Stokes-Einstein radius, effect

Figure 9.33. (a) Schematic description of the effects of ionic strength (I) and pH on the conformations of a humic molecule in solution and at a surface. Rh denotes the hydrodynamic radius of the molecule in solution and 6h denotes the hydrodynamic thickness of the adsorbed anionic poly electrolyte. (Adapted from Yokoyama et al., 1989 and O Melia, 1991). (b) The influence of ionic strength of pH on diffusion coefficient, Dl, and on Stokes-Einstein radius of a humic acid fraction of 50,000-100,000 Dalton. (From Cornel et al., 1986). 

Fig. 7. Molecular weight dependence of diffusivity. (a) The effective diffusion coefficient, D, has been plotted as a function of molecular weight for dextrans (Nugent and Jain, 1984a, b Gerlowski and Jain, 1986), albumin (Nugent and Jain, 1984a, b), and IgG (Clauss and Jain, 1990) in water, normal tissue, and tumor tissue. Symbols , dextran, aqueous O, bovine serum albumin, aqueous O, rabbit IgG, tumor , dextran, normal tissue , bovine serum albumin normal tissue , rabbit IgG, normal tissue. The half-filled symbols refer to the tumor data, (b) The effective diffusion coefficient plotted versus the Stokes-Einstein radius. Symbols as in (a) plus X, sodium fluorescein, tumor +, sodium fluorescein, normal tissue. (From Clauss and Jain, 1990, with permission.) Currently, we are measuring diffusion coefficient of molecules and particles larger than 50 A in radius.

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Knowing these functions, the mean-square radius of gyration (S2)z and the translational diffusion coefficient Dz can easily be derived eventually by application of the Stokes-Einstein relationship an effective hydrodynamic radius may be evaluated. These five... 

Figure 3.8. (a) The linear viscosity dependence of the inverse ionization rate in the reaction studied in Ref. 98. Bullets—experimental points solid line—fit performed with the generalized Collins—Kimball model, (b) The effective quenching radius for the same reaction in the larger range of the viscosity variation. Bullets—experimental points solid fine—fit performed with the encounter theory for the exponential transfer rate. The diffusion coefficient D given in A2/ns was calculated from the Stokes—Einstein relationship corrected by Spemol and Wirtz [100]. 

For simple center-of-mass diffusive motion, the mean decay rate F can be shown to be D(f, where D is the intensity-weighted average D). The use of D in the Stokes-Einstein equation, Eq. (10), yields an effective radius R. The second moment /t2 is given by... 

The values of l) >,n — the diffusivity for the Brownian motion of aerosol — are calculated from the Stokes-Einstein equation. For spherical particulates with the effective radius rp, in a gas with the dynamic viscosity p2 (nearly constant for pressures about and less than one bar), the formula is ... 

Experimental Results. Effects of pH and Ionic Strength. Experiments showing the effects of pH and ionic strength on the configuration of NOM in solution are presented in Figure 3, taken from the work of Cornel et al. (14). Experiments were conducted with the 50-100 K apparent molecular weight fraction of a humic acid (HA). Results are expressed in terms of the equivalent Stokes-Einstein or hydrodynamic radius (rh) calculated from measurements of the diffusion coefficients of the HA fraction. 

Figure 3. Effect of ionic strength and pH on the Stokes-Einstein (hydrodynamic) radius of the 50-100 K apparent molecular weight fraction of a humic acid.

This relationship is known as the Stokes-Einstein equation. Strictly speaking it should only be applied at infinite dilution to monoatomic ions. However, in practice it is applied to more complex ions and at finite ionic strengths. If the diffusion coefficient for the ion is measured experimentally, an effective radius for the ion can be estimated using the viscosity of the pure solvent. 

It is well-known that difiiision coefficients depend on the mass of the diffiising species in the gas phase and also in solids. In liquids, diffiision is nonnally described well by the Stokes-Einstein equation whidi depends on the hydrodynamic radius and on viscosity but not on the mass. There is no principal reason tiiat prevents a mass dependence in solution, and many experiments have been devised to search for isotope effects [35], mostly with no clear answer. It is obvious that a comparison between Mu, H, and D should be particularly sensitive. This question has been posed long ago [16], but the answer had to await more recent results, as outlined below. 

Even within the group of dynamic methods, one can find in the recent literature entirely different hydration numbers, for instance, those presented in Table 12.2 for biologically important ions [157]. Surprisingly, the fundamental Stokes-Einstein relationship between the hydrodynamic radius and the diffusion coefficient of the ion is being used in several different manners in the calculation of the effective hydrated radius of an ion (compare [158] and [159]). 

The function F(F) is expanded in a power series of r. The first moment (cumulant) in the expansion is the average decay constant Favg, which defines an effective diffusion coefficient A for the particle size distribution. A is converted to the effective hydrodynamic radius using the Stokes-Einstein relationship. 

In solution, D is given by the Stokes-Einstein relation which relates D to the viscosity coefficient of the solution, T], and the effective hydrodynamic radius a, where... 

Dynamic light-scattering, sometimes called quasi-elastic light scattering or photon correlation spectroscopy, can be used to measure the diffusion coefficients of polymer chains in solution and colloids, a kind of Doppler effect see Section 3.6.6. In a dilute dispersion of spherical particles, the diffusion coefficient D is related to the particle radius, a, through the Stokes-Einstein equation. 

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