Friction coefficient sphere

The diffusion coefficient D (at c 0) is related to the frictional coefficient /d [see equation (7-21). The coefficient fjy and, thus, D depend on a series of molecular quantities, as can be seen from the following reasoning. According to Stokes law, the frictional coefficient /sphere of an unsolvated sphere of homogeneous density is /sphere = Tif/iTsphere where is the solvent viscosity. In a solvated sphere, the hydrodynamically effective radius takes the place of the radius Tsphere- The deviation of the particle shape from that of an unsolvated sphere is described by an asymmetry factor /a = /nZ/sphere- Thus, the frictional coefficient fjj of a solvated particle of any given shape is, for c 0,... 

Frictional coefficients of f, solvated protein, and fo, nonsolvated sphere. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

It was suggested by Sutherland [363] that for a sphere diffusing through a medium consisting of molecules of comparable size to the diffusant, the friction coefficient is equal to... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

According to Eq. (18), the high polymer molecule should exhibit the frictional coefficient of an equivalent sphere (compare Eq. 15) having a radius proportional to the root-mean-square end-to-end distance (r ) (or to (s ) / ). Similarly, according to Eq. (23) its contribution to the viscosity should be that of an equivalent sphere (compare Eq. 16) having a volume proportional to (r ) / In analogy with Eq. (17 ), we might write... 

Theory presented earlier in this chapter led to the expectation that the frictional coefficient /o for a polymer molecule at infinite dilution should be proportional to its linear dimension. This result, embodied in Eq. (18) where P is regarded as a universal parameter which is the analog of of the viscosity treatment, is reminiscent of Stokes law for spheres. Recasting this equation by analogy with the formulation of Eqs. (26) and (27) for the intrinsic viscosity, we obtain ... 

An approximate model of the ISE can be developed with the aid of Figure 2.9. This figure shows a schematic cylindrical indenter with a conical tip being pushed into a specimen. The plastic zone is approximated by a segment of a sphere, and the diameter of the indent is 2r. The yield stress is Y, and the friction coefficient is a. 

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 1851, Stokes (well known for his pioneering work on luminescence see Chapter 1) showed that the relation linking the force exerted by a fluid on a sphere to the viscosity tj of the medium is F = Gitt/rv, where r is the radius of the sphere and v its constant velocity. In this relation, the quantity 6ra/r appears as a friction coefficient, i.e. the ratio of the viscous force to the velocity. 

Equation (4.69) for self-diffusion can be arrived at in a similar way through proper derivation of the friction coefficient. Eqnation (4.70) does not take into account hydro-dynamic interactions. It is also necessary to come np with an equivalent radius, r, for a polymer chain, which can be difficnlt, especially when the conformation is such that the chain is extended, and does not form a sphere at all. Nonetheless, a radius of gyration, rg, is often nsed to characterize polymer chains in solntion, and the resulting friction coefficient is = Anfxrg. 

Person 2 Estimate the nnsolvated Stake s sphere friction coefficient, fo, nsing the Stokes-Einstein relationship for spheres, fo = 6jrp,r. 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

R. A. Marcus In Chem. Phys. Lett. 244, 10 (1995), a very rough approximate hard-sphere model used for liquids was mentioned to relate the frictional coefficient to the pair distribution function in the cluster. 

To the extent that we can regard protein molecules as spherical we can substitute for/in Eq. 3-13 the frictional coefficient of a sphere ... 

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. 

Many soluble native proteins are compact, essentially spherical structures with frictional ratios (///min) around 1.25. (The term f/fmin represents the ratio of the measured frictional coefficient to the minimal value which could be obtained for the equivalent anhydrous sphere. This... 

A particle containing a given volume of dry material will have its smallest possible frictional coefficient, /0, in a particular liquid when it is in the form of an unsolvated sphere. The frictional ratio, flfQ (i.e. the ratio of the actual frictional coefficient to the frictional coefficient of the equivalent unsolvated sphere) is, therefore, a measure of a combination of asymmetry and solvation. 

Because the radius of a nonspherical molecule cannot be defined precisely, molecular friction coefficients and diffusion coefficients are often related to the Stokes radius (or Stokes diameter). This is defined as the radius (or diameter) of a sphere having / and D values identical to those of the molecule under consideration. 

For colloidal particles with at>h Eq. 8.22 reduces to z/zion = h la. Since the charge on a particle, zion, is proportional to its surface area, 4ira2 (if the particle is a sphere), z is proportional to radius a. Friction coefficient /is also proportional to a as shown by Stokes law, Eq. 4.42. Thus mobility... 

Consider a dilute suspension of spherical particles A in a stationary liquid B. If the spheres are sufficiently small, yet large with respect to the molecules of stationary liquid, the collisions between the spheres and the liquid molecules B lead to a random motion of the spheres. This motion is called the Brownian motion. Dilute diffusion of suspended spherical colloid particles is related to the temperature and the friction coefficient by... 

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