Swelling Kinetic Theory of Gel Networks

Right the position of a specific point after sweiiing (dotted iine indicates the position prior to sweiiing) 

In order to discuss the movement of the constitutive molecules of gels, it is necessary to describe gels as continuous bodies based on continuum mechanics. Let us consider the process for a point r on the gel network to move to point f as shown in Fig. 2. The vector defined by the following equation is called the deformation vector  

The equation of motion for the movement of a microscopic volume element of the gel is given by 

Solid line indicates gei network and black dots show crosslink points rand r indicate position vector before and after the swelling, respectively, and u Is the deformation vector. 

The component that constitutes networks, namely the partial polymer chain (for example, the partial chain between crosslinks), may not be able to move alone but may move along with other partial chains influencing each other. Therefore, the diffusion of a partial chain is not self-difliision but cooperative diffusion. The rate of this cooperative diffusion must be much slower than the self-diffusion of the solvent itself. Considering the movement of networks, the left-hand side of Eq. (5), the momentum term, is negligibly small compared with each term of the right-hand side of the equation. Therefore, 

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