Smoluchowski equation particles

The transport of dissolved species in a solvent occurs randomly through movement of the Brownian type. The particles of the dissolved substance and of the solvent continuously collide and thus move stochastically with various velocities in various directions. The relationship between the mobility of a particle, the observation time r and the mean shift (jc2) is given by the Einstein-Smoluchowski equation (in three-dimensional case)... 

The dimentionaless ica is a measure of the ratio between the particle radius and the thickness of the ionic double layer. In the limit ica (the double layer is very thin compared with particle radius f,(ica) = 3/2 and the result is the Helmholtz-Smoluchowski equation. In the limit ica + 0, f,(ica) 1 and Huckel s result is obtained. 

The mobility depends on both the particle properties (e.g., surface charge density and size) and solution properties (e.g., ionic strength, electric permittivity, and pH). For high ionic strengths, an approximate expression for the electrophoretic mobility, pc, is given by the Smoluchowski equation ... 

The central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

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. 

This important result is called the Smoluchowski equation and, as before, the zeta potential is directly related to the mobility and does not depend on either the size of the particle or the electrolyte concentration. 

Equation (39) is known as the Helmholtz-Smoluchowski equation. No assumptions are made in its derivation as to the actual structure of the double layer, only that the Poisson equation applies and that bulk values of rj and e apply within the double layer. It has been shown that this result is valid for values of kR larger than about 100 (i.e., kRs > 100 for spherical particles). 

We have now reached the position of having two expressions —Equations (27) and (39) —to describe the relationship between the mobility of a particle (an experimental quantity) and the zeta potential (a quantity of considerable theoretical interest). The situation may be summarized by noting that both the Huckel and the Helmholtz-Smoluchowski equations may be written as... 

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.)... 

It should also be noted that in the limit of kRs - 0, Equation (47a) reduces to the Hiickel equation, and in the limit of kRs - oo, it reduces to the Helmholtz-Smoluchowski equation. Thus the general theory confirms the idea introduced in connection with the discussion of Figure 12.1, that the amount of distortion of the field surrounding the particles will be totally different in the case of large and small particles. The two values of C in Equation (40) are a direct consequence of this difference. Figure 12.5a shows how the constant C varies with kRs (shown on a logarithmic scale) according to Henry s equation. 

The force in eqn. (208) on the particle k is F(rfe) due to the motion of all the N particles with respect to the unperturbed solvent. In Chap. 3, the Debye—Smoluchowski equation was derived from thermodynamic arguments. It was pointed out that the spatial gradient of the chemical potential at some point is the force acting at that point. If the potential energy is U, then the chemical potential is [cf. eqn. (40)]... 

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

Consider a system consisting of a particle diffusing in a viscous medium and under the influence of external forces. Such a system may be described2 by a Smoluchowski equation of the form... 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

In the case of dynamical interaction the pair potentials U r), (7BB(r) ar d Uab(t) should be incorporated into equation (2.3.45). It could be done using the Smoluchowski equation [27, 83, 84] for a particle drift in the external potential W (r) and expressed in terms of single particle DF (or concentration of such non-interacting particles)... 

Electrophoresis is the motion of charged particles relative to the electrolyte in response to an applied DC-electric field the field causes a shift in the particle counterion cloud, the counterion-diminished end of the particle attracts other counterions from the bulk fluid, counterions from the displaced cloud diffuse out into the bulk fluid, and the particle migrates. The particle velocity is predicted by the Smoluchowski equation. 

Technically, this result follows only if kmn < A(m + n), where A is a positive constant. If this condition is not met, cluster growth to a unit comprising all the primary particles can occur after a finite time interval (cf. Eq. 6.20b), corresponding to gel formation and a nonconstant M,. For a discussion of this point, see F. Leyvraz, Critical exponents in the Smoluchowski equations of coagulation, pp. 201-204 in F. Family and D. P. Landau, op. cit.7 and F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A 14 3389 (1981). 

When the particle sizes are much larger than the mean free path of the suspending medium, the Knudsen number (Kn = Agfa, where a and Ag are the radius of the particle and the mean free path of the medium, respectively) is extremely small. Under such conditions, the Brownian coagulation coefficient is well described by the Smoluchowski equation (1)... 

The Smoluchowski equation can also be extended to Knudsen numbers in the range 0.1 < Kn < 0.25 by introducing a slip correction factor C(Kn) in the expression for the particle diffusion coefficient. The empirical slip correction factor due to Millikan (3) has the form... 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

There is essentially a single modeling approach that has been developed, referred to here as the von Smoluchowski approach, and this method will be presented first. The von Smoluchowski approach requires analytical expressions to represent particle collision rates, to calculate collision efficiencies, and to dictate aggregate structure formation. These individual components are discussed in the subsequent sections, followed by analytical and numerical techniques of solving the von Smoluchowski equation. 

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