Stokes Einstein friction

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

To understand the dynamics of one chain in a melt, it is convenient to start from a slightly different problem. We consider one test chain of Ni monomers, embedded in a monodisperse melt of the same chemical species, with a number N of monomers per chain. We consider three types of motion for the test chain reptation, tube renewal, and Stokes-Einstein friction. We first describe tube renewal and show that this is probably negligible for most practical purposes. Then we discuss competition between reptation and Stokes-Einstein friction. 

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

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 [ ]... 

Another microscopic approach to the viscosity problem was developed by Gierer and Wirtz (1953) and it is worthwhile describing the main aspects of this theory, which is of interest because it takes account of the finite thickness of the solvent layers and the existence of holes in the solvent (free volume). The Stokes-Einstein law can be modified using a microscopic friction coefficient ci micro... 

Person 2 Estimate the nnsolvated Stake s sphere friction coefficient, fo, nsing the Stokes-Einstein relationship for spheres, fo = 6jrp,r. 

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%. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

It is interesting to compare conductance behavior with that of the shear viscosity, because conventional hydrodynamic conductance theories relate A to the frictional resistance of the surrounding medium. At first glance, one would expect from the Stokes-Einstein equation a critical anomaly of the... 

In the diffusion region the reorientational motion of the molecules is impeded by a frictional force exerted by a medium considered structureless (continuum). For a spherical molecule, the rotational diffusion coefficient, D, is given by the Stokes-Einstein-Debye equation42... 

Stokes-Einstein equation — George Gabriel Stokes (1819-1903) deduced an expression for the frictional coefficient (/s) in liquids [i] ... 

This expression relates the experimental quantity tr directly to dH provided that all other parameters are known. Alternate expressions can be obtained relating tr to the friction coefficient f or to D through use of the Stokes-Einstein equation D=kT/f giving ... 

For a binary mixture, if experimental diffusivities do not exist over the whole range of concentration, an interpolation of the diffusivities at infinite dilution D k] J is used. In calculating the diffusivities at infinite dilution by the Stokes-Einstein relation, we consider small isolated hard spheres, submerged in a liquid, that are subjected to Brownian motion The friction of the spheres in the liquid is given by the Stokes law Einstein used the Stokes law to calculate the mean-square displacement of a particle. The displacement increases linearly with time, and the proportionality constant is the Stokes-Einstein diffusivity... 

The equation reduces to the Stokes-Einstein equation for spherical particles. Since the friction coefficient for a non-spherical partiele always exceeds the friction coefficient for a spherical particle, over estimation of particle size will occur if equation (10.41) is applied. 

Hydrodynamic theory [67], based on Stokes-Einstein equation, postulates that solute is represented by a very large sphere in comparison with the surrounding small liquid phase molecules. Solute mobility, and thus its diffusion coefficient, depends on the frictional drag exerted by liquid phase molecules. For heterogeneous gels (rigid polymeric chains), Cukier [85] suggests... 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

The model was also checked by evaluating the center-of-mass friction. It was shown that hydrodynamic interactions are important for solvent-separated atoms, 8 A, but not for the diatomic with 2.66 A. The mass dep>endences of the isolated iodine and argon frictions were not consistent with hydrodynamics estimates of the Stokes-Einstein theory (Section III E). Rather, they are in agreement with the Enskog theory corrected for caging by the Herman-Adler results for hard spheres. Further studies are required which avoid the use of Eq. (5.8). 

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... 

Because of strong interactions in polyelectrolyte solutions without added salt, the use of the well-known Stokes-Einstein relation for free particle diffusion D = kBT/6Trr)Rh, where 17 is viscosity and Rh is hydrodynamic radius, is rather limited. Even at very low concentrations, where intermo-lecular interactions can be neglected due to large intermolecular separations, the friction factor contains in addition to the Stokes-Einstein contribution /SE = 67717/4 also a contribution from electrolyte dissipation, so that the total friction factor / = /SE + /eidiS, where the electrolyte dissipation term reflects the retardation of the polyion motion due to the instantaneous distortion of the surrounding ion atmosphere as the polyion moves through the solvent [27,28],... 

This is the Stokes-Einstein equation, which relates the diffusion coefficient and the frictional coefficient. Although we have derived this relation using a gravity field, it is correct for any conservative field. We derived this equation by a different route in Chapter 31 see Eq. (31.61). ... 

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