# Action of an external electric field a free and associated charges in solution

Tension distribution of the electric field E(r) and bulk density distribution charge p(r) are related by the Poisson equation [4]

If the salt the molecules of which are dissociated is dissolved in a polar dielectric fluid, then the electric field in the volume of the liquid is formed by free charge carriers - ions, forming during dissociation, with the distribution of volumetric charge density (r). The intensity distribution of this field is also described by the Poisson equation

Molecules of a dielectric fluid (solvent) are polarized in the electric field of free charge carriers. In this case, the condition [5] is fulfilled

where **P(r) **is the distribution of polarization density. Moreover, the condition is fulfilled regardless of how E(r) and **P(r) **are related.

The application of an external electric field to the solution leads to a violation of its local electroneutrality, that is, to the appearance of a nonzero bulk charge density. These charges are connected and arise solely due to heterogeneous polarization of dielectric fluid molecules. Distribution of the volume density of bound charges p_{6}(r) is determined by the relation

In the general case, the space charge consists of two components - the density of free charges and the density of bound charges, which arise in the presence of an inhomogeneous polarization of the sample **P(r).**

For simplicity, we restrict ourselves to considering the onedimensional case (when the solution is in the form of a plane parallel layer, whose transverse dimensions are much larger than its thickness). Polarization density is connected with the strength of the external electric field E , formed by the the plane-parallel electrodes, isolated from the solution, by the ratio

where E is the field strength inside the solution layer located between the electr.

The field strength inside the solution is distributed in a certain way, that is, it depends on the coordinate. Moreover, the averaged value of tension in the volume of solution *(E)* is less than the field between the electrodes *E _{0}.* The ratio

*EJE*can be estimated using equation (1.1) and the relationship between the polarization density and the field strength causing polarization of the dielectric liquids:

where e is the dielectric permittivity of the solvent.

If the electric field is formed by a charge with a bulk density p, and the polarization is caused by the electric field of this charge, then equation (1.1) can be rewritten in the form

This equation shows that at the same charge density p the

**f Je-ni**

polarization reduces the electric field strength 1 + 3 *—-J* times.

For large dielectric permeability (e > 10) the field strength in the volume of dielectric liquid *(£)* is approximately 4 times less the tension between the electrodes *E _{0}.* When the dielectric constant is close to 1, the field strength in the volume of the dielectric fluid

*(£)*is approximately 2 times less between the electrodes

*£*

_{0}.If we talk about the distribution of field strength, then from the much higher considerations it follows that the tension increases in the volume of the plane layer of the solution from the value of *(£) *to the value of *£ _{0}* when approaching the electrodes that are isolated from the solution. The macroscopic value of the density of bound charges p

_{b}(r) will be nonzero only in these near the electrode areas of the solution, since at a distance from the electrode the field strength will depend weakly on the coordinate and have the value of

*(£) ~*const. Moreover, the polarization density P also will not noticeably change, and divP(r) will be close to zero. Indeed, the macroscopic value of the density of bound charges p

_{fc}(r) due to equality (1.2), as it moves away from the electrodes, it also tends zero.

Taking into account the above reasoning, we can write the equation relating the electric field strengths in the volume of the solution with the density of free charges in the salt solution in a polar dielectric fluid:

For the solution layer situated between the electrodes plane and isolated from the solution, the equation is in the form:

or, which is the same,

# Equation of oscillations of an ion in relation to solvent molecules

We differentiate the resulting equation in time:

The right-hand side of expression (1.3), by virtue of continuity, be expressed in terms of current density *j* of free charge carriers

For a single ion in the solution, the spatial intensity distribution of the electric field created by this ion can be set as a function

where A- is a constant, *q _{ion}* is the electric charge of an ion, ,r

_{0}(/) is the coordinate of the ion, which changes in time due to the movement of the ion in an external electric field. Then

*3x (t)*

y. (?) = ' ^{ow} is the speed of the directed motion of ions. Bearing *8t*

in mind that the ion current density / = *nq* v. , where *n* is ion

*J J* ion ion ion’

density, we write equation (1.3) for the coordinate of a single ion *xft)*

in a salt solution in a polar dielectric liquid with a high dielectric constant:

Let us analyze the last equation. The numerical value of the constant *к* (proportionality coefficient in Coulomb's law) in the system of the physical quantities SI is *к* ~ 9 ■ 10^{9}. We choose a fixed point in the solution volume .r = *a,* at which the gradient of the field strength is known and its time dependence *cE _{0}{a)/8x = f(t) *In experiments with the excitation of selective drift of the solvation ions [6]

*f[t) ~ A*(I + 2 sin cot - sin 2cot), the constant is

*A*~ 4 ■ 10

^{6 }V nr

^{2}. The amplitude of oscillations of the ion Д(г) in the solution under the action of an external electric field

*(a - x*(t)) is known not to exceed 10~

^{3}m, therefore, the first term in the parentheses in the brackets of the left parts of the equation can be neglected. The equation for the oscillations of the ion Д(г) is written as:

The solution of equation (1.4) has the form [7]

where the constant *C* is determined from the condition Д(? = 0) = *a.*

Oscillation frequency v = со/2я coincides with the frequency of the external electric field and represents the frequency of the oscillations of the ion relative to the solvate shell. If this frequency coincides with its own frequency of the solvated ion, i.e., the ion-solvate shell

Fig. 1.1. **Oscillations of a calcium ion inside a sphere formed by molecules of water in the solvate shell, under the action of an external electric field.**

system, then we should expect the transition of the vibrational motion into the translational drift of the solvated ion [3]. The intensity of the translational motion will be proportional to the difference in the amplitudes of the deviation of the ion from the initial position during the positive and negative half-periods of the oscillations of the external electric field.

The results of using the obtained equation of oscillations of the central ion located inside the solvation shell are shown in Figs. 1.1 and 1.2.

The obtained equation of ion vibrations in solution for the case of ‘asymmetric electric field’ (1.5), when the amplitude and durationof the half-periods are different, shows that the ion displacement relative to the initial position for one period has an alternating character. Moreover, the absolute values of the displacements for the ‘positive’ *A _{+}* (0 < 772) and ‘negative’ Д_ (772

*< t < T)*half-periods are also different. The difference between them

*I*|Д

_{d}=_{+}| - | Д_| represents a segment of the ion drift trajectory in the direction of one of their electrodes, which form a field in a flat layer of a solution. A situation similar to the ‘two steps forward, one step backward’ pattern is realized and, as a result, the ion drifts directionally in one direction. Parameters of the trajectory formed by successive displacements of the ion in opposite directions using the resulting

Fig. 1.2. **The calculated trajectory of calcium ion in an aqueous solution under the action of external electric field.**

equation, are described only at a qualitative level. The reason is that in the formulation of the problem, the friction of the solvated ion is not taken into account with the surrounding solvent molecules. This leads to the result that the the amplitudes of displacements of the solvated ion are much lower than the amplitudes of displacements of the ion inside the solvation shell.

The application of the obtained equation is well conditioned for calculating the parameters of ion vibrations inside the solvation shell. The value of the constant *A* included in the solution is determined to a greater extent not but by the amplitude of the external electric field, but by its distribution in the solvent volume in the interelectrode gap, i.e. by the properties of a solvent as a polar dielectric. The greaterthe gradient of tension in the interelectrode gap, the more significant value is *A.*

# The consideration of friction with a solvated ion with the molecules surrounding the solvents

The resulting effect caused by the friction of the ion oscillating in the external field with the surrounding molecules can be quantified. To do this, we write the equation of motion for an ion that is affected by an external force *F,* which varies cosine in time, as well as a restoring force and force resistance (friction):

where *m* is the mass of the ion; у is the frequency of change of the external electric field creating the force; the second term takes into account the friction force proportional to the velocity; the third term takes into account the return force proportional to the displacement from the equilibrium position. The friction force, acting on a solvated ion, can be determined according to the Stokes law

where *r* is the radius of the solvated ion, q is the dynamic viscosity solvent. Thus, *X = 3nrr/m.* The coj; value is defined as co^ = *k/m, *where A- is a coefficient similar to the coefficient of rigidity for a spring, if we consider a cluster formed by a central ion and solvent molecules associated around it, as a hollow spherical shell with inner radius *R _{v}* outer radius 7?, and mass

*M.*In this case, the hollow spherical shell can rotate about an axis passing through its geometric centre. The shell is connected to this elastic spiral axis spring by stiffness

*k,*which provides a stable position of the shell. In this model, the cluster is considered as a torsion spherical pendulum. The stiffness of the spring is determined by the binding energy of the central ion with solvent molecules located on some distance from it. These molecules form the first radius solvation of

*R ,*and their number is equal to the coordination number of the ion in a given solvent.

The natural frequency of such a pendulum is determined according to the formula

where *J* is the moment of inertia of the pendulum, the value of which is determined according to the formulas

Fig. 1.3. **The occurrence of a moment of force when deviating from the equilibrium position of one of the solvated solvent molecules.**

The stiffness of the coil spring *к* is determined from the condition of equality of two works *A _{{} = A*,, where

*A*is the work that is done by the moment of the forces arising from a deviation from the equilibrium position of one of solvent molecules solvated and oriented in the ion field,

*A*is the work that the returning force

_{2}*F,*will perform, arising from spring deformation Д/. Deviation from the equilibrium position of one of the solvated molecules is schematically depicted in Fig. 1.3.

With a coordination number of 6 in a centrally symmetric field the cations are oriented and are within the first solvation radius of 6 solvent molecules. Solvation shell deformation leads to a deviation of one of the solvent molecules by an angle cp. In this case, the arising moment of forces will be *M = p ■ (E ■* sin cp), where *p* is the intrinsic dipole moment of the solvent molecule, *E* is the value of the electric field strength formed by the central ion, within the first radius of solvation. The work committed by the moment of forces returning the solvent molecule to the equilibrium position *A _{x} = M* • ф. The work done by the returning force

*F,*is d, =

*F • d*• ф, where

*d*is the length of the dipole formed by the solvent molecule. Moreover,

*F*• ф).

_{2}= к ■ Al = к ■ (dThe formula for calculating the stiffness value of a spiral spring in the SI units for the case when the solvent is water will be written as

The solution of equation (1.6) gives an expression for the amplitude of the forced ion vibrations

In the absence of friction of the solvated ion on the solvent molecule the amplitude of the forced oscillations of the ion will determine the expression

Now one can determine the coefficient of decrease in the amplitude of ion vibrations due to friction:

The equation for the oscillations of the solvated ion A*(t)* with taking into account friction with the solvent molecules is written in the form

where *g* is the coefficient of proportionality in the Coulomb law in the SI system of physical quantities (see explanations for expression (1.5)). For the case when the solvent is water, friction on the solvent molecule of the solvated ion, the radius of the solvation shell of which is 0.2 pm, leads to a decrease in the amplitude of oscillations 1400 times, i.e. *k _{f}~* 0.71 • 10~

^{3}.