Smoluchowski formula

The Smoluchowski formula, which is used in the AcoustoSizer from the measured dynamic mobility, is valid for a disperse suspension of spherical particles according to Eq. 1. 

The formula (7.73) says that the electrolyte slips along the charged surface with the velocity U. This formula is known as the Helmholtz-Smoluchowski formula. 

Now the frequency dependence and the phase lag are determined entirely by the inertia term G, and the zeta potential is calculated from a modified form of the Smoluchowski formula (41) which takes account of the inertia effect for the larger particles, especially at the higher frequencies. The determination of size and charge is particularly simple in this case. 

For a rectangular cross-section, as shown in Fig. Ic, the solution to Eq. 4 and the subsequently given velocity component were provided in a double-series form by, for example, Li [1] and later in a single series form by Wang et al, [3], The Helmholtz-Smoluchowski formula for the electroosmosis-driven flow velocity can be given of course for the thin EDL limit. The series solution for the pressure-driven flow velocity is presented, for example, by Min et al. [2] as follows ... 

Electroosmosis produces an effective slip of the liquid outside the double layer past to the solid surface. In the classical continuum model of the diffuse layer, the slip velocity is given by the Helmholtz-Smoluchowski formula ... 

FIGURE 3.5 Full calculation of the electrophoretic mobility of spheres as a function of the zeta potential for the Ka values indicated, compared with the Helmholtz-Smoluchowski formula. 

FIGURE 3.8 Full calculation of the sedimentation field ( sed) of spherical particles of 100 nm radius as compared with Smoluchowski formula, as a function of zeta potential for the na values indicated. 

In the case of an uncharged gas interface (qi = qg, qi = 0), we predict the simple Smoluchowski formula. In other words, there is no EO flow enhancement, and the flow is isotropic despite... 

The change indicated above can be obtained immediately from the Smoluchowski formula, which has the following form [92] ... 

V. Smoluchowski s formula has been tested both by an examina-, tion of the rate of decrease of the primary particles present in gold suspension undergoing coagulation and also by counting the... 

It is interesting to notice that a simple formula for the addition of transition probabilities exists only in extremely simple models of stochastic processes. An example is the Chapman-Smoluchowski-Kolmogoroff equation.30... 

The type of chosen polymer and additives most strongly influences the rheological and processing properties of plastisols. Plastisols are normally prepared from emulsion and suspension PVC which differ by their molecular masses (by the Fickentcher constant), dimensions and porosity of particles. Dimensions and shape of particles are important not only due to the well-known properties of dispersed systems (given by the formulas of Einstein, Mooney, Kronecker, etc.), but also due to the fact that these factors (in view of the small viscosity of plasticizer as a composite matrix ) influence strongly the sedimental stability of the system. The joint solution of the equations of sedimentation (precipitation) of particles by the action of gravity and of thermal motion according to Einstein and Smoluchowski leads 37,39) to the expression for the radius of the particles, r, which can not be precipitated in the dispersed system of an ideal plastisol. This expression has the form ... 

We need to determine how to account for the same effect when diffusional transfer is remote and occurs far from the contact. Even the generalized Smoluchowski approximation, substituting R for a in contact formulae, is not enough. The fitting parameter R having the same physical meaning as Rq is the radius of the black sphere around the acceptor. For the exponential transfer rate it is defined by the following condition ... 

Smoluchowski,1 treating the problem from a much more general point of view, obtained the same equation (28), and all subsequent workers have found an equation similar in form, notw ithstanding that it is universally agreed now that the double layer is not of the simple, plane parallel type used as an illustration by Helmholtz. All the formulae agree except as to the exact value of the constant. Smoluchowski found Debye and Huckel2 6 for spherical particles ... 

The value of 1/2, which is the reciprocal of the friction parameter 2, decreases as the drag exerted by the hydrogel layer on the liquid flow increases. In the limit of 1/2—> 0, Eq. (21.55) tends to the well-known Smoluchowski s mobility formula for hard particles. In other words, as 1/2 increases, the hydrogel layer on the particle becomes softer. That is, the parameter 1/2 can be considered to characterize the softness of the hydrogel layer on the particle. The observed reduction of the softness parameter 1/2 (1.2 nm at 30°C to 0.9 nm at 35°C) implies that the hydrogel layer becomes harder, which is in accordance with the observed shrinkage of the hydrogel. 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

Formulas for the Leslie viscosities, in turn, were derived from the Smoluchowski equation for hard rods by Kuzuu and Doi (1983,1984 Semenov 1987), and are given in Eqs. (10-20) with ao = 0- These formulas require as inputs values of 2,54, k, and Dr, which are functions of polymer concentration C. Reasonably reliable analytic functions for these dependencies were obtained by Kuzuu and Doi using a perturbation expansion for large order parameter, yielding... 

Formula (483) was first obtained by Albert Einstein (1879-1955) in 1905 and bears his name. Independently of Einstein, the theory of the Brownian motion was developed by Marian von Smoluchowski (1872-1917) in 1905-1906. The expression obtained by him agrees with formula (483) with a constant multiplier equal to one. 

The movement of a charged colloidal particle in an external electrical field is called electrophoretic motion and the respective phenomenon is electrophoresis. The electrophoretic velocity in the two limiting cases, of a thin and thick EDL around a spherical particle, can be calculated by von Smoluchowski 3 and HiickeF formulas ... 

All the consideration up to now implies that the dielectric permittivity and the viscosity in the EDL (at least for x > x see Figure 5.67) are equal to those of the bulk disperse medium. A more refined approach shows that for thin double layers the formulaes, stemming from the von Smoluchowski theory, may remain unaltered if the real potential ( = /(Xj)) is replaced by the quantity ... 

The anomalous surface conduction was studied extensively by O Brien and by Hunter, and they could show that the Stem layer conduction is about thirty times larger than the diffuse layer conduction at low salt concentration. This explains substantial discrepancies between the electrophoresis and the conductance estimates of zeta potential. For thin double-layer systems such as this, the zeta potential is usually calculated from the electrophoretic mobility using Smoluchowski s formula, which in O Brien s case corresponds to a zeta potential of 50 mV [8]. Complex conductivity measurements result in f = —160 mV. 

We measured the electrophoretic mobilities of crude oil droplets in alkaline solution using a Zeta Meter (20). Since the droplet sizes were larger than one micron, the zeta potentials were calculated from electrophoretic mobilities using Smoluchowski s formula. 

The basic formula for the association rate constant is given by Debye-Smoluchowski theory ... 

It was shown by Dukhin (1983) that under the conditions of strong retardation the effect of an equilibrium DL is dominant for any surface activity of the surfactant. Therefore, Smoluchowski s formula is valid at any degree of surface activity. 

A noticeable deviation of sedimentation potentials from Smoluchowski s formula takes place at large siuface concentration variation along the bubble surface. Before considering experimental data, it has to be pointed out that the validity of Smoluchowski s formula for the description of the Dorn effect at large Peclet numbers applies only to solid spherical particles. In particular, the correctness of conclusions of some papers (Dukhin, 1964 Dukhin Buikov, 1965 Derjaguin Dukhin, 1967, 1971) is experimentally confirmed by Usui et al. (1980). Sedimentation potential for four sizes of glass balls appears to be the same. Since the radii of the particles under consideration are approximately 50, 150, 250, and 350 pm, the absence of any effect of Peclet and Reynolds numbers on the sedimentation potential could be demonstrated. 

In a similar phenomenological approach to unimolecular reactions involving large-amplitude motion, the effect of friction on the rate constant can be described by a simple transition formula between the high-pressure limit of the rate constant at negligible solvent viscosity and the so-called Smoluchowski limit of... 

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