Smoluchowski equation potential

The electrophoretic mobility, /jl, can be converted to a zeta potential by using the Smoluchowski equation,... 

The Helmholtz—Von Smoluchowski equation relates the electroosmotic velocity f eof to the zeta potential in the following way ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The dependence of the velocity of the EOF (Ve,) on the zeta potential is expressed by the Helmholtz-von Smoluchowski equation [13] ... 

The Helmholtz-von Smoluchowski equation indicates that under constant composition of the electrolyte solution, the EOF depends on the magnitude of the zeta potential, which is determined by various factors inhuencing the formation of the electric double layer, discussed above. Each of these factors depends on several variables, such as pH, specihc adsorption of ionic species in the compact region of the double layer, ionic strength, and temperature. 

It has been pointed out above that electroosmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomena therefore the Helmholtz-von Smoluchowski equation based on the Debye-Huckel theory developed for electroosmosis applies to electrophoresis as well. In the case of electrophoresis, is the potential at the plane of share between a single ion and its counterions and the surrounding solution. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

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. 

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... 

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).)... 

Criticize or defend the following proposition Zeta potentials for three different polystyrene latex preparations were calculated by the Helmholtz-Smoluchowski equation from electrophoresis measurements made in different concentrations of KCl.f... 

These zeta potentials are inaccurate because the range of kRs values exceeds the range of validity for the Helmholtz-Smoluchowski equation. The nature of the error is such as to make the estimated values of f too low. 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

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)]... 

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)... 

The eigenmode expansion was also used to determine the time-dependent solution of the Smoluchowski equations for diverse bistable potentials.185... 

In particular, the analytical solution of the Smoluchowski equation for a metastable potential well depicted in Fig. 4.7b was found taking as initial condition a uniform distribution in the potential... 

Substituting a for 7 uE/e (i.e a is the apparent zeta potential calculated from the Smoluchowski equation) gives... 

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) ... 

Besides there is uncertainty in hydrodynamics inside adsorbed layer, which can be (or not) permeable for flux. Hydrodynamic profile can deviate from parabolic Pois-seuille profile, that s why in this case the Helmholtz-Smoluchowski equation does not correctly transform the streaming potential into the zeta-potential values. So we indicate calculated f potential as apparent zeta-potential ( ). 

A few corrections have been proposed to modify the Helmholtz-Smoluchowski equation (3, 16, 28). Under certain experimental conditions many of these corrections become too small to be considered (6). However, these corrections are not as important as those that have to be introduced when potentials are calculated from mobility data (4, 5, 20, 21, 32, 33). 

K.M. Hong and J. Noolandi, Solution of the Smoluchowski equation with a Coulomb potential. I. General results. J. Chem. Phys. 68(11) 5163-5171 (1978). 

Microelectrophoresis is used to measure the electrophoretic mobility or, in other words, the movement of liposomes under the influence of an electric field. From the electrophoretic mobility the electrical potential at the plane of shear or (zeta) potential can be determined (by the Helmoholtz-Smoluchowski equation). From the zeta potential values the surface charge density (o) can be calculated. 

Various equations have been derived for relating EM, u, to the zeta potential, f. Traditionally, the Smoluchowski equation,... 

This equation, known as the Helmholtz-Smoluchowski equation, relates the potential at a planar bound surface region to an induced electro-osmosis fluid velocity 6. Recall that in the previous section surface charge was related to a potential in solution. In the following section surface charge will be related to the chemistry of the surface. A model for the development of surface charge in terms of acid-base dissociation of ionizable surface groups is introduced. 

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