Translation coefficients sphere

From Eqs. (5.1.6) and (5.1.7) the mean translation coefficient for a prolate or oblate spheroid is simply 2/5, whereas for a sphere it is just the sphere radius. A quantity of interest, sometimes termed the Perrin factor, is the ratio of the mean translation coefficient (or mean friction coefficient) of a prolate or oblate... 

Y.-L. Xu, Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories. J. Comp. Phys. 139(1), 137-165 (1998a). doi 10.1006/jcph. 1997.5867 Y.-L. Xu, Electromagnetic scattering by an aggregate of spheres asymmetry parameter. Phys. 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

In 1923, Debye and Hiickel published their famous papers describing a method for calculating activity coefficients in electrolyte solutions. They assumed that ions behave as spheres with charges located at their center points. The ions interact with each other by coulombic forces. Robinson and Stokes (1968) present their derivation, and the papers are available (Interscience Publishers, 1954) in English translation. 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

Fig. 17, The correction factor to the Smoluchowski rate coefficient for reaction between an isotropic reactant, B, and an axially symmetric reactant, A, which has a cap of reactivity of the spherical surface, subtending a semi-angle of 77/9, 77/2, 577/6 at the sphere s centre. The reaction radius is R and the radius of B is rB. Translational and rotational diffusion coefficients are given by eqn. (118), for reactions with G = 1, i.e,...

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. 

The physical properties of these polymeric dendrimers have been studied to some extent. Intrinsic viscosity measurements combined with MW afford values of according to Eq. (5). Alternatively, the translational diffusion coefficient leads to Rh according to Eq. (6). These equations may well be applicable, since it is observed that Rn and Rh scale with the 1/3 power of MW in support of the equal density hard-sphere assumption [88]. 

D is the translational diffusion coefficient, which, for spheres, is given by,... 

The r-average translational diffusion coefficient l> is calculated from the equation Dj = V/q2. For a collection of identical spheres undergoing ordinary Brownian motion in solution. 

Here, kn is the Boltzmann constant, T the absolute temperature, and Dto the translational diffusion coefficient extrapolated to zero concentration. The friction coefficient for a sphere of radius r is given by the Stokes law ... 

It has recently become more widely appreciated that the presence of rotational diffusional anisotropy in proteins and other macromolecules can have a significant affect on the interpretation of NMR relaxation data in terms of molecular motion. Andrec et al. used a Bayesian statistical method for the detection and quantification of rotational diffusion anisotropy from NMR relaxation data. Sturz and Dolle examined the reorientational motion of toluene in neat liquid by using relaxation measurements. The relaxation rates were analyzed by rotational diffusion models. Chen et al measured self-diffusion coefficients for fluid hydrogen and fluid deuterium at pressures up to 200 MPa and in the temperature range 171-372 K by the spin echo method. The diffusion coefficients D were described by the rough sphere (RHS) model invoking the rotation translational coupling parameter A = 1. 

The coefficients A, B, and C were determined by using all the available information on the behavior of the function G(r) for a fluid of hard spheres. The final expression obtained for G(r), after being translated into W(r), i.e., integration of relation (N.7), is the following ... 

Translational self-diffusion and rotation of the water molecules. Since the above two models fail to explain the data, one may think of a model which combines both, and which is certainly more realistic. We h ve2tr e< such a possibility with p = 0.95 and D 1.6 x 10 cm /s. This last value is the long range self-diffusion coefficient of water in this membrane, measured by radioactive tracers. We found that no fit is possible with these values whatever D is chosen. As for the preceeding section, we find that the fiE improves considerably if we take either p or Dt as parameters. With Dt fixed, we should increase p to 3 A, as above, and with p fixed, we should increase D to 10 cm /s. These results suggest that one should think of a model which contains these two features. The simplest one is a model where the water molecules, more precisely the protons, are restricted to diffuse (diffusion goejjficient D) in a sphere of radius a, where we expect D 10 cm and a 3 S. 

Two unusual features can be observed in these plots (and, at least for the self-diffusion coefficient, this behaviour is common to all hydrogen-bonded liquids). This ratio is a function of temperature. At constant temperature and pressure, rotation and translation reveal the same isotope effect. From simple sphere dynamics one would expect the rotation to scale as the square root of the ratio of the moments of inertia (=1.38) while for translational mobility the square root of the ratio of the molecular masses ( = 1.05) should be found. This is clearly not the case, indicating that the dynamics of liquid water are really the dynamics of the hydrogen-bond network. The hydrogen bonds in D2O are stronger than those in H2O and thus the mobility in the D2O network decreases more rapidly as the temperature decreases. 

Let us consider the term of the translation diffusion. The diffusion coefficient D expresses the abihty of a molecule to change its position in solution due to chaotic translation motion. Basic evaluation of the diffusion coefficient can be obtained from the Stokes formula for a sphere in a fluid ... 

This should be compared to the result 0 = ksT/STitja3 for the sticky sphere of radius a. Perrin also determined the translational diffusion coefficients for ellipsoidal molecules.19 His result is... 

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