Self-diffusion, chain

Table 5, Average polymer chain concentration (ape), polymer swellability (S), rotational correlation times of TEMPONE (r) and self-diffusion coefficient of methanol (Zf) in the swollen 2,2% Pd catalysts.

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

In spite of the problems associated with the static structure, the coarsegrained model for BPA-PC did reproduce the glass transition of this material rather well the self-diffusion constant of the chains follows the Vogel-Fulcher law [187] rather nicely (Fig. 5.10),... 

Fig. 5.10. Plot of the inverse logarithm of the self-diffusion constant of BPA-PC, for a length N = 20 of the coarse-grained chains, vs. temperature. Straight line indicates the Vogel-Fulcher [187] fit. From [28]...

It only remains to specify the time constant, r0 [Eqs. (5.14) and (5.15)], related inversely to the attempt frequency with which the monomers attempt to cross barriers in the torsional potential (Fig. 1.2b). We have not attempted to calculate this time constant from first principles, but rather fixed it by comparison to experiment on chain self-diffusion at T = 450K [178]. This yields r0 1/50 picoseconds. This small number can be understood from the fact that because of the potentials, Eqs. (5.12) and (5.13), at T = 450 K only a few percent of the attempted hops of the effective monomers are successful the time constant for successful hops is of the order of 1 ps. These considerations... 

Fig. 5.15. Self-diffusion constant for PE chains (Cioo) plotted vs. temperature, as predicted from the coarse-grained bond fluctuation model. From [32].

Fig. 5.18. Self-diffusion constants for a bidisperse (i.e. two different chain lengths) PE melt with Mn = 20 coarse-grained monomers. Open triangles are for d = 2, filled diamonds for d = 4, open squares for d = 6 and filled circles for d = 8. There are always two symbols of the same kind shown in the figure, since the bidisperse melt contains two species of different chain length. The numbers quoted in the figure correspond to these chain lengths for a given polydisparsity d. For instance, d = 8 corresponds to Mi = 12 and M2 = 52. From [184].

Figures 8 and 9 show the dependence of the self-diffusion constant and the viscosity of polyethylene melts on molecular weight [47,48]. For small molecular weights the diffusion constant is inversely proportional to the chain length - the number of frictional monomers grows linearly with the molecular weight. This behavior changes into a 1/M2 law with increasing M. The diffusion...

Fig. 8. Self-diffusion coefficients of polyethylene chains as a function of molecular mass. The measurements were carried out at the same value of the monomeric friction coefficient. (Reprinted with permission from [48]. Copyright 1987 American Chemical Society, Washington)...

L. Garrido, J.L. Ackerman and J.E. Mark, Self-diffusion of poly(dimethylsiloxane) chains. In L.-H. Lee (Ed.), New Trends in Physics and Physical Chemistry of Polymers, Plenum, New York, 1989, p. 355. 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

It is well known in the case of self-diffusion of a linear chain polymer within the melt that Dm is in proportion to the power of M,... 

The intercept Vo and slopes B in log V against 1/AT of FCSCs were plotted against Mn in Fig. 24. This showed that Vo significantly decreased with an increase of Mn, whereas B did not, as was shown by Hoffman et al. [28] Vo and B of ECSCs showed similar Mn dependence to those of FCSCs. As Vo is related to self diffusion of polymer chains and B is related to the activation free en-... 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

The Rouse model12 that yields Eq. [6] also shows that the self-diffusion constant of the chains scales inversely with chain length... 

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as a function of temperature (open circles). The full line is a fit with the Vogel-Fulcher law in Eq. [3]. The dashed and dotted lines are two fits with a power-law divergence at the mode-coupling critical temperature. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Therefore we expect Df, identified as the fast diffusion coefficient measured in dynamic light-scattering experiments, in infinitely dilute polyelectrolyte solutions to be very high at low salt concentrations and to decrease to self-diffusion coefficient D KRg 1) as the salt concentration is increased. The above result for KRg 1 limit is analogous to the Nernst-Hartley equation reported in Ref. 33. The theory described here accounts for stmctural correlations inside poly electrolyte chains. 

Further information on the dependence of structure of microemulsions formed on the alcohol chain length was obtained from measurement of self diffusion coefficient of all the constitutents using NMR techniques (29-34). For microemulsions consisting of water, hydrocarbon, an anionic surfactant and a short chain alcohol and C ) the self diffusion... 

Thus, in summary, self diffusion measurements by Lindman et a (29-34) have clearly indicated that the structure of microemulsions depends to a large extent on the chain length of the oosurfactant (alcohol), the surfactant and the type of system. With short chain alcohols (hydrophilic domains and the structure is best described by a bicontinuous solution with easily deformable and flexible interfaces. This picture is consistent with the percolative behaviour observed when the conductivity is measured as a function of water volume fraction (see above). With long chain alcohols (> Cg) on the other hand, well defined "cores" may be distinguished with a more pronounced separation into hydrophobic and hydrophilic regions. 

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