Cooperative diffusion coefficient equation

Another way to determine the cooperative diffusion coefficient is to measure the degree of swelling of a sample with a defined geometry as function of time. The experimental data were evaluated by a solution of the diffusion equation based on the equation of motion for a given sample geometry. In the following this is demonstrated on example of swelling of a spherically symmetrical gel (shear modulus G = 0). 

The Stokes-Einstein equation relates the cooperative diffusion coefficient (Dc) to the bare dynamic correlation length ( d ) defined in terms of the temperature-dependentviscosity ofthe solvent (ri )in Eq. 12 ... 

In the linearized form, Eqs. 41 and 43 describe the time dependence of the displacement field u and the pressure field p. The Tanaka-Fillmore Eq. 27 can be obtained from these equations under the assumption of the special case that the polymer deformation velocity field and the solvent velocity field are in exactly opposite directions for every point in the whole space. The cooperative diffusion coefficient then has the form... 

Here, Dc is the cooperative diffusion coefficient of the chain if the couphng to the counterion cloud were to be absent. In dilute solutions, Dc is inversely proportional to the radius of gyration, is the velocity of the polyelectrolyte chains and ii is the electrophoretic mobility of the polyelectrolyte chain (discussed in Section 7.4). From the Poisson equation (Equation 3.9), we get... 

Self-diffusivity, cooperatively with ionic conductivity, provides a coherent account of ionicity of ionic liquids. The PGSE-NMR method has been found to be a convenient means to independently measure the self-diffusion coefficients of the anions and the cations in the ionic liquids. Temperature dependencies of the self-diffusion coefficient, viscosity and ionic conductivity for the ionic liquids, cannot be explained simply by Arrhenius equation rather, they follow the VFT equation. There is a simple correlation of the summation of the cationic and the anionic diffusion coefficients for each ionic liquid with the inverse of the viscosity. The apparent cationic transference number in ionic liquids has also been found to have dependence on the... 

Dcoop is called the cooperative difiliision coefficient. Note, ffiough Eq. 29 maffi-ematically is exactly the radial term of die diffiision equation in spherical coordinates, this equation was physically not derived fiom diffusion ffieory. Typical values of Dcoop for neutral hydrogels are lO cm /s (Gehrice 1993), and typical values for polyelectrolyte hydrogels are (510. .. 10 )cm /s (Skouri et al. 1995 Milimouk et al. 2001). 

Dc is called the cooperative or mutual diffusion coefficient, as thermodynamic correlations among all chains are used in writing the above equations. In contrast, the tracer diffusion coefficient Dt measures the diffusion of a tagged polymer. Defining the fluctuation in the local monomer density 8cp(r, f) as the deviation of Cp (r, t) from its average value Cp = c, combining Equations 7.59 and 7.60, and taking the Eourier transforms yield... 

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