Brownian motion rotary

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. 

In certain cases (e.g. iron(III) hydroxide sol) birefringence can be produced by the aligning action of electrical or magnetic fields. 

When a solution containing dipolar molecules is placed between electrodes and subjected to an alternating current, the molecules tend to rotate in phase with the current, thus increasing the dielectric constant of the solution. As the frequency is increased, it becomes more difficult for the dipolar molecules to overcome the viscous 

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However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

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 ... 

The changes have been used to provide information about the enviromnent of the fluorescent probe and to follow changes in conformation of the macromolecule. In other work the study of the fluorescence polarization properties of the attached probe under steady state illumination and the application of Perrin s equation enable calcu-latnn of the rotary Brownian motion of the polymer. This technique has been extended by Jablonski and Wahl to the use of time-resolved fluorescence polarization measurements to calculate rotational relaxation times of molecules These experiments are discussed fiilly in the fdlowing section of this review. 

Orientaticm of a spheroid is determined by the balance of hydrodynamic forces and rotary Brownian motion. Hydrodynamic forces tend to align the major axis with the flow, while Brownian motion tends to randomize the orientation. The relative importance of each is expressed in terms of the Peclet number Pe, the ratio of the time scales for Brownian motion (l/D ) to that for convective motion (1//0-... 

For the calculation of the Maxwell-constant an assembly of frozen random conformations is considered. Brownian motion is taken into account only so far as rotary diffusion of the rigid conformations is concerned. In this way a first order approximation of the distribution function with respect to shear rate is obtained. This distribution function is used for the calculation of the Maxwell-constant, [cf. the calculation of the Maxwell-constant of an assembly of frozen dumb-bell models, as sketched in Section 5.I.3., eq. (5.22)]. Intrinsic viscosity is calculated for the same free-draining model, using average dimensions [cf. also Peter-lin (101)]. As for the initial deviation of the extinction angle curve from 45° a second order approximation of the distribution function is required, no extinction angles are given. 

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

One of the important applications of Stokes law occurs in the theory of Brownian motion. According to Einstein (El a) the translational and rotary diffusion coefficients for a spherical particle of radius a diffusing in a medium of viscosity n are, respectively,... 

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] ... 

Furthermore, just as photochemistry is a clean way to cause a reaction, it may offer a clean way to cause a movement in a macroscopic object. As a matter of fact, this is an issue rarely adopted by natme, where direct conversion of light into mechanical energy is limited to a few cases in bacteria [16]. This does not preclude adopting this principle for artificial system. As an example, one may think of controlling and directing the Brownian motion of molecules in solution and to induce directional translational and rotary motion of molecules or of nano-objects. In other words, rotary and translational motors may be devised and used to power future nanodevices. For example, rotary molecular motors allow the transmission of motion in multicomponent systems as well as reaching out-of-equilibrium assemblies (see Fig. 11.5) [18]. 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

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