Equivalent hydrodynamic sphere model

FIGURE 8.6 Equivalent hydrodynamic sphere model A, solvent molecule Q, monomeric unit. 

The most frequently calculated property is the mean square unperturbed end-to-end distance, (r )o. Other properties susceptible to rapid computation include the average of the end-to-end vector, (r)o, and the mean square unperturbed radius of gyration, 5 )0. The viscosity of a dilute solution in a solvent can be estimated from 5 )0 or (r )o via the equivalent sphere model for hydrodynamic properties. Several higher even moments, (r )o and (s )o, p = 2,3,..., provide information about the shape (width, skewness) of the distribution functions for and When combined with information about the electronic charge distribution within individual rigid units, the RIS model can calculate the mean square dipole moment, (m )o-Also accessible are optical properties that depend on the anisotropy of the... 

In Simha s early model (Simha and Zakin 1962), transition from a dilute to a concentrated polymer solution was envisioned as being due to interpenetration of polymer chains that occurs when concentration lies somewhere in the region 1 < [ryjc < 10. This transition is evident from the change in the concentration dependence of viscosity in polymer solutions. The quantity [r ]c, the Simha-Frisch parameter (Frisch and Simha 1956), also sometimes called the Berry number (Gupta et al. 2005), is therefore a reasonable measure of chain overiap in solution. As Shenoy et al. (2005b), however, correctly point out, the dependency, being ultimately based on the equivalent hard sphere hydrodynamic model, is strictly applicable only at low polymer concentrations. 

The second is the hydrodynamic model of Flory and Fox (27) which represents the polymer molecule by an equivalent nondraining hydro-dynamic sphere. Assuming the Flory constant to be the same for linear and branched polymers, the degree of branching is given by the g3/2 rule ... 

Fig. 14. Dimensionless hydrodynamic thickness vs. chain length, on a log-log scale as calculated from the SF model (Cohen Stuart et al., 1984c) for y, = 1, y = 0.45,

Because it is likely that aggregates have significant internal flow through their structure, aggregate permeability must be considered. Fractal aggregates are expected to behave like objects that are smaller than equivalent spheres with reduced drag effects. Indeed, simulations of hydrodynamic friction using the Stokes model overestimate the friction of fractal objects. 

This theory has been partially confirmed by sedimentation experiment (Langevin and Rondelez, 1978). The value of the slope so far found was —0.50 0.10. We now have some evidence to believe that in the semidilute range of polymer solution the solvent is forced through in orderly fashion around the blob of radius C but still cannot penetrate the interior of the blob. Note that this theory is reminiscent of the pearl necklace model and the hydrodynamic equivalent sphere. 

In reference [19], a systematic comparison between the predictions of the SCGLE theory and the corresponding computer simulation data for four idealized model systems was reported. The first two were two-dimensional systems with power law pair interaction, u(r) = Air", with n = 50 (i.e., strongly repulsive, almost hard-disk like) and with n = 3 (long-range dipole-dipole interaction). The third one was the three-dimensional weakly screened repulsive Yukawa potential (whose two-dimensional version had been studied in reference [18]). The last system considered involved short-ranged, soft-core repulsive interactions, whose dynamic equivalence with the strictly hard-sphere system allowed discussion of the properties of the latter reference system. For all these systems G(r, f) and/or F(k, f) were calculated from the self-consistent theory, and Brownian dynamics simulations (without hydrodynamic interactions) were performed to carry out extensive quantitative comparisons. In all those cases, the static structural information [i.e., g(r) and 5(A )] needed as an input in the dynamic theories was provided by the simnlations. The aim of that exercise was to... 

---