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) ((00)\); \(c(l,t)=\bar c=\mathrm{const}\); \(c_x(0,t)=i_e/nFD\), \(nF\) 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}) ((00)\); \(c_x(0,t)=i/nFD\), \(c(l,t)=\bar c\), \(c(x,0)=c_0+(c_x)_0x+\nu(x)\), 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), (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<x0)) 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|,\
\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)