Anomalous Segment Diffusion

Because of the relatively short displacement time or length scales typically probed by NMR diffusometry, it is particularly well suited to detect anomalies in the segment displacement behavior expected on a time scale shorter than the terminal relaxation time, that is for root mean squared displacements shorter than the random-coil dimension. All models discussed above unanimously predict such anomalies (see Tables 1-3). Therefore, considering exponents of anomalous mean squared displacement laws alone does not provide decisive answers. In order to obtain a consistent and objective picture, it is rather crucial to make sure that (i) the absolute values of the mean squared segment displacement or the time-dependent diffusion coefficient are compatible with the theory, (ii) the dependence on other experimental parameters such as the molecular weight are correctly rendered, and (iii) the values of the limiting time constants are not at variance with those derived from other techniques. 

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On the other hand, if a static tube really exists as anticipated in the tube/reptation model, the predictions for anomalous segment diffusion are adequate, of course, and can be verified in experiment. This will be demonstrated by considering linear polymers confined in nanopores. 

Segment diffusion in pores suggests itself as a typical model scenario representing the premisses of the tube/reptation model. NMR diffusometry is suitable to probe the time or length scales of the Doi/Edwards limits (II)de> (III)de and beyond. Since the wall adsorption effect in the PHEMA system is expected to be negligible, one can therefore expect that the reptation features of the anomalous segment diffusion regime, especially with respect to limit (III)de> are faithfully rendered by the experiments. 

Field-gradient NMR diffusometry is suitable for recording translational displacement properties by self-diffusion. This in particular refers to the anomalous segment displacement regime. Center-of-mass diffusion is also accessible this way but must be handled with some care because of the experimental limitation of the maximum diffusion time. Furthermore the in-... 

Pick s first and second laws were developed to describe the diffusion process in polymers. Fickian or case I transport is obtained when the local rate of change in the concentration of a diffusing species is controlled by the rate of diffusion of the penetrant. For most purposes, diffusion in rubbery polymers typically follows Fickian law. This is because these rubbery polymers adjust very rapidly to the presence of a penetrant. Polymer segments in their glassy states are relatively immobile, and do not respond rapidly to changes in their conditions. These glassy polymers often exhibit anomalous or non-Fickian transport. When the anomalies are due to an extremely slow diffusion rate as compared to the rate of polymer relaxation, the non-Fickian behaviour is called case II transport. Case II sorption is characterized by a discontinuous boundary between the outer layers of the polymer that are at sorption equilibrium with the penetrant, and the inner layers which are unrelaxed and unswollen. 

The results are presented in Table 2. Parameter X is a constant related to the structure of the network, and the exponent n is related to the type of diffusion, being Fickian for n values of 0.45-0.50 (rate of diffusion of solvent is less than polymer segmental mobility), whereas 0.50< <1.0 indicates a anomalous transport, non-Fickian diffusion type [8-10, 11]. Equation 2 was applied to... 

Fig. 46. Theoretical mean squared segment displacement of a chain confined in a randomly coiled tube versus time according to the harmonic radial potential theory [70]. The tube diameter d is given in multiples of the Kuhn segment length b. The crossover tendency to free, unconfined Rouse chain dynamics with increasing tube diameter is obvious. The mean squared displacement is given in units the diffusion time t in units of the segmental fluctuation time Tj. The chain length was assumed to be N= 1,600 Kuhn segments. The three anomalous Doi/Edwards limits (see Table 1) are reproduced with finite tube diameters...

Figure 46 shows the time dependence of the mean squared segment displacement as predicted by the harmonic radial potential theory [70]. The three anomalous diffusion limits, (I)de> (H)de> and (III)de> of the tube/repta-tion model are well reproduced. Note the extended width of the transition regimes between these limits, which should be kept in mind when discussing experimental data with respect to a crossover between different dynamic limits. Increasing the effective tube diameter is accompanied by the gradual transition to Rouse-like dynamics of an unconfined chain (where entanglement effects are not considered). 

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