Stokes equation, calculation particle radii

When the particles are large (> 0.1 sedimentation under the influence of gravity is rapid enough to be observable. By the application of the Stokes equation the particle radius a, can be calculated from the rate of sedimentation p. 

From D, the particle radius R is calculated using the Stokes-Einstein equation kT... 

The particle radius R can be calculated from D using the Stokes-Einstein equation,... 

Homogeneity Distribution in Size and Shape. In 1950, Shabp and Beabd demonstrated that Stokes law could be used to determine the size of spherical latex particles of known density. Particles were sedimented in solvents with various densities and their size calculated from an equation relating the radius of the... 

For a homogeneous suspension the attenuation of the x-ray beam is proportional to the mass concentration of the suspension at the measurement radius. The size of the largest particle present in the suspension can be calculated using Stokes equation and the mass concentration undersize can be determined using the Kamack equation... 

Increasing the radius of the suspended particles, Brownian motion becomes less important and sedimentation becomes more dominant. These larger particles therefore settle gradually under gravitational forces. The basic equation describing the sedimentation of spherical, monodisperse particles in a suspension is Stokes law. It states that the velocity of sedimentation, v, can be calculated as follows ... 

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. 

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. 

Measurement of D in a solvent of known viscosity therefore permits a value of the radius r to be calculated. However, such a calculation would not be very satisfactory for macromoiecuies, for several reasons. In the first place, Stokes s law is based on the assumption of very large spherical particles and a continuous solvent, and therefore involves some error even for approximately spherical molecules. Secondly, the macromoiecuies may not be spherical, and this introduces an additional error. Furthermore, macromoiecuies are commonly solvated, and in moving through the solution they transport some of their solvation layer. In spite of these drawbacks, equation (11.66) has proved useful in providing approximate values of molecular sizes (see also pp. 99 i00). 

Obviously this probability dW is proportional to the concentration of the diffusing particles in the plane between x and x + dx at the time L If it is possible, therefore, to measure, by any experimental method, the magnitudes dTT, x and ty the coefficient of diffusion D may be calculated from equation (108). A few recent methods of performing this task will be described below it need only be added that, with certain assumptions, it is possible to estimate, from the coeflScient of diffusion, the size of the suspended particles. If we assume that the diffusing particles are spherical, or, at least, approximately spherical, and are large in comparison with the molecules of the liquid, Z>, according to Stokes, Nernst and Einstein, bears the following relation to the radius r of the particle... 

When particles are small enough to undergo Brownian motion, there is a continuous variation in the distance between the particles. As a consequence of this motion, constructive and destructive interference of the light scattered by neighboring particles yields intensity fluctuations. Following the intensity fluctuations as a function of time, the diffusion coefficient of the particles can be measured, and consequently, via the Stokes-Einstein equation, if the viscosity of the medium is known, the hydrodynamic radius or diameter of the particles can be calculated. Dynamic light scattering is therefore a very efficient method to determine the... 

---