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Random Energy Barrier Model for Ac Electrode Conductivity Fernando Batalioto, Antonio Martins Figueiredo Neto, and Giovanni Barbero J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00294 • Publication Date (Web): 28 Feb 2019 Downloaded from http://pubs.acs.org on March 1, 2019

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The Journal of Physical Chemistry

Random Energy Barrier Model for ac Electrode Conductivity †

F. Batalioto,

†

A. M. Figueiredo Neto,

and G. Barbero

∗,‡,¶

†Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, São Paulo,

05508-090, SP - Brazil ‡Dipartimento di Scienza Applicata del Politecnico di Torino,

Corso Duca degli Abruzzi 24, 10129 Torino, Italia. ¶National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),

Kashirskoye shosse 31, 115409 Moscow, Russian Federation. E-mail: [email protected]

Abstract We investigate the electric response of an electrolytic cell of water and KCl, limited by gold electrodes, in order to derive information on the role of the electrodes on the spectra of the real and imaginary parts of the electric impedance of the cell. Our experimental data can be interpreted by means of the Poisson-Nernst-Planck model with Ohmic boundary conditions for the electrodes, where the surface conductivity is consistent with that derived by a random-walk model for the charge exchange on the electrodes. By using a best-t procedure the dc conductivity and the hopping time are determined. The measurements have been performed in the absence and presence of a dc bias. In the absence of dc bias the agreement between the theoretical predictions and the experimental data is good over the full explored frequency range. On the contrary, in the presence of the dc bias, the theoretical predictions in the low-frequency

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range do not describe very well the experimental results, and the agreement decreases increasing the value of the bias. This result indicates that the hopping phenomenon describing the conduction across the electrodes has peculiar characteristics dierent from those of the bulk. The dependence of the hopping time and of the dc conductivity on the electric eld is reasonably well described by the existing models proposed for the hopping conduction in the bulk.

1. Introduction

The behaviour of the ions in electrolytic solutions subjected to external electric elds is an interesting problem not only from the fundamental, but also technological points of view. To theoretically describe the behaviour of an electrolytic cell submitted to an external electric eld, the bulk partial dierential equations relevant to the dynamical quantities characterizing the behaviour of the system under consideration have to be solved with the proper boundary conditions on the electrodes. Usually, the dynamical quantities dening the state of the system are the bulk densities of ions and the electric potential across the sample. In this case, the boundary conditions are imposed on the electric potential and on surface density currents of ionic origin, that depend on the nature of the electrodes and on its surface treatment. These boundary conditions play a fundamental role in the low frequency region, where the ions contribute to the electric response of the cell. Our goal is to investigate the charge exchange on the electrodes of an electrolytic cell submitted to an ac excitation, determining the boundary conditions on the ionic surface density currents. This is an open problem in electrochemistry, as discussed in the reference book by Atkins. Several phenomenological models have been proposed to describe 1

non-blocking character of real electrodes. Long ago, Butler and Volmer proposed the famous equation for the surface density of ionic current, based on the concept of activation energy. Butler-Volmer's equation has been adapted to electrolytic solutions in contact with 1

solid electrodes by Derfel.

2,3

Chang and Jae proposed a model similar to that used in the 2


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evaporation phenomenon, where the surface current density of charge exchange is assumed 4

proportional to the ionic bulk density variation, with respect to the value of equilibrium.

5

More recently an Ohmic model, where the surface current density of charge is proportional to the surface electric eld, has been used to interpret impedance spectroscopy experimental data. The two models are equivalent, in the sense that it is possible to map one model into 6

the other. In the models presented above the phenomenological parameters connecting the surface density of ions with the bulk density variation or with the surface electric eld are assumed to be frequency independent, and hence they are expected to work well only in the dc steady state limit, where the time does not play any role. All these models when employed to investigate the behaviour of the real and imaginary parts of the electric impedance as a function of the frequency (f ) of the external electric eld applied to the sample fail in the low-frequency range (f < 1Hz). In particular, it is experimentally observed that the real part of the electric impedance increases in the dc limit. This increasing is well described by means of the Ohmic model only for some medium, as hydrogel limited by silver electrodes. However, for other systems the theoretical predictions 7

are not in agreement with the experimental data, as in the case of KCl in water in contact with gold electrodes. This fact suggests that the behaviour of the ions in the vicinity of the electrodes depends on the frequency, and a dierent mechanism of interaction between ions and electrodes has to be considered. In a recent paper the boundary conditions proposed to describe the charge exchange 8

occurring in an ionic liquid in contact with non-blocking electrodes have been reconsidered in dynamic situation for an ionic liquid. The model presented in is based on the assumption 8

that the current related to the charge exchange is proportional variation of the the bulk density of ions with respect to the value of equilibrium. The analysis has been done in the framework of the linear analysis of a system characterized by a distribution of relaxation times.

9,10

The theoretical predictions of the model are in good agreement with the experi-

mental result reported in, where cells of ionic liquid crystal limited by FTO electrodes have 8

3



been investigated. In the present paper we apply the model developed in to an electrolytic cell of water and 8

KCl, in the shape of a slab, limited by gold electrodes submitted to an ac external electric eld. Our model contains, intrinsically, a frequency dependence of the phenomenological parameter describing the surface exchange of electric charge on the frequency of the external eld. We would expect that this random-walk model for the charge exchange on the electrodes depends on a dc bias applied between the electrodes, due to modications on the energy barriers on them, not considered in. It is our aim also verify this hypothesis. 8

Our paper is organized as follows. In Sect.2 a short review of the theoretical models proposed to describe the electric behaviour of an electrolytic cell submitted to an external eld are briey recalled. In Sect.3 the model based on the hopping mechanism for the charge exchange on the electrodes is presented. In Sect.4 we report the experimental data obtained by means of the impedance spectroscopy on an electrolytic cell limited by gold electrodes in the absence of a dc bias eld. In the same section, the measurements performed in the presence of a dc eld are also presented. From these measurements the dependence of the hopping time and of the dc conductivity on the electric eld are determined. The agreement between experimental data and theoretical description of the proposed model are discussed there. As it will be shown, for small dc bias, the dependence of the hopping time and of the dc conductivity is rather well described by the model proposed for the bulk hopping conductivity. However, increasing the bias the agreement is less and less good, indicating that the existing models for the ac conductivity in the bulk have to be modied to describe properly the electrode conductivity. Section V is devoted to the Conclusions.

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2.

Theoretical models for electric response of an elec-

trolytic cell

An electrolytic solution in an unbounded sample, in the absence of external electric eld, can be considered as a dispersion of ionic charge in an insulating medium. In this situation the ionic charge density is macroscopically homogeneous. When the solution is contained in a slab of nite thickness the interaction between the ions and the limiting electrodes can be responsible for a position dependent ionic charge density, relaxing over a surface layer of thickness comparable with the range of the surface forces. In the case where the interaction with the electrodes of the positive and negative ions is the same, the bulk density of ions remains constant across the sample. In the presence of an external electric eld, the ions move under the eect of the electric eld. If the external eld is time dependent the ions move following the time dependence of the electric eld. The presence of the limiting surfaces is responsible for an accumulation of ions, known as electrode polarization.

11,12

As it follows from this discussion, electrode polarization eects are due to the change of conduction properties in surface layers in contact with the electrodes, which is used to apply the external dierence of potential to the sample under consideration. The inuence of electrode polarization on the electric response of a cell is widely used to determine diusion coecient and ionic density in ionic conductor. The eects of the electrode polarization on 13

the electric response of the cell to an external excitation can be described assuming the cell as formed by two layers of dierent diusion coecient and dielectric constant, as proposed by Serghei et al.

11,12

The electric response of an electrolytic cell to an external electric eld is usually described by means of the Poisson-Nernst-Planck (PNP) model based on the equations of continuity for the ions and on the equation of Poisson for the actual potential across the cell.

14

These

partial dierential equations have to be solved with the boundary conditions imposed on the ionic current densities on the electrodes and on the dierence of potential imposed by the 5



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external power supply to the cell. In the case, rather important from the experimental point of view, where the electromotive force applied by the external power supply is a simple harmonic function of time ∆V (t) = V0 exp(iωt), of small amplitude V0 and angular frequency ω , PNP-model allows

the determination of the impedance, Z , of the cell.

14

This limit is known as linear approxi-

mation of the response of the cell. Let us consider rst the case where the electrodes are perfectly blocking, i.e. the ionic current densities vanish on the electrodes.

15

In this case the mathematical problem can be

easily solved. In this framework, the spectrum of R = Re[Z] is nearly frequency independent up to ωD , known as Debye's angular frequency. For ω > ωD , R tends to zero as ω −2 . On the contrary, the spectrum of χ = Im[Z], in the dc limit, diverges as ω −1 , and in the high frequency region tends to zero as ω −1 . It presents a minimum for ω = ωD , and a maximum, not well dened, for a frequency smaller than ωD , that depends on the thickness of the sample. These theoretical predictions are in agreement with the experimental data only for frequency larger than 1KHz. In the low frequency region, large deviations from the theoretical predictions are observed. This experimental result indicates that the assumption of blocking electrodes does not work well in the low frequency region. Dierent models have been proposed to describe the non blocking character of the electrodes in the framework of the PNP model. Recently, PNP model for the impedance spectroscopy of an electrolytic cell of nite thickness has been generalized to take into account fractional diusion in the presence of bulk reaction terms.

16,17

In this general framework,

the proposed boundary conditions are integral equations whose kernel is related to memory eects connected with adsorption phenomena and to charge transfer from the bulk to the external circuit. However, the general model proposed in,

16,17

if from one side looks rather

elegant from the mathematical point of view, it is of limited practical application because the number of phenomenological parameters in it is rather large. Furthermore the connection between the general boundary conditions and the physical phenomena occurring at the 6


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interface is not always transparent. For this reason we assume that the bulk equations of PNP model are the standard ones, involving usual diusion term, and propose boundary conditions based on hopping mechanism. To the best of our knowledge this approach was not considered to describe non-blocking electrodes in electrochemistry. In the pioneer paper by Chang and Jae, the ionic current densities, j , on the electrodes 5

were assumed to be proportional to the bulk variation of the ionic density with respect to the value of thermodynamical equilibrium, n−neq , j = κ(n−neq ). As stated in the Introduction, this hypothesis is similar to that used in the description of the evaporation phenomenon.

4

More recently, it has been proposed an Ohmic model based on the assumption that the ionic current densities on the electrodes is proportional to the surface electric eld, E , j = sE .

6

In both models the phenomenological parameters describing the charge exchange at the interface are assumed frequency independent. The two models are equivalent, in the sense that xing κ it is possible to nd a value for s such that the predictions of the two models coincide in all frequency range.

18

By means of the Ohmic model it is easy to evaluate, in the framework of the PNP model, the electric impedance, Z , of a cell in the shape of a slab of thickness d and surface area S , containing ions, of electric charge q , dispersed in an insulating liquid of dielectric constant ε. Indicating by n0 the bulk density of ions, in thermodynamical equilibrium, λ = εKB T /(2n0 q 2 ) the length of Debye, ωD = D/λ2 Debye's relaxation frequency, ωc = 2qs/ε p the dielectric relaxation frequency, and β = (1/λ) 1 + iω/ωD , as shown in Z is given by p

6

Z=

βd(ωc + iω) + 2(ωD − ωc ) tanh(βd/2) . βεS(ωc + iω)(ωD + iω)

(1)

The theoretical predictions of (1) are in good agreement with the experimental data relevant to hydrogels, as mentioned above. However, in the case of water containing KCl in the low 7

frequency range the agreement is rather poor. To improve the agreement between theoretical

7



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predictions and experimental data, the PNP model with Ohmic boundary conditions has to be generalized to take into account that the phenomenological parameter describing the charge exchange at the electrodes could be frequency dependent.

3. Hopping model for non-blocking electrodes.

Some years ago a model for the ac conductivity, in equilibrium conditions, in terms of a percolation path method, has been proposed.

19,20

This model can be applied both to

electronic or ionic hopping system and to a mesoscopic or macroscopic non-homogeneous media.

21,22

In the framework of this model the ac conductivity depends on the angular

frequency of the applied electric eld according to the relation σ(ω) = σ0

iωτM , ln(1 + iωτM )

(2)

where σ0 is the dc conductivity and τM the maximum hopping relaxation time. Our proposal in the present work is to generalize Ohmic boundary conditions, assuming for the s parameter a frequency dependence similar to that given by (2), along the line suggested in. This is equivalent to assume that the charge exchange at the electrodes is of hopping 8

type. The phenomenological interpretation of this mechanism is the usual one. At a given temperature, a particular ion spend most of the time close to the place where its energy is minimum, localized close to the electrode, from which it is separated by an energy barrier. Occasionally, it acquires enough energy from the surrounding heat bath to jump into another local minimum on the electrode, giving rise to a charge transfer from the bulk to the external circuit. In the case where the energy barrier is innite, the electrode is blocking, and in the external circuit the electric current is just a displacement current due to the time variation of the surface electric eld. In the following we will develop this model and will compare the theoretical predictions to the experimental results of Z from an aqueous solution of KCl. In the framework of an Ohmic model based on the hopping conduction between the bulk 8


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and the electrode, the electric impedance of the cell in the shape of a slab is given by Eq.(1) where the phenomenological parameter s is given by s(ω) = s0

8

iωτM . ln(1 + iωτM )

(3)

Note that for ω 1/τ , s(ω) → s0 , i.e. we recover the standard Ohmic model. The physical reason of Eq.(3) can be understood taking into account that the presence of the interface makes the system under consideration be similar to a solid composed of regions with dierent electrical conductivity. In this case, as is well known for solids, the overall conductivity increases with the frequency of the electric eld. The reason of this phenomenon is due to the fact that, as the frequency of the eld increases, the frequency with which the charges attempt to overcome the potential barrier increases. Hence, at high frequencies, the charges located near the interface in their motion can more easily reach regions where the potential barrier is lower, and pass into the conducting regions of the electrodes.

4. Experimental validation of the model.

To test our model we have investigated the response of electrolytic cells in the shape of slabs limited by circular electrodes of radius R = 10 mm in the frequency range 10mHz ≤ f ≤ 10MHz. The cells have dierent thicknesses d, ranging from 0.5 mm to 2.5 mm.

The electrolytic solution is milli-Q water containing 1mM of KCl. We tested the linearity of the response of the cell with respect to the amplitude V0 of the applied voltage for a cell of thickness d = 1.0 mm. The results show that the real and imaginary part of the impedance for V0 equal to 10 mV, 15mV, 20 mV, 25 mV and 30 mV are independent of the amplitude V0 . Only a small deviations appear in the dc limit, in agreement with the analysis reported

in.

23

The real and imaginary parts of the electric impedance of the cells of dierent thickness, in the frequency range 10mHz ≤ f ≤ 10MHz are reported in Fig.1a, and Fig.1b, respectively. 9



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The ts are obtained xing the values of the bulk parameters of the electrolytic solution. We used for the positive, K + , and negative, Cl− , ions the same diusion coecient in water, D = 2 × 10−9 m2 /s,

24

and the electric charge q = 1.6 × 10−19 C. The bulk ionic density

corresponding to 1mM is n0 = 6 × 1023 m−3 . Finally, the dielectric constant of the water, free of ions, is ε = 80ε0 , where ε0 is the dielectric constant of vacuum. Since the cells are the 1

same, in the t we xed the values of s0 and τM . The best ts of the spectra corresponding to the considered set of experiments performed with dierent cells' thicknesses are shown in Fig.1a and in Fig.1b, and are obtained for σ0 = qs0 = 6.2 × 10−11 1/(Ωm) and τM = 42.6 s. We have investigated also the spectra of R and χ for the electrolytic cell of thickness d = 1.0 mm in the presence of an external dc bias, Vb , of 0, 0.2, 0.4, 0.6, and 0.8 V. The

results for 0, 0.6 and 0.8V are reported in Fig.2, for the real, (2a), and imaginary parts, (2b), of Z . The best t of the experimental data by means of the Ohmic model based on the hopping conduction on the electrodes have been obtained by xing the values of the bulk parameters. As it is clear from Fig.2, the agreement between theoretical predictions of the Ohmic model with surface conductivity of hopping time decreases as the bias increases. This indicates that the hopping model developed for the bulk has to be modied to describe in a proper manner the conductivity of non-blocking electrodes. The work is in progress. The dependence of s0 and τM on Vb are shown in Fig.3. If the barrier height is U0 , the probability per unit time for a jump of the charge carrier is the attempt frequency times exp(−U0 /KB T ). Consequently the hopping time is τ = τ0 exp(U0 /KB T ). In the presence

of a dc bias voltage Vb , U = U (Vb ) = U0 − α q Vb , where α Vb is the potential drop at the interface. It follows that in the presence of bias the maximum hopping time is expected to be ∗ τM (Vb ) = τM exp(−αqVb /KB T ),

(4)

where τM∗ = τM (0). In Fig.3 with the experimental data of τM (Vb ) we show the best ts obtained by means of Eq.(4). The parameter α is expected to be of the order of α ∼ Rh /λ, where Rh is the hopping 10


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distance and λ the length of Debye introduced above. The parameters of the best ts are ∗ τM = 70 s and α = 7 × 10−2 . Assuming that the dc bias decreases in an exponential manner

over the Debye's length, Rh ∼ 6.5 × 10−10 m. Since σ ∝ 1/τ ,

19,20

the dependence of s0 on

Vb is expected to be of the functional form s0 (Vb ) = s∗0 exp(−βqVb /KB T ),

(5)

where s∗0 = s0 (0) and β is expected to be very close to α. In Fig.3 we also show the experimental data of s0 (Vb ) obtained by the best t of the data reported in Fig.2. The solid line is the best t obtained by means of Eq.(5). The best t is obtained for β ∼ 7 × 10−2 , approximately equal tothe value of α, in agreement with the theoretical expectation.

5. Conclusions

The electric response of electrolytic cells of KCl in milli-Q water of dierent thicknesses and xed doping, in the absence and presence of a dc bias, have been investigated. The experimental data have been analyzed by means of a generalization of PNP model with Ohmic boundary conditions for the ionic charge exchange at the electrodes. Our model diers from previously proposed models in the sense that the sample is assumed to be homogeneous and the presence of the electrodes, responsible for the electrode polarization, are taken into account by means of a boundary condition on the electrode. Comparing the experimental results relevant to a set of cells with dierent thickness we have shown that all the data are in good agreement wit the theoretical prediction. In this case, in the t procedure, we have xed the parameters of the bulk, namely the diusion coecient, ionic bulk density and dielectric constant, and the hopping parameters for boundary conditions. In the presence of bias, taking into account that the drop of the bias potential is mainly localized at the interfacial layer, a strong surface electric eld is expected. This eld is responsible for a bias dependence of the hopping time and of the dc conductivity. By the best t of the maximum 11



hopping time and of the dc conductivity on the bias voltage, assuming a linear decreasing of the barrier height with the bias, we have determined the order of magnitude of the hopping distance. In the presence of the dc bias, the agreement between theoretical predictions and experimental data, in the dc limit, decreases increasing of the bias. This result indicates that the existing models for the hopping conduction proposed for the bulk in the presence of the bias have to be reanalyzed to describe the surface conductivity of the electrodes.

Acknowledgements This work was supported by the MEPhI Academic Excellence

Project (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, project no. 02.a03.21.0005); Conselho Nacional de Desenvolvimento Cientíco e Tecnológico (CNPq) (465259/2014-6); Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (2008/57685-7, 2011/13616-4); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx); Núcleo de Apoio à Pesquisa de Fluidos Complexos (NAP-FCx) (2011.1.9358.1.6).

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(14) Macdonald, J. R. Theory of ac Space-Charge Polarization Eects in Photoconductors, Semiconductors, and Electrolytes. Phys. Rev. 1953, 92, 4-17. (15) Barbero, G.; Alexe-Ionescu, A. L. Role of the diuse layer of the ionic charge on the impedance spectroscopy of a cell of liquid. Liquid Crystals 2005, 32, 943-949. (16) Lenzi, E. K.; Lenzi, M. K. ; Silva, F. R. G. B.; Goncales, G.; Rossato, R.; Zola, R. S.; Evangelista, L. R. A framework to investigate the immittance responses for nite lengthsituations: Fractional diusion equation, reaction term, and boundary conditions. J.

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Figures

Figure 1: Frequency dependence of R = Re[Z], (a), and χ = Im[Z], (b), of electrolytic cells of thicknesses d = 0.5, 1.0, 1.5, 2.0 and 2.5 mm subjected to an external voltage of amplitude V0 = 30 mV. The points are the experimental data. The best ts are obtained by means of the PNP model with Ohmic boundary condition, assuming for the surface electric conductivity σ = qs and the frequency dependence predicted by a model based on the hopping mechanism for the charge exchange between the bulk and the external circuit. The bulk parameters are D = 2×10−9 m2 /s, n0 = 6×1023 m−3 , ε = 80×ε0 , q = 1.6×10−19 C (monovalent ions), xed by the experimental conditions. The mean free parameters in the best-t procedure are the hopping time τM = 42.6 s and the dc conductivity of the interface σ0 = 6.2×10−11 C(msV)−1 .

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Figure 2: Frequency dependence of R = Re[Z], (a), and χ = Im[Z], (b), of an electrolytic cells of thicknesses d = 1.0 mm subjected to an external voltage of amplitude V0 = 30 mV, in the presence of dc bias of Vb = 0, 0.6 and 0.8 V. The points are experimental data, the continuous lines are best ts obtained by means of PNP model with Ohmic boundary conditions of hopping type.

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Figure 3: ( ), bias dependence of the hopping time obtained by the best-t procedure (experimental points), and best t obtained by means of Eq.(4). The parameters of the best t are τM∗ ∼ 70 s and α ∼ 7 × 10−2 from which it follows Rh ∼ 6.5 × 10−10 m. (), bias dependence of the dc conductivity s0 obtained by the best-t procedure (points), and best t obtained by means of Eq.(5). The parameters of the best t are s∗0 = 2.8 × 108 (msV)−1 and β ∼ 7 × 10−2 .

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TOC graphic

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Random Energy Barrier Model for Ac Electrode Conductivity - The

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