Einstein relation translational

Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

A number of bulk simulations have attempted to study the dynamic properties of liquid crystal phases. The simplest property to calculate is the translational diffusion coefficient D, that can be found through the Einstein relation, which applies at long times t ... 

Measurement of the translational diffusion coefficient, D0, provides another measure of the hydrodynamic radius. According to the Stokes-Einstein relation... 

With the help of the Stokes-Einstein relation, the translational diffusion coefficient may be calculated according to... 

The translational diffusion coefficient of micelles loaded with a fluorophore can be determined from the autocorrelation function by means of Eqs (11.8) or (11.9). The hydrodynamic radius can then be calculated using the Stokes-Einstein relation (see Chapter 8, Section 8.1) ... 

The self-translational diffusion coefficient D is related to f, by the Stokes-Einstein relation and is given by... 

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

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... 

One aspect of the dynamics of micellar systems that has received a renewed interest during recent years is the translational motion of the micelles themselves. In the simplest approximation, the translational diffusion coefficient, D, of a spherical micelle is related to the hydrodynamic radius rM through the Stokes-Einstein relation... 

For temperatures below 1.2 7 , molecules on the average translate considerably further than that based on viscosity through the Stokes-Einstein relation... 

According to the Einstein relation, the diffusion coefficient is inversely proportional to the translational friction coefficient / at infinite dilution by the expression... 

Intuitively, D appears to be well-placed to capture the dynamical signature of the coupling between orientational and translational order. In the energy landscape formalism the time-dependent position rft) of a particle i can be resolved into two components rft) = Rft) + Sft), where Rft) is the spatial position of the particle i in the inherent structure for the basin inhabited at time and S ft) is the intrabasin displacement away from that inherent structure [159], It has been theoretically argued that the replacement of the real positions r (f) by the corresponding inherent structure positions in the Einstein relation yields an equivalent diffusion description [159, 160]. Such a proposition, which has been verified in simulations [159, 160], forms the foundation of the analysis presented here. 

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

Translational and rotahonal diffusion coefficient of a molecule in a liquid provides a quantitahve measure of the dynamic timescales in the liquid. These coefficients are related to viscosity by the Stokes-Einstein [2] and the Debye-Stokes-Einstein relation [3], respectively. Using the definihon of diffusion coefficient in terms of mean-square displacement [2] and the Stokes-Einstein relahon, we can estimate the time needed by a water molecule to translate a distance equal to its molecular diameter a... 

Another interesting aspect of this observation relates to another celebrated equation, the Stokes-Einstein relation between diflfirsivity and viscosity. It tells us that one can now define the viscosity in a nano-confined region (grooves of DNA), which is termed the microviscosity of that particular region [7]. The Stokes-Einstein relation, where the translational diflfirsion constant and viscosity are inversely related, provides a remarkable correlation between microviscosity and configurational entropy. 

The Dynamic Light Scattering (DLS) technique was used to measure radii of the PS latex spheres with and without adsorbed polymer brushes. We could then deduce the polymer brush hydrodynamic layer thickness by taking the difference of the radii. DLS measures the intensity autocorrelation as a function of delay time, which gives information on the diffusion constant of particles in a dilute solution. The translational diffusion coefficient, D, is related to the solution temperature T, particle radius r, and solvent viscosity ri by the Stokes-Einstein relation ... 

A number of common physical phenomena can affect and influence the autocorrelation function of a diffusion single molecule fluorescence experiment (Figure 2.1) and they are summarized in Figure 2.11. The principle component which generally dominates the autocorrelation function is diffusion (Figure 2.11(a)) [38]. The Stokes-Einstein relation describes the translational diffusion coefficient of a particle in a viscous medium,... 

The dimensions and the molecular weight of copolymer micelles can be determined by quite a number of techniques, especially scattering and hydrodynamic characterization techniques as summarized in Table 7.3. In general practice the hydrodynamic radius Ry is determined by DLS techniques. By treating the micelles as hydrodynamically equivalent spheres and using the Stokes-Einstein relation, Ry can be evaluated from the translational diffusion coefficient extrapolated to infinite dilution D -. 

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. 

The peaks in Figure 3.13a (counted from the left) are ascribed to the following modes peak 3 (dominant) to cooperative or translational diffusion, peak 4 to selfdiffusion of the copolymer chain, and peak 5 to the diffusion of clusters of a hydrodynamic radius equal to about 120 nm, as estimated via the Stokes-Einstein relation (Equation 3.27). Peaks 1 and 2 correspond to thermal diffusion and to self-diffusion of solvent molecules, respectively. 

The translational diffusion coefficient can also be expressed at infinite dilution as a function of the friction coefficient through the Einstein relation ... 

The second assumption is the spherical beads particles are used in the experiment. These particles have a small diameter compared to the molecular dimensions. Hence, the translational diffusion coefficient (Dj-.D,y) of colloidal particles in dispersions and hydrodynamic radius (Rn) can be measured with DLS technique if Stokes- Einstein relation is applied to particle [62-64]. The translational diffusion coefficient is used to calculate the particle size. [60]. Stokes- Einstein is given in Eq. (16) [11, 60, 63, 65] ... 

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 ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

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. 

A further test of the pair potentials used is the translational diffusion coefficients, D. The coefficients in Tables II, III, and IV were all evaluated using the Einstein-Kubo relation ... 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

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