Stokes-Einstein relation effects

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

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. 

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... 

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 ... 

We have made a note of the hydrodynamic interactions and other interactions to draw attention to an important fact. That is, the analysis of the DLS data is often quite complex, and a simple use of the single-exponential decay function and the Stokes-Einstein relation is not always sufficient, although many instruments available on the market use such an analysis and report an effective size for the particles in the dispersion. 

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). 

E. The Out-of-Equilibrium Generalized Stokes-Einstein Relation and the Determination of the Effective Temperature... 

In Section VI, we consider a classical particle diffusing in an out-of-equilibrium environment. In this case, all the dynamical variables attached to the particle, even its velocity, are aging variables. We analyze how the drift and diffusion properties of the particle can be interpreted in terms of an effective temperature of the medium. From an experimental point of view, independent measurements of the mean-square displacement and of the mobility of a particle immersed in an aging medium such as a colloidal glass give access to an out-of-equilibrium generalized Stokes-Einstein relation, from which the effective temperature of the medium can eventually be deduced. 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

As displayed by the out-of-equilibrium generalized Stokes-Einstein relation (203), independent measurements of the particle mean-square displacement and frequency-dependent mobility in an aging medium give access, once Ax2(z) and p(z) = p(co = iz) are determined, to T (z) and to T (co) = T (z = — iffi). Then, the identity (189) yields the effective temperature ... 

The main results of this sectionjire the out-of-equilibrium generalized Stokes-Einstein relation (203) between Ax2(z) and p(z), together with the formula (206) linking Teff(ffi) and the quantity, denoted as ( ), involved in the Stokes-Einstein relation. One thus has at hand an efficient way of deducing the effective temperature from the experimental results [12]. Indeed, the present method, which avoids completely the use of correlation functions and makes use only of one-time quantities (via their Laplace transforms), is particularly well-suited to the interpretation of numerical data. 

When the environment of the particle is itself out-of-equilibrium, as is the case for a particle evolving in an aging medium, we showed how the study of both the mobility and the diffusion of the particle allows one to obtain the effective temperature of the medium. We derived an out-of-equilibrium generalized Stokes-Einstein relation finking the Laplace transform of the mean-square displacement and the z-dependent mobility. This relation provides an efficient way of deducing the effective temperature from the experimental results. 

N. Pottier, Out of equilibrium generalized Stokes-Einstein relation Determination of the effective temperature of an aging medium. Physica A 345, 472 (2005). 

The effect of the viscosity of the crystallization medium on the nucleation rate has been described by Turnbull and Fisher. The frequency of atomic or molecular transport at the nucleus-liquid interface, v can be related to the bulk viscosity, with the Stokes-Einstein relation ... 

The Stokes-Einstein relation for free particle diffusion can be applied at infinite dilution upon partial screening of electrostatic effects by added salt. At higher added salt concentrations cs the polyion diffusion obeys the relation... 

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. 

In solution, D is given by the Stokes-Einstein relation which relates D to the viscosity coefficient of the solution, T], and the effective hydrodynamic radius a, where... 

Application of the Smoluchowski equation (1.3) requires a knowledge of the sizes and diffusion coefficients of molecules. An estimate of the effective values of r can usually be made from molecular volumes. Diffusion coefficients present more of a problem, since not very many have been experimentally determined over a range of temperature. They may, however, be eliminated from Equation (1.3) by using the Stokes-Einstein relation, which is theoretically derived from the same molecular model and is often in fairly good accord with experimental data it expresses D in terms of the viscosity of the solvent (t)) and the... 

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