Pawan P. Singh

Multiscale fluid transport theory for swelling biopolymers

Fluid flow through a (bio) polymeric matrix has multiscale characteristics and is affected by the relaxation of surrounding polymers. Models developed in the past were either single scale (Polymer (1982) 23 (4) 529; Chemical Engineering Science (1992) 47 (12) 3037) or were limited to systems with a short memory (Achanta, 1995; Moisture transport in shrinking gels during drying, Ph.D. thesis, Purdue University, West Lafayette, IN). To address these limitations, we use the generalized Darcys law equations of Singh (Effect of viscoelastic relaxation on fluid and species transport in biopolymeric materials, Ph.D. thesis) and the mass balance equations of Bennethum and Cushman (International Journal of Engineering Science (1996) 34 (2) 125) to develop a multiscale fluid transport model. The effect of viscoelastic relaxation of solid polymers on the flow of vicinal (adsorbed) fluid is considered at the mesoscale. At the macroscale two bulk fluids are incorporated, one of which is identical to the vicinal fluid. The mass balance equations for the vicinal fluid and its bulk counterpart are coupled via source/sink terms. The resulting fluid transport equation includes a novel integral term related to viscoelastic properties of the biopolymeric matrix. This term incorporates viscoelastic effects with both short and long memory. The model can describe both Darcian (Fickian) and non-Darcian (non-Fickian) modes of fluid transport. The model suggests fluid transport is Darcian in the rubbery and glassy states when the biopolymers are sufficiently far from the glass transition region. In the proximity of glass transition the flow of fluids is anomalous or non-Darcian. These predictions are in agreement with the experimental observations of Kim et al. (Chemical Engineering Science (1996) 51 (21) 4827). In spite of its multiscale characteristics, the resulting transport equation is simple and can be easily solved. The experimental parameters needed to solve the equation are the effective diffusivity, a sorption or drying curve and viscoelastic properties of the material.

Three scale thermomechanical theory for swelling biopolymeric systems

A three-scale theory for the swelling polymeric/biopolymeric media is developed via the hybrid mixture theory. At the microscale, the solid polymeric matrix interacts with the solvent through surface contact. At the mesoscale, the homogeneous mixture of vicinal fluid and solid polymers exchanges thermodynamic properties with two bulk fluids, one of which is of the same type as the vicinal fluid. The relaxation processes within the polymeric matrix are incorporated by modeling the solid phase as viscoelastic and the solvent phases as viscous at the macroscale. We obtain novel equations for the total stress tensor, chemical potential of the solid phase, heat flux and Darcys law all at the macroscale. Viscoelastic stress components in Darcys law make it applicable for both Fickian and non-Fickian fluid transport. The form of the generalized Ficks law is similar to that obtained in earlier works involving colloids. Thermoviscoelastic and thermoviscous effects are incorporated by coupling thermal gradients with strain-rate tensors for the solid phase and the deformation-rate tensors for the liquid phases.

Thermomechanics of Swelling Biopolymeric Systems

A two-scale theory for the swelling biopolymeric media is developed. At the microscale, the solid polymeric matrix interacts with the solvent through surface contact. The relaxation processes within the polymeric matrix are incorporated by modeling the solid phase as viscoelastic and the solvent phase as viscous at the mesoscale. We obtain novel equations for the total stress tensor, chemical potential of the solid phase, heat flux and the generalized Darcys law all at the mesoscale. The constitutive relations are more general than those previously developed for the swelling colloids. The generalized Darcys law could be used for modeling non-Fickian fluid transport over a wide range of liquid contents. The form of the generalized Ficks law is similar to that obtained in earlier works involving colloids. Using two-variable expansions, thermal gradients are coupled with the strain rate tensor for the solid phase and the deformation rate tensor for the liquid phase. This makes the experimental determination of the material coefficients easier and less ambiguous.

Toward rational design of drug delivery substrates: II. Mixture theory for three-scale biocompatible polymers and a computational example

In Part I of this article we focused on glassy-state biocompatible polymers (two-scale) that may possess charges. Here we extend these results to a three-scale setting for polymers that contain two liquid phases. On the microscale the three phases each behave as a continuum occupying distinct regions of space. On the mesoscale the polymer is homogenized with the sorbed liquid phase to form a particle wherein both homogenized phases are assumed to simultaneously occupy each point in space within the particle. On the macroscale, the mesoscale particles are homogenized with two bulk-phase liquids, one being the same as the sorbed liquid. Conceptually, throughout all space, each macroscale homogenized phase exists everywhere. A theory of constitution is developed at the macroscale by exploiting the entropy inequality, and the resultant constitutive equations are inserted into the macroscale field equations and simplifications made so that a solution may be obtained via finite elements. A simple imbibition pro...

Toward Rational Design of Drug Delivery Substrates: I. Mixture Theory for Two-Scale Biocompatible Polymers

To rationally design drug release substrates, much as in the rational design of the drugs themselves, the biocompatible polymer physics, chemistry, and biology must be married in a mechanistic fashion via mathematics, and the mathematics must in turn be implemented on the computer for predictive purposes. Here we take a first step in this direction by developing two-scale, thermodynamically consistent, constitutive models for swelling, glassy, biocompatible polymers. Specifically, we apply mixture theory with averaged field equations to obtain thermodynamically consistent constitutive models for both charged and uncharged viscoelastic hydrogels in the glassy state. In Part II of this article we extend these results to a three-scale setting with two fluid phases on the macroscale. We develop constitutive models, insert these into the macroscale field equations, simplify, and solve an imbibition problem by the finite element method.