Translational Brownian

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. 

Because the random fluctuations in the positions of particles in space are often translational, the kinetics of these processes can be considered comparable to the decay of a concentration gradient by translational Brownian motion. Likewise, as the orientation of any molecule undergoes similar random fluctuations in space,... 

Metzler and Klafter [86] and Metzler [85] have proposed a fractional Klein-Kramers equation (FKKE), which, according to them, corresponds to a multiple trapping picture where the tagged particle executes translational Brownian motion in accordance with the Langevin equation ... 

In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in configuration angular-velocity space for linear molecules in the same form as for fixed-axis rotators—that is, the form of the FKKE suggested by Barkai and Silbey [30] for one-dimensional translational Brownian motion. For rotators in space, the FKKE becomes... 

In previous sections, we have treated anomalous relaxation in the context of the fractional Fokker-Planck equation. As far as the Langevin equation treatment of anomalous relaxation is concerned, we proceed first by noting that Lutz [47] has introduced the following fractional Langevin equation for the translational Brownian motion in a potential V ... 

Lutz also compared his results with those predicted by the fractional Klein-Kramers equation for the probability density function/(x, v, f) in phase space for the inertia-corrected one-dimensional translational Brownian motion in a potential Eof Barkai and Silbey [30], which in the present context is... 

The difference between elastic and "quasielastic" measurements is that in the latter, small changes in the frequency due to the translational ("Brownian") movement of the scattering particles are also measured. The broadness of the intensity distribution of the emitted light for frequencies around the primary monocluomatic beam frequency is directly related to the diffusion coefficient of the particles, which can then be related to the hydrodynamic radius if a model for the particle shape is available Dynamic light scattering can thus be used to follow the kinetics of particle coagulation by following the decrease in diffusion coefficient as the particle size increases. ... 

Normal Brownian motion is a result of solvent molecules impacting on the solute particles, and these give both translational and rotational movement to the solute. An ion on its own will execute this Brownian motion. Since an ion has a charge which can interact with an external electric field, this interaction will perturb the translational Brownian motion, with a cation moving in the direction of the field while an anion will move in the opposite direction. The field will have a minor effect on the rotational Brownian motion, but this will not contribute to the translational mobility. 

Solvent separated ion pairs will also be overall uncharged and will execute Brownian motion. They will also be enclosed in a cage of solvent molecules, but since the interactions between the ions will be considerably smaller than those between ions in contact, they will separate and escape from the cage much sooner than the contact ion pairs. On the time scale of colUsions they can be considered as separating and thus not moving as a single entity. The translational Brownian motion could then be perturbed by an external field, so that, on average, the motion of a cation could be in the direction of the external field and so be able to conduct the current, and if the ion is an anion it will move in the opposite direction. 

For anisometric particles, along with translational Brownian motion one can also observe rotational Brownian motion. Studies of rotational Brownian motion indicate that the particle average square rotation angle, [Pg.341]

Let us return to the analysis of Brownian motion. For simplicity we begin by considering the continuous one-dimensional translational Brownian motion as represented by a one-dimensional random walk problem. The probability of a displacement between x and x + dx after n random steps of length I is given by the Gaussian distribution... 

Figure 5.2.2 Translational Brownian motion of a colloidal particle in water. (A) Particle motion as observed every 30 s under a microscope by Perrin (1923). (B) Numerical simulation of magnified portion of particle path observed 100 times more frequently. [After Lavenda, B.H. 1985. Brownian motion. Sci. Amer. 252(2), 70—85. Copyright 1985 by Scientific American, Inc. All rights reserved. With permission.]...

The corresponding three-dimensional solution for the particle concentration appropriate to translational Brownian motion is... 

If the particles are not spherical, even in the very dilute limit where the translational Brownian motion would still be unimportant, rotational Brownian motion would come into play. This is a consequence of the fact that the rotational motion imparts to the particles a random orientation distribution, whereas in shear-dominated flows nonspherical particles tend toward preferred orientations. Since the excess energy dissipation by an individual anisotropic particle depends on its orientation with respect to the flow field, the suspension viscosity must be affected by the relative importance of rotational Brownian forces to viscous forces, although it should still vary linearly with particle volume fraction. 

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

Rodlike polymers do two kinds of Brownian motion, translation and rotation. The translational Brownian motion is the random motion of the position vector R of the centre of mass, and the rotational Brownian motion is the random motion of the unit vector u which is parallel to the polymer. 

When the reciprocal of the scattering vector, S, is large in comparison to the size of the polymer coil, the DLS fluctuation spectrum is due to macromolecular coil center of mass motions. These motions are generated by translation (Brownian) diffusion and/or any bulk fluid flow of the solution. Unwanted scattering can be produced by slight thermal differences or vibration in the solution which can cause bulk fluid flow. 

Tandon P, Rosner D E (1995) Translational Brownian DUlusion Coeffieient of Large (Multiparticle) Suspended Aggregates. Ind. Eng. Chem. Res. 34 3265-3277... 

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