Hydrodynamically equivalent sphere

Note The size of a hydrodynamically equivalent sphere may be different for different types of motion of the macromolecule, e.g., for diffusion and for viscous flow. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

Figure 13 shows the data for the three phenolic groups of ribonuclease which ionize reversibly (Tanford etal., 1955a), based on spectrophotometric titration curves such as Fig. 11. A straight-line plot is obtained, in agreement with Eq. (14). The values of w are 0.112, 0.093, and 0.061, respectively, at ionic strengths 0.01, 0.03, and 0.15. (The salt used to produce the ionic strength was KCl, and there is evidence that neither K" nor CF is bound to an appreciable extent. The use of Zn as abscissa is therefore presumably acceptable.) Comparison with the calculated values of Table III shows that the experimental values are lower than predicted by about 20%. Such a deviation must be considered almost within the error of calculation. [If the radius of the hydrodynamically equivalent sphere (19 A) had been used as the basis of calculation, the calculated values of w would have become 0.119, 0.096, and 0.066, respectively.]...

The ionic radius r, is that of the hydrodynamically equivalent sphere consisting of both the molecular ion and its attached hydration shell. Given that the volume of a sphere is (4/3)jtr, Eqs. 15 and 18 predict that the logarithm of the iontophoretic permeability coefficient, for a given solute should be linearly related to the logarithm of its molal volume (MV) with a slope of -1/3 ... 

Table V. Estimated Size of Cellulases of Various Fungi Calculated from Given Are for Hydrodynamically Equivalent Spheres or Ellipsoids with...

On the other hand, many sharp fractions are available with several homologous series of random coil molecules. Common parameters to indicate the size of random coils are the root-mean-square of end-to-end distance, mean radius of gyration and the radius of the hydrodynamically equivalent sphere. Various discussions have been presented in the previous works with regard to the appropriate choice of the parameter or the correction factor for it (ref. 14, 25, 27, 29, 30, 31, 34). These discussions, however, have all ignored the wall effect described above and hence their significance is limited. 

The most practical parameter for the size of random coil molecules would be the radius of a hydrodynamically equivalent sphere (Stokes radius). This quantity can be directly determined through the measurements of viscosity, diffusivity or sedimentation velocity, and abundant data have been accumulated. The results of SE measurement can be conveniently expressed in terms of this quantity. This way of presentation at the same time serves as the "universal calibration curve" for SEC columns. 

Let us assume the random coil in the solution as a hard sphere of the radius / h as in the thermodynamic sphere (Figure 2.8). This hypothetical sphere is not the representative of the segment distribution, but shows the region inside the coil where the solvent flow cannot pervade. It is called the hydrodynamically equivalent sphere (Figure 2.12). Its volume is vh =4ttR /3. The radius /fn is not the same as the radius of gyration, but is... 

In the case of polymers diffusing in a solvent, we can replace the random coil by the hydrodynamically equivalent sphere of radius /fn- We find... 

The dimensions and the molecular weight of copolymer micelles can be determined by quite a number of techniques, especially scattering and hydrodynamic characterization techniques as summarized in Table 7.3. In general practice the hydrodynamic radius Ry is determined by DLS techniques. By treating the micelles as hydrodynamically equivalent spheres and using the Stokes-Einstein relation, Ry can be evaluated from the translational diffusion coefficient extrapolated to infinite dilution D -. 

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. 

If the micelles are treated as hydrodynamically equivalent spheres then the translational diffusion coefficient for the limiting case of infinitely dilute solution is given by the Stokes-Einstein relation... 

Suppose that a polymeric ion is represented by a hydrodynamically equivalent sphere. That is, suppose that a polymeric ion is represented as a sphere of radius R inside which all fixed charged groups are distributed uniformly. The density of the fixed charges in the sphere, v, is given by... 

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