Particles rotational diffusion

Scattering Function, P iq), Single Particle Translational Diffusion Coefficient, Dg, and Single Particle Rotational Diffusion Coefficient, 0, for Particles of Different Shape... 

If the diffusing bound particle is composed of many molecules, as, for example, in the case of a rigid macroscopic particle, rotational diffusion may influence the shape of the Mossbauer or neutron scattering spectra. Ofer et al. (1984) treat the case in which the particle participates in both bound translational and free rotational diffusion. The Mossbauer spectrum in this case can again be represented by a sum of Lorentzian lines. In the... 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

A possible approach to interpretation of a low-frequency region of the G ( ) dependence of filled polymers is to compare it with a specific relaxation mechanism, which appears due to the presence of a filler in the melt. We have already spoken about two possible mechanisms — the first, associated with adsorption phenomena on a filler s surface and the second, determined by the possibility of rotational diffusion of anisodiametrical particles with characteristic time D 1. But even if these effects are not taken into account, the presence of a filler can be related with the appearance of a new characteristic time, Xf, common for any systems. It is expressed in the following way... 

Chandler D. Translational and rotational diffusion in liquids. I. Translational single-particle correlation functions. J. Chem. Phys. 60, 3500-507, (1974). Translational and rotational diffusion in liquids. II. Orientational single-particle correlation functions. J. Chem. Phys. 60, 3508-12 (1974). 

The above kinetic scheme of the bimolecular reaction simplifies physical processes that proceed via the elementary bimolecular act. To react, two reactants should (a) meet, (b) be oriented by the way convenient for the elementary act, and (c) be activated to form the TS and then react. Hence, not only translational but also rotational diffusion of particles in the solution and polymer are important for the reaction to be performed. So, the more detailed kinetic scheme of a bimolecular reaction includes the following stages diffusion and encounter the reactants in the cage, orientation of reactants in the cage due to rotational diffusion, and activation of reactants followed by reaction [5,13]. 

Rotational diffusion of particles occurs in polymer much slowly than in liquids. Therefore, the observed difference in liquid (k ) and solid polymer (ks) rate constants can be explained by the different rates of reactant orientation in the liquid and polymer. The EPR spectra were obtained for the stable nitroxyl radical (2,2,6,6-tetramethyl-4-benzoyloxypiperidine-l-oxyl). The molecular mobility was calculated from the shape of the EPR spectrum of this radical [14,15], These values were used for the estimation of the orientation rate of reactants in the liquid and polymer cage. The frequency of orientation of the reactant pairs was calculated as vor = Pvrot> where P is the steric factor of the reaction, and vIol is the frequency of particle rotation to the angle equal to 4tt. The results of this comparison are given in Table 19.2. 

Fig. 3. Schematic representation of the topological space of hydration water in silica fine-particle cluster (45). The processes responsible for the water spin-lattice relaxation behavior are restricted rotational diffusion about an axis normal to the local surface (y process), reorientations mediated by translational displacements on the length scale of a monomer (P process), reorientations mediated by translational displacements in the length scale of the clusters (a process), and exchange with free water as a cutoff limit.

Size of the particle. Assuming a spherical shape, the value of the rotational diffusion constant Dp can provide an average radius using the following relationship ... 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

The rotational effect on the correction term for diffusion-limited rate coefficients is shown as the intercepts on the ordinate of Fig. 17. Here, B is a small particle and only A can be re-oriented. As the size of A decreases, it can re-orient more quickly, but the mutual diffusion coefficient decreases till rA = rB. Hence, the very fact that the correction term increases with increase of rA shows that rotational diffusion is very important. 

For simplicity we assume that the particles are magnetically hard.1 Then, the already developed formalism [Eqs. (4.90) and (4.293)] applies in full under two conditions e is identified with v and the internal relaxation time Tq is replaced by the external rotational diffusion (Debye) time x. In the modified equations, the orientation order parameter is given by (Pi)- Similarly to (4.294), we set... 

Permanent dipoles For particles whose acceleration is a negligible part of the balance of forces governing oscillation, there is only a restoring force (from rotational diffusion) and a drag term. The polarizability is of the Debye form... 

The compensation phenomena considered above are not only characterise of enzyme reactions. The compensation relationships in protein denaturation are noted for enormous ranges of Ea values (from 0 to 120 kcal/mole) and AS of (from 10 to 400 eu) (Likhtenshtein and Troshkina, 1968). These quantities have been found to be highly sensitive to to external condidion (pH, additive, moisture content, etc.) and rotational diffusion of spin labels introduced into various portions of globular proteins. They have also been observed, though to a less extend, in various processes in the condenced phase (chemical reactions, diffusion, evaporation, electrical, conduction, electron transfer, etc. The main property of all these systems, which differ from simple gas reactions, is the cooperative behavior of particle assemblies surrounding the reaction centers. 

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