Stokes-Einstein-Debye model

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

For complex ions of similar size within a homologous series, it is expected that t//tc remains relatively constant for different solvents at a given temperature, in accordance with the Stokes-Einstein-Debye model. From the slope of the plot 5iso versus y/(Az/f/2)/A av shown in Fig. 16, the correlation time constants of a series of tra 5 -[Co(acac)2XY] complexes were estimated and the results were compared favourably with the Tc obtained from the relaxation measurements of the methene carbons (see Table 5). 

Working in dilute solution (< 80 mmolP ) is recommended whenever the Stokes-Einstein-Debye model is employed for data analysis, as illustrated in the study of Co longitudinal relaxation time of Co(acac)3 in CH3CN. ... 

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),... 

Stokes-Einstein-Debye hydrodynamic theory is the most commonly used model for understanding the rotational diffusion of molecular systems [76, 77]. 

Kathmann et al. did determine reorientational correlation times for several amine bases in organic solvents based on both NMR relaxation and MD simulations as a test ability MD simulations and DFT calculations to explain the mechanism in complex reactions catalyzed by frustrated Lewis pairs (FLP) in metal free scission of H2. As the Debye-Stokes-Einstein (DSE) model gives only qualitative predictions, MD simulations are found valuable to validate the spectroscopic studies. 

These experimental findings on the rotational dynamics of anthraquinone dyes in liquid hosts are also interesting as they show deviations from hydro-dynamic theories usually used for describing the rotational diffusion of dye molecules in liquid solvents [6-12]. These theories are commonly gathered in the Uterature under the name of Stokes-Einstein-Debye theory (SED). SED models treat the solvent as a macroscopic continuum, in which the diffusional rotation of the solute is only affected by the viscosity and temperature of the hosting solvent. However, as expected, the validity of this continuum description breaks down when the size of the solute molecules approaches that of the solvent molecules or becomes smaller. In this regime that includes the important case of pure materials, the specific intermolecular interactions between the solute and the solvent molecules start to play a fundamental role in their rotational dynamics [9,11-16]. 

Schematically, theories of rotational motion in liquids may be divided into two groups, which may be called classical reorientation and jump reorientation models. For the case that the rotation of a molecule in a liquid is regarded as a solid body moving in a fluid continuum the Debye-Stokes-Einstein relation [66] should apply. Thus for the reorientation of a spherical molecule...

Whereas the Debye-Stokes-Einstein equation may be applicable for macromolecules in a low molecular weight solvent it is apparently not a realistic model for molecular motion in a neat liquid. A modificatior of the Debye-Stokes-Einstein relation was proposed by Gierer and Wirtz [67] who tried to take into account the discontinuous nature of the liquid. For a spherical molecule they obtained... 

Fig. 2.8. Comparison of the reorientational correlation time of CIO3F with the Debye-Stokes-Einstein model. The inverse of the correlation time (obtained from 35ci relaxation) is plotted as a function of the ratio of temperature to viscosity. (From Ref. [55])...

The above model has been used by Bull et al. (102) to analyse the chloride NMR relaxation data for several proteins. The overall correlation time(s) was estimated from the Debye-Stokes-Einstein equation or, when available, was taken from dielectric relaxation measurements. In order to perform the calculations, also the "true" value of the chloride quadrupole coupling constant in the site is needed. This is, however, not known and therefore Bull et al. estimated a value for Cl bound to a NH3 group of 3.6 MHz based on the electrostatic model of Cohen and Reif (103). In this way it was possible to calculate the values shown in Table 6. Obviously the model used is an oversimplification however, it is noteworthy that all internal correlation times come out with a reasonable value of about 1 ns. 

Motional factors determining the linewidths can be envisaged in terms of a frictional model (Debye-Stokes-Einstein theory) according to which... 

The proper definition of these two correlators is the fundamental issue. Very often the slow correlator is addressed to the orientational self-correlation of the single rigid molecule, see (2.33), which according to the Debye-Stokes-Einstein model (DSE) can be described as a pure Brownian diffusive process [19,42]. Hence the slow correlator becomes... 

The Dynamics of Liquid lodobenzene vs. Debye-Stokes-Einstein Model... 

As we already introduced in Sect. 2.3.2.1, the slower dynamics of simple liquids has been often described on the basis of the Debye-Stokes-Einstein (DSE) diffusion model. In our paper, Bartolini et al. [19], we checked such model on the dynamics of liquid iodobenzene as measured by HD-OKE experiments. 

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