Einstein diffusion coefficient, Brownian motion

Einstein to describe Brownian motion.5 The model can be used to derive the diffusion equations and to relate the diffusion coefficient to atomic movements. 

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... 

In suspensions, it is common to consider particles whose sizes range down to the submicron scale, where Brownian motion and colloidal forces have pronounced effects (Russel et al. 1991). The influence of Brownian motion relative to shear flow is captured through a P clet number given by Pe = ( fl )/Do = (67tTioy )/fcT, where Do = kT/ 6ny Qa) is the Stokes-Einstein diffusion coefficient, k is Boltzmann s constant, and T is the temperature. The first form shows that Pe may be interpreted as a ratio of the hydrodynamic diffusion scaling with the shear rate and particle size ya ) as well as a dimensionless function of the volume fraction not shown. It is more common, however, to interpret Pe as the ratio of a diffusive timescale u IDq, relative to the flow timescale given by When Pe = 0, a Brownian suspension will approach a true equilibrium state through its thermal motions. Interparticle forces of many sorts are possible in a liquid medium. 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

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 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. 

The motion caused by thermal agitation and the random striking of particles in a liquid by the molecules of that liquid is called Brownian motion. This molecular striking results in a vibratory movement that causes suspended particles to diffuse throughout a liquid. If the colloidal particles can be assumed to be approximately spherical, then for a liquid of given viscosity (q), at a constant temperature (T), the rate of diffusion, or diffusion coefficient (D) is inversely related to the particle size according to the Stokes-Einstein relation (ref. 126) ... 

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... 

During that century, colloidal phenomena played a pivotal role in the genesis of physical chemistry by establishing a connection between descriptive chemistry and theoretical physics. For example, Einstein provided the relationship between Brownian motion and diffusion coefficient of colloidal particles. 

Equation 6.33 states that the root-mean-square displacement is proportional to the square root of the number of jumps. For very large values of n, the net displacement of any one atom is extremely small compared to the total distance it travels. It turns out, that the diffusion coefficient is related to this root-mean-square displacement. It was shown independently by Albert Einstein (1879-1955) and Marian von Smoluchowski (1872-1917) that, for Brownian motion of small particles suspended in a liquid, the root-mean-square displacement, is equal to V(2Dt), where t is the time... 

According to the Einstein relation the ratio between the diffusion coefficients for brownian motion should be inversely proportional to the ratio of the atomic masses. The computed and theoretical values are reported in Table 2. 

Generally, mean size and size distribution of nanoparticles are evaluated by quasi-elastic light scattering also named photocorrelation spectroscopy. This method is based on the evaluation of the translation diffusion coefficient, D, characterizing the Brownian motion of the nanoparticles. The nanoparticle hydro-dynamic diameter, is then deduced from this parameter from the Stokes Einstein law. 

Einstein pointed out that the same random forces that produce Brownian motion are also in operation when the particle is dragged through the medium. Whereas diffusion is characterized by the diffusion coefficient, drag is usually quantified in terms of a mobility ju, a proportionality constant between the applied force F and the terminal velocity v, where v = fiF. In fact, the diffusion coefficient and mobility are related by... 

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) ... 

A precise determination of the frictional coefficient C in terms of the intermolecular potential and the radial distribution function at present constitutes the principal unresolved problem of the Brownian motion approach to liquid transport processes. It has been suggested by Kirkwood that an analysis of the molecular basis of self-diffusion might be a fruitful approach. The diffusion constant so calculated would be related to the frictional coefficient through the Einstein equation, Eq. 46. 

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 ... 

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

One of the important applications of Stokes law occurs in the theory of Brownian motion. According to Einstein (El a) the translational and rotary diffusion coefficients for a spherical particle of radius a diffusing in a medium of viscosity n are, respectively,... 

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 ... 

It is just because the motion is Brownian that (x ) is proportional to t cf. (6.2).) How does friction come into this picture Evidently, the greater the coefficient of friction /i for some particle, the lower will be the diffusion coefficient D, and vice versa. The exact relationship between the two was found in 1905 by Albert Einstein, and is called the Einstein relation. It states that... 

In the second part of his argument, Einstein relates the diffusion coefficient to a measurable property of Brownian motion such as the mean displacement of the particles. Let c(x, t) denote the probability for a Brownian particle to be at position X at time t. By following a probabilistic argument, the spatial and temporal changes in the concentration of the particles were shown to be related to the diffusion coefficient by... 

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