Free draining model

A plot of A versus r, the calibration curve of OTHdC, is shown in Fig. 22.2. The value of constant C depends on whether the solvent/polymer is free draining (totally permeable), a solid sphere (totally nonpermeable), or in between. In the free-draining model by DiMarzio and Guttman (DG model) (3,4), C has a value of approximately 2.7, whereas in the impermeable hard sphere model by Brenner and Gaydos (BG model) (8), its value is approximately 4.89. 

The pseudoforce associated with the dynamical projection tensor may be calculated by using dynamical reciprocal vectors to evaluate Eq. (2.205). In the simple case of a coordinate-independent mobility as in a free-draining model or a model with an equilibrium preaveraged mobility, we may use Eq. (A. 17) to express as a derivative... 

A free-draining model with equal bead friction coefficients and W = const, such as a stiff bead-spring polymer, may be efficiently simulated with... 

Free-draining models were among the first to be considered [14-18]. For flexible polymer chains of sufficient length, [77] behaves as if the polymer coil occupied a spherical volume through which the solvent cannot flow. Under these conditions,... 

What is meant by free-draining model and nondraining model in the case of viscosities of polymer solutions ... 

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by < > is performed over all macromolecules in the system. Thus, < i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory ... 

These results make it clear that the forms of t]0 — rjs and Je° are completely independent of model details. Only the numerical coefficient of Je° contains information on the properties of the model, and even then the result depends on both molecular asymmetry and flexibility. Furthermore, polydispersity effects are the same in all such free-draining models. The forms from the Rouse theory cany over directly, so that t]0 - t]s, translated to macroscopic terms, is proportional to Mw and Je° is proportional to the factor A/2M2+, /A/w. Unfortunately, no such general analysis has been made for models with intramolecular hydrodynamic interaction, and of course these results apply in principle only to cases where intermolecular interactions are negligible. 

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

Originally, Fox and Flory (121) found that the zero shear viscosity of polymer melts increases with the 3.4-th power of the molecular weight. Bueche (122) has shown that this relationship holds only above a certain critical molecular weight Mc which depends on the structure of the polymer chain. Below Mc the zero shear viscosity is found to depend on a significantly lower power of molecular weight. A theoretical interpretation of these facts has been given by the latter author on the basis of the free-draining model (Section 3.4.1.). 

The quantity of interest in connection with flow birefringence is the reduced steady-state compliance. It is easily shown that for the free-draining model eq. (3.40) can be approximated by ... 

For the calculation of the Maxwell-constant an assembly of frozen random conformations is considered. Brownian motion is taken into account only so far as rotary diffusion of the rigid conformations is concerned. In this way a first order approximation of the distribution function with respect to shear rate is obtained. This distribution function is used for the calculation of the Maxwell-constant, [cf. the calculation of the Maxwell-constant of an assembly of frozen dumb-bell models, as sketched in Section 5.I.3., eq. (5.22)]. Intrinsic viscosity is calculated for the same free-draining model, using average dimensions [cf. also Peter-lin (101)]. As for the initial deviation of the extinction angle curve from 45° a second order approximation of the distribution function is required, no extinction angles are given. 

As was the case with the free-draining model, the relaxation spectrum of the first, collective modes is flat, whereas it is unchanged with respect to the unperturbed state for more localized modes (see Figure 6). With the open chain, analytical difficulties prevent us from obtaining a closed-form solution. Numerical calculations show that the same results hold for the open chain... 

For polystyrene, the value of D obtained from Eq. (1), however, agrees very well with that based on the tube model, although Eq. (1) is presumably derived from the free draining model. 

A number of experimental methods exist that allow polymer solutions to be subjected to different shear rates or to oscillatory shear. Data obtained over a given range of shear rate, or frequency, are shifted to form a universal curve (as in the use of the Williams-Landel-Ferry equation, explained in Chapter 4). This can then be compared with the predictions of various models such as those proposed by Rouse or Zimm. The former assumes that there is minimal interaction between the solvent and the polymer, and is sometimes referred to as the free draining model. In reality, there is some interaction between the solvent and the polymer chain. This is addressed in the Zimm model, where the drag introduced by the solvent influences the motion of the chains. 

Though the diffusion model predicts very well the time-dependence of the rehealing experiments, there are still many unanswered questions, such as the absolute value of D, the roles of chain-ends and of molecular weight, the influence of the relaxing fibrils derived from the fracture event, and the nature of the physical links. The relationships of D to viscoelastic and structural parameters still rely on the macroscopic formulation such as the Buche-Cashin-Debye s equation as in Eq.(1), which was derived from the free draining model for both entangled and... 

Thus, the free draining model predicts that the exponent a of molar mass is equal to 1 for chains without excluded volume as in the Staudinger empirical formula. Except for a few cases, this model is not very realistic because it neglects hydrodynamic interactions between elements of chains. 

The rigid sphere is not an accurate picture of how a polymer molecule affects the flow of the fluid in which it is dissolved, because the fluid can penetrate within the molecule. This recognition led to the development of models in which the molecule is represented as a chain of beads, which contain all the mass of the molecule, connected by springs. In the free-draining model of Rouse [45], there is no effect of one bead on the flow pattern around other beads. This model starts from Stokes law, which gives the drag force f on a sphere in a Newtonian fluid flowing past it at the velocity U as proportional to the radius a of the sphere. In terms of the coefficient of friction f =Fj /U), Stokes law for flow past a sphere is ... 

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