The bistable, but non-oscillatory electrochemical system with the negative differential resistance (NDR) generated by the potential-dependent convection at the liquid electrode|solution interface.
Marek Orlik, Maciej T. Gorzkowski, Rafal Jurczakowski
Convection as a non-linear dynamical phenomenon is still a subject of intensive experimental and theoretical studies in both physics and chemistry. Remarkable universalities (as e.g. hexagonal cells) were found in convective motions in various systems, with different driving forces operating.
In electrochemical systems the convection can be caused either by density gradients, or by electrohydrodynamic forces, or by the surface tension gradients appearing at the surfaces of liquid electrodes, in all cases as a phenomenon accompanying the electrolysis. For the electroreduction of Hg(II) ions at the Hg electrode the self-induced convection may temporarily take a form of self-organized cellular patterns [1].
In our recent studies we investigated the electrochemical characteristics of the latter process [2] and analyzed the observed phenomena with standard techniques of nonlinear dynamics [3]. In this case the electric current is strongly enhanced by the convective transport of Hg(II) ions from the solution bulk to the electrode surface. The rate of this convection is in turn strongly dependent on the imposed electrode potential: it passes through the maximum, and when it decreases, the region of the N-shaped negative differential resistance (N-NDR) develops on the current-potential characteristics.
Since the NDR systems are expected to exhibit bistability and oscillations, which were however never studied before for this particular system, we undertook respective experimental and theoretical investigations. The destabilization of the N-NDR systems requires appropriate ohmic drops, i.e. inserting of adjustable external ohmic resistance in the electric circuit of the working electrode.
We studied the dynamical instabilities in the Hg(II) + 2e = Hg process, for varying external voltages U and serial resistances Rs, with the intention to construct the experimental bifurcation diagram. In spite of certain drift of the steady-state characteristics we were able to detect the hysteresis of the current I vs. cyclic variations of the voltage U (i.e. bistability) , but no spontaneous oscillations of the current were observed [2].
In order to find out whether the observed characteristics are reliable, we constructed the theoretical model of the process studied. The following assumptions were made [3]:
(i) the faradaic current of the Hg(II) electroreduction is approximated by the dependence invoking the concept of the linear (Nernst) diffusion layer of a thickness delta;
(ii) for the electrode potentials applied, the concentration of Hg(II) at the electrode surface was equal to zero: c_ox(0,t) = 0;
(iii) as a consequence, the observed I=f(E) dependences are interpreted in terms of variation of only the Nernst diffusion layer delta with E
(iv) the effective electrode potential E and the thickness of the diffusion layer delta were chosen as the two dynamical variables.
Based on the charge conservation principle and quantitative balance of a transport of Hg(II) in the diffusion layer, the two equations describing the dynamics of our electrochemical systems were derived [3]. The stability analysis was performed [3] for these linearized equations. The trace Tr(J) and the determinant Det(J) of the corresponding Jacobi matrix J showed that in terms of our simple model the condition Tr(J)=0 is never met and thus the oscillations originating through the Hopf bifurcation are not possible. The latter conclusion is thus in line with the experimentally observed lack of oscillations. Formally, this situation occurs because the a_11 element of the Jacobi matrix, being the d/dE[dE/dt] expression, is always negative, so the electrode potential E in this system is NOT an autocatalytic variable, contrary to the systems in which the current is controlled not only by the rate of transport, but also by rate of the proper electron-transfer step at the electrode|solution interface.
The theoretical bifurcation diagram based on Det(J)=0 was constructed and exemplary nullclines corresponding to the bistable behavior, were plotted.
A relative importance of the presented analysis may be summarized in the following way. The N-NDR electrochemical systems usually exhibit oscillations and bistability, for the appropriate range of external voltages U and serial resistance Rs. The system described in this presentation is a RARE example of the N-NDR system, the dynamical characteristics of which do not allow oscillatory behavior through the Hopf bifurcation.
References:
[1] R. Aogaki, K. Kitazawa, K. Fueki, T. Mukaibo, Electrochim. Acta, 23 (1978) 867 - 874
[2] M. T. Gorzkowski, R. Jurczakowski, M. Orlik, J. Electroanal. Chem., 615 (2008) 135-144
[3] M. Orlik, M. T. Gorzkowski, J. Electroanal. Chem., 617 (2008) 64-70