Stokes-Einstein hydrodynamic radius

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.

Measurement of the translational diffusion coefficient, D0, provides another measure of the hydrodynamic radius. According to the Stokes-Einstein relation... 

The translational diffusion coefficient of micelles loaded with a fluorophore can be determined from the autocorrelation function by means of Eqs (11.8) or (11.9). The hydrodynamic radius can then be calculated using the Stokes-Einstein relation (see Chapter 8, Section 8.1) ... 

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 use of the Stokes-Einstein relation with the above value of the average diffusion coefficient leads to a hydrodynamic radius of roughly 30 nm, which is consistent with the specification of the manufacturer. ... 

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... 

Second, an equivalent hydrodynamic radius can be defined from the diffusion coefficient via a Stokes-Einstein relationship... 

In polymer solutions, DLS is used to determine the hydrodynamic radius of the constituent particles using the Stokes-Einstein equation... 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

Diffusions NMR spectroscopy (e.g. PGSE = Pulsed Gradient Spin Echo STE = Stimulated Echo DOSY = Diffusion Ordered Spectroscopy) is a straightforward and accurate method for determination of the self-diffusion coefficient of a molecule. Its principal use in dendrimer chemistry is for size determination of dissolved dendrimers since the self-diffusion coefficient is directly correlated with the hydrodynamic radius of the molecule via the Stokes-Einstein equation [24]. Although one-dimensional and multidimensional diffusion NMR experiments can thus make an important contribution to structural characterisation of dendrimers, they have been used comparatively rarely until recently [25, 26]. 

One aspect of the dynamics of micellar systems that has received a renewed interest during recent years is the translational motion of the micelles themselves. In the simplest approximation, the translational diffusion coefficient, D, of a spherical micelle is related to the hydrodynamic radius rM through the Stokes-Einstein relation... 

D has been sometimes expressed in terms of the hydrodynamic radius. However, we consider that hydrodynamic radius is not the proper term to describe the change of D, because this is not a well-defined radius such as the radius of gyration, which is clearly defined to show the molecular size. The hydrodynamic radius has just the same meaning as D as long as the Stoke-Einstein relation holds good. 

When the solute is spherical, or close to be so, its radius is easily obtained otherwise, estimations can be made on the basis of the geometry and arrangement of the constituting atoms or ions. For solutes having a complex stucture (e.g., micelles), a distinction should be made between the hydrodynamic radius (which appears in the Stokes-Einstein equation of the diffusion coefficient) and the reaction radius [98]. For Ps, RPs should represent the bubble radius. However, as shown in Table 4.4, the experimental data are systematically very well recovered by using the free Ps radius, RPs = 0.053 nm using the bubble radius results in a calculated value of kD (noted kDb) that is too small by a factor of 2 or 3. Table 4.4 does not include such cases where k kD, as these do not correspond to purely diffusion-controlled reactions. 

The key link between ELS experiments and particle electrostatic properties is the theoretical model of colloidal electrohydrodynamics. The required model is considerably more complicated than the one needed in the interpretation of DLS data. DLS relies upon a relatively simple colloidal hydrodynamic model to relate the measured particle diffusivity to particle radius via the Stokes-Einstein Eq. (39). The colloidal electrohydrodynamic model for ELS must account for the complex physical/chemical/electrical structure of the particle surface as well as the distortion of the diffuse part of the electrostatic double layer due to the motion of the particle through the medium. 

The hydrodynamic radius can be obtained from D by substitution in the Stoke-Einstein equation (4)... 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

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). 

The refractive index of the ethane/propane mixture, needed for calculation of the scattering vector, was determined from experimental density measurements and an empirical Lorentz-Lorenz relationship (12). Incorporating the errors from the viscosity and refractive index calculations into the Stokes-Einstein relation results in a maximum error in the hydrodynamic radius of approximately 5%. 

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. 

Measurements of the diffusion coefficients of globular protein molecules in solution yield values for molecular size that are greater than the corresponding radii determined by x-ray crystallography. The apparent hydrodynamic radius can be calculated from the Stokes-Einstein relation ... 

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