Einstein equations for spherical

The equation reduces to the Stokes-Einstein equation for spherical particles. Since the friction coefficient for a non-spherical partiele always exceeds the friction coefficient for a spherical particle, over estimation of particle size will occur if equation (10.41) is applied. 

The values of l) >,n — the diffusivity for the Brownian motion of aerosol — are calculated from the Stokes-Einstein equation. For spherical particulates with the effective radius rp, in a gas with the dynamic viscosity p2 (nearly constant for pressures about and less than one bar), the formula is ... 

Kholodenko, A.L. and Douglas, J.F., Generalized Stokes-Einstein equation for spherical-particle suspensions, Phys. Rev. E 51,1081-1090 (1995). 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

According to a simple theory due to Einstein (1906) for spherical particles at very low concentrations, the relative viscosity, T)/r]s, where r]s is the viscosity of the pure liquid medium, is related to the volume fraction, [Pg.115]

The Stokes-Einstein equation for binary molecular diffusion coefficients of dilute pseudo-spherical molecules subject to creeping flow through an incompressible Newtonian fluid is (see equation 25-98) ... 

A polymer molecule dissolved in a solvent can be envisioned as a necklace comprising spherical beads connected by a string [23]. The polymer molecules are separated and only interact through the solvent. The Stokes-Einstein equation for the diffnsion coefficient of the polymer can be used for a Flory theta solvent. The root-mean-sqnare... 

Einstein (1911) Equations for Spherical Molecules Einstein, in a study of the viscosity of a solution of suspension of particles (colloids), suggested that the specific viscosity Tj p is related to a shape factor v in the following way ... 

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Thus, the Stokes-Einstein equation is expected to be valid for colloidal particles and suspensions of large spherical particles. Experimental evidence supports these assumptions [101], and this equation has occasionally been used for much smaller species. 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

The Stokes-Einstein equation gives a good approximation of the molecular radius (r) for spherical or nearly spherical molecules ... 

The reorientational correlation time can be predicted for spherical rigid particles, according to the Stokes Einstein equation (75-77) ... 

Albert Einstein derived a simple equation for the viscosity of a solution of spherical particles, and from this result it is obvious that if we could make the polymer in small colloidal-sized balls, then the solution would be much less viscous. Also, if we could use surfactants to stabilize (e.g. by charging) the polymer particles in water, then there would be no need for organic solvents. Both these conditions are neatly obtained in the emulsion polymerization process, which is schematically explained in Figure 5.3. A polymer latex is produced by this process and can contain up to 50% polymer in the form of 0.1-0.5 im size spherical particles in water. A typical starting composition is ... 

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

The form of Eq. (4.84) should look familiar. For large, spherical particles in a low-molecular-weight solvent, f = 671 fir, where fi is the viscosity of the pure solvent and r is the large particle radius, and Eq. (4.84) becomes Equation (4.70) a form of the Stokes-Einstein equation, which gives the binary diffusion coefficient, Dab ... 

It should be noted that this derivation contains no assumptions about the shape of the particles. However, when the particles are assumed to be spherical, we can substitute Equation (8) for/, and the resulting equation for the diffusion coefficient is the well-known Stokes-Einstein relation. 

Solution Equation (4.41) gives the Einstein relationship between [r/] and , the volume fraction occupied by the dispersed spheres. The volume fraction that should be used in this relationship is the value that describes the particles as they actually exist in the dispersion. In this case this includes the volume of the adsorbed layer. For spherical particles of radius R covered by a layer of thickness 8R, the total volume of the particles is (4/3) + 4ttR2 8R. Factoring out the volume of the dry particle gives Vdfy(1 + 38RJRS), which shows by the second term how the volume is increased above the core volume by the adsorbed layer. Since it is the dry volume fraction that is used to describe the concentration of the dispersion and hence to evaluate [77], the Einstein coefficient is increased above 2.5 by the factor (1 + 36/Vfts) by the adsorbed layer. The thickness of adsorbed layers can be extracted from experimental [77] values by this formula. ... 

According to the theory developed by Smoluchowski and by Einstein, if a spherical particle of radius r rotates in a liquid of viscosity i), in a short time A/, by an angle Aa, then the mean value of angular rotation A is given by the Brownian equation for rotational motion ... 

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 term k in this metric is a constant that determines the spacial curvature of the cosmology. For k = 1 the cosmology is a closed spherical universe, for k = 0 the cosmology is flat, and for k - — 1 the cosmology is open. The Einstein field equations give a constraint equation and a dynamical equation for the rate the radius changes with time. If we define a velocity as v = (R/R)H(t)r, where H (t) is the Hubble parameter, a constant locally, the constraint equations is... 

Application of the Stokes-Einstein equation requires a value for the solute radius. A simple approach is to assume the molecule to be spherical and to calculate the solute radius from the molar volume of the chemical groups making up the molecule. Using values for the solute radius calculated this way along with measured and known diffusion coefficients of solutes in water, Edward [26]... 

For dilute dispersions of spherical particles, the diffusion coefficient can be related to the hydrodynamic diameter of the particles by the Stokes-Einstein equation... 

When the solute is spherical, or close to be so, its radius is easily obtained otherwise, estimations can be made on the basis of the geometry and arrangement of the constituting atoms or ions. For solutes having a complex stucture (e.g., micelles), a distinction should be made between the hydrodynamic radius (which appears in the Stokes-Einstein equation of the diffusion coefficient) and the reaction radius [98]. For Ps, RPs should represent the bubble radius. However, as shown in Table 4.4, the experimental data are systematically very well recovered by using the free Ps radius, RPs = 0.053 nm using the bubble radius results in a calculated value of kD (noted kDb) that is too small by a factor of 2 or 3. Table 4.4 does not include such cases where k kD, as these do not correspond to purely diffusion-controlled reactions. 

The hydrodynamic behavior of complex particles in solution is similar to that of suspensions of solid spheres. Applying to the solutions of the PMAA-poly(ethylene glycol) complex, Einstein s equation for the viscosity of suspension of spherical particles, t]Sp/c = 2.54 q> (where tp is the volume fraction of dissolved substance) the solvent content in complex coils has been estimated33. It is about 75 vol%, i.e. the complex particles contain comparatively small quantities of the solvent in comparison with a usual random coil in solution which contains about 97-99 vol% of solvent34. ... 

---