Einstein equation, Brownian particle

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

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... 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

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... 

The original theory of Brownian motion by Einstein was based on the diffusion equation and was valid for long times. Later, a more general formulism including short times also, has been developed. Instead of the diffusion equation, the telegrapher s equation enters. Again, an indeterminacy relation results, which, for short times, gives determinacy as a limit. Physically, this simply means that a Brownian particle s... 

Even without solving this equation one can draw an important conclusion. It has the same form as the diffusion equation (IV.2.8) and in fact it is the diffusion equation for the Brownian particles in the fluid. Consequently a2 is identical with the phenomenological diffusion constant D. On the other hand, a2 is expressed in microscopic terms by (2.4) or by (1.6). This establishes Einstein s relation... 

After the work of Einstein and Smoluchowski an alternative treatment of Brownian motion was initiated by Langevin.Consider the velocity of the Brownian particle, as in VIII.4. When the mass is taken to be unity it obeys the equation of motion... 

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... 

The excellent review of Chandrasekhar provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers, who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential. This problem allows us to explain what we mean by a systematic version of the AEP. We can state the problem as follows. Let us consider the motion of a free Brownian particle described by the one-dimensional counterpart of Eq. (1.2),... 

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. 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

If the central particle is also in Brownian motion, the diffusion constant, D, should describe the relative motion of two particles. The relative displacement is given by Jt — xj, where a / and xj are the displacements of the two particles in the x direction measured from a given reference plane. The diffusion constant for the relative motion can be obtained from the Einstein equation for the diffusion coefficient (Chapter 2) ... 

Three mechanisms are important for the backtransport of particles from a membrane. For small solutes and submicron colloids. Brownian Diffusion (determined by the Stokes-Einstein equation (Sethi and Wiesner (1997)) dominates backtransport of the colloids from the membrane into the bulk solution. Inertial Lift, which is caused by the presence of a wall is important for large particle sizes and high shear rates. Shear Induced Diffusion is an orthokinetic mechanism also more important for larger colloids. 

A method for measuring the size of aggregates in aqueous environments is dynamic light scattering (DLS). This technique uses scattered light to measure diffusion rates (Brownian motion) of particles in stable suspensions to determine a size based on the Stokes-Einstein equation ... 

DLS (dynamic light scattering)—in dynamic light scattering laser light is scattered by the nanoparticles. Due to the Brownian motion of the particles, a time-dependent fluctuation is imparted to the scattered light intensity. Analysis of the signal intensity yields information about the diffusional motion of the particles, which is in turn related to the hydrodynamic size via the Stoke-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 ... 

When monochromatic light hits the particles, time-dependent fluctuation in the scattered hght intensity is observed as long as the particle size is smaller than the wavelength of the incident light. This is due to the Brownian motion of the particles and can be described using the Stokes-Einstein equation shown in the succeeding text ... 

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 — / ) ... 

Diffusion, as indicated above, is a related problem, differing however in the fact that molecules of different sizes are involved. Many years ago, Einstein (5), in considering the Brownian movement of colloids where the particles are very large in comparison to the molecules of the solvent and assuming that Stokes law described the motion of the particles, arrived at what is known as the Stokes-Einstein equation ... 

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