Self diffusion constants

The components of a microemulsion constantly self-diffuse. The process is intimately related to the internal structural state of the system. The surfactant and cosurfactanl molecules diffuse back and forth between the interfacial layer and the bulk, the water and oil molecules self-diffuse in the medium, and the droplets self-diffuse in the continuum. This dynamic process is influenced by the self-association or clustering of the droplets. Self-diffusion studies in microemulsions are therefore of considerable importance. 

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

We finish this section by comparing our results with NMR and incoherent neutron scattering experiments on water dynamics. Self-diffusion constants on the millisecond time scale have been measured by NMR with the pulsed field gradient spin echo (PFGSE) method. Applying this technique to oriented egg phosphatidylcholine bilayers, Wassail [68] demonstrated that the water motion was highly anisotropic, with diffusion in the plane of the bilayers hundreds of times greater than out of the plane. The anisotropy of... 

There are a number of NMR methods available for evaluation of self-diffusion coefficients, all of which use the same basic measurement principle [60]. Namely, they are all based on the application of the spin-echo technique under conditions of either a static or a pulsed magnetic field gradient. Essentially, a spin-echo pulse sequence is applied to a nucleus in the ion of interest while at the same time a constant or pulsed field gradient is applied to the nucleus. The spin echo of this nucleus is then measured and its attenuation due to the diffusion of the nucleus in the field gradient is used to determine its self-diffusion coefficient. The self-diffusion coefficient data for a variety of ionic liquids are given in Table 3.6-6. 

Mechanisms of micellar reactions have been studied by a kinetic study of the state of the proton at the surface of dodecyl sulfate micelles [191]. Surface diffusion constants of Ni(II) on a sodium dodecyl sulfate micelle were studied by electron spin resonance (ESR). The lateral diffusion constant of Ni(II) was found to be three orders of magnitude less than that in ordinary aqueous solutions [192]. Migration and self-diffusion coefficients of divalent counterions in micellar solutions containing monovalent counterions were studied for solutions of Be2+ in lithium dodecyl sulfate and for solutions of Ca2+ in sodium dodecyl sulfate [193]. The structural disposition of the porphyrin complex and the conformation of the surfactant molecules inside the micellar cavity was studied by NMR on aqueous sodium dodecyl sulfate micelles [194]. 

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].

Let us consider another situation where a force (or forces) is not compensated on a time average. Then the particles upon which the force is exerted become transported in the medium. This translocation phenomenon changes with time. Particle transport, of course, also occurs under equilibrium conditions in homogeneous media. Self-diffusion is a process that can be observed and its velocity can be measured, provided that a gradient of isotopically labelled species is formed in the system at constant composition. 

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...

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]. 

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

Figure 5 Relationship among loci of structural, dynamic, and thermodynamic anomalies in SPC/E water. The structurally anomalous region is bounded by the loci of q maxima (upward-pointing triangles) and t minima (downward-pointing triangles). Inside of this region, water becomes more disordered when compressed. The loci of diffusivity minima (circles) and maxima (diamonds) define the region of dynamic anomalies, where self-diffusivity increases with density. Inside of the thermodynamically anomalous region (squares), the density increases when water is heated at constant pressure. Reprinted with permission from Ref. 29.

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