Excluded volume effect, scaling laws

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Since Nk, the number of steps in the chain, is proportional to the polymer mo-leculaj weight, Eq. (3-2) implies that the root-mean-square end-to-end separation distance, R )q, of an undistorted coil, scales with molecular weight M according to a power law, R fJ oc M , with V = 0.5. Although this result applies in the melt state and for concentrated solutions, note that in dilute solutions with a good solvent the value of v is as large as 0.6 because of excluded-volume effects, as discussed in Section 2.2.3.3. 

For high concentrations the dynamics of the chains are the same as in melts. Excluded volume effects and the hydrodynamic interaction are not important. However, both the radius of the tube and the friction coefficient strongly depend on the concentration. Because the depedence of on G is nontrivial, there are no scaling laws relating x ax nnd r o lo the concentration. For the plateau modulus, it was found experimentally that (8)... 

Such behaviour is reasonable when the chains have a large overlap. However, this formula ignores the modification of critical exponents resulting from excluded volume effects and therefore it cannot lead to proper scaling laws. 

For a particular solvent at the theta temperature 0, the excluded volume effect can be neglected, and the polymer will behave like an ideal chain with a scaling exponent of v = 0.5. This temperature can be considered analogous to the Boyle temperature in a gas. At the Boyle temperature, a nonideal gas will obey the ideal gas law, and the effects of molecular volume can be neglected. 

Our considerations qualitatively explain the results of Fig. 3.16 The dissolution of salt in addition to the polyelectrolyte suppresses the osmotic pressure contribution by the counter-ions and transforms the stiffened polyelectrol3de chain into a much more flexible quasi-neutral chain. In the absence of these polyelectrolyte characteristics, one recovers the behavior of neutral systems and, therefore, in the semidilute range the associated scaling law Eq. (3.41). Equation (3.136) correctly describes the general tendencies, but is not an accurate expression. First, in view of the complex structures in polyelectrolytes with shell formations and screening effects, equilibria have to be expressed in terms of activities rather than concentrations. Furthermore, the Donnan expression is not the only contribution to the second virial coefficient. There exists another part, AA2, which accounts as usually for excluded volume effects, the quality of the solvent and the peculiar ordering phenomena found in polyelectrolyte solutions. Therefore, in general, one has to write... 

A more direct calculation has been proposed by Muthukumar and Edwards, including excluded volume effects directly and using mode coupling arguments in equation (72), and leads to the same scaling laws for... 

Let us discuss briefly the changes in the pair correlation function g(r) that occur when we incorporate the effects of excluded volume. First, g(r) and its Fourier transform g (q) follow simple scaling laws. For instance... 

Excluded volume interactions become effective for distances larger than f We may account for this behavior in the model by assuming that the subunits cannot interpenetrate each other. The chain of subunits then displays the properties of an expanded chain, and we express its size with the aid of the scaling law Eq. (2.83), identifying op with and N with Ngu-... 

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