Dynamic Relaxation in Polymer Solutions

The density at all distances from a given monomer is as uniform and constant as in a small-molecule liquid. The chains are then simple random walks at all length scales much larger than a monomer. The osmotic pressure in this state is difficult to define, since all the solvent has been removed. It is progressively more difficult to remove the last traces of solvent in this sense the osmotic pressure becomes large on the scale of kT per monomer. 

In a quiescent dilute solution the motion of polymers is a random walk the mean squared distance (rc ) covered is the diffusion constant C, times the elapsed time t, as noted in Sect. 8.5. The various chains are far apart and thus do their random motions independently. As the concentration approaches 0, this ceases to be true. The random currents moving one polymer are also felt by its neighbors, so that nearby chains have similar motions. To characterize the motion we must specify two diffusion coefficients the self-diffusion coefficient and the cooperative diffusion coefficient. The self-diffusion coefficient Cs is defined by the motion of an individual chain as introduced above. 

Neighboring chains impede the random currents near a given chain thus C, decreases as the concentration increases. The other important aspect of diffusion is the spreading of extra local concentration in the solution. The extra material spreads over a distance x whose square is proportional to time. The cooperative diffusion coefficient Cc gives the constant of proportionality Cc = x /t. 

A small region of extra concentration is under extra osmotic pressure. This extra pressure, due to the repulsive interaction between the polymers, tends to spread the chains in the concentrated region apart faster than they would otherwise spread. Thus the interactions increase ( c- And thus as the concentration increases from zero, Q increases from its limiting value of just as Q decreases. 

The mean-squared center-of-mass position x ) is the average of those of its blobs (x ) = x )/K. Thus the diffusion constant of the center of mass is reduced from that of a blob by a factor of the number of blobs in a chain K. 

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