Smoluchowski equation correction

The von Smoluchowski equation must be corrected when the partners are ions to account for attractive or repulsive forces. They can be approximated by an electrostatic model. The quantity by which Eq. (9-10) or (9-13) is to be multiplied is... 

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... 

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). 

Corrections to the Smoluchowski equation (1.13) to order have been evaluated by Titulaer, using a Chapman-Enskog expansion. 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

For large Ka or a both Du s go to zero, meaning that then surface conduction may be neglected. For the electrophoretic mobility this means that the Helmholtz-Smoluchowski equation [4.3.41 also remains the correct limit for high Ka if surface conduction does occur. 

According to general experience the Influence of sol concentration Is absent when the volume fraction remains of the order of a few percent. This may be concluded from older experiments and from recent electroacoustic studies, discussed in sec. 4.5d. Experiments involving mlcro-electrophoresls are not suited to stud3dng the volume fraction effect because the required extreme dilution may readily lead to spurious adsorption on the particles. Reed and Morrison studied theoretically the hydrodynamic interactions between pairs of different particles in electrophoresis they corrected the Helmholtz-Smoluchowski equation for various distances between the (spherical) particles and values of the electroklnetlc potentials of the particles, and... 

Editor s Note A peculiarity of the telegraph equation, when used to approximately include inertial effects, is that it yields the correct result for the variance while yielding a poorer approximation for the distribution function than the Smoluchowski equation [see Ref. 248 and Risken (loc.cit.).]... 

It should be emphasized that EOD is most attractive when the water is trapped between fine clay particles (i.e., small pores or water transport chaimels) and caimot be further removed efficiently by the application of pressure or vacuum, etc. If the ionic concentration of the trapped water is low, the thickness of the electrochemical double layerio i can become comparable to or even exceed the pore size, thus requiring corrections to the zeta potential approach based on the simple Helmholtz-Smoluchowski equation such corrections cannot, however, be carried out accurately despite many attempts. " ... 

In all the 2BSM calculations presented here, the diffusion coefficient Dj equals 1, which defines the unit of frequency (inverse time) whereas the diffusion coefficient for the solvent, Dj varied from 10 (very fast solvent relaxation) to 1, 0.1, 0.01 (very slow solvent relaxation). In the Dj = 10 case, one finds that the reorientation of the solute is virtually independent of the solvent a projection procedure could easily be adopted in this case to yield a one-body Smoluchowski equation for body 1 with perturbational corrections from body 2. The temporal decay of the first and second rank correlation functions is then typically monoexponential. When the solvent is relaxing slowly (i.e., Dj is in the range 1-0.01), the effect of the large cage of the rapid motion of the probe becomes... 

Note that g(r) is a function of the constant fall speed U from the Smoluchowski equation (1), and that at small Pe the 0(Pe) problem must be solved to get the correction to the fall speed.]... 

Figure 14.4 Semi-logarithmic plot of normalized fluorescence decay of excited HPTS. Points are experimental data = 375 nm, = 420 nm) in water acidified by HCIO, after lifetime correction. The geminate recombination data (pH = 6) is fitted by a numerical solution ofthe Debye-von Smoluchowski equation convoluted with the instrument response function after lifetime correction. (Adapted from Ref [125].)...

The zeta potential of the virus at 0.02 I was measured by zonal electrophoresis in a sucrose gradient. In its evaluation we corrected the retarded, relaxed Smoluchowski equation (33) for the sucrose-dependent viscosity gradient (8). Again, the Hunter and Wright correlation was used to estimate the value of the zeta potential at 0.305 I. 

In other studies, Jonah et al. [116] measured the rate of reaction of the hydrated electron with Cd and Cu cations. They noted a decreasing rate coefficient with increasing ionic strength. In all cases, the rate was slower than that based on the Debye- Smoluchowski equation [68], eqn. (51), but greater than or equal to the corrected rate coefficient using the Bronsted- Bjerram correction [eqn. (58)]. In fact, Jonah et al. found that the rate coefficient for reaction of hydrated electrons with pure Cd(C104 )2 or Cu(C104 )2 follows that predicted by Coyle et al. [94] where no ionic atmosphere has developed around e q. Jonah et al. pointed out that such a situation was improbable (see Sect. 1.6). Furthermore, no hydrodynamic correction was made to the rate coefficient, which would lower the expected value by 20%. Jonah et al. [119] showed that the observed rate for reaction of e q with HsO was about one third of the expected Debye—Smoluchowski diffusion-limited rate (see the Debye... 

Equation (8.1) is correct only for 1. To discuss the general case, we have to study the Smoluchowski equation for the rotational Brownian motion. This equation can be derived straightforwardly according to the Kirkwood theory described in Section 3.8. Such a derivation is given in Appendix 8.1. Here we derive it by an elementary method to clarify the underlying physics. 

Streaming potential, streaming current and Helmholtz-Smoluchowski equation approach is correct and valid when electrolyte solution is forced through a narrow slit formed by two similar measured surfaces. This ensures that the thickness of the electrochemical double... 

This equation corrects R 9) for the Einstein-Smoluchowski solvent scattering. The factor of 2 appearing in the second term, RHS of equation (3.44) arises from first multiplying equation (3.41) by c, then differentiating as shown in equation (3.42). If the particles are very small compared to the wavelength of the radiation, R 9) reduces to 3xl 6n, where t is the turbidity in Beer s law ... 

The electrophoretic mobilities of the latex particles in the presence of various concentrations of SDS, sodium ions, and magnesium ions were determined, employing a micro-electrophoresis apparatus at an applied field of 3.16 volts/cm. The Zeta potential of the particles was calculated using the Smoluchowski equation (6), applying only the correction of Henery (6,7) to account for the distortion of the electric field created by the presence of the particle electric double layer. 

We have reviewed models of unimolecular and bimolecular reactions in solution We have shown that, if the Smoluchowslci equation is used to describe the spatio-tenqporal evolution of the probability density, rather single analytical expressions can be derived where the influence of the solvent is taken into account through macroscopic parameters like viscosity or dielectric constant. However, we have shown the limits of the Smoluchowski equation for reactions occurring at short times and/or distances. Time- or frequency - dependent rate constsmts have been derived, but the theory suffers from the difficulty of correctly expressing the response of the solvent as a function of frequency. 

A potential is simply the work done in bringing a point charge from infinity to the particle surface. Potentials are always relative to ground (i.e. at an infinite distance from the surface). The surface potential is very important and is approximated by the so-called zeta potential, which can be estimated by (micro)electrophoresis experiments, at least for some special cases (small and large particles via the so-called Hiickel and Smoluchowski equations). In the general case, the zeta potential is calculated from values of the electrophoretic mobility, using graphical solutions or the Henry equation, which requires a correction factor. 

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. 

As the value of kR is neither below 0.1 (sometimes less than 1 is sufficient) or above 100, we cannot use the Hiickel or Smoluchowski equations for estimating the smface or zeta potential. We have to use the Henry equation (see Equation 10.10) with a correction parameter/, which we can obtain from the available data (by interpolation) ... 

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