Quasi-Periodic Noise Oscillations in Oxyhydrates of Rare-Earth Elements

The present paper dwells upon the chemical foundations of formation of a gel nanostructure (by the example of zirconium oxyhydrate) at the time of formation of oxyhydrate polymer chain and develops electrophoretic (diffusional) conceptions related to the phenomenon of self-organization of non-equilibrium oxyhydrate gel. Adduced are the experimental data (measurements of spontaneous current induced by the alterations in capacity of the polarized double electrical layer of the gel) that permit to claim that the mathematical model of the phenomenon is not in contradiction with the experiment. On the ground of the experimental data and calculations it is concluded that the constant magnetic field alters the character of geometrical configuration of chemical interaction of the oxyhydrate gel fragments with each other, which affects the value of nano-current measured in the course of the experiment.


Introduction
Let us contemplate general colloid conceptions first in order to further consider behavior of the ion fluxes in colloidal matter under conditions that are far from equilibrium. All macromolecular colloid structures of oxyhydrates are surrounded with a polarized diffuse double electrical layer [1]. A gel often contains an electrolyte, e.g., a solution of its own salt, and, as it was formulated in [2], the oxyhydrate gel interacts with both that solution of its own salt and the water molecules. The interaction may lead to formation and absorption of a certain number of ions. The double electrical layer is formed because some of the ions are absorbed by the gels' matrix (potential-determining layer), while ions of the opposite sign stay in the intermicelle liquid to form the double electrical layer around the macromolecule.
The macromolecules of an oxyhydrate gel, as it was proven by quantum chemical calculations [2,4], form a great number of varied conformers, and their shapes and structures can change under the effect of the surrounding medium even at room temperature [2]. At that phenomena of destruction and polymerization play not a small role as they also create a variety of conformers. That is why the macromolecular structures are capable of nearly instantaneous realignments at the temperature 298 К as a consequence of low potential barrier.
There are many oxyhydrate forms of zirconium having the shared formula ZrO 2 (Н 2 О) n that can be a monomer link of an oxyhydrate systems' nonequilibrium polymer chain. The detailed analysis of possible links carried out by means of the ZINDO/1, a semi-empirical quantum chemistry method, proved that the dependence of the ΔН h hydration enthalpy on the n hydration degree had an extreme nature, as shown on Fig. 1. This dependence is not contrary to the experimental data. It is a generally accepted belief [4], that it is zirconic acid ZrO(OH) 2 (or ZrO 2 (Н 2 О)), or its hydrated form ZrO(OH) 2 Н 2 О (or ZrO 2 (Н 2 О) 2 ) that makes the basis for the zirconium oxyhydrate polymer structures. The water is valence-bound in those compounds; and it is the coordination sphere that another water molecule is added to. Formation of a polymer chain is an interaction between couples of monomer links. In this case, one should consider three types of interaction. The first type may be produces as follows: 1_1. 2 ZrO 2 (Н 2 О) → (ZrO 2 (Н 2 О)) 2 -105.89

1_2.
ZrO 2 (Н 2 О) 2 + ZrO 2 (Н 2 О) → ZrO 2 (Н 2 О) 2 •ZrO 2 (Н 2 О) -58.75 1_3. 2 ZrO 2 (Н 2 О) 2 → (ZrO 2 (Н 2 О) 2 ) 2 -17.01 1_4. ZrO 2 (Н 2 О) 2 + ZrO 2 (Н 2 О) 3 → ZrO 2 (Н 2 О) 2 •ZrO 2 (Н 2 О) 3 -49.01 The continued growth of the polymer chains goes through a series of processes listed below: In these reactions, the hydration is likely to compete with the growth of the trimers, as it is in dimerization. The hydration is represented by the equation: Enthalpies of the monomers hydrations are given in Fig.1. Enthalpies of the competing dimer chain growth and the cumulative hydration of the monomers and the dimers are listed in Table 1. It is obvious that the trimerization process competes with the hydration reaction while interaction of the (ZrO 2 (Н 2 О) 3 ) 2 with the monomer links from the trihydrate to the hexahydrate (that is, from the ZrO 2 (Н 2 О) 3 to the ZrO 2 (Н 2 О) 6 ). In other cases, it is more energetically advantageous for the dimer and the monomer to be in the solution in the shape of independent hydrated compounds. The joining of the hexa-and tetrahydrate forms of the monomer to the dimer are the most energetically advantageous. Nevertheless, the tetrahydrate and the hexahydrate forms are not among the compounds that are commonly found in the solution.
The joining of the predominant hydrate form -a pentahydrate (reaction 2_4) is the most thermodynamically probable in this case, but it leads to the formation of a metastable (that is, not the most probable) product (ZrO 2 (Н 2 О) 3 ) 2 ⋅ZrO 2 (Н 2 О) 5 .
The hydrate water is located in the inner bindings region, between the zirconium oxide molecules, providing the bindings between the monomer links. While the hydration of the compound (ZrO 2 (Н 2 О) 3 ) 2 ⋅ZrO 2 (Н 2 О) 5 , the water molecule cannot go straight into the inner region, that is why a hydration of such a chain is not equivalent to a simple joining of hexahydrate to a dimer. A complicated hydration like this entails an isomerization including an opening of the inner region where the water's molecule is supposed to go to, and its subsequent closing. This process is too complicated and therefore unlikely to happen.
A more probable process is that including the destruction of the (ZrO 2 (Н 2 О) 3 ) 2 ⋅ZrO 2 (Н 2 О) 5 trimer together with the hydration of the dimer and the pentahydrate form of the polymer. This process only takes 6.60 kcal/mole, a reaction contrary to 2_4. Further on, the freshly formed hexahydrate joins the dimer and emits 28.68 kcal/mole, process 2_5. In this case, the enthalpy of the destruction of (ZrO 2 (Н 2 О) 3 ) 2 ⋅ZrO 2 (Н 2 О) 5 can be regarded as the activation energy for the formation of a more advantageous chain (ZrO 2 (Н 2 О) 3 ) 2 ⋅ZrO 2 (Н 2 О) 6 . The value of the energy barrier (6.60 kcal/mole) is low and is equal to the activation energy of the diffusion.
The high kinetic probability of the pentahydrate zirconium oxide's joining therefore leads to the formation of thermodynamically metastable chains. As the chain grows longer, its instability increases, thus increasing the probability of a relaxation of the metastable state, that is, the destruction of the chain. The chain is expected to break apart as soon as it reaches a certain length. The two resulting fragments are more stable and may resume their growth separately.
Upon disruption of the polymer chains structural water rises to the surface to have a share in formation of the double electrical layer. Clear is the recurrent and oscillatory nature of these phenomena. Thus, the successive growth of the polymer chain fragments and their destruction both contribute to the temporal periodicity of the gel's properties as well as energetically to the pulsating power noise.
Gel as it is has a reticulate structure that tends to change, probably due to the polymerization of gel's fragments. Not all of gel fragments are capable of moving around freely within the gel space. There are formative conditions for osmotic pressure that affects the charged particles.
Therefore, there is a complex system of cluster motions -both linear traveling and oscillatory rotations -as well as motions of the sizeable oxoolic molecules (microelectrophoresis [3][4][5][6][7][8][9]). Meanwhile electrical fields appear spontaneously provided that presumptions of osmotic influence of the nanoclusters' charged particles are valid [3]. Dynamical systems like that evolve due to the polymerization processes going on in the zirconium oxyhydrate, e.g., destruction and oxolation. It is remarkable that geometry of gel (that is shape of the electrochemical cell) considerably, may be totally, determines the value of electric potential difference in the gel system, as the linear velocity of ion flow depends on the regular cross-section of the cell.

Experimental
Zirconium oxyhydrate gels had been synthesized from zirconium oxychloride salts by adding caustic liquor to the solution upon certain conditions, e.g., рН = 9.25; quantity of added zirconium being n = 0.00094 mole.
The current monitoring electrolytic cell consists of a hollow glass tube with round platinum electrodes (R=0.4 cm) attached at either end, and an electrical unit that registers current alterations on the electrodes. The cell was filled with a fresh gel oxyhydrate. The span between the electrodes altered within the range of 1 to 70 mm. The gelfilled cell was factually short-circuited, that is the output resistance was very low. The current formed inside the system was registered by the digital equipment [5,8] at the report rate 5 times per second. The experiment lasted for 6 hours. For the instrument layout see Fig. 2.
The tube filled with the oxyhydrate was placed in a system of annular stationary magnets (the field's intensity in the system of magnets А was Н А = 900 Oe, in the system if magnets В it was Н В = 600 Oe), then the cell was fastened to an oscillatory-type mechanical agitator. The experiment was thermostated (Т = 303 К). All the current-carrying bus bars have been protected from external electromagnetic blasts.

Analysis of the results
It was experimentally revealed (Fig. 3 a, b, c), that there was a spontaneous pulsating current formed between the electrodes. In oxyhydrate gel systems of zirconium oxyhydrate, as it has already been mentioned in [1][2][3][4][5], oscillatory rotary ionic molecular flows are caused by a great number of relatively rare stochastic transformations of the gel macromolecules (conformer transitions at the temperature 298К) that, while remaining almost motionless within their coordinates (their mass centers coordinates that is) inside the cell space (due to their enormous masses), continuously pumped the repeated oscillatory diffused ionic flows into the extended gel space. Those flows were formed by the double electrical layers that surrounded the conformers. This is the meaning of the Arnold diffusion's mechanism in gel systems [9]. Those transitions could be viewed as periodical impacts or perturbations of the gel system under the conditions that are far from equilibrium.
It is common knowledge [11,12] that any dynamical system generally has an irremovable stochasticity region in the phase space in all cases. In fact, those are Langevin's classical ideas [11]. The Hamiltonian function of such an oscillatory rotary ionic molecular colloid chemical system can be represented as: where I, θ -N 0 are for the measurement vectors.
Such a system has N = N 0 + 1/2 degrees of freedom, 1/2 of them belongs to the t variable (time). Generally, some extent of the H 0 (I).
Hamiltonian has separatrixes, that is why a εV perturbation, destroying them, creates stochastic layers at any ε . The stochastic trajectories are further on defined by the weak chaos topology of the colloid chemical system in the phase space.
In practice, the hyperbolic growth of the polarization current is registered in the samples placed into the magnet that has the highest intensity ( Table 2). Zirconium hydroxide samples placed into a lower intensity magnet demonstrate the same hyperbolic current increase and surge, but in the latter case, the polarization current starts to decrease rapidly as soon as it reaches a certain maximum. The current sign changes to the opposite, and there is a charge exchange between the electrodes or in the gel phase double electric layer (Fig. 3c). The charge exchange effect is much stronger in the double electric layer of the gel that was not exposed to the magnetic effect.
The quantitative evaluation of a current surge in the zirconium gel oxyhydrate required the calculation of the ΔI value, microamperes, the upper and the lower currents surge values, as shown on the diagrams and Table. 2. Thus, the range of the ΔI value grows in microamperes as the intensity of the magnetic field increases. The phenomenon is probably caused by the orientation effect of the mesophase-like regions of the gel exposed to a magnetic field, which leads to a one-way ejection of charged particles connected to the double electrical layer and producing the current. Therefore, we see an ionic surge effect. Alternatively, we see extended bound dipoles of the gel fragments that have a considerable polarization in the double electric layer.
The extended current surges are shown on Fig.4. Let us attempt to analytically picture the phenomenon of current expulsions by making the algorithmic model of the given problem, meanwhile without the magnetic field.  Fig.3. Alterations in the gels polarization current in zirconium hydroxide that were synthesized in three parallels, where a) the gels were studied in a magnetic field with the intensity 900 Oe, b) in a magnetic field with the intensity 600 Oe, and c) with the magnetic field off; the sample was synthesized at the рН = 9.25; length of the rube L = 70 mm; the quantity of the zirconium oxyhydrate in the tube n = 0.00094 moles.

Mathematical simulation
Let there be an electrolyte, e.g., a solution of the gel's own salt, added to a fresh gel. Opposite sign ions will be subsiding on large fragments of colloid particles (or rather, they will be absorbed by them), canceling each other's charges. The number of the freely traveling ions will thus be limited, or the effect of the electrical field on the movement of the charged particles will not be noticeable. What we are interested in is the very possibility of the phenomenon, not a precise quantitative result, nor the measurements of the parameters. Therefore, we will be considering all the parameters related to the movement of the ions as either constant or as linear (linearized). That is why we are going to confine our discourse to a simple linear model. As it will be proven below, this approach makes some forecasting sense.
According to [8,11], the I + and the Icharge flows in the absence of temperature gradients and chemical potentials are defined by the relations: 11 12 11 12 where ϕ is for the electrical potential, P is for the osmotic pressure that forms due to the reticulate structure of the gel, L 11 is for the conductivity coefficient for the flow of freshly charged '+' particles, L 12 is for the electroosmotic flow of positively charged particles, and 11 % L and 12 % L are the respective negatively charged ions' coefficients.
The I + and the Icurrents can be of various natures, and it can lead to an accumulation of a charge.
The continuity equation of the concentrations with a separated diffusion looks like: where + r j and − r j are for the ionic currents' density with the respective signs, DΔ n is for the contribution made by the diffusion, D is for the diffusion coefficient, Δ is for the Laplace operator, and n ± is for the concentration of the positive and negative particles.
It is remarkable that the J's and the I's dimensions are not the same. So let us divide the I + and the Iby the sections' area that we will consider to be constant, and, from (3) we will derive the relations: where all the coefficients are in relation to the area of the cross section. Now let us substitute the currents density expressions (4) to the continuity equations (3): 11 12 l l P D n t n l l P D n t (5) The osmotic pressure P is defined by the formula P = inRT, where i is a number, n is for the concentration of the substance that forms the osmotic pressure, R = k B N A is for the universal gas constant, and T is for the temperature. Only the concentration changes in the course of the experiment, consequently, we can consider the pressure proportional to the remainder of the ions' concentrations, P = α (n + -n -).
( ) ( ) 11 12 21 22 l l n n D n t n l l n n D n t (6) where the constant factor α is present in the new n l n n l n n D n t n l n n l n n D n t (7) Let us now subtract the equations of the system (2.3.6) from each other and introduce notation u = n + -n -. The result will be the expression 11 21 12 22 Adding the boundary conditions and the initial conditions to this equation, we will derive a closed boundary-value problem.
We will regard the ionic flow as moving towards the electrode and forming an instantaneous potentials difference on it (within the closed circuit of first-class conductors, Fig.2).
Consequently, we are to stipulate the boundary convection condition on the region's boundary -a third-class condition: is for a number, for we believe that the electrode is located at the 0 x = . The whole region where we are going to solve the problem will be regarded as an unlimited one, x [0; + ∞]. It is convenient to ascribe such geometrical dimensions to a limited region because all the phenomena listed in the introduction and that we are able to register are onlyconfined to a narrow laminar layer adjacent to the electrode that has the thickness 10 -5 m. The geometrical dimensions of the double electrical layer region of the macromolecules is approximately 10 -7 m [8,9]), that is the dimensions of the double electric layer are small in comparison with the infinitely large near-wall laminar region. Talking of the initial condition, it is something that we already know and it is represented by the function u| t = 0 = u 0 (x). This is how the problem looks now: Now let us accept D -b = B. It deserves a mention that in the (8), the difference D -b can be either positive or negative. The positive difference means that the diffusion of the ions suppresses the charges separation due to the osmotic pressure. The negative ones means that the osmotic pressure is stronger than the diffusion, and if there is even a slightest difference between the charges that is different from zero, then that difference will be only increasing.
We will confine our studies to an instance whenD -b > 0. The problem presupposes that the a number is positive. The problem (8) has been studied inside and out. The uniqueness theorem and the existence theorem, and the solutions can be found in [10].
The flow is the concentrations difference gradient: At the region's boundary To begin with, let us consider the modeled problem: let the initial deflection be u 0 =Aδ (x-x 0 ), where x 0 > 0. Then the whole problem (8) will look like: We will be seeking a solution that looks like: u = ν exp (-at), where ν is for a novelty unknown function. The solution (9) for the ν function can be represented as [10]: We can only register currents when the x = 0, consequently, in order to draw a comparison with the experiment, we are to find Calculating the integral, we will derive: ( ) Let us note that the second component rapidly decreases, as there is a growth in time. The component can be neglected, at least to make simple and rough evaluations easier, seeing as the experiment's time is normally not too short.
In terms of the experimental curves diagrams (Fig. 3, 4) that represent the dependences of the registered currents or the ionic cluster flow on time and considering it as only containing the "aftereffects" of the charge fluctuation, we can calculate the a and the coefficients.
Let us represent them as 2 0 1 x K 4B = and K 2 = -a respectively, and resort to the least square technique to fit them. The K 2 coefficient can hardly be negative, for it looks like a case when the charges traveling in an electric field (that is moving towards their compensation) start to separate. The positive K 1 coefficient is for the case when the charges separate, affected by the osmotic pressure.
An instance of the K 1 , K 2 coefficients computations by means of the least square technique is represented on Fig. 5.
The second one corresponds to the formula ( ) Both the calculations above and the model itself are far from being precise, yet they provide correct evaluations of, e.g., the diffusion coefficients; as it was stated in [1,2], their value was x 0 ≈10 -5 cm. Meanwhile, the ions' diffusion ceofficient in the gel is evaluated to be the D ≈ 10 -8 cm2/sec value, which is not contrary to the data provided by the scholars [3].
There are several types of charge separations. In the case in question, we can seek the u 1 that looks like.
( ) The dependence calculations for 2 N = were made too. The precision we attained in this case was to a certain degree higher, but still we failed to reach any noticeable improvements. The evaluated parameters' values were little or no different from the those on Fig. 5 (sec).
The calculations above only meet a certain averaged asymptotic non-oscillatory instance's conditions, Fig.5. The real experimental picture showed a whole range of currents surges in time, that could not have been deduced from the statements above and required a separate description different from the those calculations. It was a physico-chemical phenomenon of a different nature.
When moving towards an electrode, the charges flow sort of "subsides" on graphite or platinum electrodes (or rather, it gets absorbed by them). Then the flow creates a potentials difference, that is, an electrical current, and "trickles back" to the dispersion. The potentials difference is "dissolved," in other words, it simply disappears. This is how oscillatory motions in the ionic flows of the double electric layer appear, forming the general picture of the ion currents surges.
Considering this phenomenon, the problem's representation will be more convenient (9) if put down in a way as follows: where ε is for a small quantity, an oscillations parameter, ω is for the frequency of the nearelectrode oscillations, and ϕ is for their phase.
We will not go beyond the simplest description of the oscillations -by means of a harmonic function -though the phenomenon deserves a separate and a more detailed consideration.
If λ > ε, then, using the approach [7,10], it will be easy to prove that if the solution (10) exists, then it is the unique one. Let us try to develop a solution, and thus prove that the solution (10) does exist. To do so (10), we carry out a Laplace transform for the positive part of the OX axis, solve the derived equation in relation to the t, and then invert the Laplace transform. In the derived integral relation, we will go over to the x → 0 limit. As a result, we will get the following integral equation: The solution to this equation at the λ>ε does exist, and it is the unique one [13]. The solution for the case ε = 0 can be found in [10]. When the ε values are small, a solution can be developed by means of iterations. Taking   The data obtained correspond to the results of an immense number of experiments that we have carried out. The evident modeled approximation of the consideration requires a more detailed consideration in terms of cluster nanophenomena.

Conclusions
For the first time the pulsating character of the current surges in the zirconium oxyhydrate is described as determined by conformational rearrangements of the oxyhydrate matrix. Consequence of the conformational-peptizating instability is the alternation of polarization of the double electrical layer of the gel particles, which manifests itself in the form of periodical electrical current surges. Mathematical model reckons ejective periodical disturbances of the gel system, which determines extended current expulsions а peak character. The vastest range of the ΔI values is observed for those gels that were exposed to the most intensive magnetic field. The phenomenon is probably caused by the orientation effect of the mesophase-like regions of the gel exposed to a magnetic field, which leads to a one-way ejection of charged particles connected to the double electrical layer and producing the current.