Debye-Stokes Einstein relationship

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)... 

S nuclear quadrupole coupling constants have been determined from line width values in some 3- and 4-substituted sodium benzenesulphonates33 63 and in 2-substituted sodium ethanesulphonates.35 Reasonably, in sulphonates R — SO3, (i) t] is near zero due to the tetrahedral symmetry of the electronic distribution at the 33S nucleus, and (ii) qzz is the component of the electric field gradient along the C-S axis. In the benzenesulphonate anion, the correlation time has been obtained from 13C spin-lattice relaxation time and NOE measurements. In substituted benzenesulphonates, it has been obtained by the Debye-Stokes-Einstein relationship, corrected by an empirically determined microviscosity factor. In 2-substituted ethanesulphonates, the molecular correlation time of the sphere having a volume equal to the molecular volume has been considered. 

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 Co longitudinal relaxation rates of tris(acetylacetonato)cobalt(lIl) in dichrolo-methane and benzene were measured over the concentration range between 20 and 110 mol m 3 at several temperatures. The dependence of the relaxation rate on the temperature and the complex concentration is primarily attributable to the change in the viscosity of solutioa The values of eQqlh were calculated from the concentration dependence of the Co relaxation rate by using the Debye-Stokes-Einstein equation and the Einstein relationship between molar volume and viscosity B coefflcienL... 

Similarly, the rotational diffusion coefficient at temperature r,Dr(T), is predicted to be coupled to rj(T) by the Debye-Stokes-Einstein (DSE) relationship ... 

It was established that CDs labelled with TEMPO derivatives, Le. spin labelled CDs 32-34 undergo interaction with PEG 600 (PEG = polyethylene glycol) and with PPG 425 (PPG = polypropylene glycol) in concentrated aqueous solutions. The EPR spectra of 32-34 are changed when they are complexed with PEG or PPG. It was observed that the relationship between rotational correlation times (t) and solvent viscosity caimot be described by the Debye-Stokes-Einstein equation, this fact being due to self-aggregation of alkylene glycols in concentrated solutions. However the use of the fractional Debye-Stokes-Einstein equation, i.e. the relationship between relative x values and relative viscosity is in accordance with the experimental data [77]. 

Determination of the PSD requires knowledge of the relationship between Do and particle size analogous to the Stokes-Einstein Eq. (39). The number distribution of particle size also requires an expression for P(q). For roughly spherical particles that do not absorb light at the laser wavelength, the Raylcigh-Debye-Gans (RDG) (43,44) approximation for spheres... 

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