Stokes friction coefficient

C[, corresponding to the Stokes diffusional process, can be written as the product of the Stokes friction coefficient multiplied by a correcting factor fr taking into account the finite thickness of the solvent layers... 

Solvent friction is measured by the Stokes friction coefficient = 6 r)is H- The interparticle forces = — d/dr, U ( rj ) derive from potential interactions of particle i with all other colloidal particles U is the total potential energy. The solvent shear-flow is given by v ° (r) = yyx, and the Gaussian white noise force satisfies (with a,j8 denoting directions)... 

An interpolation formula is often used to cover the entire range of values of the Knudsen number 2ip/dp) from the continuum to the free molecule regimes. It is introduced as a correction to the Stokes friction coefficient ... 

Here u is the particle velocity, U/ i.s the local fluid velocity, and / is the Stokes friction coefficient. We call particles that obey this equation of motion Stokesian particles. The use of (4.2S) is equivalent to employing (4.19), neglecting the acceleration terms containing the gas density. Because (4.19) was derived for rectilinear motion, the extension to flows with velocity gradients and curved streamlines adds further uncertainty to this approximate method. 

The factor /a describes, first, the relationship between the radius of gyration and the radius most suitable in describing the molecule, and, second, all deviations from the Stokes frictional coefficient of an unsolvated sphere. The relationship = (3/5)r is valid for spheres. Thus, /< = (5/3) for unsolvated spheres. Consequently, P takes on a value of 24.34. The Einstein equation allows to be calculated for spheres ... 

Another means of eliminating the frictional coefficient comes from viscosity measurements. According to equation (7-25), the frictional coefficient fj) is related to an asymmetry factor and the Stokes frictional coefficient for spheres. On extending equation (7-25) with we obtain the form... 

When the effective size, due to adsorption, increases, this results in a larger friction as follows from the Stokes friction coefficient / = 6ntf R. We stress that any study on colloid-polymer mixtures should be preceded with an analysis of whether the polymers adsorb or not. Analysis of the composition of the two phases can be used to verify whether depletion interaction is responsible for demixing. 

For spherical particles, the friction coefficient can be identified with the Stokes friction coefficient j6 j61. Two cases are of interest. 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

In this equation, % is a proportionality factor known as the bead-solvent friction coefficient which purports to account in some kind of average way for the complex molecular interactions as the polymer segments (schematized by the bead) move about in the solvent. Following Stokes law of drag resistance, this friction coefficient is usually given as = 67trisa, with a equal to the bead radius. 

Rothe, GM, Determination of Molecular Mass, Stoke radius. Frictional Coefficient and Isomer-Type of Non-denatured Proteins by Time-Dependent Pore Gradient Gel Electrophoresis, Electrophoresis 9, 307, 1988. 

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 ... 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Figure 9b shows the friction constant as a function of a. For large a the friction coefficient varies linearly with ct in accord with the prediction of the Stokes formula. The figure also shows a plot of (slip boundary conditions) versus ct. It lies close to the simulation value for large ct but overestimates the friction for small ct. For small ct, microscopic contributions dominate the friction coefficient as can be seen in the plot of (m. The approximate expression 1 = + 071 interpolates between the two limiting forms. Cluster friction... 

If we know the friction coefficients a, we thus have an explicit formula for the conductivity coefficient. In particular, it is often assumed that a is determined by the well-known Stokes formula ... 

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. 

Another microscopic approach to the viscosity problem was developed by Gierer and Wirtz (1953) and it is worthwhile describing the main aspects of this theory, which is of interest because it takes account of the finite thickness of the solvent layers and the existence of holes in the solvent (free volume). The Stokes-Einstein law can be modified using a microscopic friction coefficient ci micro... 

The hydrodynamic radius is the equivalent spherical radius obtained from the Stokes law for the chain friction coefficient, Eq. (41), with substituted by R[j. It is easily verified that... 

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

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by < > is performed over all macromolecules in the system. Thus, < i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory ... 

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