Preaveraging

The evaluation of the generalized mobility matrix jj.(Q) follows the lines outlined in Section 5.1.1. The elements n1p(Q) can be approximated to high accuracy [100] by sums of a preaveraged term... 

The calculation of g for Gaussian uniform star chains was carried out by Zimm and Kilb (ZK) [83]. They used a modified version of the dynamic Rouse theory including preaveraged HI (in the non-draining limit) that considers the particular connectivity of units consistently with the star architecture. This ap-... 

Numerical calculations inspired in the ZK method for stars have also been applied for the description of the dynamics of model dendrimers. La Ferla [232] used a freely-rotating model, including a topology-dependence stiffness parameter and preaveraged HI. With this model, he obtained a complex analytical expression for the mean size. Cai and Chen [233] used a Gaussian model without HI and performed a detailed analysis of the relaxation motions. They investigated the diffusion of the center of mass, the relaxation of the center of mass position relative to the core monomer, and also the rotational and internal modes. 

Equation (51) is highly nonlinear. One of the key approximations made to enable analytical calculations is the preaveraging approximation, whereby the diffusion tensor D[R(i) - R s )] is replaced by its average over chain configurations. 

By making the above approximations (preaveraging and uniform expansion), the Langevin equation for the polymer segment can be written as... 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

Assuming that R, is the same as the center-of-mass velocity R for all i and invoking the preaveraging approximation, we get... 

Using the Rouse mode variable q and employing the preaveraging approximation, we obtain... 

In obtaining Eqs. (217)-(219), we have employed the preaveraging approximation and assumed that solvent motion is instantaneous in comparison to the motion of poly electrolytes. For a solution of polyelectrolytes, the effective medium theory for the equilibrium properties gives... 

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 probably more correct treatment has recently been given by Fixman (91) and by Pyun and Fixman (92). These authors avoid the preaveraging of the Oseen-interaction tensor [eq. (3.20)]. A comparison of results will be given at the occasion of the discussion of eigen values. Fortunately, Zimm s results appear to be only slightly different. 

Fig. 4. The dynamic scattering function < >g for a pair of elements (j, k) after averaging over all orientations and distance fluctuations in the hydrodynamic pre-average approximation, Eq. (B.49), and without this approximation, Eq. (B.45). The line labeled exact gives the exact deviation of the preaverage approximation, the dotted line represents the approximation of Eq. (B.50)R2)...

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

Since it is the dynamics of the system that is of interest, it would be convenient to preaverage over the environment variables and obtain an equation of motion for ps(t), the system component of the density matrix. Formal work of this kind [161,. .. 162] yields the so-called generalized master equation. Deriving the generalized f master equation, and extracting the various approximations utilized, goes well "astray of the central focus of this book. For this reason we just sketch the models id direct the reader to suitable review articles [161, 162] that provide an appropriate pview. 

Figure 3.2 Trouton ratio, Tr, of uniaxial extensional viscosity to zero-shear viscosity jq after start-up of steady uniaxial extension at a rate of 1 sec i for a Boger fluid consisting of a 0.185 wt% solution of flexible polyisobutylene (Mu, = 2.11 x 10 ) in a solvent composed mostly of viscous polybutene with some added kerosene (solid line). The dashed line is a fit of a multimode FENE dumbbell model, where each mode is represented by a FENE dumbbell model, with a spring law given by Eq. (3-56), without preaveraging, as described in Section 3.6.2.2.I. The relaxation times were obtained by fitting the linear viscoelastic data, G (co) and G"(cu). The slowest mode, with ri = 5 sec, dominates the behavior at large strains the best fit is obtained by choosing for it an extensibility parameter of = 40,000. The value of S — = 3(0.82) n/C(x, predicted from the...

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