Hydrodynamic preaveraged

Burchard W, Schmidt M, Stockmayer WH. Influence of hydrodynamic preaveraging on quasi-elastic scattering from flexible linear and star-branched macromolecules. Macromolecules 1980 13 580-587. 

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. 

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... 

In the preaveraged approximation for the Oseen hydrodynamic tensor [19, 20], the linear Langevin equation may be written as... 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

The possibility to calculate hydrodynamic quantities (eg, [i ]) both with preaveraging Ozeeii s tensor and directly (without any preaveraging) proves to be a serious advantage of the calculational procedure of the renormalization group approximation. This allowed researchers to estimate the error of determination of the hydrodynamic quantities caus[Pg.744]

In the Zimm model (see Fig. 2A) the hydrodynamic interactions are included by employing the Oseen tensor Him the tensor describes how the mth bead affects the motion of the /th bead. This leads to equations of motion that are not Unear anymore and that require numerical methods for their solution. In order to simplify the picture, the Oseen tensor is often used in its preaveraged form, in which one replaces the operator by its equiUb-rium average value [5]. For chains in -solvents, this leads for the normal modes to equations similar to the Rouse ones, the only difference residing in the values of the relaxation times. An important change in behavior concerns the maximum relaxation time Tchain> which in the Zimm model depends on N as and implies a speed-up in relaxation compared to the Rouse model. Accordingly, the zero shear viscosity decreases in the Zimm model and scales as Also, in the Zimm model the diffusion coefficient... 

The first term Sij/C is the local Rouse term the Oseen tensor describes the hydrodynamic interactions. A frequently used approximation, introduced first by Kirkwood and Riseman, averages the Oseen tensor over the configurations of the polymer chain in its equilibrium Gaussian state. This replaces the Oseen tensor by the preaveraged Oseen tensor... 

Simulations of Gaussian chains, i.e, polymers with the bond potential (81), can be compared with analytical calculations based on the Zimm approach [6,107]. Note, however, that the simulations are not performed in the Zimm model. The Zimm approach relies on the preaveraging approximation of hydrodynamic interactions, whereas the simulations take into account the configurational dependence of the hydrodynamic interactions, and therefore hydrodynamic fluctuations. Hence, the comparison can serve as a test of the validity of the approximations employed in the Zimm approach. 

The Kirkwood formula neglects hydrodynamic flucmations and is thus identical with the preaveraging result of the Zimm approach. When only the hydrodynamic part is considered, the Zimm model yields the diffusion coefficient... 

Considerable further computational work on star polymers has been undertaken by Freire and his collaborators [63,83-87]. This is summarized in Table 6. The latest development, simulation on chains with thermodynamic interactions and including non-preaveraged hydrodynamic interactions, modulated by Brownian motion, yields values of are in agreement with... 

The first indication that the preaveraging technique has a substantial effect on the calculation of the friction coefficients of star pol3miers is found in the calculation of the g-dependence of Dq, i.e., the contributions of internal modes [90,91]. Zimm showed that preaveraging of the hydrodynamic interactions in the Kirkwood-Riseman model was largely to blame for the discrepancy between ggp and experimental values of gjj [57]. Further refinements of the Monte Carlo simulations with thermodynamic interactions to represent 0-solvent conditions or good solvent conditions combined with non-preaveraged hydrodynamic interactions lead to close agreement between calculated and... 

It has been clearly shown that the hydrodynamic properties of star polymers cannot be compared with those of linear polymers within the Kirkwood-Riseman model with preaveraged hydrodynamic interactions. In order to match calculated with experimental results, non-preaveraging of hydrodynamic interactions becomes increasingly more important as /increases. It is interesting to note that non-preaveraging increases but decreases from the calculated preaveraged values. 

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