Diffusion coefficient semidilute solution

Thus in salt-free semidilute solutions, the fast diffusion coefficient is expected to be independent of both N and c, although the polyelectrolyte concentration is higher than the overlap concentration. This remarkable result is in agreement with experimental data [31, 33, 34] discussed in the Introduction. Upon addition of salt, Df decreases from this value as given by the above formulas. 

Mechanistic Ideas. The ordinary-extraordinary transition has also been observed in solutions of dinucleosomal DNA fragments (350 bp) by Schmitz and Lu (12.). Fast and slow relaxation times have been observed as functions of polymer concentration in solutions of single-stranded poly(adenylic acid) (13 14), but these experiments were conducted at relatively high salt and are interpreted as a transition between dilute and semidilute regimes. The ordinary-extraordinary transition has also been observed in low-salt solutions of poly(L-lysine) (15). and poly(styrene sulfonate) (16,17). In poly(L-lysine), which is the best-studied case, the transition is detected only by QLS, which measures the mutual diffusion coefficient. The tracer diffusion coefficient (12), electrical conductivity (12.) / electrophoretic mobility (18.20.21) and intrinsic viscosity (22) do not show the same profound change. It appears that the transition is a manifestation of collective particle dynamics mediated by long-range forces but the mechanistic details of the phenomenon are quite obscure. 

The diffusion coefficient D in semidilute solutions decreases as a power law in concentration ... 

The semidilute diffusion coefficient can be written in terms of the Zimm diffusion coefficient of the chain Dz [Eq. (8.23) valid for diffusion in dilute solutions] and the overlap concentration (f> [Eq. (5.19)] ... 

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. 

The correlation length in semidilute solution can be experimentally determined by measuring the diffusion coefficient of very dilute colloidal spheres of various sizes, provided that the spheres do not interact with the polymers. Consider diffusion of a non-interacting sphere in a semidilute unentangled solution. 

The diffusion coefficient in semidilute polymer solutions is determined from the fact that the chain diffuses a distance of order of its own size in its reptation time ... 

Consider a solution of polystyrene with molar mass M= 0°g moP in cyclohexane at 35 °C (0-solvent with viscosity ris = 7.6 x 10 Pa s). Estimate the relaxation time, plateau modulus, viscosity, and diffusion coefficient as functions of concentration in semidilute solution. 

The diffusion coefficients that can be measured with the PGSE method cover the range from fast diffusion of small molecules in solutions with D values typically around 10" m /s to very slow diffusion of, for instance, polymers in the semidilute concentration regime, where D values down to 10 m /s can be measured [19]. Measurements of such very slow diffusion requires gradients of extreme magnitudes and places severe demands on the actual experimental setup [9]. What often limits the lowest value of D that can be measured is the value of spin-spin relaxation time, T2. As a general rule, slow diffusion is often found in systems that also show rapid transverse relaxation. As a consequence, the echo intensity gets severely damped by T2 relaxation in such systems. For microemulsion systems, such problems are virtually nonexistent for the solvents, while for the surfactant molecules the accuracy is often reduced because of T2 effects. 

Furthermore, the concentration dependence of the apparent diffusion coefficient has been investigated. The determination of the hydrodynamic radius from the diffusion coefficient is valid only when pure self-diffusion is measured, that is, when the experiment is performed in the dilute concentration range. In the semidilute or concentrated regime, interactions between individual solute molecules have to be considered. Whether interactions... 

The van t Hoff Law for osmotic pressure (Equation 5.19) depends explicitly on the molecular weight of the solute. One of the most remarkable properties of semidilute solutions is that the osmotic pressure is observed to be independent of the molecular weight of the macromolecules. Another property fliat is observed to be independent of molecular weight in semidilute solution is the mutual-diffusion coefficient, DJ c). These phenomena are discussed in Section 6.2 and explained in more detail in Section 6.3. 

The mutual-diffusion coefficient in semidilute polymer solutions can be expressed as ... 

Application of the model of a semidilute solution as a collection of regions of size can be made to both the osmotic pressure and the mutual-diffusion coefficient. It has been proposed by deGermes that the osmotic pressure should scale as the number density of screening regions ... 

In Section 3.2, we learned that dynamic light scattering (DLS) measures the mutual diffusion coefficient and that it increases with an increasing polymer concentration in the good solvent. We extend it here to the semidilute solution. Figure 4.27... 

Figure 4.27. Diffusion coefficient measured in DLS for various concentrations of a polymer in a good solvent is schematically shown as a function of polymer concentration c. In the low concentration limit, it is Dq> the diffusion coefficient of an isolated chain. With an increasing concentration, the mutual diffusion coefficient increases linearly, followed by a sharp upturn to a crossover to the cooperative diffusion coefficient Dc p in the semidilute solution. The latter increases in a power law with an exponent close to 3/4.

In the semidilute solution, the hydrodynamic interactions are shielded over the distance beyond the correlation length, just as the excluded volume is shielded. We can therefore approximate the dynamics of the test chain by a Rouse model, although the motion is constrained to the space within the tube. In the Rouse model, the chain as a whole receives the friction of N, where is the friction coefficient per bead. When the motion is limited to the curvilinear path of the primitive chain, the friction is the same. Because the test chain makes a Rouse motion within the tube, only the motion along the tube survives over time, leading to the translation of the primitive chain along its own contour. The one-dimensional diffusion coefficient for the motion of the primitive chain is called the curvilinear diffusion coefficient. It is therefore equal to Dq of the Rouse chain (Eq. 3.160) and given by... 

In the previous section, we describe the general method for measuring diffusion coefficients of a polymer solution by laser light scattering. We now list three techniques for measuring diffusion coefficient specifically for polymers in semidilute solutions. 

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