Einstein relation rotational

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 [ ]... 

The nature of rotational motion responsible for orientational disorder in plastic crystals is not completely understood and a variety of experimental techniques have been employed to investigate this interesting problem. There can be coupling between rotation and translation motion, the simplest form of the latter being self-diffusion. The diffusion constant D is given by the Einstein relation... 

A spectroscopic technique that probes membrane fluidity can either directly measure mobility and order parameters for membrane constituents (NMR) or use probes (ESR, fluorescence). Some fluorescent and ESR probes are shown in Fig. 4. The connection between the rotational correlation time of a membrane embedded probe and the membrane fluidity can be illustrated using the example of a simple isotropic liquid, in which fluidity is merely a reciprocal viscosity ri and the rotational correlation time Xc for a molecule with a hydrodynamic volume V is given by the well-known Debye-Stokes-Einstein relation Xc = r VlkT, where k is the Boltzmann constant and T is the... 

The orientation autocorrelation function P2[cos 0(t)] is given by r(t) and reflects the motion undergone by the fluorescent chromophore (2,14). A number of models for Brownian motion have been proposed (14) but in the simple case of a rigid sphere, r(t) is described by a single exponential decay where Tf., the rotational correlation time is related to the hydrodynamic volume of the sphere and the viscosity of the medium through the Stokes-Einstein relation (14,16). More complex motions of fluorophores necessitate the development of models which fit the functional form of r(t) experimentally obtained (14). 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

Both the SRLS and the FT inertial models were discussed in the context of the Hubbard-Einstein relation, that is, the relation between the momentum correlation time Tj and the rotational correlation time (second rank) for a stochastic Brownian rotator [39]. It was shown that both models can cause a substantial departure from the simple expression predicted by a one-body Fokker-Planck-Kramers equation ... 

In the Einstein relation for induced radiation processes the ratio of absorption and emission is proportional to the population density in the two energy levels involved however, we have to account for the degeneracy factor g. This is of particular importance in vibration-rotation transitions with the selection rules AJ=-1 and AJ=+1 for P-branch and R-branch transitions, since there the J-values and hence the degeneracy factors of the two levels involved are different. The relation is... 

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...

More specifically, the detailed spectroscopic analysis by Agmon found that the timescale of water molecule rotation, which takes 1-2 ps at room temperature, is similar to proton hopping times determined from the analysis of the resonance in NMR studies by Meiboom (1961). Using the hopping time of Tp = 1.5 ps and a hopping length of Ip = 2.5 A, an estimate of proton mobility in three-dimensional networks can be obtained from the Einstein relation, Dh+ = Ip/6rp = 7.0 10 cm s . This estimate is close to the experimental value of 9.3 10 cm s . From this closeness, Agmon concluded that proton mobility in water is an incoherently Markovian process. 

G. H. Koenderinck and A. P. Philipse. Rotational and translational self-diffusion in colloidal sphere suspensions and the applicability of generalized Stokes-Einstein relations. Langmuir, 16 (2001), 5631-5638. 

A comparison of rotational and translational diffusion results obtained in l-octyl-3-imidazolium tetrafluoroborate, [omim][BF4], and in 1-propanol and isopropyl benzene has been given for TEMPONE. Measurements at different temperatures and concentrations indicate that rotational motion can be described by isotropic Brownian diffusion only for the classical organic solvents used, but not for the IL. Simulation of the EPR spectra fit with the assumption of different rotational motion around the different molecular axes. Rotational diffusion coefficients >rot follow the Debye-Stokes-Einstein law in all three solvents, whereas the translational diffusion coefficients do not follow the linear Stokes-Einstein relation D ot versus Tlr ). The activation energy for rotational motions Ea,rot in [omim][BF4] is higher than the corresponding activation energies in the organic solvents. 

Viscosity is a useful quantity, in that both rotational and translation mobility of molecules in solution are viscosity dependent and can be related to viscosity through the Stokes-Einstein equation ... 

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

Existence of a high degree of orientational freedom is the most characteristic feature of the plastic crystalline state. We can visualize three types of rotational motions in crystals free rotation, rotational diffusion and jump reorientation. Free rotation is possible when interactions are weak, and this situation would not be applicable to plastic crystals. In classical rotational diffusion (proposed by Debye to explain dielectric relaxation in liquids), orientational motion of molecules is expected to follow a diffusion equation described by an Einstein-type relation. This type of diffusion is not known to be applicable to plastic crystals. What would be more appropriate to consider in the case of plastic crystals is collision-interrupted molecular rotation. 

Here r is the shear viscosity, (xc) is the mean rotational correlation time, and rs the spherical radius of the probe molecule. On the other hand, the translational diffusion coefficient, Dt, of the probe molecule is given by the Stokes-Einstein (SE) relation [167-171],... 

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 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... 

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