Entanglement theories

Molecular theories, based in large part on ideas about chain entanglements, have been constructed to explain certain of these observations. The theories must still be regarded as tentative and incomplete. They are based, first of all, on reasonable but still incompletely accepted ideas about chain organization in concentrated solutions and melts. Secondly, they deal with the response of individual chains or pairs of chains in a smoothed medium, rather than with an entire interacting ensemble. Finally, they circumvent the deep mathematical difficulties of the central problem, interaction between mutually uncrossable sequences of chain elements, by approximations which are not easy to evaluate. The purpose of this review is to summarize the present status of entanglement theories and the data upon which they are based. 

Molecular Entanglement Theories of Linear Viscoelastic Behavior... 

Graessley.W.M. Molecular entanglement theory of flow behavior in amorphous polymers. J. Chem. Phys. 43, 2696-2703 (1965). 

This relation between y from recoil and the zero-shear-rate properties j0 and 60 from steady shear flow has also been obtained by Lodge [(46), p. 141] for his elastic liquid model which is derivable from a network entanglement theory his model does not, however, give the higher... 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

C2/C1 decreased with interchain distance (expressed as cross-sectional area per unit chain), appears consistent with both an entanglement theory and a local order theory. In the paper, arguments for both theories are critically examined. The suggestion given above that the relevant modulus should be C2/A would favour the entanglement theory rather than the local order theory. The reason is that an increase in A would decrease C2/A but would be expected to increase local order. 

This equation suggests that a unique curve should result if the steady-shear viscosity data of entangling polymers is plotted as versus yd. This is, indeed, found to be the case for solutions of narrow-molecular-weight-distribution polystyrene dissolved in n-butyl benzene these data are displayed in Figure 14.18 [30]. Note that the entanglement theory has also been extended to polydisperse polymers [30]. 

As is suggested frequently , this term might well result from the restriction of the hydrogen bonding possibilities experienced by the water molecules in the first hydration shell. For each individual water molecule this is probably a relatively small effect, but due to the small size of the water molecules, a large number of them are entangled in the first hydration shell, so that the overall effect is appreciable. This theory is in perfect agreement with the observation that the entropy of hydration of a nonpolar molecule depends linearly on the number of water molecules in the first hydration shell ". ... 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

Figure 2.13 Model of several orders of coupling through entanglements according to Bueche theory.

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

Inspection of Fig. 3.9 suggests that for polyisobutylene at 25°C, Ti is about lO hr. Use Eq. (3.101) to estimate the viscosity of this polymer, remembering that M = 1.56 X 10. As a check on the value obtained, use the Debye viscosity equation, as modified here, to evaluate M., the threshold for entanglements, if it is known that f = 4.47 X 10 kg sec at this temperature. Both the Debye theory and the Rouse theory assume the absence of entanglements. As a semi-empirical correction, multiply f by (M/M. ) to account for entanglements. Since the Debye equation predicts a first-power dependence of r) on M, inclusion of this factor brings the total dependence of 77 on M to the 3.4 power as observed. 

Vincent analyzed the tensile fracture stress o, of a broad range of polymers as a function of the number of backbone bonds per cross sectional area ( 2) and found a nearly linear relation, o 2, as shown in Fig. 12. 2 is related to via the theory of entanglements for random walk chains as [74]... 

Appropriately, this was called the Folded Chain Theory and is illustrated in Fig. A.ll. There are several proposals to account for the co-existence of crystalline and amorphous regions in the latter theory. In one case, the structure is considered to be a totally crystalline phase with defects. These defects which include such features as dislocations, loose chain ends, imperfect folds, chain entanglements etc, are regarded as the diffuse (amorphous) regions viewed in X-ray diffraction studies. As an alternative it has been suggested that crystalline... 

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