Unperturbed real chain

Monte-Carlo simulations are applied to estimate the characteristic ratios and p parameters from the RIS models for PE, POM, polybutadiene, and polyisoprene. Here the p parameter is defined as the ratio of the radius of gyration to the hydrodynamic radius. The p parameters of these real chains in the unperturbed state show only a slight dependence on the microconformation in the limit of large molecular weights and are found close to 1.504, which is the value for an idealized Gaussian chain. The estimated p parameters of the real chains appear to be correlated to the chain stiffness and increase with the characteristic ratios. 

The Unperturbed Real Chain The Self-Consistent Approach... 

In the next section we shall consider the equilibrium properties of some typical models of unperturbed chains with an increasing degree of complexity. They are (i) the bead-and-spring phantom chain (ii) the phantom chain with nearest-neighbor correlation and (iii) the unperturbed real chain... 

In the preceding, oc iq) (without the tilde) is the mean-square strain ratio of the q mode with respect to the unperturbed real chain with screened interactions (i.e., at r = 0), not with respect to the phantom chain accordingly, cco(q) s 1. In agreement with our assumption of a small solvent strength, will be proportional to T—, and we shall write... 

It should be commented that two distinct readjustments of the reference 0 state are implicit in Eqn. (2.2.4), both relating to the change of the lower limit of integration from k to pN. When applied to the integral of the screened interactions [last integral in Eqn. (2.2.4)], this change reflects the adoption of the unperturbed real chain instead of the phantom chain. When applied to the three-body integral [see Eqn. (2.2.7)], it implies a small shift of the 0 temperature from the phantom chain value... 

Kuhn monomers with Kuhn length h. Recall that the unperturbed size of the real chain confined to the air-water interface in good solvent is hA [Eq. (3.54)],... 

Similarly the RMS radius of gyration for perturbed dimensions in real chains, is greater than that for unperturbed dimensions. 

The Gaussian chain expanded in this way is then the best Gaussian approximation to the real chain. This equation shows that as du tends to zero, a tends to 1, so that the chain retains its unperturbed value of /"nns i-S-the best Gaussian approximation is then the unperturbed Gaussian. On the other hand, if du becomes large, a becomes proportional to and /"nns becomes proportional to n, in agreement with equation (3.12). A solvent at a temperature T = 0 is called a theta-solvent. 

The idea that a chain in a melt is effectively a free chain without selfinteraction was first clearly expressed by Flory and is often called the Flory theorem. At first sight it is paradoxical how can the presence of other chains allow a chosen chain to take up conformations that it could not take up if the other chains were absent The answer is that it cannot. The unperturbed chain is only equivalent to the real chain in terms of its Gaussian statistics. This is nevertheless very important, because it is the statistical properties of the chains and their link with the entropy that largely determine some of the properties of the corresponding materials. 

Sepai ation of chain ends in both the Gaussian and real chains is much lower than the value Nb (1) and, therefore, the most probable spatial arrangement of the individual disordered macromolecule may exist it is the random coil. Real unperturbed chains possess less flexibility and, hence, longer end-to-end distance, compared to the mty-chain. A real chain is characteris-... 

Obviously, the temperature dependent parameter indicates the difference between the real chain and rwm structure. A review of the Cpf values for a great number of polymers can be found in reference [11]. The unperturbed real chain exhibits swelled conformations (in contrast to the rwm approximation) due to the iutra-molecular short-range interactions and almost fixed bond angles. This consequently leads to the temperature dependent end-to-end distance and other peculiarities such as the non-zero energy term in the rubber elastic response of real polymers [12]. 

A fully extended chain, without distortion of bond angles or deformation of bonds, is represented in Figure 9(b). In this conformation, the value of the end-to-end distance is nZp, where Zp is the length of the bond vector projected on the chain axis. If the equivalent random chain is now required to have the same end-to-end distance in full extension and also the same mean-square displacement length as the real chain, then equations (33) and (34) result. The maximum extension ratio of unperturbed macromolecules may be defined as the ratio of the fully extended length nZp to the root-... 

The simplest model of this type is called the freely jointed chain, and is illustrated in Figure 2.21. In it, the skeletal bonds are joined end to end, but are completely unrestricted in direction. This is clearly a situation not found in a real polymer (bond angles in real polymers are relatively fixed). It is also assumed that the chains have zero cross-sectional area, that is that the chains are unperturbed by excluded-volume effects. These effects arise because atoms of a chain exclude from the space they take up all other atoms from all other chains. They are related to excluded-volume effects occurring even in systems as simple as real gases. The expression for the mean-square end-to-end distance of such an idealized chain is particularly simple ... 

Figure 10. Real and imaginary part of complex modulus, G and G —cut/j, vs. loor for atactic polystyrene in dilute solution Oj- is shift factor. Experimental points are from refs. 102 and 103 best-fit continuous lines [from Eqn. (34) of ref. 12] superimposed on experimental points after rigid, parallel shift. [Model assumptions and parameters unperturbed periodic chain, N = 8000 (left) and N = 1300 (right), to/to = 47, R ff = 0.125 A.] (Reprinted with permission from ref. 12, Copyright 1981, American Chemical Society.)...

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