The Equivalent Hydrodynamic Sphere

The classical thermodynamic and kinetic model is that of a rigid sphere impenetrable by water. A spherical geometry has been observed in many polysaccharide systems, notably hyaluronic acid-protein complexes (Ogston and Stainer, 1951), dispersed gum arabic (Whistler, 1993), and spray-dried ungelatinized starch granules (Zhao and Whistler, 1994). Spherulites of short-chain amylose were obtained by precipitation with 30% water-ethanol (Ring et al., 1987), and spherulites of synthetic polymers were obtained 

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FIGURE 12-32 Schematic diagram of the equivalent hydrodynamic sphere. 

O is the so-called Flory s constant, a is the expansion factor of the polymer molecule, which depends from the thermodynamic quality of the solvent (a = 1 in ideal solvent), ( o> is the mean-square radius of gyration, is Avogadro s number, and is the volume of the equivalent hydrodynamic sphere. 

Figure 12. Logarithmic plot of the molecular radius of the equivalent hydrodynamic sphere against molecular weight for cellulose of Myrothecium verrucaria (77) and other globular proteins. This relationship was used to estimate the probable size of cellulolytic enzymes of other fungi as shown in Table V...

Equation 12.67 predicts that the specific viscosity is proportional to the volume of the equivalent hydrodynamic sphere. The Einstein viscosity relation was derived for rigid spherical particles in solution. However, real polymer molecules are neither rigid nor spherical. Instead the spatial form of the polymer molecule in solution is regarded as a random coil. Theories based on this characteristic form of polymer molecules have resulted in the expression... 

Fig. 6 Radius of the equivalent hydrodynamic sphere (Stokes radius) for dextran and poly(ethylene oxide) in water. Calculated from the collected data of viscosity and diffusivity. The scales for two lines are displaced from each other for clarity, (Reproduced from ref. 32 with permission.)...

The radius of gyration may be related to an equivalent hydrodynamic sphere, which is conceptualized as a solid sphere of radius Re.138 This sphere... 

The nice thing about the assumption of an equivalent hydrodynamic sphere is that it allows you to do two things first, use the Einstein relationship for the viscosity of a dilute solution of solid particles, given previously in Equation 12-43. Then we can use the definition of a volume fraction, (Equation 12-51) ... 

Several size parameters can be used to describe the dimensions of polymer molecules radius of gyration, end-to-end distance, mean external length, and so forth. In the case of SEC analysis, it must be considered that the polymer molecular size is influenced by the interactions of chain segments with the solvent. As a consequence, polymer molecules in solution can be represented as equivalent hydrodynamic spheres [1], to which the Einstein equation for viscosity may be applied ... 

Continuous viscometer detectors are not subject to the same limitation as LALLS instruments. The reasons for this are as follows. If one represents a polymer molecule in solution as an equivalent hydrodynamic sphere (4), then the intrinsic viscosity of the solution [r ] is defined according to 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. 

A is a reciprocal of the equivalent hydrodynamic diameter of a sphere. Table 5.3 lists A values for several common shapes, where x = 2b/L, with b and L being the half-length of the short axis and the length of the long axis, respectively. If one knows or assumes the axial ratio or one of the axis lengths for all particles, then the distribution q(L) can be obtained from q(Dx). This is relatively easy for thin rods or disks. For many products the rod or disk thickness is either a constant or much smaller than the corresponding rod length or disk diameter. 

A similar formula (Eq. 5.30) as that given in Eq. 5.29 exists for the rotational diffusion coefficient. In Eq. 5.30, B is a reciprocal of the equivalent hydrodynamic diameter of a sphere and is listed in Table 5.3 for several common shapes. The mean value for the rotational diffusion coefficient Dg an be obtained from the angular dependence of the mean characteristic decay f in a multiangle PCS experiment following Eqs. 5.19 and 5.20. If another independent experiment, such as depolarized PCS or transient electric birefringence can be performed to determine Dr, a combination of Dt and Dr... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

However, according to Little (1969), polyethyleneoxide solutions of different molecular weights gave the same drag reduction when their concentration was proportional to the critical concentration at each molecular weight (i.e., the computed concentration for the polymer coils to touch each other). Kinnier obtained similar results but used the concept of equivalent concentrations . He found that to have equal drag reduction for different molecular weight polymer solutions, one has to have equal volumes of polymer based upon the hydrodynamic sphere considerations. 

A nondraining polymer molecule, also referred to as the impermeable coil, can be represented by an equivalent impermeable hydrodynamic sphere of radius R. The frictional coefficient of this sphere which represents the frictional coefficient of the non-draining polymer coil can thus be written,... 

Equations (1.28) to (1.30) do not apply to flexible chain molecules, which are not rigid and can exhibit fluctuations in conformation. Here, one approach is to ignore shape anisometry and to make the assumption v = 2.5 in Eq. (1.30). This enables determination of an impermeable sphere-equivalent hydrodynamic volume for flexible chain macromolecules from [ ], provided that M is known. As noted above, the Mark-Houwink-Sakurada equation [Eq. (1.23)] is often used to relate [ j] to M when dealing with flexible coils. 

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