PHYSICAL CHEMISTRY

A. Ya. Gokhshtein

ON THE STABILITY OF STATIONARY STATES OF ELECTROLYTIC SYSTEMS

(Presented by Academician A. N. Frumkin, 27 X 1962)

Nonstationary phenomena, in particular periodic phenomena, associated with the loss of stability of a stationary state have recently been discovered in electrolytic systems free from passivation \((^{1-5})\). In the circuit of such a system the current \(i\) passes successively through the electrolyte (active resistance \(r\)) and the boundary \((S)\) electrolyte—electrode, at which a substance \(A\) is discharged; in the simplest case this substance practically does not participate in the transfer of current through the electrolyte and is delivered to the electrode by diffusion (coefficient \(D\)) occurring in a layer of thickness \(l\); \(c(x,t)\) is the concentration of \(A\) at a distance \(x\) from an electrode of unit area at time \(t\) \((0<x<l,\ t>0)\); \(c(l,t)=\bar c=\mathrm{const}\); \(c_x(0,t)=i_e/nFD\), \(nF\) is the expenditure of electricity per 1 mole of \(A\); the discharge current \(i_e\) depends on the jump \(\varphi\) of the potential at the boundary \(S\): \(i_e(\varphi)=c p(\varphi)\); \(p(\varphi)>0\) is a single-valued differentiable function, \(c=c(0,t)\). The double electric layer at the boundary \(S\) has capacitance \(q(\varphi)\). \(i=i_e+i_q\), where \(i_q=q\,d\varphi/dt\). The voltage at the ends of the circuit \(v=\mathrm{const}\); \(ir+\varphi=v\).

1. Canonical form of the equation of stationary states. The stationary state \(O(\varphi_0,c_0,i_0)\) (Fig. 1) is determined as follows \((^3)\): \(i_q=0\), \(i_0=i_e=(v-\varphi_0)/r=G(\bar c-c_0)\), where \(G=nFD/l>0\); \(\varphi_0\) is a root of the equation \(i_0(\varphi)r+\varphi=v\), where \(i_0(\varphi)=G\bar c/[1+G/p(\varphi)]\); \(c_0=c(\varphi_0)\), where \(c(\varphi)=(v-\varphi)/rp(\varphi)\). Fixing the stationary state \(0\), introduce the quantity \(f\)

\[ f=-\frac{i-i_0}{c-c_0} =-\frac{(v-\varphi)/r-(v-\varphi_0)/r}{c(\varphi)-c(\varphi_0)} =\frac{1}{r}\,\frac{\varphi-\varphi_0}{c(\varphi)-c(\varphi_0)}, \tag{1} \]

which will subsequently be denoted by \(f(u)\), where \(u=c-c_0\), or \(f(\varphi)\). It is not difficult to verify that all stationary states of the system coincide with the roots of the equation \(u f(u)=Gu\). Therefore, if the stationary state is unique, then \(f(u)\ne G\) for any \(u\) \((\varphi)\) not equal to \(0\) \((\varphi_0)\). From (1)

\[ f(\varphi_0)=\frac{1}{r}\,\frac{1}{c'(\varphi_0)} =-\frac{p(\varphi_0)}{1+c_0p'(\varphi_0)r} =\frac{p(\varphi_0)}{\alpha}, \]

where \(\alpha=-(1+c_0p_0' r)\).

The condition \(f(\varphi_0)>0\) is necessary and sufficient in order that, in the neighborhood \(\mathfrak{B}\) of \(\varphi_0\) \((\mathfrak{B}=\{\varphi:\ c'(\varphi)>0\})\), the unique stationary state \(\varphi_0\) satisfy \(f(\varphi)>G\). Sufficiency. On the basis of (1) the existence is established of such \(\varphi_m\) and \(\varphi_n\) for which \(f(\varphi)\) is continuous on \((\varphi_m,\varphi_n)\) and \(f(\varphi)\to+\infty\) as \(\varphi\to\varphi_m+0,\ \varphi\to\varphi_n-0\); from the continuity of \(f(\varphi)\) on \((\varphi_m,\varphi_n)\) it follows that \(f(\varphi)>G\) for \(\varphi\in\{(\varphi_m,\varphi_n)\setminus\varphi_0\}\): otherwise \((f(\varphi)\leqslant G)\) there would be at least one value \(\varphi\in(\varphi_m,\varphi_n)\) not equal to \(\varphi_0\), for which \(f(\varphi)=G\), which contradicts the uniqueness of \(\varphi_0\); the rest follows from \(\mathfrak{B}\subset(\varphi_m,\varphi_n)\). Necessity. If \(f(\varphi)>G\) in a neighborhood of \(\varphi_0\), then \(\lim f(\varphi\to\varphi_0)\geqslant G>0\). If \(\varphi_0\) is not unique, then it is possible that \(0<f(\varphi_0)<G\). According to (1), \(f=f(\varphi)\) is single-valued; \(f=f(u)\) may be multivalued, but as long as \(\varphi\in\mathfrak{B}\), the functions

\(f=f(\varphi)\) there corresponds a single-valued branch of the function \(f=f(u)\), \(u\in\{u:\varphi\in\mathfrak{B}\}\) \((u'(\varphi)=c'(\varphi)>0\) for \(\varphi\in\mathfrak{B})\). This branch \(f(u)\) is used in Sec. 2.

2. Distributed system with a diffusion layer.

Consider the case \(q=0\), which is feasible in physical models of electrolytic systems. The problem
\(c_t=Dc_{xx}\) \((0<x<l,\ t>0)\); \(c_x(0,t)=i/nFD\), \(c(l,t)=\bar c\), \(c(x,0)=c_0+(c_x)_0x+\nu(x)\), where \(\nu(x)>0\) is a small perturbation, by the substitution \(u(x,t)=c(x,t)-c_0-(c_x)_0x\) and taking (1) into account, is reduced to the form
\(u_t=Du_{xx}\), \(u_x(0,t)=-u f(u)/lG\), \(u(l,t)=0\), \(u(x,0)=\nu(x)\). Using the corresponding source function \(Q(x,\xi,t)\), we form an integral equation with respect to \(u=u(0,t)\):

\[ u=\int_0^l \nu(\xi)Q(0,\xi,t)\,d\xi - D\int_0^t u_x Q(0,0,t-\tau)\,d\tau = \]

\[ =\eta(t)+\frac{D}{l}\int_0^t u\,\frac{f(u)}{G}\,K(t-\tau)\,d\tau, \tag{2} \]

\[ K(t)=\frac{1}{\sqrt{D\pi t}} \left[ 1+2\sum_{k=1}^{\infty}(-1)^k\exp\left(-\frac{(kl)^2}{Dt}\right) \right], \quad 0<\eta(t)<\max \nu(\xi),\quad \eta(t\to\infty)\to 0. \]

Consider the solution of the auxiliary problem
\(\tilde u_t=D\tilde u_{xx}\), \(\tilde u_x(0,t)=-h\tilde u(0,t)\), \(\tilde u(l,t)=0\), \(\tilde u(x,0)=\nu(x)\). The eigenvalues \(\mu\) to which the separation-of-variables method leads here are determined by the roots of the equation \(h\tan lz=z\); \(\sqrt{\mu}=z\). For \(hl>1\), along with real roots, it has purely imaginary ones: \(z=\pm i\rho\), \(\mu=-\rho^2\), which corresponds to an unbounded increase of \(\tilde u(0,t)\). Let \(hl=1+\sigma\), \(\sigma>0\), and
\(\nu(x)=\varepsilon[\exp \rho(2l-x)-\exp \rho x]/[\exp 2\rho l-1]\le \varepsilon\). Then
\(\tilde u(x,t)=\nu(x)\exp D\rho^2t\), \(\tilde u(0,t)=\varepsilon\exp D\rho^2t\). At the same time, \(\tilde u(0,t)\) satisfies the equation

\[ \tilde u=\eta(t)+(1+\sigma)\frac{D}{l}\int_0^t \tilde u K(t-\tau)\,d\tau . \]

Fig. 1. \(a\)—\(\delta,\ l=\mathrm{const},\ \alpha\ne\mathrm{const};\ \sigma\)—\(\alpha,\ \delta=\mathrm{const},\ l\ne\mathrm{const}\)

Compare it and (2) as equations with kernels \((1+\sigma)K\) and \((f/G)K\). From the representation of their solutions by resolvents it follows that \(f/G\ge 1+\sigma>1\) is necessary and sufficient for \(u\ge\tilde u\), i.e., the fulfillment of \(f(\varphi)>G\), where \(\varphi\in\mathfrak{B}\setminus\varphi_0\), guarantees not only the instability of \(\varphi_0\), but also the departure of \(\varphi\) beyond the limits of \(\mathfrak{B}\). If, however, \(f(\varphi)\le G\) in some neighborhood of \(\varphi_0\), then the state \(\varphi_0\) is stable. Hence, also from the property of \(f(\varphi)\) found in the preceding item, it follows: the unique stationary state \(\varphi_0\) is stable for \(\alpha<0\) and unstable for \(\alpha>0\). For \(\alpha=0\), either is possible; the question is resolved analogously, by comparing \(f(\varphi)\) and \(G\).

3. Distributed system with a diffusion layer and lumped parameters.

A. A system with a diffusion layer of thickness \(l\) and a capacitance \(q\ne0\), lumped at the point \(x=0\) (the conclusions do not change for a larger number of parameters, for example capacitance and inductance),

is described by the equation \(c_t(x,t)=Dc_{xx}(x,t)\) \((0<x<l,\ t>0)\) with the conditions
\(c_x(0,t)=c(0,t)p(\varphi)/nFD,\ r[q(\varphi)\varphi_t(t)+c(0,t)p(\varphi)]+\varphi(t)=v,\ c(l,t)=\bar c,\)
\(c(x,0)=c_0+(c_x)_0x+v(x)\). Let
\(p(\varphi)=p_0+p_0'(\varphi-\varphi_0)+p_0''(\varphi-\varphi_0)^2+\cdots\) and
\(1/q(\varphi)=1/q_0+(1/q)_0'(\varphi-\varphi_0)+\cdots\). In the new variables
\(u=c(x,t)-c_0-(c_x)_0x,\ m=\varphi-\varphi_0,\ \tau=t/rq_0,\ y=x/\sqrt{rq_0D},\)
\(\lambda=l/\sqrt{rq_0D}\), where \(\tau,y\) are dimensionless, the problem takes the form:

\[ \begin{gathered} dm/d\tau=\alpha m+\beta u+O_1(um)\big|_{y=0},\qquad \partial u/\partial y=\gamma m+\delta u+O_2(um)\big|_{y=0},\\ \partial u/\partial \tau=\partial^2 u/\partial y^2\qquad (0<y<\lambda,\ \tau>0); \end{gathered} \tag{3} \]

\[ \alpha=-(1+c_0p_0'r),\qquad \beta=-rp_0,\qquad \gamma=\sqrt{rq_0D}\,c_0p_0'/nFD, \]

\[ \delta=\sqrt{rq_0D}\,p_0/nFD>0. \]

We shall denote the linearized problem (without the terms \(O_1\) and \(O_2\)) by (3L).

It is not difficult to establish the following fact: in the case of an electrolytic (or similar) system the coefficients \(\alpha,\beta,\gamma,\delta\) are connected by the relation: \(\beta\gamma-\alpha\delta=\delta\). Transform the expression for \(\delta\), multiplying and dividing it by \(l(\bar c-c_0)\):
\[ \delta=\{p_0c_0[(\bar c/c_0)-1]/[nFD(\bar c-c_0)/l]\}\sqrt{rq_0D}/l. \]
Since \(p_0c_0=i_0=nFD(\bar c-c_0)/l\), it follows that \(\delta=(\chi-1)/\lambda\), where \(\chi=\bar c/c_0\).

B. Let us investigate the stability of the trivial solution of (3L). We use the integral representation ((2), first equality), in which one should put \(t=\tau,\ l=\lambda,\ D=1\). We apply to it and to the first two equations of (3L) the Laplace transform. Eliminating \(\mathcal L\{m(\tau)\}\) and finding \(\mathcal L\{K(\tau)\}=\operatorname{th}(\lambda\sqrt s)/\sqrt s\), we obtain

\[ a(s)=\mathcal L\{u(0,\tau)\}= \frac{(s-\alpha)\xi(s)-\gamma m(+0)\operatorname{th}(\lambda\sqrt s)/\sqrt s} {\dfrac{\delta}{\sqrt s}(s+\omega)\left(\dfrac{\sqrt s}{\delta}\dfrac{s-\alpha}{s+\omega}+\operatorname{th}\lambda\sqrt s\right)} = \]

\[ =\frac{\tilde b(s)}{\tilde w(\sqrt s,\lambda)} =\frac{b(s)}{w(\sqrt s,\lambda)}. \]

where
\[ \tilde w(z,\lambda)=\frac1z\left(\frac z\delta\,\frac{z^2-\alpha}{z^2+\omega}+\operatorname{th}\lambda z\right), \qquad w(z,\lambda)=\frac1z\,[g(z)e^{\lambda z}-g(-z)e^{-\lambda z}], \]
\[ \omega=(\beta\gamma-\alpha\delta)/\delta,\qquad g(z)=z^3+\delta z^2-\alpha z+\omega\delta;\quad \alpha,\delta,\omega,\lambda\ \text{are real},\quad \xi(s)=\mathcal L\{\eta(t)\}. \]
\(a(s)\) is single-valued and is a sum of transforms; for sufficiently large \(\operatorname{Re}s>N\),
\[ \left|\sqrt s(s-\alpha)/(s+\omega)\delta\right|>|\operatorname{th}\lambda\sqrt s|, \]
therefore \(a(s)\) is regular for \(\operatorname{Re}s>N\); \(a(s)\to0\) uniformly with respect to \(\arg s\) for \(|s|=(k\pi/\lambda)^2\) and \(k\to\infty\). Then \({}^{(6)}\)
\[ u(0,\tau)=\cdots+\operatorname{res}_{s_k}a(s)\exp s_k\tau+\cdots, \]
where the residues are taken at \(s_k\), which are zeros of \(w(\sqrt s,\lambda)\). \(u(0,\tau)\) is bounded if \(\operatorname{Re}s_k\le0\), i.e. \(z_k=\sqrt{s_k}\in\mathfrak P=\{z:\ |\arg z|<\pi/4\}\). Thus, for the stability of the solution \(u(y,\tau)\equiv0,\ m(\tau)\equiv0\) of the system (3L), it is necessary and sufficient that the domain \(\mathfrak P=\{z:\ |\arg z|<\pi/4\}\) contain no zeros of \(w(z,\lambda)\). Replacing \(m(\tau)\) in system (3) by a vector function, and also changing the type of condition for \(u\) at the boundary \(y=0\), is reflected in the degree of the polynomial \(g(z)\); \(-g(-z)\) coincides with the characteristic polynomial introduced by A. N. Tikhonov when considering the case \(\lambda=\infty\) \({}^{(7)}\).

C. Together with a zero \(z=z_0\), zeros of the function \(w(z,\lambda)\) are also \(-z_0,\ \bar z_0,\ -\bar z_0\). The equation \(w(z,\lambda)=0\) in implicit form specifies the dependence \(z=z(\lambda)\), and \(dz/d\lambda\) has a discontinuity at the \(\lambda\)’s corresponding to multiple zeros of \(w(z,\lambda)\). Let \(z_g\) be one of the zeros of \(g(z)\), \(\operatorname{Re}z_g>0\) (if there are no such zeros, then the solution \(u\equiv0\) is certainly stable). Denote by \(z(\lambda)\) (or \(z(\lambda,\alpha)\)) the zero of the function \(w(z,\lambda)\) corresponding to \(z_g\): \(z(\lambda)\to z_g\) as \(\lambda\to\infty\). For \(0<\lambda<\infty\), besides the zeros \(z(\lambda)\), \(w(z,\lambda)\) has purely imaginary zeros, which do not affect the stability of the solution \(u\equiv0\). For

for small \(\lambda\)

\[ z(\lambda)\sim \sqrt{(\alpha-\delta\omega\lambda)/(1+\delta\lambda)}; \qquad (dz/d\lambda)_{z=\sqrt{\alpha}}=-\delta(\alpha+\omega)/2\sqrt{\alpha}; \]

\(z(\lambda)\to\sqrt{\alpha}\) as \(\lambda\to 0\). Thus, for \(\alpha>0\) and sufficiently small \(\lambda\), \(z(\lambda)\in\mathfrak{P}\), and the solution \(u\equiv 0\) is unstable. The position of the curve \(z(\lambda)\) in the \(z\)-plane depends on the parameters \(\alpha\), \(\delta\), and \(\omega\). Further, let \(\delta>0\), \(\omega=1\), in accordance with (3) and the relation \(\beta\gamma-\alpha\delta=\delta\). For various fixed \(\alpha\), consider the change of \(z(\lambda)\) as \(\lambda\) decreases from \(\infty\) to \(0\), \(z=z(\lambda,\alpha)\) (Fig. 2, \(\delta=1/30\), \(\omega=1\); the trajectories of the zeros of \(w(z,\lambda)\) are shown by heavy lines). Segments of the curve \(z(\lambda,\alpha)\) that fall into the region \(\mathfrak{P}\) (hatched) correspond to those values of \(\lambda\) for which \(u\equiv 0\) is unstable. 1) \(\alpha=0\); as \(\lambda\) decreases from \(\infty\) to \(0\), the curve \(z(\lambda,0)\) first unwinds asymptotically about the point \(z(\infty,0)\), then (\(\lambda_m=2.90\)) reaches the imaginary axis and descends along it to \(z(0,0)=0\) (\(\lambda_0=\alpha/\omega\delta=0\)); for \(\alpha<0\), \(u\equiv0\) is asymptotically stable for any \(0<\lambda<\infty\). 2) \(\alpha=0.1\); the point \(z(\infty;0.1)\) and part of the spiral \((7.3<\lambda<\infty)\) are located in \(\mathfrak{P}\); at \(\lambda=7.3\), \(z(\lambda)\) leaves \(\mathfrak{P}\), reaches the imaginary axis (\(\lambda_m=3.66\)), descends along it to \(z=0\) (\(\lambda_0=\alpha/\omega\delta=3.00\)) and, having entered \(\mathfrak{P}\) again, along the real axis reaches \(z(0;\alpha)=\sqrt{\alpha}=0.316\); thus, for \(\alpha=0.1\), \(u\equiv0\) is unstable for \(0<\lambda<3.0\) and \(7.3<\lambda<\infty\), while \(u\equiv0\) is asymptotically stable for \(3.0<\lambda<7.3\). As the point \(z(\infty,\alpha)\) approaches, with changing \(\alpha\), the boundary of \(\mathfrak{P}\) (the line \(\operatorname{Im} z=\operatorname{Re} z\)), the number of alternating intervals of stability and instability in the region \(0<\lambda<\infty\) increases without bound. 3) \(\alpha=0.2\); the entire curve \(z(\lambda;0.2)\)—from the asymptotic point \(z(\infty;0.2)\) to \(z(0;0.2)=\sqrt{0.2}=0.447\)—is located in \(\mathfrak{P}\); \(u\equiv0\) is unstable for any \(0\leqslant\lambda<\infty\) and remains so under a further increase of \(\alpha\). Thus, there exists an intermediate region of values of the parameters \(\alpha\) and \(\delta\) in which, as the thickness of the near-electrode layer \(l\) decreases, the stability of the fixed stationary state \(u\equiv0\) alternates with its instability.

Fig. 2.

\[ w(z,\lambda)=\frac{1}{z}\,[g(z)e^{\lambda z}-g(-z)e^{-\lambda z}]=0; \]

\[ g(z)=z^{3}+\delta z^{2}-\alpha z+\omega\delta;\quad \omega=1; \]

\[ \delta=1/30;\quad \alpha=0,\;1/10,\;2/10 \]

Some differences between systems without capacitance (Sec. 2) and with capacitance (Sec. 3): a) in the former, the stationary state can be stabilized by decreasing \(l\) (\(u\equiv0\) is stable if \(f(0)<G=nFD/l\)); b) for complex \(z(\infty,\alpha)\in\mathfrak{P}\) and sufficiently large \(l\), in the system with capacitance (Sec. 3), \(u\) leaves the stationary state by oscillating.

Institute of Electrochemistry
Academy of Sciences of the USSR

Received
27 X 1962

CITED LITERATURE

1. A. Ya. Gokhshtein, A. N. Frumkin, DAN, 132, 388 (1960).
2. A. N. Frumkin, O. A. Petrii, N. V. Nikolaeva-Fedorovich, DAN, 136, 1158 (1961).
3. A. Ya. Gokhshtein, DAN, 140, 1114 (1961).
4. A. Ya. Gokhshtein, A. N. Frumkin, DAN, 144, 821 (1962).
5. A. Ya. Gokhshtein, DAN, 148, 148 (1963).
6. M. A. Lavrent’ev, B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Moscow, 1958.
7. A. N. Tikhonov, Matem. sborn., 26, 35 (1950).