Booth’s equation

The variations in the intrinsic viscosity predicted by the primary electroviscous effect are often small, and it is difficult to attribute variations in the experimentally observed [17] from the Einstein value of 2.5 to the above effect since such variations can be caused easily by small amounts of aggregation. Nevertheless, Booth s equation has been found to be adequate in most cases. Further discussions of this and related issues are available in advanced books (Hunter 1981). 

This broken-down region near the ion was the subject of mathematical discussion by Webb as early as 1926, by Conway et al., and by Booth, whosepaper also can be considered seminal. Grahame made an attempt to simplify Booth s equation for the dielectric constant as a function of field strength, and a diagram due to him is shown in Fig. 2.27. 

Direct comparison between theoretical predictions either from the Smoluchowsky-Krasny-Ergen or Booth s equations and experimental data is not possible, as it is difficult to fulfill the requirements of the Smoluchowsky-Krasny-Ergen theory, and the Booth s equation requires the knowledge of the individual mobilities of all ions. Under any approximation, a significant overestimation from both equations were found [56-58J. Note that Eq.(95) and (96) don t contain the electric field strength, which is only built up within the electric double layer. The value of Kr thus has a critical effect on the viscosity of suspensions. Many other equations or theories [59-61J are only valid in certain Kr ranges and very... 

In both cases the electroviscous effect is about 100 times smaller than that calculated from cq. (7) and of the order of magnitude following from Booth s equation (8). In Table 2 wc show the data obtained by Bull compared with the two different theoretical estimates. The interpretation of the measured viscosities is based upon the assumption that the increase in viscosity above the value at the iso-electric point is the electroviscous contribution. In the I.B.P. the viscosity is about 2 times the value ex-... 

Booth73 used Frolich s108 modification of the Onsager expressions101 for the cavity field in non-associated polar liquids, and corresponding modifications of Kirkwood s equation for associated polar media. Booth s assumptions in deriving Eq. (28) for the Kirkwood case are important to determine the validity of his final expressions. He used the Onsager-Frolich cavity field ratio... 

This expression may determine the relationship between ea as a function of Aa in intense fields, e.g., for x > 3, when the approximation L(x) A = 1 - l/xA applies. However, it assumes classical statistics, so it will require a quantum correction for x values in this range, assuming that the Langevin function is still applicable. For smaller values of xA, Eq. (48) may be solved graphically. A more transparent expression for small xA values would be useful. Equation (42) with m = 1/2 leads to a quartic equation in ea which may be simplified by expansion to a cubic expression. A solution is the use of Booth s approximation ea and e0 n2, with substitution ofEq. (47) into Eq. (42). This gives a simple and useful approximation ... 

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

As at the moment of writing this chapter details of the treatment of Booth were not yet available, we shall now give a short description of Overbeek s results. In the provisional paper by Booth it is mentioned that the results of these two authors are in fair agreement. Overbeek found that the counter E.M.F. of relaxation is in the first approximation proportional to C for unsymmetrical electrolytes and to for symmetrical electrolytes. The resulting electrophoresis equation is of the form... 

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