Dynamics of a Non-equilibrium Electrochemical System

Abstract

As was shown in Chap. 4, the behaviour of real electrochemical systems on long timescales cannot often be predicted from short-term observations. In terms of the approach considered in 1.6, this happens because of the onset of positive feedbacks destabilising the stationary state of the system. Several possible reasons exist for this event to take place. Among them, most significant, however very rarely considered, is the coupling between the electrochemical processes at separate electrodes. To understand how it works, we shall first consider the process of electrochemical decomposition of an electrolyte with mixed ion–electron conductivity.

Irreversible phenomena are much more stubborn. However even they tend to enter the general order.

H. Poincarè

Appendices

Appendix A. Derivation of the Differential Equations (5.11) and (5.12)

Because of the mixed conductivity of the anodic film, the Faradaic process at the anode splits into two, i.e. at the inner junction (metal/film) and the outer one (film/electrolyte).

Let us denote the ionic current part as α. Then, at current I and in a lime interval of dt the quantity of deposit

$$ \mathrm{ d}{\mu_{{\operatorname{int}}}}=\frac{{\alpha \cdot I}}{{i\cdot F}}\mathrm{ d}t $$

(1a)

is formed at the inner junction.

At the outer junction, (1 – α)Idt Coulombs of electricity are consumed at the same time to oxidise the portion of the film substance dμ _ext, and the portion of the intermediate transferred from the bulk by the flux J _c is

$$ (1-\alpha )\cdot I\cdot \mathrm{ d}t=-(N-i)\cdot F\cdot \mathrm{ d}{\mu_{\mathrm{ ext}}}+{S_{\mathrm{ a}}}\cdot {J_{\mathrm{ a}}}\cdot \mathrm{ d}t $$

(2a)

To determine the flux, Nernst model of stationary diffusion with linear distribution of the concentration across the layers of the constant thickness is used, and the concentration of the intermediate is assumed to be zero, i.e. the anode discharge proceeds at limit current condition. Then

$$ {J_{\mathrm{ a}}}=\frac{D}{\delta}\cdot {C_i} $$

(3a)

The differential equation (Eq. 5.11) is obtained by combination of the equation for the material balance of the film substance

$$ {\mu}^{\prime}=\frac{{\mathrm{ d}\mu }}{{\mathrm{ d}t}}=\frac{{\mathrm{ d}{\mu_{\mathrm{ int}}}}}{{\mathrm{ d}t}}+\frac{{\mathrm{ d}{\mu_{\mathrm{ ext}}}}}{{\mathrm{ d}t}} $$

(4a)

with Eqs. (1a)–(3a).

Equation (5.12) can be obtained from the balance of the intermediate in the electrolyte, which is supported by the cathodic flux J _c to the bulk and the anodic flux J _a from the bulk

$$ V\cdot {C_i}={S_{\mathrm{ c}}}\cdot {J_{\mathrm{ c}}}-{S_{\mathrm{ a}}}\cdot {J_{\mathrm{ a}}} $$

(5a)

Apart from the above-mentioned Nernst diffusion, the following assumptions and approximations were made in the derivation of Eq. (5.12).

Intervalent equilibrium between polyvalent species at the cathode surface is assumed to establish. A convenient notation of this equilibrium, according Chap. 2, is

$$ \left( {1-\frac{i}{N}} \right)\cdot E(0)+\frac{i}{N}\cdot E(N)=E(i), $$

(6a)

where the symbols E(0), E(i), and E(N) correspond to polyvalent metal compounds, oxidation numbers being zero, i, and N, respectively.

The activity of E(0) is regarded to be 1 (solid pure metal). Further, the concentration C _N of the high-valence compound E(N) at the surface of the cathode and in the bulk is assumed to be equal. This is acceptable for not very dilute solutions. We can now express the cathodic surface concentration of the intermediate by

$$ {C_{\mathrm{ i}\rm{,}\mathrm{ cath}}}=k\cdot C_N^{i/N } $$

(7a)

Further, the charge balance equation in the form

$$ N{C_0}=N\cdot {C_N}+i\cdot {C_i}+\frac{{i\cdot \mu }}{V} $$

(8a)

is used, where C _o is the total initial concentration of the electrolyte. This means that the electrolyte at the initial time condition contains only the high-valence compound E(N) with a concentration of C _o, and no additions of salts to the melt during the process are made.

Now the cathodic concentration C _i,cath required for the determination of the flux

$$ {J_{\mathrm{ c}}}=\frac{D}{\delta}\cdot ({C_{{i\rm{,}\mathrm{ cath}}}}-{C_i}) $$

(9a)

can be obtained by determining C _N from Eq. (8a) and substituting it into Eq. (7a). The obtained power function may be expanded into a series and, after neglecting the second and higher order members, may be written in the form

$$ {C_{{i\rm{,}\mathrm{ cath}}}}=k\cdot C_0^{i/N}\cdot \left( {1-\frac{{\mu \cdot {i^2}}}{{{N^2}\cdot {C_0}\cdot V}}} \right) $$

(10a)

This linearising simplification is sufficiently correct when the condition is fulfilled

$$ {C_i} < < \frac{\mu }{V} < < \frac{N}{i}{C_0} $$

(11a)

Finally, Eq. (5.12) is obtained by substitution of Eqs. (3a), (9a), and (10a) into Eq. (5a).

Appendix B. Approximate Solution of the Non-linear Second-Order Differential Equation by Small-Parameter (Krylov–BogoIiubov) Method [20]

If the non-linear term of the differential equation may be thought of as a small disturbance, the above-mentioned method can then be applied to derive an approximate solution. Equation (5.35) is presented as follows:

$$ {y}^{\prime\prime}+{w^2}\cdot y+\gamma \cdot f(y,{y}^{\prime})=0 $$

(12a)

where

$$ {w^2}=a-\gamma $$

(13a)

and non-linear function

$$ f(y,{y}^{\prime})=-\gamma \cdot {y}^{\prime}\cdot \left( {\frac{{\gamma -1}}{\gamma }-{y^2}} \right) $$

(14a)

The solution is sought in the form

$$ y=r(\tau )\cos \varphi (\tau ) $$

(15a)

by a common technique described elsewhere [19, 20].

Finally, the approximate periodic solution of the limit cycle type is obtained in the form

$$ {y_{\mathrm{ L}}}=2\cdot \sqrt{{\frac{{\gamma -1}}{\gamma }}}\cos (w\cdot \tau +{\varphi_0}) $$

(16a)

where the dimensionless frequency w is approximately equal to $\sqrt{{a-\gamma }}$. The dimensionless period $T=2\pi /w$ of the oscillation, then, is equal to ${{{2\pi }} \left/ {{\sqrt{{a-\gamma }}}} \right.}$. Therefore, the formula of Eq. (5.36) for the real timescale period is accurate to a constant coefficient. This coefficient is of the order 1 and should increase as the current I and concentration C ₀ increase. This is hard to verify experimentally because of the very long duration of the tests.