Stokes friction

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... 

Charged compounds undergo the influence of the electric force (Egl) nd the Stokes frictional force (Es) in an electric field. When these two forces (in kg cm s ) are in equilibrium at steady-state conditions, the following equations can be written ... 

The combined viscosity and diffusion measurements on concentrated systems by Bueche et al. (33), were described in Section 2. The results suggest that viscosity behavior above Mc can be explained simply in terms of an enhanced Stokes friction for the molecules. Bueche (7, 224) suggested that since polymer molecules in concentrated systems are looped through the coils of neighboring molecules, relative motion must be resisted by the need for these loops to slip around one another. Each molecule is assumed to participate in an average of E such temporary couples, creating (for E > 1) a temporary network... 

Mobility — The (ionic or electric) mobility u of an ion is given by the drift velocity v (the velocity of an ion at equilibrium between the accelerating effect of the electric field and the decelerating effect of the viscous medium (Stokes friction)) of an ion and the effective electric field E v... 

This force makes the ion move at velocity v. Since this is equal to the Stokes frictional force, 6ntjr0v, the flux, J = -cv, is given by... 

The coefficient k(h/a) has been introduced by H. Brenner 2[ and represents the modification of the Stokes friction due to the bottom wall located at a distance /( from the particle. If hja — oc, then k h/a) = I and we recover the usual Stokes law for an isolated particle. On the other hand, if the particle comes very close to the plane (h/a — I), then k(h/a) — oc and the particle will never reach the plane, owing to the presence of lubrication forces. The sedimentation velocity follows from Hq. (2) ... 

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. 

FIGURE 20.2 Forces acting on a charged particle. The particle is negatively charged and surrounded by a positively charged ionic atmosphere, indicated by the dashed circle. Fi is the electrical force, F2 is Stokes frictional drag, F3 is electrophoretic retardation, and F4 is the relaxation effect. 

This equation can also be used approximately for microscopic particles like the molecules and ions mentioned above. We expect that the diffusion coefficient >b of a substance B will be the smaller, the more viscous the medium is in which it migrates. Let us consider a simple example of a rigid, spherical particle with the radius r. The diffusion force [expressed by Eq. (20.1)] is counteracted by the Stokes frictional force [Eq. (20.18)] ... 

Another means of eliminating the frictional coefficient fs comes from viscosity measurements. According to Equation (7-19), the frictional coefficient fo is related to an asymmetry factor fA and the Stokes frictional... 

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 ... 

As far as qualitative aspects of the ion-size dependence of the friction coefficients are concerned, which has a minimum with increasing ionic radius, both models (the solventberg and dielectric friction models) explain the experimental observations to a certain extent. Then one might ask which model of the two is more faithful to the real physics of the ionic friction and/or how do the two effects interplay if both are coexisting For instance, use of the same parameter for the effective ionic radius will not be justified since increase in the effective radius should give rise to increase in the Stokes friction but decrease in the dielectric friction. The question should be answered by microscopic theory, not by treatment based on the continuum model. 

In Wolynes treatment Chh is identified as the Stokes friction and is assumed to be given by the Stokes law with the hydrodynamic radius equal to the crystallographic radius. A second approximation is to ignore the cross term (hs- Wolynes theory then focuses on the time dependence... 

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. 

This deformation is caused by volumetric forces such as gravitational force and Stokes-friction, and surface tensions (normal and tangential ones). The equilibrium of forces and surface tensions can be calculated although there is no generic analytical method for solving these equations [60]. 

The dielectric constant at angular frequency to is e (ai) = e (u>) — ie" u>). The limiting dielectric constants arc e, (sometimes written e ) and e, (sometimes written e,). In Debye s model the relaxing clement (which represents a polar molecule) is a sphere of radius a containing an electric dipole of dipole moment p = qd (see Fig. 4.36). The sphere is immersed in a liquid of viscosity q. Under an electric field E the torque on the dipole is pE sin 0. The rotation of the dipole under this torque is resisted by the Stokes frictional torque Xnt/cr tl. The dipole will follow the field for low U biit not for high. The relaxation time is... 

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

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