Collective Diffusion of Gel Networks

Gels usually consist of small amount of polymer as a network and a lai amount of solvent. Therefore when we discuss the dynamics erf polymer gels, we are tempted to deal with these Is from the stand point of the dynamics of polymer solutions. However, since the polymer chains in a gel are connected to each other via chemical bonds and/or some kinds of sj cific interaction, sudi as, hydrogen bonding or hydrophobic interaction, the gel has to be treated as a continuum. In addition, gels behave as an assembly of springs due to the entropy elasticity of polymer chains between the crosslink points. Therdbre, the dynamics of polymer gels is well described in terms of the theory of elasticity 

Let us introduce a displaconent v or u(r, t), which represents the displacement of a point r on the network from its average position at time L The time average u(r, t) t is always zero. A small deformation of a unit volume of the gel having the mass density, p, is given by 

This is the fundamental equation to describe the kinetics and dynamics of polymer networks in a liquid. The left-hand side of Eq. (3.4) represents the acceleration term, whereas the first two terms of the right-hand side represent the elastic term. The last term of the right-hand side is the contribution of the friction between the network and solvent molecules. In most cases, however, the acceleration term is much smaller than the other terms. Thus one obtains 

This equation of motion has three solutions corresponding to one longitudinal and two transverse modes of which can be expre ed by the following diffusion equations  

Polymer molecules in a solution undergo random thermal motions, whidi give rise to space and time fluctuations of the polymer concentration. H the concentration of the polymer solution is dilute enough, the interaction between individual polymer molecules is negligiUe. Then the random motions of the polymer can be described as a three dimensional random walk, which is diaractoized by the diffusion coefficient D. Light is scattered by the density fluctuations of the polymer solution. The propagation of phonons is overdamped in water and becomes a simple diffusion process. In the case of polymer networks, however, such a situation can never be attained because the interaetion between diains (in 

As discussed in Sect. 3.1, a gel obeys the diffusion equations given in Eq. (3.5). The time-space correlation of a gel network is expressed in terms of the displacement vector as follows 

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