Diffusion constant Einstein relation

A is the wavelength of the laser in vacuum and q is the magnitude of the so called scattering vector. In turn, for spherical particles, the diffusion constant is related to the particle diameter through the Stokes-Einstein equation ... 

The conductivity of an electrolyte describes the transport of ion.s within an electrical field in the solution. Ions and molecules can also move in the solutions via diffusion. This becomes important in the electrode reactions of molecules or ions added to the supporting electrolyte. Provided that the concentration of these ions is much smaller than that of the supporting electrolyte, then the electric field does not affect the movement of the ions, i.e. these ions, as well as uncharged molecules, reach the electrodes only by diffusion. If the rate of the electrode process becomes large then the concentration of the reacting species decreases and that of the generated species increases near the electrode, which leads to a concentration profile. The corresponding diffusion process can be described by Fick s law, as will be discussed in Section 7.1.2. The diffusion process is essentially characterized by the diffusion constant D. Similarly to electrons and holes, the diffusion constant is related to the mobility by the Einstein relation (see Eq. 1.39). We have then... 

The diffusion constant is related to the friction of the particle with the media. Let t be the friction constant of the diffusing particle. Einstein found that the relation... 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

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. 

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 ... 

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

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). 

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... 

Einstein loG, oit.) showed further that the actual mean distance X travelled by a particle in a short time t under the influence of random molecular collisions was related to the diffusion constant jD, by the following equation ... 

The nature of rotational motion responsible for orientational disorder in plastic crystals is not completely understood and a variety of experimental techniques have been employed to investigate this interesting problem. There can be coupling between rotation and translation motion, the simplest form of the latter being self-diffusion. The diffusion constant D is given by the Einstein relation... 

In Fig. 3, the orientational diffusion time constants ror of the first solvation shell of the halogenie anions CD. Br, and D are presented as a function of temperature. From the observation that ror is shorter than rc, it follows that the orientational dynamics of the HDO molecules in the first solvation shell of the Cl ion must result from motions that do not contribute to the spectral diffusion, i.e. that do not affect the length of the O-H- -Cl hydrogen bond. Hence, the observed reorientation represents the orientational diffusion of the complete solvation structure. Also shown in Fig. 3 are fits to the data using the relation between ror and the temperature T that follows from the Stokes-Einstein relation for orientational diffusion ... 

Now in the Rouse model the diffusion constant is given by the Einstein relation... 

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... 

The result is visible in the Brownian movement of microscopic particles suspended in a fluid. If an individual particle is followed, it is seen to undergo a "random walk," moving in first one direction then another. Albert Einstein showed that if the distances transversed by such particles in a given time A t are measured, the mean square of these Ax values A2 can be related by Eq. 9-24 to the diffusion constant D (which is usually given in units of cmV1). 

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) ... 

The diffusion constant Di of a particle in a solvent is related to the viscosity of the solvent by the Stokes-Einstein relation known from hydrodynamics ... 

Once the base resistance problems are solved, the base transit time will be mi obstacle for high frequency operation due to the small diffusion constant, as estimated from the small mobility and the Einstein relation. A theoretical study on base transit time on GaN/InGaN HBTs [7] showed that a base transit time of less than 0.1 ps can be achieved with an appropriate base design by employing composition grading and/or doping grading. 

The ability to correctly reproduce the viscosity dependence of the dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other theories. As already pointed out (Section II.C 22), existing theories invoking repulsive interactions severely misrepresent the viscosity dependence at high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity dependence that is not unreasonable on first impression. The frequency correlation time is determined by the diffusion constant D, which can be estimated from the viscosity and molecular diameter a by the Stokes-Einstein relation ... 

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