Einstein diffusion regime

Where u < 1 (u = 1 for the Einstein diffusion regime). At very short times (i < 1 ps), the MSD may be quadratic iv n time ( = 2) which is characteristic of free flight as may occur in a pore or solvent cage prior to collision with the pore or cage wall. The result of anomalous diffusion, which may or m not occur in intermediate time scales, is to create a smaller slope at short times, resulting in a larger value for the diffusion coefficient. At sufficiently long times (the hydrodynamic limit), a transition from anomalous to Einstein diffusion ( = 1) may be observed [71]. 

FIG. 2 Mean-square displacement (MSD) of helium atoms dissolved in polyisobutylene. There is a regime of anomalous diffusion (MSD a followed by a crossover at 100 ps to normal (Einstein) diffusion (MSD a r) [24],... 

Let us now briefly recall the main features of the behavior of D c °°(r) as a function of t for t > 0 [25]. First, the limiting value at infinite time of D ,c, 00(f) is, at any nonzero temperature, the usual Einstein diffusion coefficient kT/rp Above the crossover temperature Tc as defined above, ) e, 00(f) increases monotonously toward its limiting value. Below the crossover temperature (i.e., T < T( ).D"] (t) first increases, then passes through a maximum and finally slowly decreases toward its limiting value. Thus, in the quantum regime, I)" x(t) can exceed its stationary value, and the diffusive regime is only attained very slowly, namely after times t fth- At T - Q.If x(Vj can be expressed in terms of exponential integral functions ... 

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... 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

The slowdown of molecular motion with the growth of the shear viscosity across the supercooled regime has been extensively investigated with a variety of experimental techniques. The Stokes-Einstein (SE) relation describes a coupling between the translational diffusion coefficient Dt(T) and the shear viscosity ri(T) at a temperature T ... 

In the continuum regime the Brownian diffusivities are given by the Stokes-Einstein for- ... 

DLS is certainly the foremost method to obtain the diffusion coefficient D of colloidal particles in the dilute regime. Since the measurements are performed at high dilution, a possible influence of mutual interaction of the particles can be safely dismissed. Hence, the diffusion coefficient D may directly be converted into the hydrodynamic radius Rh by the well-known Stokes-Einstein relation ... 

The hopping mobility (m) in the high-temperature regime can be approximated from the electron transfer rates by considering the Einstein relation for diffusive motion [20, 23],... 

The finding of anomalous diffusion in gas transport through amorphous polymers by TSA was bom out completely by MD. For a favourable case of fast diffusion, MD could detect the crossover from short-time motion inside cavities to the anomalous diffusion, and then to the Einstein regime. Anomalous diffusion must be expected for all transport of light gases through amorphous polymers. 

Measurements of transport properties would provide another means to explore the metastable regime. In particular, studies based on simulation focused on the supercooled regime [104] correlate the breakdown of the Stokes-Einstein relation Dv = constant) with the Widom line w P), locus of the correlation length maxima emanating down from the proposed liquid-liquid critical point toward lower pressures (Fig. 3b). Measurements of diffusivity could be performed at negative pressure by NMR on static samples (e.g., via MVLE or inclusions) and viscosity could be measured by capillary rheometry with the MVLE method. 

The influence of polydispersity on the analysis of quasi-elastic lightscattering data is considerable. For non-interacting particles in the Stokes-Einstein regime, the effective diffusion coefficient is ... 

In these theories the dynamics of the polyions is assumed to be very simflar to the dynamics of neutral polymers. For neutral polymers in good solvents, different concentration regimes with different static and dynamical properties are predicted [43]. Below the overlap concentration the diffusion of individual chains is measured, which obeys a simple Stokes-Einstein relation... 

The photomicrographic measurements refer directly to polymer motion under the influence of an external force. However, measurements of migration velocity v as a function of applied electrical field E show that some of these electrophoretic measurements were made in a low-field linear regime, in which the electrophoretic mobility jx is independent of E. Linear response theory and the fluctuation-dissipation theorem are then applicable they provide that the modes of motion used by a polymer undergoing electrophoresis in the linear regime, and the modes of motion used by the same polymer as it diffuses, must be the same. This requirement on the equality of drag coefficients for driven and diffusive motion was first seen in Einstein s derivation of the Stokes-Einstein equation(16), namely thermal equilibrium requires that the drag coefficients / that determine the sedimentation rate v = mg/f and the diffusion coefficient D = kBT/f must be the same. 

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