Rotational Brownian diffusion coefficient

Capillary and extrudate diameter, respectively Droplet deformability in extensional flow Rotational Brownian diffusion coefficient Steady-state tensile compliance Volume-to-surface average particle diameter Elasticity of the interphase Interaction energy Tensile, or Young s, modulus Electron... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Tao T 1969 Time-dependent fluorescence depolarization and Brownian rotational diffusion coefficients of macromolecules Biopolymers 8 609-32... 

When the regular motion is simply uniform rotation of the absorption and emission dipoles with angular velocity to around the helix axis, one has p(t) - p(0) = cot. For the corresponding random motion, one might have m)2> = 2Dt, where D is the effective diffusion coefficient for Brownian rotation of the transition dipole around the helix axis. When these expressions are incorporated in Eqs. (4.31) and (4.24), the latter becomes a generalization of a relation recently derived using a more cumbersome approach/104-1... 

The second piece of evidence in distinguishing rods in a magnetic field to those out of the magnetic field was the rotational diffusion coefficient of the rod. It was the rotational diffusion coefficient that revealed the effect that an applied magnetic field had on a nanorod moving non-Brownian outside a field (2000 ° /s) and in it (70 ° /s). 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

How rapidly diffusion occurs is characterized by the diffusion coefficient D, a parameter that provides a measure of the mean of the squared displacement x of a molecule per unit time f. For diffusion in two dimensions such as a membrane, this is given by = 4Ht. The Saffman-Delbrtlck model of Brownian motion in biologic membranes describes the relationship between membrane viscosity, solvent viscosity, the radius R and height of the diffusing species, and D for both lateral and rotational diffusion of proteins in membranes (3, 4). This model predicts for example that for lateral diffusion, D should be relatively insensitive to the radius of the diffusing species, scaling with log (1/R). 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

As seen from Ref 9, data for the rotational diffusion coefficient also shows a far milder dependence wit temperature than its translational counterpart. However the observed decoupling from the SE and SED behaviors is there seen to follow a far less drastic behaviour than that here found. On such a basis we deem that the presence of a strong directional interaction cannot account for the higher mobility observed by experiment if compared to the Brownian dynamics estimates of SE and SED. Finally it is worth to emphasize that the observed breakdown of both SE and SED approximations only appear for the miscible phase below Tl but not after the re-entrance above Tu into the high temperature, miscible phase. Such fact is thus suggestive of the existence of phenomena additional to those responsible of the re-entrant behavior being... 

Here Vp is used to denote that the gradient operates with respect to the orientation space of p,32 and D is the diffusion coefficient for rotational Brownian motion. 

Rotational Brownian motion results in the disordering of anisometric particles previously oriented in some particular way owing to the flow of the dispersion medium (see Chapter IX) or the application of an electric field. From the time of the disordering one can determine the rotational diffusion coefficient, and, for known particle size and shape, also Avogadro s number. If the particles are able to undergo co-orientation, they usually are of substantially anisometric shape, and their translational and rotational diffusion coefficients differ from those obtained for spherical particles. For example, for prolate ellipsoids of revolution with a ratio of their main diameters d] d2-= 1 10, the diffusion coefficient, D, is about 2/3 of the value obtained for spherical particles of the same volume. 

The self-diffusion coefficients of toluene in polystyrene gels are approximately the same as in solutions of the same volume fraction lymer, according to pulsed field gradient NMR experiments (2fl). Toluene in a 10% cross-linked polystyrene swollen to 0.55 volume fraction polymer has a self-diffusion coefficient about 0.08 times that of bulk liquid toluene. Rates of rotational diffusion (molecular Brownian motion) determined from NMR spin-lattice relaxation times of toluene in 2% cross-linked ((polystytyl)methyl)tri-/t-butylphosphonium ion phase transfer catalysts arc reduced by factors of 3 to 20 compai with bulk liquid toluene (21). Rates of rotational diffusion of a soluble nitroxide in polystyrene gels, determined from ESR linewidths, decrease as the degree of swelling of the polymer decreases (321. 

A similar analysis of Brownian rotation yields for the rotational diffusion coefficient ... 

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. 

In summary, two mechanisms are responsible for stress overshoots after rest. The first is based on randomization of the orientation imposed by Brownian motion and relaxation of the matrix, whereas the second assumes that a three-dimensional structure is broken by shearing and re-forms under quiescent conditions. The former mechanism is expected to be applicable to LCP and exfoliated PNC, where platelets are still able to rotate freely. The second mechanism dominates the intercalated systems, especially those with large low-aspect-ratio stacks. The probability of the Brownian force contribution might be assessed from the rotary difiusivity coefficient and the diffusion time [Larson, 1999] ... 

We have seen that molecules in solution show translational movement caused by the Brownian motion of the solvent. In addition to this translational movement, each solute molecule rotates relative to its centre of mass. This motion is known as rotational diffusion and is described in terms of rotational diflusion coefficient . It has the units of reciprocal seconds and expresses,. 

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