Diffusion coefficient cooperative

As seen in the preceding section, the counterions play a crucial role in the mobility of the polyelectrolyte molecules. Even in the absence of an external electric field, the counterions exert an induced electric field in the immediate environment of a charged segment which in turn significantly modifies the collective diffusion coefficient of the polymer. This additional contribution is absent for uncharged polymers, where the cooperative diffusion coefficient Dc is given by the Stokes-Einstein law in dilute solutions. 

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... 

The cooperative diffusion coefficient in the salt-free limit is thus strongly 7 -dependent. In this limit the equilibrium scattering function g k) exhibits a peak atk =. Approximating g k) in Eqs. (282) and (285) by its value at the peak... 

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. 

Here D is cooperative diffusion coefficient of the gel. Such a relationship applies to the random or diffusive motions of molecules in a fluid for example, ink molecules in water. It is interesting that the same relation holds for a polymer network even though all the polymers are connected into a single network. 

However, for non-dilute systems, the diffusion coefficient obtained from the low q time dependence of S q, t) may not be the diffusion coefficient of the polymers. For example, in semidilute solutions the dominant decay in S q, t) corresponds to correlations disappearing at the scale of the correlation length. In such cases, the diffusion coefficient is called the cooperative diffusion coefficient. 

In the semi-dilute solution where the molecules overlap, the diffusion coefficient may be the cooperative diffusion coefficient of entangled chain and may be related to the correlation length, by... 

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). 

Table 1 Cooperative diffusion coefficients determined from the time dependence of gel geometry...

This is followed by a change of the degree of swelling (second step), which is governed by the above discussed cooperative diffusion. Typically, the cooperative diffusion coefficient Dcoop is in the order of 10 cm /s, i.e., about one hundredth of the mass transfer. 

The swelling/shrinking rate dQ/dt and hence the response time t for any application during volume phase transitions of smart hydrogels depends on the cooperative diffusion coefficient Dcoop and on the square of their characteristic dimension /,... 

It is convenient to describe the average decay rate of g " t) by its first cummulant r which is correlated to the cooperative diffusion coefficient D oop of the network chains ... 

Table 2 Procedure for calculating the cooperative diffusion coefficient D oop hydrogel...

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 ... 

Body force of component i Polymer mass concentration Macroscopic diffusion coefficient Cooperative diffusion coefficient Stokes-Einstein diffusion coefficient Rouse diffusion coefficient Self diffusion coefficient... 

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