Polymer diffusion coefficient related

JL Duda, YC Ni, JS Vrentas. An equation relating self-diffusion and mutual diffusion coefficients in polymer-solvent systems. Macromolecules 12 459-462, 1979. 

Dynamic light-scattering, sometimes called quasi-elastic light scattering or photon correlation spectroscopy, can be used to measure the diffusion coefficients of polymer chains in solution and colloids, a kind of Doppler effect see Section 3.6.6. In a dilute dispersion of spherical particles, the diffusion coefficient D is related to the particle radius, a, through the Stokes-Einstein equation. 

Duda, J. L., Y. C. Ni, and J. S. Vrentas, An Equation Relating Self-Diffusion and Mutual Diffusion Coefficients in Polymer-Solvent Systems, Macromolecules, 12, 459 62, 1979. 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

The kinetics of transport depends on the nature and concentration of the penetrant and on whether the plastic is in the glassy or rubbery state. The simplest situation is found when the penetrant is a gas and the polymer is above its glass transition. Under these conditions Fick s law, with a concentration independent diffusion coefficient, D, and Henry s law are obeyed. Differences in concentration, C, are related to the flux of matter passing through the unit area in unit time, Jx, and to the concentration gradient by,... 

By combining these two expressions, one can relate the diffusion coefficients to the polymer volume fraction ... 

Shimizu, T. and Kenndler, E., Capillary electrophoresis of small solutes in linear polymer solutions Relation between ionic mobility, diffusion coefficient and viscosity, Electrophoresis, 20, 3364, 1999. 

Polymer gels and ionomers. Another class of polymer electrolytes are those in which the ion transport is conditioned by the presence of a low-molecular-weight solvent in the polymer. The most simple case is the so-called gel polymer electrolyte, in which the intrinsically insulating polymer (agar, poly(vinylchloride), poly(vinylidene fluoride), etc.) is swollen with an aqueous or aprotic liquid electrolyte solution. The polymer host acts here only as a passive support of the liquid electrolyte solution, i.e. ions are transported essentially in a liquid medium. Swelling of the polymer by the solvent is described by the volume fraction of the pure polymer in the gel (Fp). The diffusion coefficient of ions in the gel (Dp) is related to that in the pure solvent (D0) according to the equation ... 

Although these examples demonstrate the feasibility of using calculated values as estimates, several constraints and assumptions must be kept in mind. First, the diffusant molecules are assumed to be in the dilute range where Henry s law applies. Thus, the diffusant molecules are presumed to be in the unassociated form. Furthermore, it is assumed that other materials, such as surfactants, are not present. Self-association or interaction with other molecules will tend to lower the diffusion coefficient. There may be differences in the diffusion coefficient for molecules in the neutral or charged state, which these equations do not account for. Finally, these equations only relate diffusion to the bulk viscosity. Therefore, they do not apply to polymer solutions where microenvironmental viscosity plays a role in diffusion. 

Various diffusion coefficients have appeared in the polymer literature. The diffusion coefficient D that appears in Eq. (3) is termed the mutual diffusion coefficient in the mixture. By its very nature, it is a measure of the ability of the system to dissipate a concentration gradient rather than a measure of the intrinsic mobility of the diffusing molecules. In fact, it has been demonstrated that there is a bulk flow of the more slowly diffusing component during the diffusion process [4], The mutual diffusion coefficient thus includes the effect of this bulk flow. An intrinsic diffusion coefficient, Df, also has been defined in terms of the rate of transport across a section where no bulk flow occurs. It can be shown that these quantities are related to the mutual diffusion coefficient by... 

The permeation technique is another commonly employed method for determining the mutual diffusion coefficient of a polymer-penetrant system. This technique involves a diffusion apparatus with the polymer membrane placed between two chambers. At time zero, the reservoir chamber is filled with the penetrant at a constant activity while the receptor chamber is maintained at zero activity. Therefore, the upstream surface of the polymer membrane is maintained at a concentration of c f. It is noted that c f is the concentration within the polymer surface layer, and this concentration can be related to the bulk concentration or vapor pressure through a partition coefficient or solubility constant. The amount... 

In general, mass transfer processes involving polymer-penetrant mixtures are generally analyzed by using a mutual diffusion coefficient. Therefore, a relationship between the mutual diffusion coefficient, D, and self-diffusion coefficients, ZVs is needed. Vrentas et al. [30] proposed an equation relating D to D, for polymer-penetrant systems in which Dx is much larger than Dr. 

For hydrophilic and ionic solutes, diffusion mainly takes place via a pore mechanism in the solvent-filled pores. In a simplistic view, the polymer chains in a highly swollen gel can be viewed as obstacles to solute transport. Applying this obstruction model to the diffusion of small ions in a water-swollen resin, Mackie and Meares [56] considered that the effect of the obstruction is to increase the diffusion path length by a tortuosity factor, 0. The diffusion coefficient in the gel, )3,i2, normalized by the diffusivity in free water, DX1, is related to 0 by... 

Although many different processes can control the observed swelling kinetics, in most cases the rate at which the network expands in response to the penetration of the solvent is rate-controlling. This response can be dominated by either diffu-sional or relaxational processes. The random Brownian motion of solvent molecules and polymer chains down their chemical potential gradients causes diffusion of the solvent into the polymer and simultaneous migration of the polymer chains into the solvent. This is a mutual diffusion process, involving motion of both the polymer chains and solvent. Thus the observed mutual diffusion coefficient for this process is a property of both the polymer and the solvent. The relaxational processes are related to the response of the polymer to the stresses imposed upon it by the invading solvent molecules. This relaxation rate can be related to the viscoelastic properties of the dry polymer and the plasticization efficiency of the solvent [128,129],... 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

The solubility of ethylene in freshly prepared polyethylene, and its diffusion out of the latter were studied in relation to the formation of explosive ethylene-air mixtures in storage. Explosive mixtures may be formed, because the solubility of ethylene in its polymer (e.g. 1130 ppm w/w at 30°C) considerably exceeds the concentration (30 ppm at 30°C) necessary to exceed the lower explosive limit above the gas-containing polymer in closed storage, and the diffusion coefficient is also 30% higher than for aged polymer samples. 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

Here Tq is — C2 and is a prefactor proportional to which is determined by the transport coefficient (in this case at the given reference temperature. The constant B has the dimensions of energy but is not related to any simple activation process (Ratner, 1987). Eqn (6.6) holds for many transport properties and, by making the assumption of a fully dissociated electrolyte, it can be related to the diffusion coefficient through the Stokes-Einstein equation giving the form to which the conductivity, <7, in polymer electrolytes is often fitted,... 

For comparison purposes, the proton mobility. Do (for Nafion solvated with water), which is closely related to the self-diffusion coefficient of water, is also plotted. At low degrees of hydration, where only hydrated protons (e.g., H3O+) are mobile, it has a tendency to fall below the water diffusion coefficient (this effect is even more pronounced in other polymers), which may be due to the stiffening of the water structure within the regions that contain excess protons, as discussed in Section 3.1.1. . Interestingly, the proton mobility in Nafion solvated with methanol (Da(MeOH) in Figure 14a) is even lower than the methanol self-diffusion (Z ieon). This may... 

Further, the electron-transport properties of the polymer 7 were enhanced by extending the separation between the redox center and backbone from a single Os—amino linkage to one that extends over 17 bonds. The goal was to provide mobility of the redox center independently of backbone motion, which is necessarily restricted by cross-linking. The mobility of the redox center can be characterized by an apparent diffusion coefficient, Z app- According to the relation proposed by Blauch and Saveant ... 

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