hold for any medium which assumes the same set (or subset of) independent variables as constitutive unknowns, e.g. swelling biopolymers