Mathematical description of «polythiophene paradox» for potentiostatic electropolymerization of electrochemically modified thiophenes

The phenomenon of "polythiophene paradox" for insoluble polythiophenes based on electrochemically modified monomers has been described theoreti-cally. The corresponding mathematical model was examined using linear theory of stability and bifurcation analysis. Stability conditions of steady state, monotonic and oscillatory instabilities have also been obtained.

Polythiophene (PT) is one of the most used conductive polymers (PP) of heterocyclic compounds, since the potential polymerization of its monomer is lower than that of unsubstituted benzene, and its stability in different environments is higher than that of pyrrole and furan. [1][2][3][4][5][6][7][8]. It is also worth mentioning that PT was the first conducting polymer, commercially used in electrostatic brushes produced by the company "Xerox".
In the synthesis of the PP the choice of the monomer is guided by the properties of the resulting polymer and sometimes it is more convenient to obtain monomers by electrochemical methods than chemically [9][10][11][12]. Moreover, some electrochemical transformations of organic compounds do not have the purely chemical analogies.
PT can be obtained either chemically or electrochemically and the latter method has some advantages over the first -for example, the best conductivity of the polymer and coplanarity between the monomer fragments in it. However, the electrochemically synthesized polythiophene may undergo partial overoxidation due to the fact that its potential of overoxidation may be equal or lower than the potential of polymerization of the corresponding monomer. This phenomenon discovered in 1989 was named "polythiophene paradox" and is manifested in the appearance of oxygen-containing functional groups and the decrease in conductivity in the polymer, due to changes in the configu-ration of the conjugated system [13][14]. Further researches showed that the "polythiophene paradox" is also possible for some of polypyrroles [15][16][17], making the explanation of this phenomenon is important not only to determine the exact mechanism of electropolymerization of thiophenes, but in general for the synthesis of polymers of five-membered heterocyclic compounds. This phenomenon is manifested among other things and in electrochemical instabilities -fluctuations in current or potential or the multiplicity of the stationary states [15][16][17].
The phenomenon of "polythiophene paradox" has been described by many experimental methods and electrochemical instabilities basically obtained a phenomenological interpretation. The main disadvantage of this explanation is that despite the fact that it can be derived from the point of view of purely logical reflections, it is not based on the solid theoretical basis, which can only be given through development and analysis of mathematical models able to adequately describe the processes in the system.
We have attempted to describe the processes of overoxidation (including "polythiophene paradox") for polymers of monomers present in the solution [18][19][20]. Now the aim of our work is to describe the same phenomenon for the monomers, obtained electrochemically, the study of the behavior of the given process and comparison with the general case "polythiophene paradox".

The system and its model
With the purpose of the mathematical description of potentiostatic (U > U mod > > U pol > U ov ) electropolymerization of thiophene, electrochemically modified with the help of the substances present in solution in excess, we introduce two variables: -Θ 1 -the degree of coverage of the electrode by modified monomer; -Θ 2 -the degree of coverage of the electrode by nonoveroxidation polythiophene.
To simplify the model we assume that non-modified monomer at the initial moment of the reaction covers the surface of the anode completely (Θ 0 t = 0 = 1). The modified monomer is obtained by electrochemical oxidation of the starting monomer and then electropolymerizes. In turn, the polymer overoxidizes at the same time with their own synthesis. Thus, the balance equations for their concentrations will be written as: where r 1, r 2 and r 3 are speeds of modification, polymerization and overoxidation which can be described as: (3)(4)(5) where k 1 , k 2 , k 3 are the rate constants of the corresponding reactions, z 1 , z 2 , z 3 is the number of electrons transferred in each of them, φ0 is the potential jump relative to the zero charge potential, ξ is the reaction order of electropolymerization on the monomer, f (θ 2 ) is a function of the possible autocatalytic growth of the polymer chain, rising to the order of reaction with respect to the polymer. Autocatalytic growth of the chain occurs due to the acceleration of electrochemical reaction at transition from monomer through oligomers to the polymer.
Mathematically equations (1-2) remind of equations (2-3) of general model for polythiophene paradox for the case of the presence of monomer in solution [19] that shows the similarity of the systems. However, due to the fact that in this case the process occurs completely on the surface and also in view of the presence of the greater number of electrochemical steps the system behavior will be slightly different from the general case that will be described and discussed below.

Results and discussion
To analyze the behavior of the system with "polythiophene paradox" under electropolymerization of electrochemically generated polymers we analyze the system of differential equations (1-2) using linear stability theory. Functional Jacobian matrix whose elements are calculated for the stationary state can be shown as: a a a a 11 12 a ar r r f a r r Here a x and b x are parameters describing the effect of the electrochemical reactions in the electric double layer (EDL).
The necessary conditions for the oscillatory behavior described by Andronov-Hopf bifurcation are Tr J = 0, Det J > 0 (simultaneously); where Tr J = a 11 + a 22 is the trace of the Jacobian matrix and Det J is its determinant. Since the second condition is satisfied in most cases, the first one is the main at calculation which for this system are written as: This condition can be satisfied only in case of presence in the main diagonal of the Jacobian matrix of positive elements, describing the positive feedback. You can see that the elements a x r x and b x r x , where x is the phase number can be positive (or negative under the sign minus), depending on the impact of the electrochemical stages of the reaction on the capacity of DES and the strength of oxidising substances as reducing agents. This factor is common to all systems with electropolymerization [18][19][20].
The second factor responsible for the oscillatory behavior can be autocatalytic chain growth, the impact of which is described by the positivity of the elements containing the derivative of the function f that describes this growth. Thus, in this system self-oscillatory behavior is caused by two factorselectrochemical and autocatalytic. There are temporal dissipative structures in this system the existence of which maintained by steady "supply" of the initial monomer and by the excess of the modifying substance in the solution (input entropy) and by the overoxidation of the polymer formed by (output entropy). The factor of interaction of particles on the surface of the electrode by adsortion -desorption, which is the reason of self-oscillations in such systems [18][19][20] does not apply here owing to the purely superficial nature of the process.
For two-dimensional systems the conditions of stability of stationary states are described as: Tr J < 0, Det J > 0. Herewith the latter condition is main. For analysis of the determinant of the Jacobian matrix without cumbersome expressions we introduce new variables so that the determinant of a matrix was written as: Opening the straight brackets and solving the inequality Det J>0 relative to R1, we obtain the condition of stability of stationary state for the system in the form: This inequality is satisfied, in the case of the growth of effects of modification of the monomer and overoxidation of the corresponding polymer (increase R 1 and R 3 ) on DES and the fall of the effects of electropolymerization on it (drop R 2 ). Also the stability of the stationary state is determined by the stability of the polymer (X 2 >X 3 ). The process is controlled by adsorption of unmodified monomer as in the similar case of electrooxidation of procarbazine [21][22].
The critical value of the parameter R 1 which is on topological limit of stability of stationary state corresponds to the multiplicity of steady states, the condition of which is: It appears by N-shaped plot of voltamperogramme and explains the equality of the stabilizing and destabilizing influences of the electrochemical processes at DEL.
The reaction in galvanostatic and potentiometric modes is described by system based on the described above, while the variable introduces in the SDR that describes the change in charge density of the anode. The behavior of such system is more complex and will be described in future works.
Effect of pH is an important factor in the behavior of the system, because the process of electropolymerization and the process of overoxidation are highly dependent on pH. This system describes the electrosynthesis of polythiophene at neutral pH. In the case of more acidic pH third variable introduces in the above described DES that describes the behavior of protons. Addition to the above factors it can also be responsible for the appearance of electrochemical instabilities in this system.

Conclusion
From the analysis of the system with polythiophene paradox under electropolymerization of electrosynthesized monomers, we can conclude that: -As for all similar systems with "polythiophene paradox" the temporal dissipative structures present in this system, the existence of which is maintained by steady "supply" of the initial monomer and by an excess of the modifying substance in the solution and by overoxidation of the formed polymer.
-The stability of the stationary state is determined by the distribution of the ef-fects of electrochemical processes on the DEL so that the polymer remained stable. The process is controlled by the adsorption of initial monomer.
-In case of equality of stabilizing and destabilizing influences in DEL, the monotonic instability, manifested in the multiplicity of the stationary states realizes.
-The oscillatory instability in this system can be caused by the action of the factor of influence of electrochemical processes on the DEL and autocatalytic factors.