Self Einstein relation

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

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. 

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... 

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

Noting from the Einstein relation that the self-diffusion coefficient can be expressed as... 

The Nernst-Einstein relation shows the dependence between the self-diffusion coefficient Dt and the equivalent conductivity A of molten salts ... 

This equation was deduced in Section 4.4.8. It is of interest to inquire here about its degree of appiicabiiity to ionic liquids, i.e., fused salts. To make a test, the experimental values of the self-diffusion coefficient D and the viscosity tj are used in conjunction with the known crystal radii of the ions. The product D r//T has been tabulated in Table 5.22, and the plot of D tj/T versus 1/r is presented in Fig. 5.31, where the line of slope k/6n corresponds to exact agreement with the Stokes-Einstein relation. ... 

The classical MD simulations performed in task I provide self-diffusion coefficients for water and also for hydronium ions, which is strictly the vehicular component of the proton diffusivity. These diffusion coefficients are calculated from the mean square displacement of H2O and HsO using the Einstein relation. The numerical values for Nation and SSC membranes at the four hydration levels are hsted in Table 5 along with the experimental values. ... 

The molecular dynamics calculations were used to calculate the self-diffusion coefficient (D) of the guest molecule using the mean square displacements (MSD) of the center of mass with respect to time (Figure 6). Using the Einstein relation... 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

According to the Einstein relation the self-diffusion coefficient is... 

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

One property that is particularly amenable for calculation with MD is the self-diffusivity, Ds, which can be calculated via the Einstein relation... 

We have above introduced concentrations and relations between concentrations of species of interest The area-specific flux (flux density) of a species, i, resulting from a driving force, F, is proportional to its concentration and to its mechanical mobility (ease of movement) j j = Cj Bj Fj as we come back to in the next section. First we briefly recall from textbooks that for species with an activated diffusion, the self-diffusion coefficient, D, mechanical mobility, B, charge mobility, u, and conductivity, a, are linked through the Nemst-Einstein relation (1.14) ... 

The limiting ionic conductivity of a rigid ion is inversely proportional to the self-diffusion coefficient of the ions. This dependence goes by the name of Nemst s law of electrochemistry. The Einstein relation relates the diffusion coefficient to the friction coefficient of the ion (Cion)- In simple terms we have the following relations. 

An alternative approach to investigating particle size and shape is to monitor the self-diffusion of the colloidal particle itself The self-diffusion coefficient is, in the absence of interactions (i.e., at infinite dilution), given by the Stokes-Einstein relation. 

The diffusion coefficient Dj of solute 1 in solvent 2 at infinitely dilute solution is a fundamental property. This is different from the self-diffusion coefficient Dq in pure liquid. Both Di and Dq are important properties. The classical approach to Di can be done based on Stokes and Einstein relation to give the following equation... 

In the earlier NMR studies the self-diffusivity was derived indirectly from measurements of the spin-lattice relaxation time over a range of temperatures. The correlation time which is roughly equivalent to the average time between successive molecular jumps, may be derived from such information and the self-diffusivity is then estimated from the Einstein relation using an assumed mean square jump length A. For an isotropic cubic lattice... 

We now focus on the Stokes Einstein relation, which relates the self-diffusion coefficient D, viscosity t], and temperature T as D cc T/t] and which is known to be accurate for normai and high temperature liquids. Since (rr) is proportional to the viscosity, the relationship between D and (tt) is examined in the inset of Fig. 10, which shows quantity D(vy)/T as a function of T. 

The dynamics of supercooled silicon, in addition to the intermediate scattering function, has been characterized by the self-diffusion coefficient. The self-diffusion coefficient or the diffusivity D is obtained in simulations from the mean square displacement using the Einstein relation... 

Residence Times. The dynamic behavior of water is frequently characterized by the self diffusion coefficient (sdc) D, which can be calculated from the particle mean square displacements via the Einstein relation or from the velocity autocorrelation functions (acf) via the Kubo relation. Near an interface this quantity D is not the self diffusion coefficient, since there are no free boundary conditions for the surface layer. Sonnenschein and Heinzinger [52] calculate a property called residence autocorrelation function... 

Figure 3 shows the mean square displacement of the chain center of mass, Remit) — i cm(0)) ), in time for the longer-chain systems, Cise, C200, and C250- From the linear part of these curves the self-diffusion coefficient D can be obtained using the Einstein relation. 

By analogy with the Einstein relation for self-diffusion, the shear viscosity can be calculated using the mean-square displacement of the time integral of the shear components of the stress-tensor (Mondello and Grest 1997). 

The self diffusivity is obtained from the trajectory by means of the Einstein relation (3.5.4-1). 

The multitude of transport coefficients collected can thus be divided into self-diffusion types (total or partial conductivities and mobilities obtained from equilibrium electrical measurements, ambipolar or self-diffusion data from steady state flux measurements through membranes), tracer-diffusivities, and chemical diffusivities from transient measurements. All but the last are fairly easily interrelated through definitions, the Nemst-Einstein relation, and the correlation factor. However, we need to look more closely at the chemical diffusion coefficient. We will do this next by a specific example, namely within the framework of oxygen ion and electron transport that we have restricted ourselves to at this stage. 

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