Stokes-Einstein equation Brownian diffusion coefficient

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

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... 

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

To measure the droplet size distribution of the primary emulsion (W/O in W/O/W or O/W in O/W/O) that has a micron range (with an average radius of 0.5-1.0 pm), a dynamic light-scattering technique (also referred to as photon correlation spectroscopy PCS) can be apphed. Details of this method are described in Chapter 19. Basically, the intensity fluctuation of scattered light by the droplets as they undergo Brownian diffusion is measured from this, the diffusion coefficient of the droplets can be determined, and in turn the radius can be obtained by using the Stokes-Einstein equation. 

In a QELS experiment, a monochromatic beam of light from a laser is focused on to a dilute suspension of particles and the scattering intensity is measured at some angle 0 by a detector. The phase and the polarization of the scattered light depend on the position and orientation of each scatterer. Because molecules or particles in solution are in constant Brownian motion, scattered light will result that is spectrally broadened by the Doppler effect. The key parameter determined by QELS is the diffusion coefficient, D, or particle di sivity which can be related to particle diameter, d, via the Stokes-Einstein equation ... 

Another method for determining 5h is to apply dynamic light scattering, referred to as photon correlation spectroscopy (PCS). For this purpose, dilute monodisperse particles must be used. From measurements of the intensity fluctuations of scattered light by the particles as they undergo Brownian diffusion, one can obtain the diffusion coefficient D, which can be used to obtain the hydrodynamic radius by using the Stokes-Einstein equation (equation (20.19)). By measuring D for the particles, both with and without the polymer layer, one can obtain / h and / , respectively. One should make sure that the bare particles are sufficiently stable 8 is then equal to (/ h — / ) ... 

The diffusion of NPs in blood flow can be due to (1) Brownian diffusion caused by the bombardment of fluid molecules and (2) shear-induced diffusion due to the presence of red blood cells (RBCs) in shear flow. The Brownian diffusion coefficient, Dpr, can be calculated using the Stokes-Einstein equation ... 

Brownian motion is the random thermal motion of a particle suspended in a fluid. This motion results from collisions between fluid molecules and suspended particles. For time intervals At much larger than the particle inertial response time, the dynamics of Brownian motion are independent of inertial parameters such as particle and fluid density. The Brownian diffusion coefficient D is given by the Stokes-Einstein equation as... 

Dynamic light scattering (photon correlation spectroscopy, PCS) can also be applied to obtain the hydrodynamic radius of the micelle. By measuring the intensity fluctuation of scattered light by the micelles (when these undergo Brownian diffusion), one can obtain the diffusion coefficient of the micelles D, from which the hydrodynamic radius R can be obtained using the Stokes-Einstein equation ... 

Brownian motion of a panicle is a result of the thermal motion of the molecular agitation of the liquid medium. Much stronger random displacement of a particle is usually observed in a less viscous liquid, smaller particle size, and higher temperature. A particle of size larger than 1 pm doesn t show a remarkable Brownian motion. There is much literature available on Brownian motion [7-9], and the Brownian motion is regarded as a diffusion process. For an isolated particle, i.e., there is no intcrparticlc action, the diffusion coefficient D , can be expressed as the Stokes-Einstein equation ... 

If this sedimentation velocity is small compared to the average velocity from Brownian motion of the particles, then the particles will remain in suspension. As we saw in Chapter 1, the diffusion coefficient of a particle in solution can be given by the Stokes-Einstein equation ... 

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... 

In the absence of hydraulic or wind forces, the water becomes quiescenf but natural or free convection processes remains operative. Driven by bottom residing thermal or concentration gradients. Equations 12.14 and 12.15 may be used for estimating these low-end MTCs. The chemical diffusion coefficient in the porewaters of the upper sediment layer is the key to quantifying the sediment-side MTC. Use Archie s law. Equation 12.18, to correct the aqueous chemical molecular diffusivity for the presence of the bed material. Bed porosity is the key independent variable that determines the magnitude of the correction factor. See Table 12.7 for typical porosity values in sedimentary materials. Eor colloids in porewaters. Equation 12.18 applies as well. The Stokes-Einstein equation (Equation 12.19) is recommended and some reported particle Brownian diffusion coefficients appear in Tables 12.9 and 12.10. Under quasisteady-state conditions, Equation 12.23 is appropriate for estimating the bed-side MTCs. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Similar discrepancies were noted by Blatt et al32 for colloidal suspensions such as skimmed milk, casein, polymer latexes, and clay suspensions. Actual ultrafiltration fluxes are far higher than would be predicted by the mass transfer coefficients estimated by conventional equations, with the assumption that the proper diffusion coefficients are the Stokes-Einstein diffusivities for the primary particles. Blatt concluded that either (a) the "back diffusion flux" is substantially augmented over that expected to occur by Brownian motion or (b) the transmembrane flux is not limited by the hydraulic resistance of the polarized layer. He favored the latter possibility, arguing that closely packed cakes of colloidal particles have quite high permeabilities. However, this is not a plausible hypothesis for the following reasons ... 

The diffusion coefficient. Dp, that appears in the transport equations in the diffusion boundary layer, was defined by treating the disentangling chains in the boundary layer as Brownian spheres. Thus, a Stokes-Einstein type diffusivity... 

As expected, the larger the diffusion coefficient, the lower the drag force. Of course, Einstein s diffusion law can be combined with Stokes equation for/ and the resulting equation is called Stokes-Einstein law (Problem 8.1). Together with the equation for the Brownian displacement, it was used by Perrin for early, rather accurate calculations of the Avogadro number. 

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