Diffusion cooperative

The diffusion coefficient Dq in the dilute solution limit gives the hydrodynamic radius of the isolated polymer chain in solution. The mutual diffusion coefficient 

As f decreases with an increasing concentration, the cooperative diffusion becomes 

The difference in the expression of the diffusion coefficient between the singlechain diffusion (Eq. 3.54) and the cooperative diffusion is only and The motional unit of size gives way to the blob size fas the concentration exceeds c.  

The reason why Aoop is given by Eq. 4.43 is as follows. In Section 3.2.7, we learned that the hydrodynamic radius of a linear chain polymer is given as the reciprocal of the average of where r is the distance between two monomers on the chain. We used the definition to estimate Re for a chain with a Gaussian chain conformation. We can use the same formula to calculate D oop for the cooperative dynamic mode of the blob. It is given by 

The decay rate E in the autocorrelation function gi(T) for this cooperative mode is proportional to k. It means that the monomer density fluctuation within the blob is diffusional. There is a good reason for this dynamic mode to be called the cooperative diffusion. 

---

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. 

Here the first term is the usual diffusive current, with Dc being the usual cooperative diffusion constant of the polymer molecule. The second term is a convective current due to the presence of induced electric field arising from all charged species in the system, p is the electrophoretic mobility of the polymer molecule derived in the preceding section. From the Poisson equation, we obtain... 

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. 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

The Anderson—Hyde dislocation model differs from the earher model based on the cooperative diffusion mechanism described by Andersson and Wadsley (1966), in which CS planes, e.g. in rutile, were diought to be produced by cation migration during reduction.The reduced oxygen potential at die surface means an enhanced Ti-potential and dierefore die Ti ions diffuse cooperatively into the crystal down diis Ti-potential gradient. However, diere is no experimental evidence to support this hypodiesis. This mechanism is also less hkely since diis would involve a large number of cations. 

Lastly, Callaghan and Pinder believed to have detected cooperative diffusion in their experiment48). Their polystyrene specimen of M = 2x 10 displayed a two-component echo attenuation, interpreted in terms of two time-dependent diffusion coefficients. The more rapid component was not observable at long diffusion times, and at short diffusion time (t = 16 ms) scaled with polymer concentration (2, 4, and 8 %) in the manner... 

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. 

K. Ratzke, P.W. Huppe, and F. Faupel. Transition from single-jump type to highly cooperative diffusion during structural relaxation of a metallic-glass. Phys. Rev. Lett., 68( 15) 2347-2349, 1992. 

This result has been quoted in the latest paper by Swalin (30) as a support for a cooperative diffusion theory. The physical image is also in agreement with the liquid transport model recently derived by Rahman (27) from computer experiments. 

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... 

Fig. 11 The dynamic structure factor C(, r) of polybutadiene star 12880 (nominally f = 128, Ma = 80kgmol ) in cyclohexane at ci = 0.016gmL and q = 0.035nm , along with the fit (solid line) from the ILT analysis. The corresponding relaxation distribution function L(ln(T)) (shown here for f i and q = 0.023gmL ) embraces the cooperative diffusion (1), the collective apparent diffusion (2), and the self-diffusion (3). The slowing-down of the middle structural mode (2) and the increase of its intensity with q are shown in the upper inset whereas the lower cartoon illustrates the liquid-like ordering [43,189]. The core regions are drawn out of scale (larger) for clarity...

---