Colloids Smoluchowski equation

In this result, the condition of small particles means that the actual size of the particles (which is often difficult to obtain) is not required. For reasons to be discussed later, we will call the potential obtained by this method the zeta potential (Q rather than the surface potential. In the following section we consider the alternative case of large colloidal particles, which leads to the Smoluchowski equation. 

A major remaining problem is that many systems of interest in colloid chemistry do not correspond to either of these two limiting cases. The situation is summarized in Figure 12.4, which maps the particle radii Rs and 1 1 electrolyte concentrations that correspond to various kRs values. Clearly, there is a significant domain of particle size and/or electrolyte concentration for which neither the Huckel nor the Helmholtz-Smoluchowski equations can be used to evaluate f from experimental mobility values. The relationship between f and u for intermediate values of kRs is the topic of the following section. 

FIG. 12.4 The domain within which most investigations of aqueous colloidal systems lie in terms of particle radii and 1 1 electrolyte concentration. The diagonal lines indicate the limits of the Hiickel and the Helmholtz-Smoluchowski equations. (Redrawn with permission from J. Th. G., Overbeek, Quantitative Interpretation of the Electrophoretic Velocity of Colloids. In Advances in Colloid Science, Vol. 3 (H. Mark and E. J. W. Verwey, Eds.), Wiley, New York, 1950.)... 

FIG. 12.8 Plot of rju/e versus f/0, that is, the zeta potential according to the Helmholtz-Smoluchowski equation, Equation (39), versus the potential at the inner limit of the diffuse part of the double layer. Curves are drawn for various concentrations of 1 1 electrolyte with / = 10 15 V-2 m2. (Redrawn with permission from J. Lyklema and J. Th. G. Overbeek, J. Colloid Sci., 16, 501 (1961).)... 

University, Krak6w [i]. He described Brownian molecular motion independently from Einstein considering the collisions explicitly between a particle and the surrounding solvent molecules [ii], worked on colloids [iv-v], and obtained an expression for the rate with which two particles diffuse together (-> Smoluchowski equation (b)) [iii-v]. He also derived an equation for the limiting velocity of electroosmotic flow through a capillary (-> Smoluchowski equation (a)). 

The movement of colloidal particles in an electric field is termed electrophoresis. Most commercially available zetameters are designed to measure the electrophoretic mobility of particles. The potential is not measured directly, but is calculated from u. The Smoluchowski equation... 

In describing the mechanical response of microstructured fluids, e.g., polymers, emulsions, colloidal dispersions, etc., one needs to determine the pair distribution function - the probability density P(r) for finding a particle at a position r given a particle at the origin in suspensions, or the probability density of the end-to-end vector in polymers, or a measure of the deformation of drops in an emulsion. This probability density satisfies an advection-diffusion or Smoluchowski equation of the following (when suitable approximations have been made) form ... 

The simplest model of this process leads to the following Smoluchowski equation to determine the concentration distribution of the background colloidal particles ... 

N. J. Wagner. The smoluchowski equation for colloidal suspensions developed and analyzed through the generic formalism. J. Non-newtonian Fluid Mech., 96 177-201, 2001... 

Sze A, Erickson D, Ren LQ, Li DQ (2003) Zeta-potential measurement using the Smoluchowski equation and the slope of the current-time relationship in electroosmotic flow. J Colloid Interface Sci 261(2) 402-410... 

The study of the rates of aggregation of colloidal particles owing to Brownian motion has a long history and was initiated by von Smoluchowski in 1916. Several solutions to the Smoluchowski equation have been attempted since then and two will be sketched here. The problem is to establish the time evolution of the number and size of the agglomerates in terms of the number of monomer particles. The expressions are typical for simplified models as is usual in physical chemistry. 

In the first process, the Brownian motion velocity and presence of electrolytes influences the increase in viscosity of the immobilizing media, the coagulation velocity, the domain of attraction forces, and the concentration of colloidal solution. Consequently, from the Smoluchowski equation (Pomogailo and Kestelman 2005), the rate constant of particle coagulation, k, is inversely proportional to the viscosity of the media, r ... 

The frequency, k, has the units of a second order rate constant, M s", N is Avogadro s number and D, r, are the diffusion constants (in cm" s" ) and the radii of reactions (in cm), respectively, of the two reactants A and B. In keeping with common practice we shall call this the Smoluchowski equation. It is difficult to ascertain the first use of the equation in this form. It has its origins in Smoluchowski s (1916) mathematical treatment of the kinetics of the coagulation of colloidal particles in solution. He derived the equation for the time course of the formation of aggregates of increasing size. His theoretical predictions were found to to fit the results obtained by colloid chemists at that time. Smoluchowski s... 

Notice that, unlUce the case of the Henry equation, in the simple cases of Htickel and Smoluchowski equations the size of particles (radius) is not required. There are alternative graphical ways for correcting for the size of particles (Pashley and Karaman, 2004). Although the Htickel and Smoluchowski equations are very useful, it can be shown that they only cover a very small part of the colloidal domain. In most cases, corrections are needed, e.g. via the use of Henry equation or other graphical methods where the correction factor, f, can be estimated. Negative values of y/o can be obtained if p is negative. 

In more recent work, Cheng et al. (2002) reported measurements of the low-shear viscosity for dispersions of colloidal hard spheres up to (f) = 0.56. Nonequilibrium theories based on solutions to the two-particle Smoluchowski equation or ideal mode coupling approximations did not capture observed viscosity divergence (Cheng et al. 2002), although the Doolittle and Adam-Gibbs equations still appeared to hold. 

Diehl, A. and Y. Levin. 2006. Smoluchowski equation and the colloidal charge reversal. Journal of Chemical Physics 125 (5) 054902-1-054902-5. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

A more quantitative measure of stability, known as the stability ratio, can be obtained by setting up and solving the equation for diffusive collisions between the particles. Quantitative formulations of stability, known as the Smoluchowski and Fuchs theories of colloid stability, are the centerpieces of classical colloid science. These and related issues are covered in Section 13.4. 

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