Double-solution theory

From that time on, many causal theories were developed. In this work, we shall only refer to the causal theory proposed by de Broglie [2] and known as the double-solution theory. This theory, as well as Bohm s theory, are the most developed of all causal theories they are both able to explain and predict, practically all quantum phenomena. 

Even if de Broglie never proposed a nonlinear master equation for the quantum mechanics, he always thought that quantum mechanics needs, for a more complete description, a nonlinear approach, and his double solution theory is, as we have seen, a good example of it. Here we shall propose a possible way to tackle this problem by means of an example. Let us select... 

J. P. Vigier, Particular solutions of a non-linear Schrodinger equation carrying particle-like singularities represent possible models of de Broglie s double solution theory, Phys. Lett. A 135(2), 99-105 (1989). 

Fig. V-5. The repulsive force between crossed cylinders of radius R (1 cm) covered with mica and immersed in propylene carbonate solutions of tetraethylammonium bromide at the indicated concentrations. The dotted lines are from double-layer theory (From Ref. 51).

Studies of the adsorption of surface active electrolytes at the oil-water interface provide a convenient method for testing electrical double layer theory and for determining the state of water and ions in the neighborhood of an interface. The change in the surface amount of the large ions modifies the surface charge density. For instance, the surface ionic area of 100 per ion corresponds to 16, /rC/cm. The measurement of the concentration dependence of the changes of surface potential were also applied to find the critical concentration of formation of the micellar solution [18]. 

The surface sites and complexes lie in a layer on the mineral surface which, because of the charged complexes, has a net electrical charge that can be either positive or negative. A second layer, the diffuse layer, separates the surface layer from the bulk fluid. The role of the diffuse layer is to achieve local charge balance with the surface hence, its net charge is opposite that of the sorbing surface. Double layer theory, applied to a mixed ionic solution, does not specify which ions make up the diffuse layer. 

Since the sorbing surface holds a charge, its electrical potential differs from that of the solution. The potential difference between surface and fluid is known as the surface potential T and can be expressed in volts. The product e 4 is the work required to bring an elementary charge e from the bulk solution to the sorbing surface. According to one of the main results of double layer theory, the surface potential is related to the surface charge density by,... 

Double layer emersion continues to allow new ways of studying the electrochemical interphase. In some cases at least, the outer potential of the emersed electrode is nearly equal to the inner potential of the electrolyte. There is an intimate relation between the work function of emersed electrodes and absolute half-cell potentials. Emersion into UHV offers special insight into the emersion process and into double layer structure, partly because absolute work functions can be determined and are found to track the emersion potential with at most a constant shift. The data clearly call for answers to questions involving the most basic aspects of double layer theory, such as the role water plays in the structure and the change in of the electrode surface as the electrode goes frcm vacuum or air to solution. 

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

It was recently ascertained that the behavior of the adsorbed film of two surfactants in equilibrium with their micelle can be explained by assuming both the surface region and the micelle particle to be mixtures of the surfactants (1 - ) - Further, the application of the regular solution theory to the mixtures was shown to be useful to describe the nonideal behavior of ionic surfactants ( - ) However, the above treatments are incomplete from the thermodynamic viewpoint, because they do not consider the dissociation of surfactants and ignore the presence of solvent (T). In addition, it is impossible to suppose that the regular solution theory is applicable to both the adsorbed film and the micelle of ionic surfactants accompanied by the electrical double layer ( ). 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

When transient techniques are employed for fundamental research on these and other subjects, the effect of double-layer charging has to be accounted for in the analysis procedures. It has been observed frequently that at solid—solution interfaces, this process does not obey the capacitive behaviour predicted by double-layer theories. For example, the doublelayer admittance, Fc, cannot be represented by Yc = jciCd, but rather follows the relation [118]... 

Une tentative d interpreration console et non lintaire de la mecanique ondulatoire la theorie de la double solution. Gauthier-Villars. Paris, 1956. 

Application of the electric double layer theory to soil minerals at a quantitative level is difficult because soil mineral surfaces at the microscopic scale are not well defined, that is, they are neither perfectly spherical nor flat, as the double layer requires. However, application of the double layer theory at a qualitative level is appropriate because it explains much of the behavior of soil minerals in solution, for example, dispersion, flocculation, soil permeability, and cation and/or anion adsorption. When equilibrium between the counterions at the surface (near the charged surface) and the equilibrium solution is met, the average concentration of the counterions at any... 

Many methods have been used to determine the value of the PZC on solid electrodes. The one that seems to be most reliable, and relatively easy to perform, is based on diffuse-double-layer theory. Measurement of the capacitance in dilute solutions (C < 0.01 M) should show a minimum at , as seen in Eq. 15G and Fig. 4G. Lowering the concentration yields better defined minima. Modem instrumentation... 

Complex adsorption models based on double-layer theory, which take a mechanistic and atomic-scale approach to adsorption, can be used to model and predict these observations. Such models have been called electrostatic adsorption models or surface complexation models. They can consider simultaneously such important system properties as changes in pH, aqueous complex formation and solution ionic strength (solution speciation), and the acid-base and complexing properties of one or more sites on several sorbing surfaces simultaneously. 

Before I began writing. I considered the pos.sibility of a general text covering small particle behavior in both gases and liquids. Much of the theory of physical behavior is the same or very similar for both aerosols and hydrosols. almost as much as in the fluid mechanics of air and water. The differences include double layer theory in the case of aqueous solutions and mean free path effects in gases. There are other important, specifically chemical differences. 

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