Recent developments in non-Fickian diffusion : a new look at viscoelastic materials

Abstract

The aim of this dissertation is to fully understand from a mathematical point of view the two coupled processes of sorption of a fluid by a viscoelastic material and the successive or simultaneous desorption of the fluid with solved molecules of a chemical compound which is dispersed in the material. These two coupled processes have a central role in several areas of Life Sciences and Material Sciences namely in Controlled Drug Delivery. When a penetrant fluid diffuses into a viscoelastic material, such as a polymer, it is well known that the process cannot be completely described by Fick’s classical law of diffusion. The reason lies in the fact that as the fluid diffuses into the material, it causes a deformation which induces a stress driven diffusion that act as a barrier to the fluid penetration. Thus a modified flux must be considered, resulting from the sum of the Fickian flux and a non-Fickian flux. We propose a new interpretation of this non-Fickian mass flux as being related to a convective field which represents an opposition of the polymer to the incoming penetrant fluid. To study the complete problem of sorption coupled with desorption, we progressively address more complex models. We begin by studying the process of sorption in Chapter 1, we generalize the model to a more abstract formulation in Chapter 2, and we study a numerical method for the abstract formulation in Chapter 3. The complete problem of sorption coupled with desorption is addressed in Chapter 4. The first sorption model studied is based on an integro-differential equation, coupled with initial and boundary conditions. The non-linear dependence between strain and the incoming fluid concentration is considered and introduced in a Boltzmann integral with a kernel computed from a Maxwell-Wiechert model. To illustrate the behavior of the model we solve it numerically on a general nonuniform grid in space and a uniform grid in time. We exhibit numerical simulations that give some insight of the dependence of the solution on the different parameters that describe the viscoelastic properties of the polymer. This lead us to a generalization of this model by considering a class of integro-differential equations of Volterra type. We establish the well posedness, in the Hadamard sense, of the initial boundary value problem. The stability analysis is separated in two cases, non-singular kernels and weakly singular kernels. An implicit explicit difference scheme, which can be seen as a fully discrete piecewise linear finite element method, is proposed to discretize the general model. Stability and convergence results for the method are established showing that it is second order convergent in space and first order convergent in time. The numerical analysis of the method does not follow the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. A new approach, that enable us to reduce the smoothness required for the theoretical solution, is used. The results are established for both non-singular and weakly singular kernels. A tridimensional model of the whole process of sorption and desorption is presented in Chapter 4. A viscoelastic matrix with a dispersed drug, or a chemical compound, is considered. The model is based on a system of partial differential equations coupled with boundary conditions over a moving boundary. We combine non-Fickian sorption of a penetrant fluid , non-Fickian desorption of the fluid with dispersed drug, with non-linear dissolution of a drug agent and polymer swelling. An Implicit-Explicit numerical scheme is used to numerically solve the model and some plots are presented to illustrate the behavior of the approximations. Experimental rheological information of the polymer-solvent matrix system can be easily introduced in the models studied in this dissertation because all the parameters can be measured or estimated according to well-known theories of viscoelastic materials. This makes the models suitable for both data fitting and quantitative prediction of drug release kinetics, opening new routes of research in Material Science.