Stokes-Einstein behavior

Fig. 5. Variation of rR with molecular weight for compounds in Fig. 4, illustrating the approximate Stokes-Einstein behavior...

Looking at Fig. 5.32, it can be seen that there is a fairly significant fit. The anions, particularly those of the group n halides, are not very consistent with the Stokes-Einstein relation. However, their poorer fit is offset by the better Stokes-Einstein behavior of the cations. The relatively good fit of the cations tempts one to conclude that there is a particular reason for the deviations of the anions. Some attempts have been made... 

Won J, Onyenemezu C, Miller WG, Lodge TP (1994) Diffusion of spheres in entangled polymer solutions a return to stokes-einstein behavior. Macromolecules 27(25) 7389-7396... 

D. E. Dunstan and J. Stokes. Diffusing probe measurements in polystyrene latex particles in polyelectrolyte solutions deviations from Stokes-Einstein behavior. Macromolecules, 33 (2000), 193-198. 

Because a depends on M > with modestly different exponents y for different transport coefficients, non-Stokes-Einsteinian behavior such as Ds c)t c) Ds(0)rj(0) should be common. Indeed, non-Stokes-Einstein behavior is observed for Ds and Dp. On the other hand, models that yield simple relationships between Ds and r], e.g.,... 

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)... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

The discrepancies between the experimental data and the behavior predicted using the Smoluchowski-Stokes-Einstein model for ksenfluor and ksenphos likely arise from the inadequacies of the simple Smoluchowski-Stokes-Einstein analysis for application to the anthracene/diaryliodonium salt molecular system. For example, the Smoluchowski analysis assumes that the reacting molecules are spherical in... 

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... 

It is interesting to compare conductance behavior with that of the shear viscosity, because conventional hydrodynamic conductance theories relate A to the frictional resistance of the surrounding medium. At first glance, one would expect from the Stokes-Einstein equation a critical anomaly of the... 

Here, R is the radius of the sphere, q is the coefficient of viscosity, kB is Boltzmann s constant, and T is temperature. Equation (4) implies that rR should vary linearly with volume, or mass, in the range where the Stokes-Einstein equation is valid. Figure 5 shows the roughly linear behavior of rR for these compounds, and illustrates why polymeric conjugates of Gd3+ chelates remain a very attractive method of modulating both rR and the intravenous retention time (t1/2) of BPCAs. 

Electron spin resonance (ESR) studies of radical probe species also suggest complexity. Evans et al. [250] study the temperature dependence of IL viscosity and the diffusion of probe molecules in a series of dissimilar IL solvents. The results indicate that, at least over the temperature range studied, the activation energy for viscous flow of the liquid correlates well with the activation energies for both translational and rotational diffusion, indicative of Stoke-Einstein and Debye-Stokes-Einstein diffusion, respectively. Where exceptions to these trends are noted, they appear to be associated with structural inhomogeneity in the solvent. However, Strehmel and co-workers [251] take a different approach, and use ESR to study the behavior of spin probes in a homologous series of ILs. In these studies, comparisons of viscosity and probe dynamics across different (but structurally similar) ILs do not lead to a Stokes-Einstein correlation between viscosity and solute diffusion. Since the capacities for specific interactions are... 

For diffusion in amorphous polymers at temperatures above their glass point, Tg, one can assume a behavior with some analogy to a liquid. On the other hand the Stokes-Einstein Eq. (6-4) for liquids was derived under the assumption that the diffusing particle is much larger in size than the matrix particles. If we let the matrix be a... 

Chang (W8) based on the Stokes-Einstein equation. In this formula the association parameter.r allows for differences in solvent behavior . r = 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for benzene, ether, heptane, and other unassociated solvents. The average error for systems surveyed by the authors was about 10% the relationship cannot be used when a complex is formed between solute and solvent. For amyl alcohol. 

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