Zeta potential, numerical values

Thus far, these models cannot really be used, because no theory is able to yield the reaction rate in terms of physically measurable quantities. Because of this, the reaction term currently accounts for all interactions and effects that are not explicitly known. These more recent theories should therefore be viewed as an attempt to give understand the phenomena rather than predict or simulate it. However, it is evident from these studies that more physical information is needed before these models can realistically simulate the complete range of complicated behavior exhibited by real deposition systems. For instance, not only the average value of the zeta-potential of the interacting surfaces will have to be measured but also the distribution of the zeta-potential around the mean value. Particles approaching the collector surface or already on it, also interact specifically or hydrodynamically with the particles flowing in their vicinity [100, 101], In this case a many-body problem arises, whose numerical... 

The numerical solution for the surface potential as a function of pH is compared in Fig. 7, for various NaCI concentrations, with the experimental results provided by Li and Somasundaran [32], The potentials — ]i,s = — ]i(0) and — tyd= — t)t(t/E) are plotted as functions of distance, since the zeta potential determined by electrophoresis is not defined at the surface, but at an unknown location, the plane of slip . The magnitude of <-s is always larger than that of i >d, since the potential decays with the distance. The value dE=4 A, which is provided by the dependence of the surface tension of water on the NaCI concentration at high ionic strengths was employed. For the equilibrium constant, the value K ou= () 10 M, which is consistent with the experimental data for pH values between 3 and 6, was selected. A reasonable agreement with the data (which have a rather large error) was obtained by selecting A=5.0X 1016 sites/m2 and W1 = 0.5kT. 

The numerical optimization methods do not require additional assumptions of the temporal constancy, or even neglect some physical constants, for example surface potential. Used for the optimization of the edl parameters (surface hydroxyl group reaction constants, capacity and density of adsorption sites) the numerical methods allow us to find the closest values to the experimentally available data (surface charge density, adsorption of ions, zeta potential, colorimetric measurements). Usually one aims to find the parameters, accepted from physical point of view, where a function, that expresses square of the deviation between calculated and measured values will be the smallest. 

The numerical results show that the polarization effect of the double layer impedes particle s migration because an opposite electric field is induced in the distorted ion cloud, which acts against the motion of the particle. For a given ica, the electrophoretic mobility increases first, reaches a maximum value and then decreases as the absolute zeta potential is increased. This maximum mobility arises because the electrophoretic retarding forces increase at a faster rate with zeta potential than does the driving force. 

With this boundary condition, the Navier-Stokes equations for longitudinal F, and radial V, components of EOF velocity were solved numerically, and the calculated EOF profiles were used to simulate the solute peak shapes. The situation where the part of the capillary length was modified to the zero value of the zeta potential and the part of capillary was not modified + 0)... 

Henry s calculations are based on the assumption that the external field can be superimposed on the field due to the particle, and hence it can only be applied for low potentials (f < 25 mV). It also does not take into account the distortion of the field induced by the movement of the particle (relaxation effect). Wiersema, Loeb and Overbeek [19] introduced two corrections for the Henry s treatment, namely the relaxation and retardation (movement of the hquid with the double layer ions) effects. A numerical tabulation of the relationship between mobility and zeta-potential has been provided by Ottewill and Shaw [20]. Such tables are useful for converting u to f at aU practical values of kR. 

It is difTicult to find in the literature details of experiments that relate contact angle to the efficiency of the dispersion process. A correlation between contact angle and flotation rate for quartz in aqueous solutions of dodecylammonium acetate, with adsorption from solution and electrokinetic (zeta potential) data, was reported by Fuerstenau". In a later review Fuerstenau discussed the relevance of the adsorption process to flotation technology illustrating the direct relationship between the amount of surfactant adsorbed and the contact angle . Numerous values of 0 are published in the literature. Most have been measured on flat surfaces and even with these problems arise due to surface roughness" . For powders the measurement is more difficult, although a number of methods have been reported" and reviewed . 

Fig. 3 Electroosmotic flow velocity distributions in a charged mictDcapillary packed with charged microspheies, for different values of zeta potential ratio w/ p- The results are obtained rai the basis of (a) the numerical solution of Eq. 13 and (b) the analytical solution based on Eq. 30...

In general, the linearization is accurate when i/r is small. Figure 2 shows that the LPM results agree perfectly with multigrid solutions at all zeta potentials and with the analytical solution of the linearized equation when the absolute value of the surface zeta potential C is small, less than about 30 mV. This validates the accuracy of the LP-M. When the absolute value of the zeta potential is large (>30 mV), the numerical results of the LPM depart from the linearized analytical solutions as expected. 

A careful account of the problem can be found in Ref. [95]. Ohshima et al. [96] first found a numerical solution of the problem, valid for arbitrary values of the zeta potential or the product Ka. In the same paper, they dealt with the problem of finding the sedimentation potential and the DC conductivity of a suspension of mercury drops. The problems are solved following the lines of the electrophoresis theory of rigid particles previously derived by O Brien and White [18]. The liquid drop is assumed to behave as an ideal conductor, so that electric fields and currents inside the drop are zero, and its surface is equipotential. The main difference between the treatment of the electrophoresis of rigid particles and that of drops is that there is a velocity distribution of the fluid inside the drop, Vj, governed by the Navier-Stokes equation with zero body force (in the case of electrophoresis), and related to the velocity outside the drop, v, by the boundary conditions ... 

Let us finally mention that other approaches have been proposed that numerically solve the problem for arbitrary values of the parameters of interest, specifically, zeta potential and ionic strength. Details can be found in Refs. [99,100]. The problem of the dielectric spectroscopy of suspensions of this type of particles has been analyzed in Refs. [101,102]. 

PBE equation was also solved numerically assuming water and ion-permeable hollow spheres treating specific ion adsorption using a Volmer isotherm [49]. The calculations suggest that the distribution of ions in the internal aqueous of (DODA)X vesicles is measurable and that the value of the electrical potential, ij/, at the vesicle center is not negligible at moderately low salt concentration. As could be expected the value of j/ at the vesicle center decreases with vesicle size and vesicle concentration (Fig. 4). Using realistic parameters, for a 265 nm (DODA)X vesicle, the value of the potential at the vesicle center can reach 100 mV [49]. The calculations yielded results that are consistent with measured values for external ion dissociation and zeta potentials of vesicles of synthetic amphiphiles. 

The pH at which the zeta-potential is zero is called isoelectric point (lEP). Because of specific ion adsorption, the lEP may deviate from the PZC, which is a purely material property. For this reason, the lEP is less a parameter of the particle phase but rather a characteristic parameter of the suspension. In a couple of papers, Kosmulski reviewed experimentally determined PZC and lEP-values for numerous materials in varying environments (Kosmulski 2002, 2004, 2006, 2009). 

Wiersema, Loeb and Overbeek [25] introduced two corrections to Henry s treatment, namely the relaxation and retardation (movement of the liquid with the double layer ions) effects and Ottewill and Shaw have compiled a numerical tabulation of the relation between mobility and zeta potential [26]. Such tables are useful for converting u into C at all practical values of kR. 

Ff may be estimated from experimental measurements of the zeta potential [173] using numerical solutions of the Poisson-Boltzmann equations [174]. F i is, of course, positive and its contribution to the total AGJ value is generally small (see Table 3.11). 

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