Influence of Ion Generation–Recombination on Dielectric Relaxation Time in Electrolytes

Abstract

The well-known Poisson–Nernst–Planck model is a classical approach usedto describe ion transport in liquids. Extended versions of this model account for thegeneration–recombination of ions at equilibrium. In this paper, we investigate the influenceof the generation–recombination term on dielectric relaxation in an electrolytic cell shapedlike a slab, bounded by two parallel blocking electrodes. We show that in the adiabaticlimit—which holds when the reaction time is much longer than the dielectric relaxationtime—the electric current in the external circuit does not follow a simple relaxation mechanism.Instead, it is characterized by two distinct relaxation times: a short relaxationtime associated with dielectric relaxation and a longer relaxation time related to the iondissociation–association process. Conversely, this information could be used to assess thepresence and/or significance of the generation–recombination effect in an electrolytic cell.

1. Introduction

The Poisson–Nernst–Planck (PNP) model is a standard framework for describing charge transport in liquids, biological tissues, and semiconductors [1,2,3,4,5,6]. It is based on the drift-diffusion continuity equations and Poisson’s equation, which relates the electric field to the electric charge of ionic origin. The PNP model has numerous applications, including electrochemical and electrobiochemical systems and semiconductor devices [7,8,9,10]. Dynamics and relaxation phenomena in an electrolytic cell subjected to an electric field have been extensively studied using the PNP model [11,12,13,14,15,16,17]. In particular, the diffuse-charge dynamics in electrochemical systems were reconsidered by Bazant et al. [17], who showed that, for small applied voltages of the order of the thermal voltage, the evolution of the bulk ion density and the potential is simply exponential for point-like ions with equal anionic and cationic mobilities.

Relaxation depends on many parameters, including the ambipolar effect [18], adsorption phenomena [19], ion generation and recombination [20,21,22], ion size [23,24], confinement effect [25], electric field amplitude, and curvature effects [26]. The generation–recombination term [20] in the continuity equations is typically considered when determining the impedance spectroscopy of cells subjected to periodic excitations of small amplitude [21,22,27,28]. However, this term is often neglected in the analysis of relaxation phenomena in electrolytic cells subjected to a step-like external electric field.

The goal of the present paper is to investigate the role of the generation–recombination term in dielectric relaxation in an electrolytic cell with blocking electrodes. We assume that ions are generated through the decomposition of neutral particles via the reaction mechanism $A\rightleftharpoons B^{+}+C^{-}$, with dissociation and recombination coefficients described by standard kinetic models. This reaction could be optically induced, as in semiconductors, or driven by another external energy source. Our focus is on the relaxation of the electric current in the circuit, which is the experimentally observable quantity.

The paper is organized as follows: Section 2 examines the time dependence of bulk ion density resulting from the dissociation of neutral particles. Section 3 explores the time dependence of the Debye length, Debye time, and dielectric relaxation time, along with a discussion on the validity of the adiabatic approximation. Section 4 focuses on the analysis of current relaxation in a circuit containing an electrolytic cell with blocking electrodes. Finally, Section 5 presents the conclusions.

2. Model

Let us consider an insulating liquid that, at $t=0$, contains a uniform distribution of neutral impurities capable of producing ions via the reaction $A\rightleftharpoons B^{+}+C^{-}$. This reaction could be initiated, for example, by exposure to light, heat, or interactions with another external source of energy. The evolution of the bulk density of neutral, $n_n$, positive, $n_p$, and negative, $n_m$, particles is described by the continuity equations [20]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial n_n}{\partial t}}& =& -\nabla {\mathbf{j}}_n-k_{d}n+k_a{n}_p{n}_m\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial n_p}{\partial t}}& =& -\nabla {\mathbf{j}}_p+k_{d}n-k_a{n}_p{n}_m\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial n_m}{\partial t}}& =& -\nabla {\mathbf{j}}_m+k_{d}n-k_a{n}_p{n}_m\hfill \end{array}$$

(3)

where $k_d$ and $k_a$ are the dissociation and association coefficients, respectively. The dissociation coefficient is expected to depend on the energy of the source inducing the decomposition of neutral particles. In (1,2,3), the current densities ${\mathbf{j}}_n$, ${\mathbf{j}}_p$, and ${\mathbf{j}}_m$ are given by

$$\begin{array}{ccc}\hfill {\mathbf{j}}_n& =& -D_n\nabla n_p\hfill \\ \hfill {\mathbf{j}}_p& =& -D_p\nabla n_p+{\mu }_p{n}_p\mathbf{E}\hfill \\ \hfill {\mathbf{j}}_m& =& -D_m\nabla n_m-{\mu }_m{n}_m\mathbf{E}\hfill \end{array}$$

(4)

In Equation (4), $D_n$, $D_p$, $D_m$ denote the diffusion coefficients of the particles in the insulating liquid, while ${\mu }_p$ and ${\mu }_m$ denote the mobilities of the positive and negative ions, respectively. The local electric field $\mathbf{E}$ is related to the bulk ion densities through the Poisson equation

$$\nabla ·\mathbf{E}={\displaystyle \frac{1}{\epsilon }}q(n_p-n_m)$$

(5)

where $\epsilon$ is the dielectric constant of the solvent in which the ions are dispersed. For a thick slab-shaped cell, if we limit our analysis to an infinite medium in equilibrium, drift-diffusion phenomena can be neglected. In this case, the medium remains both globally and locally electrically neutral. Therefore, assuming $n_p=n_m=p$ and writing $n_n=n$, Equations (1)–(3), are cast in the form

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dn}{dt}}& =& -k_{d}n+k_a{p}^2\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dp}{dt}}& =& k_{d}n-k_a{p}^2\hfill \end{array}$$

(7)

We assume that at $t=0$, the impurities have an initial bulk density $N_0$, and that the system evolves toward an equilibrium state as $t\to \infty$ under the influence of the external source inducing the decomposition of neutral particles. This implies that

$$n\left(0\right)=n_0,p\left(0\right)=0,\underset{t\to \infty }{lim}n\left(t\right)=N,\underset{t\to \infty }{lim}p\left(t\right)=P$$

(8)

From Equations (6) and (7), it follows that

$${\displaystyle \frac{d(n+p)}{dt}}=0,\hspace{1em}n\left(t\right)+p\left(t\right)=N_0$$

(9)

Consequently, from Equation (7), we obtain

$$n\left(t\right)={\displaystyle \frac{1}{k_d}}\left({\displaystyle \frac{dp}{dt}}+k_a{p}^2\right)$$

(10)

By substituting Equation (10) into Equation (6), we obtain

$${\displaystyle \frac{d^{2}p}{d{t}^2}}+2k_{a}p{\displaystyle \frac{dp}{dt}}+k_d{\displaystyle \frac{dp}{dt}}=0$$

(11)

From the latter, it follows that the quantity

$${\displaystyle \frac{dp}{dt}}+k_a{p}^2+k_{d}p=c$$

(12)

is time-independent. Hence, for Equation (8), the integration constant c can be expressed in terms of the equilibrium value of p, denoted by P. Thus, we have

$$c=k_a{P}^2+k_{d}P$$

(13)

and the ordinary differential Equation (12) becomes

$${\displaystyle \frac{dp}{dt}}=k_a(P-p)\left[P+p+(k_d/k_a)\right]$$

(14)

Integrating Equation (14) with the initial condition $p\left(0\right)=0$, we obtain

$$p\left(t\right)=P\left[P+(k_d/k_a)\right]{\displaystyle \frac{1-exp(-t/\tau )}{(k_d/k_a)+P[1+exp(-t/\tau )]}}$$

(15)

where the relaxation time $\tau$ is given by

$$\tau ={\displaystyle \frac{1}{k_a[2P+(k_d/k_a)]}}$$

(16)

The bulk densities of neutral particles and ions in the equilibrium state, N and P, respectively, are given by Equations (6) and (7). Considering the condition $n\left(t\right)+p\left(t\right)=N_0$, these equations can be rewritten as

$$\begin{array}{ccc}\hfill -k_{d}N+k_a{P}^2& =& 0\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill N+P& =& N_0\hfill \end{array}$$

(18)

with the solution

$$P=-{\displaystyle \frac{k_d}{2k_a}}+\sqrt{{\left({\displaystyle \frac{k_d}{2k_a}}\right)}^2+{\displaystyle \frac{k_d}{k_a}}{N}_0}$$

(19)

The relaxation time (16), in terms of the dissociation and association coefficients $k_d$ and $k_a$, is given by

$$\tau ={\displaystyle \frac{1}{\sqrt{k_d(k_d+4k_a{N}_0)}}}$$

(20)

Within the framework of the present model, the dissociation parameter, defined as $\rho =P/N_0$, is

$$\rho =-{\displaystyle \frac{k_d}{2N_0{k}_a}}+\sqrt{{\left({\displaystyle \frac{k_d}{2N_0{k}_a}}\right)}^2+{\displaystyle \frac{k_d}{N_0{k}_a}}}$$

(21)

It is possible to express $k_a$ and $k_d$ in terms of $\tau$ and $\rho$. A straightforward calculation yields

$$k_a={\displaystyle \frac{1-\rho }{\rho \tau (2-\rho )N_0}},\hspace{1em}\mathrm{and}\hspace{1em}{k}_d={\displaystyle \frac{\rho }{\tau (2-\rho )}}$$

(22)

In terms of $\rho$ and $\tau$, the time evolution of the ionic density is

$$p\left(t\right)=\rho N_0{\displaystyle \frac{1-exp(-t/\tau )}{1+(1-\rho )exp(-t/\tau )}}$$

(23)

The time evolution of neutral (not-dissociated) particles is given by $n\left(t\right)=N_0-p\left(t\right)$. Using Equation (23), it follows that

$$n\left(t\right)=N_0{\displaystyle \frac{1-\rho +exp(-t/\tau )}{1+(1-\rho )exp(-t/\tau )}}$$

(24)

It may be of interest to define the time $t^{\ast }$ at which $n\left(t^{\ast }\right)=p\left(t^{\ast }\right)$. A straightforward calculation yields

$$t^{\ast }=\tau log\left({\displaystyle \frac{\rho +1}{2\rho -1}}\right)$$

(25)

From this, it follows that $t^{\ast }$ exists only if $\rho >1/2$, as expected. In Figure 1, the time dependence of $p/N_0$ and $n/N_0$ is shown for $\rho =0.8$ and $\rho =0.3$, with the same $\tau$.

Figure 1. Reduced concentrations, $p/N_0$ and $n/N_0$, as a function of the reduced time, $t/\tau$, for two values of the dissociation parameter $\rho$. Red curves, $\rho =0.7$, blue curves, $\rho =0.2$. The characteristic time of the chemical reaction $\tau$ is assumed the same for both $\rho$ values. The vertical line indicates $t^{\ast }$.

3. Debye Length and Dielectric Relaxation Time

The Debye length [29] for the simple case of a single type of mobile ion with bulk density p is defined as $\mathrm{\Lambda }=\sqrt{\epsilon v_{\mathrm{th}}/\left(pq\right)}$, where $v_{\mathrm{th}}=k_{B}T/q$ is the thermal voltage. In standard problems, $\mathrm{\Lambda }$ is on the order of the thickness of the layer where the ions are confined when the cell is subjected to a small external potential difference, of the order of $v_{\mathrm{th}}$. The Debye relaxation time is defined as ${\tau }_D={\mathrm{\Lambda }}^2/D$ and the dielectric relaxation time [17] as ${\tau }_{\mathrm{diel}}=\mathrm{\Lambda }d/\left(2D\right)$. The quantity ${\tau }_{\mathrm{diel}}$ describes the relaxation of the initial ion distribution under the influence of an external field and is particularly important for applications. In the present case, since $p=p\left(t\right)$, these quantities are time-dependent. We denote

$$f\left(t\right)={\displaystyle \frac{1-exp(-t/\tau )}{1+(1-\rho )exp(-t/\tau )}}$$

(26)

and rewrite Equation (23) as $p\left(t\right)=Pf\left(t\right)$. The quantities of interest are

$$\mathrm{\Lambda }\left(t\right)={\displaystyle \frac{{\mathrm{\Lambda }}_{\mathrm{eq}}}{\sqrt{f\left(t\right)}}},{\tau }_D\left(t\right)={\displaystyle \frac{{\tau }_{D,\mathrm{eq}}}{f\left(t\right)}},{\tau }_{\mathrm{diel}}\left(t\right)={\displaystyle \frac{{\tau }_{\mathrm{diel},\mathrm{eq}}}{\sqrt{f\left(t\right)}}}$$

(27)

where

$${\mathrm{\Lambda }}_{\mathrm{eq}}=\sqrt{{\displaystyle \frac{\epsilon v_{\mathrm{th}}}{Pq}}},\hspace{1em}{\tau }_{D,\mathrm{eq}}={\displaystyle \frac{{\mathrm{\Lambda }}_{\mathrm{eq}}^2}{D}},\hspace{1em}{\tau }_{\mathrm{diel},\mathrm{eq}}={\displaystyle \frac{{\mathrm{\Lambda }}_{\mathrm{eq}}d}{2D}}$$

(28)

are the usual quantities evaluated for the equilibrium density of mobile ions at equilibrium, P.

The bulk density of mobile ions, $p\left(t\right)$, evolves over time t with a characteristic time $\tau$, while their confinement occurs on a timescale given by ${\tau }_{\mathrm{diel}}\left(t\right)$ [30,31,32]. If ${\tau }_{\mathrm{diel}}\left(t\right)\ll \tau$, the system adiabatically follows the variation of $p\left(t\right)$. This implies that p can be considered time-independent in the analysis. This assumption holds for $t\gg t_c$ where

$$t_c=\tau log\left({\displaystyle \frac{(1-\rho )+{(\tau /{\tau }_{\mathrm{diel},\mathrm{eq}})}^2}{{(\tau /{\tau }_{\mathrm{diel},\mathrm{eq}})}^2-1}}\right)$$

(29)

In the case where $\tau \gg {\tau }_{\mathrm{diel},\mathrm{eq}}$, $t_c$ is very small, allowing the use of the adiabatic approximation for analyzing relaxation phenomena in a real system. In Figure 2, the time dependence of the dielectric relaxation time is shown in the presence of the considered generation–recombination term.

Figure 2. Time dependence of the dielectric relaxation time ${\tau }_{\mathrm{diel}}$ in units of $\tau$. The vertical line indicates the critical time $t_c/\tau$, beyond which $t\gg t_c$ the adiabatic approximation holds. The horizontal line marks ${\tau }_{\mathrm{diel}}=\tau$. The physical parameters used for the simulation are $D={10}^{-10}$ m²/s, $N_0={10}^{21}$ m⁻³, $\rho =0.1$, $\tau =120$ s, and $\epsilon =2.4\times {\epsilon }_0$, which are typical for magnetic particles in kerosene.

4. Electric Current Relaxation in a Cell Limited by Blocking Electrodes

We recently investigated dielectric relaxation in an electrolytic cell with blocking electrodes [33] containing a bulk density of fully dissociated impurities. According to the analysis presented in [33], when the cell is subjected to a small dc external potential difference of amplitude $V_0$, the equations of the PNP model can be linearized. The equilibrium distributions are practically reached after a few relaxation times, ${\tau }_{\mathrm{diel}}$. In particular, the electric potential profile tends to

$$V\left(z\right)={\displaystyle \frac{1}{2}}{V}_0{\displaystyle \frac{sinh(z/\mathrm{\Lambda })}{sinh[d/(2\mathrm{\Lambda }\left)\right]}}$$

(30)

For $t\gg {\tau }_{\mathrm{diel}}$, the time dependence of the potential profile is well described by

$$V(z,t)=V\left(z\right)\left(1-e^{-t/{\tau }_{\mathrm{diel}}}\right)$$

(31)

In the adiabatic approximation, Equations (30) and (31) remain valid; however, $\mathrm{\Lambda }$ and ${\tau }_{\mathrm{diel}}$ must be replaced with their time-dependent counterparts, $\mathrm{\Lambda }\left(t\right)$ and ${\tau }_{\mathrm{diel}}\left(t\right)$, as given by Equation (27). Consequently, $V(z,t)$ depends on t both directly and via the time-dependent quantities $\mathrm{\Lambda }\left(t\right)$ and ${\tau }_{\mathrm{diel}}\left(t\right)$. In the external circuit, if the electrodes are blocking, the electric current is just a displacement current, which is related to the time variation of the surface electric field $E_s$. From Equation (30), the profile of the electric field is given by

$$E(z,t)=-{\displaystyle \frac{V_0}{2\mathrm{\Lambda }}}{\displaystyle \frac{cosh(z/\mathrm{\Lambda })}{sinh[d/(2\mathrm{\Lambda }\left)\right]}}$$

(32)

In particular, the surface electric field is

$$E_s\left(t\right)=E(d/2,t)=-{\displaystyle \frac{V_0}{2\mathrm{\Lambda }}}coth\left({\displaystyle \frac{d}{2\mathrm{\Lambda }}}\right)$$

(33)

The related displacement current is then

$$i=\epsilon {\displaystyle \frac{d{E}_s}{dt}}$$

(34)

Figure 3a shows the time dependence of the electric current in the external circuit containing the electrolytic cell. As evident from the figure, the current exhibits two distinct relaxation times: a short relaxation time associated with dielectric relaxation and a longer relaxation time related to the generation–recombination term, $\tau$. For $t\gg {\tau }_{diel}$, an apparent plateau is observed. The corresponding current values are proportional to the applied potential difference, as evident from Equations (33) and (34).

Figure 3. Time dependence of the displacement current, i, in an electrolytic cell with a generation–recombination term, evaluated in the adiabatic approximation. Panel (a) shows results for two values of the applied potential difference: red curve, $V_0=60$ mV, and blue curve, $V_0=30$ mV. Panel (b) presents the relaxation of the current on a logarithmic scale, revealing the presence of two distinct relaxation times. The physical parameters are the same as in Figure 2, while the geometrical parameters of the cell are $d={10}^{-4}$ m and $S={10}^{-4}$ m².

5. Conclusions

We have investigated the role of the generation–recombination term in the dielectric relaxation of an electrolytic cell with blocking electrodes. In our analysis, we assumed that the characteristic time of the chemical reaction governing the decomposition of neutral particles into ions is much longer than the dielectric relaxation time at/toward equilibrium. Within this framework, we applied the adiabatic approximation, where the system is always considered to be near equilibrium for a given ion density. Using this approach, we evaluated the electric field profile and the corresponding displacement current, which arises from the time variation of both the bulk ion density and the dielectric relaxation time.

Our results indicate that the relaxation of the electric current in the external circuit is governed by two distinct relaxation times, which differ significantly and correspond to two separate current sources. Specifically, we expect our findings to be relevant to a wide range of electrochemical problems, where the dynamics of ionic redistribution in the presence of an external electric field play a fundamental role.

In the future, several directions for further investigation remain, following an approach similar to the one presented here. First, we plan to extend our results to cases where the diffusion coefficients of anions and cations differ, as well as to scenarios involving ohmic electrodes, porous media, and other related contexts.