# Self-consistent electric field in the volume of the salt solution

## Fluctuations in the polarization charge in volume of the solution; solvated ion sizes and frequencies excitations of electroinduced drift

We assume that each cation (anion) causes polarization of the surrounding solvent (water): around each cation (anion), an ‘atmosphere’ is formed with an excess of polarized water molecules, screening the field of the cation (anion). The generalized electron shell of each polarized water molecule is deformed relative to the unperturbed configuration when the total spin of the molecule is zero. The disturbance is caused by an electric field, of the cation (anion). The deformation of the shell leads to the fact that part of the charge of the nuclei or electrons that make up the water molecule will be uncompensated in a certain part of the region of space occupied by the unperturbed electron shell. This uncompensated part of the charge is the polarization charge of a water molecule in the inhomogeneous electric field of a charged particle - a cation (anion). In this case, the polarization charge of the water molecules is determined by the charge of the particle that they shield. The sum of the polarization charges of all molecules associated around one particle is equal to the charge carrier (in absolute terms). In addition, the polarization charge is positive if the water molecules screen the anion, and negative if they screen the cation. This ‘atmosphere’ (‘fur coat’) with an excess of polarized water molecules is a solvation shell.

Consider the simplest model of electron polarization of a solvent molecule. In this case, a single molecule can be represented as a mechanical oscillator, when a charge having a mass, under the influence of electric forces, carries out a forced oscillatory motion near the centre of mass. In this case, the deviation of the polarizable charges from the equilibrium position is determined by the magnitude of the electric field and the coefficient of elasticity characterizing the elasticity of the binding forces of charges in the molecule. These quantities are related by

where r_{ra}, p are parameters, the first of which is the deviation of the charges from the equilibrium position, and the second is the elasticity coefficient characterizing the elasticity of the electric forces of the binding of charges in the molecule. We introduce the concept of resonance frequency related

charges

then from (3.1) we can obtain the relation

It is seen that in relation (3.3), as a parameter, the frequency of natural vibrations, which includes the mass of the charge, is already present. This suggests that the inertial properties of oscillating charges will affect the vibrational processes of polarized atoms and molecules.

Inertial properties play the role of inductance in the electrical circuit. We call this ‘kinetic inductance’. This determines the dependence of the magnitude of the polarization vector on the frequency of the electric field. We introduce the polarization vector

The dependence of the polarization vector on the frequency is associated with the presence of mass of links and their inertia does not allow this vector to precisely follow the electric field, reaching its static value at each moment of time. When the field frequency in the dielectric volume coincides with the resonant frequency of an individual molecule the polarization vector tends to infinity. This means there is resonance at this frequency. Since electric induction is determined by the relation

then the second Maxwell equation takes the form or

where *j* is the total current flowing through the volume of the dielectric.

In this expression, the first term on the right-hand side is the bias current in vacuum, and the second is the current associated with the presence of bound charges in the dielectric molecules. In expression (3.7), the specific kinetic inductance of the charges participating in the vibrational process *L _{kd} = m* appeared again

This kinetic inductance determines the kinetic inductance of the bound charges. This is natural, since the oscillating charges also have masses, therefore, have inertia.

With this in mind, relation (3.7) can be rewritten in the form

or

or

where

Relation (3.13) determines the so-called ‘plasma’ frequency for the series that are part of the dielectric molecules, if these charges were free.

These results can be obtained without introducing the polarization vector. This is easy to do by calculating current densities. In our consideration, it is assumed that the molecules are in vacuum. In fact, the molecules form a liquid (dielectric solvent). In fact, the density of molecules is very high, but vacuum is still between the two, and bias currents (although they are much smaller than currents associated with the movement of charges) will still be present.

We write the total current density as the sum of current densities bias and conduction current:

Using relation (3.13) to find the speed of oscillating charges, we obtain

Substituting this expression into relation (3.14), we obtain

or

Relations (3.10) and (3.17) completely coincide. Thus, a conditional possibility arises to name the value facing the derivative in the relation (3.12) as the dispersing (frequency-dependent) dielectric constant of the dielectric, which is done currently in all existing literature, which discusses the issues of frequency dispersion in dielectrics. But is such a definition correct? To answer this question, we consider the structure of currents flowing in a dielectric represented by the right-hand side of relations (3.17) and (3.10). The first member of the right-hand side, as was already said, represents the bias current in the vacuum in which the molecules are located. The second term represents the currents associated with the presence in the vacuum of molecules of the dielectric itself. This current is completely identical to the current that occurs in a conventional electrical series oscillatory circuit. Therefore, the equivalent circuit of the unit volume of the dielectric, in which the current distribution can be considered homogeneous, can be represented as a sequential oscillatory circuit with the only difference that the kinetic inductance of the charges should be taken as the circuit inductance. If you take into account the current displacement in vacuum, then in parallel with the circuit it is also necessary to include a capacitance equal to the dielectric constant of the vacuum.

Thus, as in the case of plasma, the coefficient before the derivative in relation (3.12), is not the dielectric constant, but is a composite parameter and now it immediately includes three parameters independent of the frequency. In addition to the dielectric constant of the vacuum of the mind, two more characteristic frequencies enter into it. The frequency co_{Q} (let's call it proper) is an individual characteristic of each molecule, it is assumed that it does not depend on the density of filling the space with molecules, and in the space of the molecule located at such a distance that their mutual influence on each other is absent. On the contrary, copd (let's call it the plasma frequency of the dielectric) depends on the packing density of the molecules in the composition of the dielectric fluid. And here arises an important question: what if the composition of a liquid dielectric includes various groups of dissimilar molecules? In this case, each such group will be characterized by its own resonant frequency, while the plasma frequency will be characterized by the density of molecules that make up the dielectric. Now you can imagine which variety of different resonances can be observed in multicomponent systems. This determines the variety of colors that we see around, because at the resonant frequencies there is a maximum reflection or absorption of a signal of a given frequency of electromagnetic waves, which sets off a given colour of the visible us facility. And the purer we see the light, for example in ruby or sapphire, the better the resonance of an atom or molecule whose frequency we observe.

We consider two limiting cases. If со co_{0} then from (3.12) we obtain

In this case, the coefficient facing the derivative does not depend on the frequency and represents the static dielectric constant of the dielectric. As you can see, it depends on the natural frequency of the oscillations and on the plasma frequency. This result is clear. The frequency in this case is so small that the inertial properties of the charges do not affect and the magnitude of the polarization vector almost reaches its maximum static values. From the point of view of equivalent electrical circuits, the unit volume of such a dielectric is a capacitance whose value is equal to the coefficient facing the derivative in the relation (3.14).

From the formula (3.18) we can draw another obvious conclusion. The tougher the bonds in the molecule, i.e., the higher the natural frequency, the lower the static dielectric constant. At the same time, the greater their density in space, the higher the static dielectric constant. Immediately she recipe for creating dielectrics with maximum dielectric constant. To achieve this, it is necessary to pack the maximum number of molecules with the most soft bonds between the charges inside the molecule in a given volume of space. Very indicative is the case when со » co_{0}. Then

which corresponds to the transition of the dielectric to the conducting state (plasma), because the obtained relation exactly coincides with the case of the plasma. It was this coincidence that prompted Landau to think that there is no difference between plasma scattering and the behavior of dielectrics at very high frequencies [150]. However, it is not. Indeed, at very high frequencies in dielectrics, due to the inertia of the charges, the amplitude of their vibrations is very small and the polarization vector is also small. At the same time, as in plasma, it is always identically equal to zero regardless of the frequency of oscillations. This consideration showed that such a parameter as the kinetic inductance of charges characterizes any oscillatory processes in material media, be it conductors or dielectrics. It has the same fundamental value as the dielectric and magnetic permeability of the medium. Why has not been seen so far, and why was not it given the proper place? This (again) is due to the fact that physicists are accustomed to think in mathematical categories, without much understanding of the essence of physical processes themselves. From relation (3.3) it is seen that in the case of the equality со = co_{0}, the oscillation amplitude is infinity. This means there is resonance at this point. An infinite amplitude of oscillations occurs due to the fact that we did not take into account losses in the resonance system, while its quality factor is equal to infinity. In some approximation, we can assume that significantly below the indicated point we are dealing with a dielectric in which the dielectric constant is equal to its static value. Above this point, we are actually dealing with metal, in which the density of current carriers is equal to the density of atoms or molecules in a dielectric. Now, from an electrodynamic point of view, one can consider the question of why the dielectric prism decomposes polychromatic light into monochromatic components. In order for this to take place, it is necessary to have a frequency dependence of the phase velocity (dispersion) of electromagnetic waves in the environment in question. If we add the first Maxwell equation to relation (3.12)

where p_{0} is the magnetic permeability of the vacuum, then from the relations (3.20) and (3.21) it is easy to find the wave equation

Given that

where *c* is the speed of light, no one will doubt that during the propagation of electromagnetic waves in a dielectric their frequency dispersion will be observed. But this dispersion will not be connected with the fact that such a material parameter as dielectric constant depends on the frequency. In the formation of such a dispersion, three frequency-independent physical quantities will take part at once, namely: the intrinsic resonance frequency of the molecules, the plasma frequency of the charges, if they are considered free, and the dielectric constant of the vacuum.

In accordance with the concept of a self-consistent field [151], there is such a distribution of the electric field in the system of interacting charged particles that creates a particle distribution that in turn excites this field. A salt solution can be considered as a system of interaction between acting cations, anions, positively and negatively polarized water molecules. Of course, there are also unpolarized water molecules in the solution, but their distribution (to a first approximation) does not affect the distribution of charged particles.

To find a self-consistent field, we still use the Poisson equation

and the Boltzmann distribution

where *n _{k}* is the concentration of particles with charge number

*Z*at a point with potential (p. For electrons, for example, Z = -1. But there are no free electrons in the solution. For cations,

_{k}*Z = m,*and for anions, Z =

*-in,*where ?/; is the valency of the metal whose salt is dissolved;

*n*in distribution (3.25) is the concentration of particles with a charge number

_{k}*Z*at a point with zero potential equal to the average concentration of these particles over the entire volume of the solution.

_{k}Further, by the index *‘i’* we will denote cations, by the index ‘я’ - anions, *‘p ^{+}’* - polarized and water molecules located around the anions,

*‘p~’*- water molecules around the cations.

Average concentrations *n _{k}* satisfy the quasineutrality condition

which reflects the fact that the solution as a whole (from the outside) is neutral.

The following conditions also apply:

in which it was taken into account that the charge numbers of the cation and anion are equal in absolute origin, but opposite, and the charge numbers of polarized water molecules are determined by what they are associated around: around anions *Z _{p}>* 0, and around cations

*Z <*0.

Thus,

The following relationships also occur:
where *N* is the number of water molecules associated around one

*о*

anion; *N* is the number of water molecules associated around one cation; Z is the valency of a metal whose salt is dissolved; *n _{m}* is the concentration of salt molecules in solution, if we assume that they are not dissociated.

The volume charge in this way

where *e* is the electron charge modulus (in the SI system 1.6 • 10~^{19} C). Relation (3.30) is very similar to expression (3.5) with the significant difference that it contains the value of the concentration of particles at the point at which the potential of the self-consistent field is determined.

The Poisson equation is rewritten in the form

Using the expansion of the exponential function in a series and taking into account the relations (3.27) and (3.29), we can obtain

Even if nothing more than secondary hydration takes place, more than 10 water molecules will be concentrated in the hydration shell. Therefore, we can assume that 2 » 1 *IN.* and 2 » 1 *IN*. With this in

*i a*

mind, the Poisson equation takes the form

Solution (3.33) for a spherically symmetric potential distribution around a point charge (cation or anion) has the form

where the screening constant is

Inverse value *I = l/%* is called the screening length, and we can assume that its value determines the value of the radius of the sphere within which polarized molecules of the solvent line up, and allows us to estimate the value of the radius of the solvated cation. The polarized solvent molecules located within the scope of the electric field of the ion shield it.

Thus, the value of the cluster radius (solvated cation or anion) can be estimated using the relation

Table 3.1. Frequencies for Y^{3+}, Ce^{3+}, La^{3+} ions at a temperature of 298 К

Cation |
Solvate shell radius r |
Number of H,0 molecules in solvate shell, xlO |
Cooperative rotational movement of H,0 molecules combined into a solvation shell, relative to an axis passing at a distance |
Cooperative rotational movement of H,0 molecules combined into a solvation shell relative to the centre of inertia, |
Frequency of the rotational motion of the cluster as a whole, kHz |
The frequency of the transition of the vibrational motion to the rotational one, kHz; |

Y |
12,73 |
2.891 |
0,496 |
1.73 |
0.86 |
0.87 |

Ce |
13.86 |
3.732 |
0.266 |
0.93 |
0.46 |
0.47 |

La |
13.84 |
3.716 |
0.268 |
0.94 |
0.46 |
0.47 |

Table 3.1 shows the frequency values corresponding to various components of the rotational-translational motion of cationic aquacomplexes, and the frequency values corresponding to the transition of vibrational movements into rotational ones. In this case, the cluster radii were determined by the relation (3.36) at a temperature of 298 К and a salt concentration in water of 2 g/1.

It can be seen that, in these approximations, manifestations of the effect of the electroinduced selective drift of cationic aquacomplexes should be expected at electric field frequencies not exceeding units of Hz. Moreover, as follows from relation (3.13), the cluster size is inversely proportional to the square square of the concentration of the salt in the water. The values of frequencies, in turn, are inversely proportional to the value of the moment of inertia of the cluster.

The moment of inertia is proportional to the reduced mass of the cluster, i.e., the number of water molecules in the solvation shell is *g = (rjr)* and the quadrature radius of the cluster is *r ^{2}v* It turns out that the moment of inertia

*I ~ r~**and the values of the excitation frequencies of the various components v ~ r

_{v}^{-5}.

Thus, the frequencies v ~ *n ^{2}m^{5}.*

It should be expected that with an increase in the salt concentration by a factor of 3-5, the values of the excitation frequencies of the various components of motion, determined in the approximation of existence of self-consistent field in the solution, increase 15-60 times.

The performed experiments prove the possibility of using the previously discovered phenomenon of electroinduced selective drift of solvated ions in salt solutions under the action of the asymmetric electric field for organizing the technological process of enrichment of solutions for the target metal. The experimental results confirm the previously obtained theoretical positions, according to which, at high salt concentrations (of the order of 0.1 g//), the effect induced by an external electric field of selective drift of solvated ions in an aqueous solution is excited at relatively low frequencies. In this case, the frequency interval in which the effect manifests itself at high concentrations is located between the interval characteristic of low salt concentrations [87] and the interval whose boundaries are determined using the radii of solvated ions calculated

by the relation (3.36). It remains to be assumed that the motion of solvated ions, excited by the action of an external asymmetric electric field, is more complex or that the clusters are formed not by one ion and solvent molecules associated with it, but by several ions.