The mathematical description of the electrosynthesis of composites of oxy-hydroxycompounds cobalt with polypyrrole overooxidazed

The electrosynthesis of the composite with of the overoxidized polypyrrole with cobalt oxy-hydroxide in strongly acidic media has been described mathematically, using linear stability theory and bifurcation analysis. The steadystates stability conditions and oscillatory and monotonic instability requirements have been described too. The system ́s behavior was compared with behavior of other systems with overoxidation, electropolymerization of heterocyclic compounds and electrosynthesis of the cobalt oxy-hydroxides.


Introduction
Electrically conducting or conducting polymers (CP) occupy their place among the most studied materials during the last 50 years [1][2][3][4], due to their ability to combine the properties of plastics (flexibility and corrosion resistance) with metallic conductivity, so they are called synthetic metals. Another advantage that they possess is the ease of modification, giving them a wide and rich range of applications, from anticorrosive coatings to sensors and biosensors [5][6][7][8][9][10][11][12].
On the other hand cobalt is one of the most commonly used metals [13][14]. Its oxy-hydroxycompounds, in the form of films, can be used as anticorrosion coatings, as well as the surfaces for photoelectrocatalysis, due to to their intensively black color [15][16], therefore, their composites with conducting polymers should also exhibit interesting properties.
However, the oxidation of compounds of bivalent cobalt to trivalent occurs at relatively high potentials (from +1.6 V), № 1 | 2016 Chimica Techno Acta in which some conducting polymers, such as polypyrrole [17][18], undergo a process of so-called "overoxidation", known as reconfiguration of the conjugated system in the polymer according to the following mechanism.
So far, such phenomena has gained only phenomenological interpretation, which, though it may proceed from a logical conviction, has no solid theoretical justification, that can be given only by analyzing mathematical models, can adequately describe the process, which will be done in this work.

System and model
For the mathematical description of potentiostatic electrosynthesis of the composite of the overoxidized polypyrrole with CoO(OH), from neutral polypyrrole in strongly acidic media without fluorine ions, we introduce three variables: cconcentration of ions of bivalent cobalt in the surface layer; θ -degree of surface filling overoxidized polypyrrole; h is the concentration of protons in the surface layer.
To simplify the model, we suppose that the reactor is intensively stirring, and the supporting electrolyte is in excess, allowing neglect the influence of convective and migration flows. The thickness of the surface layer, the concentration profile of cobalt salts and protons, assumed to be linear, is constant and equal to δ.
The protons diffuse in surface layer by their diffusion and are formed as a result of the process of overoxidation, their concentration in the surface layer decreases due to proton attack of overoxidative polypyrrole. Thus, considering the first law and Fick coefficient modeling related to electric double layer (EDL), the balance equation of the concentration of protons will look like: where h 0 is the concentration of protons in the bulk solution, r o is the speed of the process of overoxidation, r a is the rate of proton attack of overoxidative polymer. The conventional conductive polymer, occupying in the beginning of the reaction the entire electrode, overoxidizes. Thus, overoxidated polymer interacts with the cobalt salt, forming a composite with the product of its oxidation, and is attacked by protons. Thus, the balance equation for the concentration of the overoxidative polymer can be written as (1)(2) The mathematical description of the electrosynthesis of composites of oxy-hydroxycompounds cobalt with polypyrrole overooxidazed In which r c is the reaction rate of the composite formation, and Г is the polymer maximum surface concentration.
The cobalt salt diffuses to the anode, and then oxidized in the overoxidative polypyrrole with the formation of the composite and the overoxidation product. Thus, the balance equation of its concentration in the subsurface layer can be written as: being C 0 is the concentration of the cobalt salt in the solution bulk. The rate of the correspondent reactions can be expressed as: where the k parameters represent rate constants of respective reactions, parameters n is the number of electrons transferred in electrochemical processes, φ 0 is the potential drop of the EDL, in comparison with the potential of zero charge F = N A · e is the number of Faraday. Differential equations (1-3) resemble the systems of equations, described for the case of polypyrrole overoxidation in a strongly acidic media [33], and for the case of the electrosynthesis of oxyhydroxycompounds of cobalt [34], so it is possible to confirm the third theorem of similarity. However, in order to prove the behavior of such systems, it is necessary to compare the investigation of the mentioned systems with the system of equations (1-3), as will be illustrated below.

Results and discussion
In order to research the behavior of the system with potentiostatic electrosynthesis of composite of CoO(OH) -the overoxidative polypyrrole in strongly acidic media, it is necessary to investigate the system of differential equations (1-3), taking in account the algebraic relations (4-6), using linear stability theory. Functional Jacobi matrix, the elements of which are computed for a stationary state is represented in the form: where: The Oscillatory instability in this system is possible, because the necessary conditions of Hopf bifurcation may be satisfied. They are performed in the case of the main diagonal of the Jacobian positive elements: In this system there are three positive elements: in case of strong effects of the process of overoxidation on DEL, leading to the autocatalytic filling of the electrode surface with the overoxidated polypyrrole. This can happen if the polypyrrole derivative is a strong reducing agent and thus can form fluctuations because of the possibility of autocatalytic formation of protons in the process of overoxidation of polypyrrole. This factor was considered as a factor self-oscillating behavior of the overoxidative polypyrrole in [33,[35][36], and this system it also presents.
n the case of strong influences on the DEC process of oxidation of cobalt salts to hydroxy-hydroxycobalamine, which, incidentally, has already been described in [15,16,34].
In this case because of the greater possibility of positive feedback, the probability of self-oscillations in this system will be higher than in case of simple potentio-static overoxidation of polypyrrole or the anodic oxidation of cobalt salts.
The stability of the stationary state we analyzed according to the criterion of Routh-Hurwitz. To avoid cumbersome expressions, we introduce new parameters so as the determinant of the Jacobian matrix can be written as: Using the condition Det J < 0, which can be derived from the criterion of Routh-Hurwitz, we obtain the condition of stability of stationary state in the form of inequalities that can be resolved relative to the diffusion coefficient of cobalt salt κ 2 : Or: Thus, the stability of the stationary state, in this case, is supported by the more rapid diffusion of cobalt salts than protons and by high oxidation rate, which shifts the right part of inequality towards more negative values (in this case, the left part of the inequality is positive). So, at high concentrations of cobalt salts, the pH-dependence of stability of the stationary state decreases drastically. Thus, for an optimal synthesis of the composite, it is necessary to use a concentration of cobalt salts with a small margin relative to the concentration of pyrrole and pH, but the composite will contain a quite significant proportion of the oxy-hydroxy cobalt compound.
The pH-decreasing leads to an increase in the value of the fraction in the right part of the inequality and, at relatively low concentrations of cobalt salts, also reduces the stabilization trend of a stationary state, even including the system leave from the stationary state, which is displayed at the point of a monotonic instability, occurring in the condition of: represented by a N-shaped part of the voltamperogram. In a common system with potentiostatic overoxidation of polypyrrole, contrarily to this system, it is impossible. DEL factors also strongly affect the stability of the stationary state in this system. For example, in case of strong effects of electrochemical oxidation of cobalt salts on the structure of the DEC, the parameter Λ will have negative values which will lead to the exit of the stationary state of the field of stability. This factor is also present in other similar systems.
The presence of fluoride ions. As known, oxy-hydroxycompound of cobalt is unstable in the presence of fluoride ions, and dissolves with the formation of the complex [CoF 6 ] 3-, according to the total equation: In this case, the model isn´t capable to describe the behavior of the composite yet. The stability of the composite in the presence of fluoride ions will be analyzed in one of our next works.

Conclusions
1. The steady-state stability for this system is maintained by relatively high pH and a high concentration of cobalt ions in the surface layer and common to all such systems the absence of strong effects of electrochemical processes on DEL.
2. The oscillatory behavior of this system is possible, being caused not only by the autocatalytic protons´ formation, as for the general case of overoxidation of the conducting polymers, but the influence on the electrochemical reactions of DEC -overoxidation and oxidation of cobalt salts.
A monotonic instability for this system is realized at relatively low pH and the strong effects of electrochemical reactions on the DEC.