Debye-Stokes-Einstein equation

The time dependence of the anisotropy r(t) depends on the underlying dynamics of reorientational motion. For rotational diffusion (tumbling) of a spherical object, the expected anisotropy decay is exponential with a rotational diffusion time given in the hydrodynamic limit by the Stokes-Einstein-Debye equation. For nonspherical molecules, more complex time dependence may be detected. (For more on these topics, see the book by Cantor and Schimmel in Further Reading.)... 

Objects rotate in media slower than in vacuum because of viscous friction, which may be quantified using the Stokes-Einstein-Debye equation. " For a sphere... 

Results of the EPR study of rotational diffusion (Tc) in viscous ionic hquids BmimBF4 and BmimPFe with proxyl radical (R5COOH) in a molecular and anionic forms were analysed in (Miyake et al., 2009). The experimental average values and activation energies for rotational diffusion disagreed with those calculated from fractional Stokes-Einstein-Debye equations. 

The Debye-Stokes-Einstein (DSE) equation (60) predicts that the orientational correlation time of a spherical object in a continuum liquid is... 

The rotational diffusion coefficient, Dr, of a probe molecule in a glass-former follows the temperature dependence of the Debye-Stokes-Einstein (DSE) equation [167-171],... 

Baxendale and Wardman (1973) note that the reaction of es with neutrals, such as acetone and CC14, in n-propanol is diffusion-controlled over the entire liquid phase. The values calculated from the Stokes-Einstein relation, k = 8jtRT/3jj, where 7] is the viscosity, agree well with measurement. Similarly, Fowles (1971) finds that the reaction of es with acid in alcohols is diffusion-controlled, given adequately by the Debye equation, which is not true in water. The activation energy of this reaction should be equal to that of the equivalent conductivity of es + ROH2+, which agrees well with the observation of Fowles (1971). 

On the other hand, if the rate constant for the quenching step exceeds that expected for a diffusion-controlled process, a modification of the parameters in the Debye equation is indicated. Either the diffusion coefficient D as given by the Stokes-Einstein equation is not applicable because the bulk viscosity is different from the microviscosity experienced, by the quencher (e.g. quenching of aromatic hydrocarbons by O, in paraffin solvents) or the encounter radius RAb is much greater than the gas-kinetic collision radius. In the latter case a long-range quenching... 

Osborne and Porter have found that the Debye equation probably underestimates kiU because of a failure in the Stokes-Einstein dependence of diffusion coefficients on viscosity.185 The coefficient of sliding friction has always been assumed to be infinity. If it is zero, the Debye equation is increased by 50%. 

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... 

Since the rotation of the complex molecule will be isotropic as in the cases of several tiis(did iate)cobalt(III) complexes, and the mdecule can be regarded as a sphere, we assume that the rotational correlation time is expressed by the Debye-Stokes-Einstein equation > ... 

Thus, the combined SE and the DSE equations predict that the product Dtxc = (A Tc)sedse should equal 2r /9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product Dtr can be much larger than this value, revealing a breakdown of the Stokes-Einstein (SE) relation and the Debye-Stokes-Einstein (DSE) relation. There is an enhancement of probe translational diffusion in comparison with rotational diffusion. The time dependence of the probe rotational time correlation functions tit) is well-described by the KWW function,... 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

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... 

The assumption of a diffusive motion entails also the validity of hydrodynamics for the reorientation. If this is the case, then we are justified to some extent to use the Debye-Stokes-Einstein equation ... 

The correlation time data can be further rationalized by the use of the Debye-Stokes-Einstein equation 35, as displayed in figure 13. 

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. 

Intcrmolecular dipole-dipole relaxation depends on the correlation time for translational motion rather than rotational motion. Intermolecular dipole-dipole interactions arise from the fluctuations which are caused by the random translational motions of neighboring nuclei. The equations describing the relaxation processes are similar to those used to describe the intramolecular motions, except is replaced by t, the translation correlation time. The correlation times are expressed in terms of diffusional coefficients (D), and t, the rotational correlation time and the translational correlation time for Brownian motion, are given by the Debye-Stokes-Einstein theory ... 

A number of attempts have been made to account for the discrepancies between correlation times calculated from the Debye-Stokes-Einstein equation and the correlation times observed by magnetic resonance spectroscopy. A micro viscosity correction factor (/ ) has been introduced in the Debye-Stokes-Einstein equation in which rj is reduced by... 

Ti measurements performed on cyclic dipeptides in dimethyl-sulfoxide-dg revealed a correlation between the molecular weight of the dipeptide and the value of the a-carbon of optically active amino acid residues. These are shown in Figure 26 (Deslauriers et al, 1975b). According to the Debye-Stokes-Einstein equation, the correlation plotted in dotted lines is to be expected. The discrepancy between observed and calculated values is worse for the lower-molecular-weight dipeptides. For dipeptides... 

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