Einstein relation diffusion

A famous equation by Einstein relates diffusion and mobility ... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

The oxide solid elecU olytes have elecuical conductivities ranging from lO Q cm to 10 cm at 1000°C and these can be converted into diffusion coefficient data, D, for die oxygen ions by the use of the Nernst-Einstein relation... 

Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

The flux by diffusion is described by the diffusivity Di and the migration by the conductivity cr-. The conductivity is proportional to the product of the mobility and the concentration of the mobile species. The diffusivity and mobility are related by the Nernst-Einstein relation [3J. The flux is in general given by... 

A number of bulk simulations have attempted to study the dynamic properties of liquid crystal phases. The simplest property to calculate is the translational diffusion coefficient D, that can be found through the Einstein relation, which applies at long times t ... 

Since thermal agitation is the common origin of transport properties, it gives rise to several relationships among them, for example, the Nemst-Einstein relation between diffusion and conductivity, or the Stokes-Einstein relation between diffusion and viscosity. Although transport... 

Since the diffusion coefficient is the infinite-time integral of the velocity correlation function, we have the Einstein relation, D = kBT/Q. 

The mobility of eh was determined by measuring the equivalent conductance following pulse irradiation (Schmidt and Buck, 1966 Schmidt and Anbar, 1969). After correcting for the contribution of H30 and OH ions, they found the equivalent conductance of eh = 190 10 mho cm2. From this, these authors obtained the mobility p(eh) = 1.98 x 10"3 cm2/v.s. and the diffusion coefficient D(eh) = 4.9 x 10-5 cm2/s using the Nernst-Einstein relation, with about 5% uncertainty. The equivalent conductance of eh is the same as that for the OH - ion within experimental uncertainty. It is greater than that of the halide ion and smaller than that of eam... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

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... 

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

With the help of the Stokes-Einstein relation, the translational diffusion coefficient may be calculated according to... 

Einstein s work on the diffusion of particles (1906) led to the well known Stokes-Einstein relation giving the diffusion coefficient D of a sphere ... 

Various modifications of the Stokes-Einstein relation have been proposed to take into account the microscopic effects (shape, free volume, solvent-probe interactions, etc.). In particular, the diffusion of molecular probes being more rapid than predicted by the theory, the slip boundary condition can be introduced, and sometimes a mixture of stick and slip boundary conditions is assumed. Equation (8.3) can then be rewritten as... 

Changes in fluidity of a medium can thus be monitored via the variations of Jo/J — 1 for quenching, and Ie/Im for excimer formation, because these two quantities are proportional to the diffusional rate constant kj, i.e. proportional to the diffusion coefficient D. Once again, we should not calculate the viscosity value from D by means of the Stokes-Einstein relation (see Section 8.1). 

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) ... 

A unified understanding of the viscosity behavior is lacking at present and subject of detailed discussions [17, 18]. The same statement holds for the diffusion that is important in our context, since the diffusion of oxygen into the molecular films is harmful for many photophysical and photochemical processes. However, it has been shown that in the viscous regime, the typical Stokes-Einstein relation between diffusion constant and viscosity is not valid and has to be replaced by an expression like... 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

The self-translational diffusion coefficient D is related to f, by the Stokes-Einstein relation and is given by... 

Breaking and reforming of such cross-links relate to the renewal time Tr of the DDH model [321] of the conductivity. Here the mean squared displacement without renewal events saturates after a short time to a value of r (°°)) until a restructuring establishes the start of a new diffusion step. Then D(0)=(r (oo))/(6TR) and via the Nernst-Einstein relation ... 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Equation (2.158) is the Einstein relation relating the mobility and diffusivity tensors. 

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 ... 

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