# A theory of coupled diffusion and large deformation in polymeric gels

（Entry by Jianyu Li, AP225, Fall, 2010）

## Contents

## Abstract

The deformation of gels involves two kinds of behaviors, one is mass diffusion of small solvent molecules filled within the polymer network, another is large deformation of the polymer network. To construct an thorough analysis of the unique behavior, this paper formulates a theory of coupled diffusion and large deformation in gels. In analysis, they applied a Lagrange multiplier to define the compressibility of the system, producing the term to describe the osmosis pressure or the swelling stress.

## Summary of Theory

### Kinematics

To describe the deformation of the network, they attach marker <math>X</math> to each differential element. At time t, the marker <math>X</math> moves to a new position <math>x</math>. The deformation gradient is defined as, <math>F_{iK}(X,t)=\frac{\partial x_i (X,t)}{\partial X_K}</math>.

### Mass Conservation

<math>C(X,t)</math> refers to the nominal concentration of the solvent in the gel in the current configuration, while <math>J_K(X,t)</math> the nominal flux. As showed in figure 2, they assume a pump connected to the gel, injects <math>r(X,t)dV(X)</math> of the solvent molecules into an element of volume per unit time, accordingly, <math>i(X,t)dA(X)</math> is the number of the solvent molecules into an element of area per unit time. Without any chemical reaction, we have <math>\frac{\partial C(X,t)}{\partial t}+\frac{\partial J_K(X,t)}{X_K}=r(X,t)</math>, <math>J_K(X,t)N_K(X)=-i(X,t)</math>.

### Conditions of Equilibrium

They assume that the rearrangement process is instantaneous, by neglecting the viscosity, the nominal stress is, <math>s_iK=\frac{\partial W(F,C)}{\partial F_iK}</math>. The chemical potential is given by <math>\mu=\frac{\partial W(F,C)}{\partial C}</math>.

### Kinetic Law

The kinetic of migration is taken to be the linear to the gradient of the chemical potential, <math>J_K=-M_{KL}(F,\mu)\frac{\partial \mu(X,t)}{\partial X_L}</math>, <math>M_{KL}</math> is the mobility, a symmetric and positive-definite tensor.

### Material Model

According to the theory of Flory and Rehner, they take the form, <math>W(F,\mu)=1/2NkT[F_{iK}F_{iK}-3-2log(det F)]-\frac{kT}{\nu}[(detF-1)log(\frac{det F}{det F-1})+\frac{\chi}{detF}]-\frac{\mu}{\nu}(det F-1)</math>. The mobility tensor is <math>M_{KL}=\frac{D}{\nu kT}(detF-1)H_{iK}H_{iL}</math>, here <math>H_iK</math> is the transpose of the inverse of the deformation gradient, <math>H_{iK}F_{iL}=\sigma_{KL}</math>. Based on the theory, they manage to describe the configuration of the gel. Let's summarize the related items in the following: 1. the initial conditions at particular time, <math>x(X,t_0)</math>, <math>\mu(X,t_0)</math>, 2. the applied force <math>B_i(X,t)</math>, 3. the rate of injection of solvent, <math>r(X,t)</math>, 4. <math>i(X,t), \mu(X,t)</math> on the surface, 5. the free energy function <math>W(F,C)</math>