Equivalent sphere hydrodynamic

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).

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. 

It appears that one can develop a qualitative understanding of the simple flow properties at moderate concentration without going beyond concepts which are already inherent either in the dilute solution theory of polymers or in the properties of particulate suspensions. The dependence of viscosity on c[i ] is believed to reflect a particle-like or equivalent sphere (127) hydrodynamics in solutions of low to moderate concentration. In particular, the experimental facts do not force the consideration of effects which might arise from the permanent connectedness of the polymer backbones. Situations conducive to the entangling of molecules may be present, e.g., overlap of the coils, but either entanglement contributions are small, or else they are overwhelmed by the c[f ] interactions. 

Fig. 49. A sketch of the segment density profile for a microgel. The various bars indicate the position of the radius of gyration, the expected equivalent sphere radius and the hydrodynamic radius, respectively1881...

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... 

Being able to determine [r ] as a function of elution volume, one can now compare the hydrodynamic volumes Vh for different polymers. The hydrodynamic volume is, through Einstein s viscosity law, related to intrinsic viscosity and molar mass by Vh=[r ]M/2.5. Einstein s law is, strictly speaking, valid only for impenetrable spheres at infinitely low volume fractions of the solute (equivalent to concentration at very low values). However, it can be extended to particles of other shapes, defining the particle radius then as the radius of a hydrody-namically equivalent sphere. In this case Vjj is defined as the molar volume of impenetrable spheres which would have the same frictional properties or enhance viscosity to the same degree as the actual polymer in solution. 

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 ... 

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,... 

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

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. 

Finally, because SEC separates molecules according to some function of size, one might ask which parameter is most appropriate. Hydrodynamic volume is certainly a reasonable choice, but other parameters related to size have also been considered (44,45), allowing alternative bases for constructing universal calibration curves. The product [TiJMis related to the radius of an equivalent sphere i , by the equation... 

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 most appropriate particle size to use in equations relating to fluid-particle interactions is a hydrodynamic diameter, i.e. an equivalent sphere diameter derived from a measurement technique involving hydrodynamic interaction between the particle and fluid. In practice, however, in most industrial applications sizing is done using sieving and correlations use either sieve diameter, Xp or volume diameter, Xy, For spherical or near spherical particles Xv is equal to Xp. For angular particles, Xy l.lSXp. 

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... 

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 -. 

The equivalent diameters of spheres with the same particle density (500kgm ) and with the same rates of descent as the reed particles were calculated from a balance of forces on a single particle, that is, the weight of a particle (less the almost negligible lifting force) equals the hydrodynamic resistance force (Section 3.4.1.2). The calculated equivalent sphere diameters are shown in Figure 6.2.20. 

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 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. 

It states that the friction coefficient of a sphere scales with the radius R. When dealing with particles which are isotropic on average but have otherwise an arbitrary structure, it is sometimes convenient to replace them by an equivalent sphere . The replacement implies that we assign to the particle a hydrodynamic radius i h, defined by... 

---