Diffusion coefficient constraint release

For 100 < P < 1000, the measured diffusion coefficients for N = P no longer follow the N 2 reptation prediction. In the same range of N values, D remains proportional to N 2 if P N, i.e. if the motion of the chains surrounding the test chain are frozen down during the diffusion time of the test chain. The comparison of the data obtained with N = P and with N P clearly puts into evidence the acceleration of the dynamics associated with the matrix chains, similarly to what has yet been observed with other polymers [11, 12, 42 to 44] or in solutions [10]. This acceleration, by a factor close to three, can be attributed to the constraint release mechanism [7, 8, 13], the effects of fluctuations of the test chain inside its tube [9] being a priori the same in the two situations P = N and P N. 

This combination is equivalent to saying that the overall diffusion coefficient is the sum of the coefficients of the two processes. Watanabe and Tirrell [18] suggested that the rate constant of constraint release depends on the tube configuration, but the comparison between the two models [19] does not allow us to conclude in favor of either one of the models. Thus we wiQ use the simplest one given by relation (5-2). 

The center of gravity of a linear chain now moves by two uncorrelated processes, reptation and constraint release, so the diffusion coefficient is just the sum of the individual contributions. Equation 21 gives the reptation contribution. Equation 18 gives the constraint release contribution with

[Pg.99]

The ratio of reptation contribution to constraint release contribution is thus 4ANs/ jAJ, so, with A(z) fixed, the reptation contribution dominates for sufficiently large N in the case of self diffusion (N = N). Reptation should always dominate, and D should be independent of the matrix, when the diffusing species is smaller than the matrix (N < Nj). The di sion coefficient should vary as (or M" M7 ) when the diffusing... 

If constraint release were the only process for conformational rearrangement, the initial path motions would be the same as the chain motions of the N-element Rouse model. Equation 4 relates the longest relaxation time Ti to the diffiision coefficient for Rouse chains. The diffusion cxieffident from constraint release is gjven by Eq. 90 with Ns = N. With Eqs. 1, 2, 4 and 9,... 

On the other hand, if the matrix chains are short enough (small A ) constraint release controls the terminal dynamics of the P-chains [Eq. (9.85)] and the diffusion coefficient of the P-mers depends strongly on N ... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

Here, the effect of a very large, or in the present model infinite, transverse friction coefficient is to retard the lateral diffusion over length scales larger than the mean spacing between entanglements. Additional non-reptative processes, such as the release of constraints, can lead to transverse diffusion for p < p. These effects can be included in this model by modifying Eq. (73) appropriately. 

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