Coupled diffusion coefficient, polyelectrolyte

The Dependencies of Radius of Gyration Rg, Static Correlation Length Hydrodynamic Screening Length Viscosity r, Self-Translational Diffusion Coefficient D, Cooperative Diffusion Coefficient Dc, Coupled Diffusion Coefficient Df, and Electrophoretic Mobility p on c and N for Various Regimes of Polyelectrolyte and Salt Concentrations... 

Therefore, remarkably, the coupled diffusion coefficient becomes independent of N and c in the Rouse regime of salt-free polyelectrolyte solutions. This is to be... 

Combining Equations 7.79 through 7.82 in the limit of ik 0 (corresponding to very large probe lengths), the coupled diffusion coefficient of the polyelectrolyte chain in dilute solutions becomes... 

But p decreases with salt concentration with an apparent exponent of k which changes from 0 at low salt concentration to — at high salt concentrations. The N-independence of p arises from a cancellation between hydrodynamic interaction and electrostatic coupling between the polyelectrolyte and other ions in the solution. It is to be noted that the self-translational diffusion coefficient D is proportional to as in the Zimm model with full... 

We have identified three diffusion coefficients. These are the self-translational diffusion coefficient D, cooperative diffusion coefficient Dc, and the coupled diffussion coefficient fly. fl is the cooperative diffusion coefficient in the absence of any electrostatic coupling between polyelectrolyte and other ions in the system, fly is the cooperative diffusion coefficient accounting for the coupling between various ions. For neutral polymers, fly and Dc are identical. Furthermore, we identify fly as the fast diffusion coefficient as measured in dynamic light scattering experiments. The fourth diffusion coefficient is the slow diffusion coefficient fl discussed in the Introduction. A satisfactory theory of flj is not yet available. 

When the characteristic time for charge diffusion is lower than the experiment timescale, not all the redox sites in the film can be oxidized/reduced. From experiments performed under these conditions, an apparent diffusion coefficient for charge propagation, 13app> can be obtained. In early work choroamperometry and chronocoulometry were used to measure D pp for both electrostatically [131,225] and covalently bound ]132,133] redox couples. Laviron showed that similar information can be obtained from cyclic voltammetry experiments by recording the peak potential and current as a function of the potential scan rate [134, 135]. Electrochemical impedance spectroscopy (EIS) has also been employed to probe charge transport in polymer and polyelectrolyte-modified electrodes [71, 73,131,136-138]. The methods... 

From a theoretical point of view, charge propagation in films containing space-distributed redox centers can be achieved either by the physical displacement of the sites or by the transference (hopping) of electrons from neighboring reduced to oxidized sites or by the combination of both processes. In the case of free diffusing couples immobilized in oppositely charged polyelectrolytes, both processes occur and an apparent diffusion coefficient can be defined and measured [136, 142, 143] ... 

The behavior of polyelectrolyte molecules cannot be interpreted as arising solely from the polymer. The coupled behavior of the chain and its counterion cloud must be treated together. The counterions exert an electric field, even in the absence of any externally imposed electric field, in the immediate environment of a charged segment. This field in turn modifies the collective diffusion coefficient of the polyelectrolyte. The polyelectrolyte molecule must be treated not as an individual particle but as a quasiparticle along with its counterion cloud. 

Therefore, the coupling of polyelectrolyte chain to the counterion cloud, which is directly responsible for the aN term in the above equation, dominates the experimentally observed diffusion coefficient. Since Dc URg l/N, for... 

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