Coil model viscosity

Fig. 9.— Double Log Plots of (a) Intrinsic Viscosity, (b) the Reciprocal of the Diffiision Coefficient, and (c) Sedimentation Coefficient Data versus Molecular Weight for Human Cervical Mucins. [Key and O, whole mucins and , subunits and A, T-domains. Molecular weights determined from Zimm plots (filled symbols) or the Svedbeig equation using QLS (open symbols). Values for the slopes are in all cases consistent with a random-coil model and not with a rigid sphere or a rod.]...

To describe the intrinsic viscosity of wormlike coils in the absence of excluded volume, Yamakawa and co-workers developed theoretical descriptions based, first, on the KP model [Yamakawa and Fujii, 1974] and subsequently on its later adaptation, the HW model [Yoshizaki et al., 1988]. Using the cylindrical wormlike coil model, Yamakawa and Fujii [1974] obtained the following expressions, with L expressed in... 

A major advantage of the random coil model, interestingly, is its simplicity. By not assuming any particular order, the random coil has become amenable to extensive mathematical development. Thus, detailed theories have been developed including rubber elasticity (Chapter 9) and viscosity behavior (Section 3.8), which predict polymer behavior quite well. By difference, little or no analytical development of the other models has taken place, so few properties can be quantitatively predicted. Until such developments have taken place, their absence alone is a strong driving force for the use of the random coil model. 

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

From the weak dependence of ef on the surrounding medium viscosity, it was proposed that the activation energy for bond scission proceeds from the intramolecular friction between polymer segments rather than from the polymer-solvent interactions. Instead of the bulk viscosity, the rate of chain scission is now related to the internal viscosity of the molecular coil which is strain rate dependent and could reach a much higher value than r s during a fast transient deformation (Eqs. 17 and 18). This representation is similar to the large loops internal viscosity model proposed by de Gennes [38]. It fails, however, to predict the independence of the scission yield on solvent quality (if this proves to be correct). 

Before discussing details of their model and others, it is useful to review the two main techniques used to infer the characteristics of chain conformation in unordered polypeptides. One line of evidence came from hydrodynamic experiments—viscosity and sedimentation—from which a statistical end-to-end distance could be estimated and compared with values derived from calculations on polymer chain models (Flory, 1969). The second is based on spectroscopic experiments, in particular CD spectroscopy, from which information is obtained about the local chain conformation rather than global properties such as those derived from hydrodynamics. It is entirely possible for a polypeptide chain to adopt some particular local structure while retaining characteristics of random coils derived from hydrodynamic measurements this was pointed out by Krimm and Tiffany (1974). In support of their proposal, Tiffany and Krimm noted the following points ... 

Although shear rate effects are more pronounced in good solvents, the intrinsic viscosity decreases with shear rate even in 0-solvents, where excluded volume is zero (317,318). The Zimm model employs the hydrodynamic interaction coefficients in the mean equilibrium configuration for all shear rates, despite the fact that the mean segment spacings change with coil deformation. Fixman has allowed the interaction matrix to vary in an appropriate way with coil deformation (334). The initial departure from [ ]0 was calculated by a perturbation scheme, and a decrease with increasing shear rate in 0-systems was predicted to take place in the vicinity of / = 1. 

Thus, the observations made at the beginning of this section [see eqs. (5.10) and (5.11a)] with respect to coil molecules and rigid rods, are confirmed for the behaviour of the dumb-bell models. In particular, a comparison of eqs. (5.18) and (5.22) shows that, other than for the intrinsic viscosity, the Maxwell constant can only be calculated when, besides (hi) also (hi) is known. This remark will be of importance for the next section, where theories for short chain molecules will be discussed. 

As the results of Schwander and Cerf (188) and of Leray (180,181), which were obtained on samples of DNA from calf thymus, have been reproduced already in several review articles (1,3), the present discussion can be kept rather short. When the viscosity of the solvent (1.0 molar aquous solution of sodium chloride) is increased by replacing part of the water by glycerol, the behaviour of the initial slope of the extinction angle curve follows the qualitative pattern, as given by Fig. 5.10. At low solvent viscosities the molecules seem to behave like frozen molecules, at high solvent viscosities they seem to become flexible coils exhibiting internal friction. These results have been considered to prove the correctness of Cerf s theoretical model. 

Theories of the frictional properties of polymer solutions, from which the relationship between viscosity and molecular weight must ultimately be derived, can proceed in various ways, but we will mention just two models. In the first, it is assumed that the velocity of the solvent is barely affected by the presence of the polymer, so that it streams or freely drains through the coil in a largely unperturbed fashion (Figure 12-31). 

The concept of polymer entanglements represents intermolecular interaction different from that of coil overlap type interaction. However, it is difficult to define the exact topological character of entanglements. The entanglements concept was aimed at understanding the important nonlinear rheological properties, such as the shear rate dependence of viscosity. However, viscoelastic properties could not be defined quantitatively as is possible with the reptation model. Because an entanglement should be... 

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