Einstein relation, expression

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

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... 

We can determine from Eq. (8.2) the values of D for solvated metal ions. The value of D changes with changes in solvent or solvent composition. The viscosity (rj) of the solution also changes with solvent or solvent composition. However, the relation between D and tj can be expressed by the Stokes-Einstein relation ... 

In order to evaluate this expression, we need to know the force v / that is responsible for producing the molecular flux. It could be an external force such as an electric field acting on ions. Then evaluation of Eq. 18-48 would lead to the relationship between electric conductivity, viscosity, and diflusivity known as the Nernst-Einstein relation. 

In this expression, the primary environmental determinant is the viscosity in the denominator. Note that the exponential is slightly larger than in the Stokes-Einstein relation (Eq. 18-52). Since viscosity decreases by about a factor of 2 between 0°C and 30°C, D,w should increase by about the same factor over this temperature range. Furthermore, the influence of the molecule s size is also stronger in Eq. 18-53 than in 18-52 (note r, = constant V 173). In Box 18.4 experimental information on the temperature dependence of D,w is compared with the theoretical prediction from Eqs. 18-52 and 18-53. 

In the following table the different models are applied to CFC-11. Note the excellent correspondence between the temperature variation calculated by the Stokes-Einstein relation (Eq. 3) and the expression by Hayduk and Laudie (Eq. 4), although both models overestimate the temperature effect compared to the activation model derived from the experimental data (Eq. 2). 

As for Illustrative Example 18.2a (diffusivity of CFC-12 in air), these values agree fairly well with each other, except for the Stokes-Einstein relation, which was not meant to be a quantitative approximation but an expression to show qualitatively the relationship between diffusivity and other properties of both molecule and fluid. 

Other than dynamical correlations, transport properties have also been derived using hydrodynamic theory. In hydrodynamics the diffusion of a tagged particle is defined by the Stoke-Einstein relation that is given by the following well-known expression ... 

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

This is known as the Stokes-Einstein relation and is independent of the charge of the species. Using this expression, diffusion coefficients can be estimated from viscosity measurements, so long as Stokes Law is applicable. It is used particularly for macromolecules. 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

Interestingly, each one of the two FDTs can be formulated in two equivalent ways, depending on whether one is primarily interested in writing a Kubo formula for a generalized susceptibility %(co) [namely, in the present case, p(m) or y(o))], or an expression for its dissipative part [namely, the Einstein relation... 

The Einstein relation (159) or the expression (157) of the dissipative part ffiep(m) of the mobility constitute another formulation of the first FDT. Indeed they contain the same information as the Kubo formula (156) for the mobility, since p(co) can be deduced from 9ftep(oo) with the help of the usual Kramers-Kronig relations valid for real co [29,30]. Equation (156) on the one hand, and Eq. (157) or Eq. (159) on the other hand, are thus fully equivalent, and they all involve the thermodynamic bath temperature T. Note, however, that while p(oo) as given by Eq. (156) can be extended into an analytic function in the upper complex half-plane, the same property does not hold for D(co). 

D has been sometimes expressed in terms of the hydrodynamic radius. However, we consider that hydrodynamic radius is not the proper term to describe the change of D, because this is not a well-defined radius such as the radius of gyration, which is clearly defined to show the molecular size. The hydrodynamic radius has just the same meaning as D as long as the Stoke-Einstein relation holds good. 

According to the Einstein relation, the diffusion coefficient is inversely proportional to the translational friction coefficient / at infinite dilution by the expression... 

The hole-theory expression for viscosity is known. It is Eq. (5.94). Let this be introduced into Stokes-Einstein relation [Eq. (5.95)]. Using Eq. (5.44) the result is... 

The expression a = nDI2/3 for the ionic conductivity a follows directly from the comparison with the phenomenological theories [2, 8] and indeed gives the well-known Nernst-Einstein relation connecting the conductivity with the mutual diffusion coefficient. Note that D = lirri, , 0 D(k, z) is the hy-... 

D may again be substituted by the viscosity rj using Stokes-Einstein relation D = RT / (3nNAdiffusing species and is roughly same as a in the earlier expression. Thus... 

For particles 0.1 //ni in radius, the characteristic time (n,- + fl ) /( A + A) about 10 sec, and the use of the steady-state solution is justified in most cases of practical interest. When the Stokes-Einstein relation holds for the diffusion coefficient (Chapter 2) and dp )S> f, this expression becomes... 

Both the SRLS and the FT inertial models were discussed in the context of the Hubbard-Einstein relation, that is, the relation between the momentum correlation time Tj and the rotational correlation time (second rank) for a stochastic Brownian rotator [39]. It was shown that both models can cause a substantial departure from the simple expression predicted by a one-body Fokker-Planck-Kramers equation ... 

Using Stokes-Einstein relation (D = kBTI6mrj), the expressions for characteristic displacements induced by random Brownian motion in one dimension is given by [7]... 

The masses of particles may be expressed as given in Table 1.1 in terms of energy through the Einstein relation... 

Ln-L distance, energy transfer occurs as long as the higher vibrational levels of the triplet state are populated, that is the transfer stops when the lowest vibrational level is reached and triplet state phosphorescence takes over. On the other hand, if the Ln-L expansion is small, transfer is feasible as long as the triplet state is populated. If the rate constant of the transfer is large with respect to both radiative and nonradiative deactivation of T, the transfer then becomes very efficient ( jsens 1, eqs. (11)). In order to compare the efficiency of chromophores to sensitize Ln - luminescence, both the overall and intrinsic quantum yields have to be determined experimentally. If general procedures are well known for both solutions (Chauvin et al., 2004) and solid state samples (de Mello et al., 1997), measurement of Q is not always easy in view of the very small absorption coefficients of the f-f transitions. This quantity can in principle be estimated differently, from eq. (7), if the radiative lifetime is known. The latter is related to Einstein s expression for the rate of spontaneous emission A from an initial state I J) characterized by a / quantum number to a final state J ) ... 

In the same vein, the variation of the diffusion coeflBcient of species across the lipid membrane cannot be explained by employing hydrodynamic expressions, such as the Stokes-Einstein relation. Here one would need to consider the free-energy barrier for entrance into the layer for each species, charged (positive or negative) and neutral the free-energy barrier is expected to be different even for same-sized species. The lipid bilayer diffusion series (LBDS) given by Eq. (12.2) is a manifestation of such microscopic effects. 

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