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Electrochimica Acta 55 (2009) 363–373

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

The effect of IR compensation on stationary and oscillatory patterns indual-electrode metal dissolution systems

Swati Jain a, István Z. Kiss b, Jennifer Breidenich a, John L. Hudson a,∗

a Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22904, United Statesb Department of Chemistry, Saint Louis University, St. Louis, MO 63103, United States

a r t i c l e i n f o

Article history:Received 30 October 2008Received in revised form 26 February 2009Accepted 27 February 2009Available online 14 March 2009

Keywords:IR compensationElectrode arraysNegative couplingAmplitude deathSwitching entrainment ratios

a b s t r a c t

We report an experimental and model based study on the effect of negative coupling, induced by addingIR compensation, on bistability, and synchronization behavior of a dual-electrode metal dissolution elec-trochemical system. We show that, unlike the case of a single electrode, IR compensation cannot beused to remove bistability; with a large IR compensation the electrodes do not exhibit uniform steadystates and patterned surface develops. In the case of oscillatory system, addition of IR compensationproduces aperiodic time series that are characterized by switching between oscillations with 1:1, 1:2,and 2:1 entrainment ratios. For higher negative coupling strengths (i.e., larger magnitude of IR com-pensation) amplitude death occurs and either coexistence of oscillations with steady state or multipleanti-symmetric steady states are observed.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Interplay among electrode kinetics, transport of ions, potentialdrop in the electrolyte, and external circuitry can lead to a variety ofnonlinear phenomena in electrochemical systems including bista-bility [1], oscillations [2], phase locked entrainment [3–9], standingwaves [10,11], rotating waves [12], clusters [5,13,14], acceleratingfronts[15], and stationary patterns [16]. In classical electro-kineticstudies such “cell instabilities” are usually avoided by minimizingthe potential drop either by placing the reference electrode to theclose proximity of the working electrodes, or by electronic meanswith IR compensation [17]. It has been shown, however, that placingthe reference electrode in close proximity to the electrode surfaceinduces negative coupling [18,4] that can cause stationary spatialpatterns [19,20], remote triggering of waves [21], and oscillatorycluster formation [22]. Krischer et al. have shown that external IRcompensation can introduce a negative global feedback in the cir-cuit whose effect is equivalent to moving a reference electrode closeto the working electrode [6].

The investigation of the effect of negative coupling induced byIR compensation on the behavior of electrochemical cells is com-plicated because IR compensation not only changes the amount ofnegative coupling among the reaction sites, but also changes thelocal dynamics through affecting the overall cell potential drop. For

∗ Corresponding author. Tel.: +1 434 924 6275; fax: +1 434 982 2658.E-mail address: [email protected] (J.L. Hudson).

example, in a study with two oscillatory iron electrodes [9] at inter-mediate reference-to-working electrode distances, a ‘gray’ regionwas found in which it was difficult to identify in-phase or anti-phase entrainment typically seen with distant and close electrodeplacements, respectively.

In this paper, we investigate the effect of negative couplinginduced by IR compensation on stationary and oscillatory patternformation. We utilize a dual working electrode setup in whichthe negative coupling of IR compensation can be studied with-out changing the local dynamics by carrying out the experimentsat fixed total resistance. We carry out a systematic study over awide, but finely tuned, range of negative coupling strength withoutchanging other conditions such as driving force for the reaction. Thestationary and oscillatory patterns are analyzed as a function of theimposed negative coupling with nickel and iron electrodissolution,respectively. Complementary numerical simulations are performedto confirm the experimental findings and to explore the nature ofdynamical transitions leading to pattern formation.

2. Experimental

A schematic of the experimental set up as well as a representa-tive equivalent circuit are shown in Fig. 1. The two (nickel or iron)working electrodes are embedded in epoxy. The reference electrodeis Hg|H2SO4|K2SO4 and the counter electrode is made of a Pt–Nbmesh. The counter electrode was wrapped around the circumfer-ence of the cell. The reference electrode was kept farthest from theelectrodes but in the center of the line joining them. A PAR 273A

0013-4686/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.electacta.2009.02.105


364 S. Jain et al. / Electrochimica Acta 55 (2009) 363–373

Fig. 1. (a) Experimental set-up. (b) Equivalent circuit for the two electrodes.

potentiostat was used to add IR compensation, Rcomp < 0. In manyof the experiments two parallel resistors, Rp1 = Rp2 = Rp, are alsoadded as shown in Fig. 1. Rc is the collective resistance in series withthe electrodes; it models the coupling effect of potential drop in theelectrolyte (Rc > 0) or negative coupling induced by IR compen-sation (Rc < 0). Because of the presence of parallel and collectiveresistances in the equivalent circuit (Fig. 1(b)) we can introduce anequivalent resistance, Req = Rp + 2Rc , that characterizes the overallresistance in the cell.

2.1. Experimental methodology in the bistable region; nickelelectrodissolution

The electrodes were made of nickel and have a diameter of 1 mm.The electrolyte was 3 M sulfuric acid. The solution was stirred toavoid the formation of a diffusional boundary layer.

2.2. Experiment methodology in the oscillatory region; ironelectrodissolution

Iron electrodes of diameter 1 mm were used for the study. Thedistance between the electrodes was set to 2.8 cm. The electrolytewas 0.5 M sulfuric acid and the series solution resistance betweenthe working and the reference electrodes is 40 �, as measured usingAC impedance. The experiments were performed by first passivat-ing the iron electrode by scanning the circuit potential (at a rateof 5 mV/s) from −934 mV to 150 mV. After passivation, the cir-cuit potential was scanned backwards from 150 mV to −202 mVwhere the current oscillations were recorded at a given compen-sated resistance, Rcomp. To maintain the total resistance of the cell

constant during IR compensation, two external parallel resistorsRp = −2Rcomp were applied (see Fig. 1).

3. Model

3.1. Bistable region

The electrical coupling between the two electrodes can be mod-eled by the presence of a combination of parallel and collectiveresistors shown in Fig. 1(b). Thus the double layer potentials ofthe two electrodes (e1 and e2) in the presence of coupling vary as[23,24,9,25]:

de1

dt= v − e1

Req− iF (e1) + K

2(e2 − e1) + A�1 (1)

de2

dt= v − e2

Req− iF (e2) + K

2(e1 − e2) + A�2 (2)

A forward or a backward scan can be made at a rate s.

dvdt

= s (3)

The coupling strength K is given by [23]

K = �1 − �

1Req

(4)

where � = 2Rc/Req is the collective resistance fraction. Note that�/(1 − �) = 2Rc/Rp. IR compensation applied to both electrodesintroduces negative coupling: Rc = Rcomp < 0 thus K < 0.

The term A�i has been added to the system to account forunavoidable small noise that exists in the experimental system. Thisnoise eliminates long, spurious transients that can occur with twocoupled identical systems. A is noise amplitude and was set to a lowvalue of 0.05.

�(t) = ran(1) − 0.5

ran(1) is a uniform random number between 0 and 1.The reaction current iF (ei) is approximated by a cubic equation.

iF (ei) = 1.1e3i − 2.3e2

i + 1.4ei + 0.013

The approximation simulates a negative differential resistanceof the electrochemical system: an initial increase is followed by adecrease of Faradaic current with electrode potential.

3.2. Oscillatory region

A model that describes the relaxational active–passive oscilla-tions for a single electrode was proposed by Koper and Sluyters [24];the model incorporates variables for electrode potential (e), surfacehydrogen ion (h) and iron(II) ion (u) concentrations. This modelhas been modified here to describe an IR compensated system oftwo relaxational oscillators coupled through the electrolyte with aseries (collective) resistance Rs and IR compensation Rcomp < 0 inseries. Therefore, in Fig. 1(b) Rc = Rs + Rcomp. To maintain the systemat constant equivalent resistance, Req = Rs, we introduce parallelresistances Rp1 = Rp2 = Rp = −2Rcomp. In this way, we can investi-gate the effect of negative coupling exerted by the IR compensationwithout changing the dynamics through changing Req. The modelin dimensionless form:

(1 + �)de1

dt= v − e1

req− qk∗

1(1 − �i) + �(e2 − e1)

2(5)

dh1

dt= d(hb − h1) − th1

d1/2

q

(v − e1

req+ �

(e2 − e1)2

)(6)


S. Jain et al. / Electrochimica Acta 55 (2009) 363–373 365

du1

dt= −˛du1 + d1/2k∗

1(1 − �1) − tu1 d1/2

q

(v − e1

req+ �

(e2 − e1)2

)

(7)

(1 − �)de2

dt= v − e2

req− qk∗

2(1 − �2) + �(e1 − e2)

2(8)

dh2

dt= d(hb − h2) − th2

d1/2

q

(v − e2

req+ �

(e1 − e2)2

)(9)

du2

dt= −˛du2 + d1/2k∗

2(1 − �2) − tu2 d1/2

q

(v − e2

req+ �

(e1 − e2)2

)

(10)

�i is the fractional coverage of the ith electrode with a passive oxidelayer.

�i = 11 + (k2/k−2)hi

� is the electric coupling strength

� = Rcomp + Rs

RsRp(DCd)

Note that � plays the same role as K in Eqs. (1), (2), and (4)because the Req = 2Rs and Rc = Rcomp + Rs. Rcomp < 0 is the externalcompensation. Cd is the capacitance, D is the diffusivity, and theterms 1 + � and 1 − � account for the difference in the capacitanceof the two electrodes.

The current is calculated as

I1 = (v − e1)req

+ �(

e2 − e1

2

)(11)

I2 = (v − e2)req

+ �(

e1 − e2

2

)(12)

Other parameters and variables in the model are describedbelow.

v, ei, r, q are the dimensionless quantities given by

v = VappF

RT

req = 2DCd(Rs) and

ei = EiF

RT

q = F2A1/2

2RTCd

d = A

ı2

Vapp is the applied potential, Ei is the potential drop across the dou-ble layer of the i-th electrode, hi is the local concentration of thehydrogen ions near the surface and ui is the local concentration ofmetal ions near the surface.

R is the universal gas constant, T is the temperature, F is Faradayconstant. � is the fractional coverage of the surface by the oxide. ıis the thickness of boundary layer.

˛ = is the ratio of diffusivities of Fe2+ ions and H+,

tui = ˛ui

hi(1 + ˛) + 2˛ui

thi = hi/(hi(1 + ˛) + 2˛ui) transference numbers for the metal ionsand the hydrogen ions.

k∗i = ki

(1 − ui

uci

)

ki = k∗o exp

(ei − eo

12

)

k2

k−2= ko

2ko

−2(exp(ei − eo

2))

uci = − 12

hi +(

14

h2i + Ksol

)(1/2)

Eqs. (5) and (8) are the result of application of the Ohms law andKirchoff law to the electrochemical circuit. Eq. (6), (7), (9), and (10)are the equations describing the local kinetics and molar diffusionin the system. Note that although the coupling between the elec-trodes is electric, the coupling strength � also appears for the massbalance of hydrogen and iron(ii) ions because they transfer a por-tion of the current, and, as shown in Eqs. (11) and (12), the currentsof the electrodes depend on �.

3.3. Data analysis: reconstruction of phase

To interpret the synchronization behavior of the two oscillatorsin the presence of coupling we will use a phase difference measurefor m:n entrainment [26]. An approach based on Hilbert transform[27] is used to assign a phase to every point on the limit cycle. TheHilbert transform shifts the time series signal by an angle of �/2 andthe phase can be calculated with respect to the shifted signal. Theinstantaneous phase pi(t) for a current signal I(t) can be obtainedas the arc tangent of a complex function (t); where,

(t) = I(t) + jH(I(t)) (13)

I(t) = I − mean(I)

H(I(t)) is the Hilbert Transform of I(t).The frequency fi (i=1,2) of the signal from the i-th electrode is

given by

fi = 12�

⟨dpi

dt

⟩(14)

If the oscillators are m:n entrained, the phase difference,�mn = np1 − mp1, should be a constant. This is equivalent to thefrequency difference (nf1 − mf2 = (d�mn/dt)) being zero.

4. Results and discussion

4.1. Experiments

4.1.1. Bistable regionFig. 2(a) shows the forward and backward polarization curves for

the two nickel electrodes in 3 M sulfuric acid with no added exter-nal resistance. The electrodes are monostable at these conditionsbut exhibit negative differential resistance for 1.06 V < V < 1.14V.Fig. 2(b) shows the result of adding an external resistance, Rp =602�, to each electrode. As a result of the presence of the resistors,bistability can be observed for 1.16 V < V < 1.22 V. This bistabilitydevelops because of the negative slope on the polarization curve inFig. 2(a).

The added external resistance can be compensated by addingIR compensation. The polarization curves for a fully compen-sated external resistance Rcomp = −Rp/2 = −301� are plotted inFig. 2(c). Now the total added resistance is zero, however, insteadof obtaining the polarization curve in Fig. 2(a) we observed asymmetry broken state in a potential range where the negativeslope of the polarization curve starts to develop: one electrodeexhibits larger, the other electrode lower current than it wouldbe expected from the polarization curve without added resistance.



Fig. 2. Polarization scan curves of nickel electrodes at different values of added external resistance, Rp and IR compensation Rc; forward scan rate: = 1 mV/s and backwardscan rate: = −1 mV/s. (a) Rp = 0, Rc = 0. (b) Rp = 602�, Rc = 0. (c) Rp = 602�, Rc = −301�.

Apparently, because of the global nature of the IR compensationsymmetry broken patterns develop in the two-electrode sys-tem.

4.1.2. Oscillatory regionThe effect of IR compensation in an oscillatory system is inves-

tigated with the oscillatory iron electrodissolution.The conditions of the experiments were chosen such that the

current signal from the two electrodes without adding IR com-pensation exhibits relaxational active–passive oscillations (Vapp =−202mV , Rcomp = 0, and Rp = 0).

The current oscillations for single electrodes of the two-electrode setup are shown in Fig. 3(a). (The currents from thesetwo separate experiments are overlaid for better comparison of theamplitudes and frequencies.) The two electrodes exhibit a largefrequency (28%) and small peak-current (3%) difference. Fig. 3(b)shows the phase difference, �11, between the uncoupled oscilla-tors (time series in Fig. 3(a)). Since the oscillators are not entrained,the phase difference changes almost linearly with time.

Fig. 3(c) shows the time series for the two oscillators that areconnected simultaneously to the circuit with no IR compensationor an external resistor. The oscillators are 1:1 in-phase synchronized

Fig. 3. Current time series (electrode 1: solid line and electrode 2: broken line) and phase difference for iron electrodes at Vapp = −202 mV, Rcomp = 0, and Rp = 0. (a) Currentfor uncoupled electrodes. f1 = 0.192 Hz and f2 = 0.24642 Hz. (b) Phase difference (�11 = p1 − p2) as a function of time for current in (a). (c) Current for the simultaneouslyconnected electrodes. (d) Phase difference for 1:1 entrainment for current in (c).



because the electrodes are coupled through the potential drop inthe electrolyte. The phase difference, �11, (Fig. 3(d)) is nearly aconstant with time with a mean value close to zero.

4.1.2.1. Effect of adding negative coupling between the oscillators.Negative coupling was now added by applying IR compensation,Rcomp. The coupling strength was increased by increasing the mag-nitude of Rcomp and the time series of the current were recordedat each Rcomp. Because IR compensation adds a negative resistor inseries with the electrodes, we add Rp = −2 × Rcomp parallel resis-tors to each electrode. Thus the total resistance of the cell is notchanged and the negative coupling induced by IR compensation canbe studied without changing the local dynamics of the oscillators.

Fig. 4(a) shows the current time series for very weak added nega-tive coupling, Rcomp = −2.5�. The oscillators are not synchronized.The phase difference plot in Fig. 4(b) indicates a drifting behaviorwith a frequency difference (28%) that corresponds to that of theindividual (uncoupled) oscillators.

For weak negative coupling strengths, i.e., 5� ≤ |Rcomp| ≤ 15�,the oscillators were out of phase 1:1 synchronized with someintermittent phase slips. The current time series and the phasedifference at Rcomp = −10� are shown in Fig. 4(c) and (d), respec-tively.

At moderate added negative coupling strength, 20� ≤ |Rcomp| ≤55�, the time series were aperiodic; snapshots of the time seriesfor Rcomp = −30� and Rcomp = −40� are shown in Fig. 5(a) and (c),respectively. In each time series there are long time intervals forwhich the oscillators are either entrained with a ratio of 1:1 , 1:2 , or2:1. (For example, see the phase difference plots for 1:1 entrainment(�11), 1:2 entrainment (�12), and 2:1 entrainment (�21) areshown in Fig. 5(c) and (d) for Rcomp = −30� and Rcomp = −40�,respectively). Repeated switching between these entrainments wasseen, however, there was no obvious trend in switching times orlength of the time interval of a particular type of entrainment in atime series.

At a stronger added negative coupling, Rcomp = −75�, the oscil-lators are 1:1 entrained and are anti-phase with respect to eachother (see Fig. 6(a) and (b)).

To check the robustness of the oscillations and possible exis-tence of other dynamic states a perturbation was applied to thesystem by quickly turning the potentiostat off and on. On apply-ing the perturbation at high added negative coupling strengths, theoscillations were replaced by steady states that are anti-symmetricwith respect to each other, i.e., one electrode was active and thesecond electrode was passive; as an example current time seriesfor Rcomp = −75� is shown in Fig. 7 (a) and (b). This phenomenonof disappearance of oscillations is referred to as amplitude death.

No oscillations were seen at IR compensation larger than 75�.Anti-symmetric steady states are seen at Rcomp = −100� (Fig. 7(c)).If a perturbation was applied to this state the electrode that wasactive before applying the perturbation became passive and thepassive electrode was now active; the time series after perturbationis shown in Fig. 7(d).

4.1.2.2. Characterization of the trend of appearance of differententrainment ratios as a function of coupling strength. The phase dif-ference, �mn, is constant with respect to time when the oscillatorsare m:n synchronized. This property of the phase difference is usedhere to estimate the length of the time for which the oscillatorsare m:n entrained in a time series at a fixed coupling strength.Fig. 8 shows the proportion of the time for which the oscillatorsare entrained with a particular entrainment (1:1, 1:2, or 2:1) ratioin a time series, as a function of Rcomp.

The percentage of 1:1 entrained oscillations in a time seriesdecreased from 100% at Rcomp = 0� to a minimum at Rcomp =−2.5�, the point where the oscillators are drifting with respect toeach other and there is no synchronization (time series in Fig. 4(a)).The proportion of 1:1 entrained oscillations rises again to a valuenear one at weak added coupling strengths (time series in Fig. 4(c)).

Fig. 4. Current time series (electrode 1: solid line and electrode 2: broken line) and phase difference for two iron electrodes for very weak added negative coupling strengthsat Vapp = −202 mV. (a) Current for Rcomp = −2.5� and Rp = −2Rcomp = 5�. (b) Phase difference, �11, as a function of time for current in (a). (c) Current for Rcomp = −10� andRp = −2Rcomp = 20�. (d) Phase difference, �11, as a function of time for current in (c).



Fig. 5. Current time series (electrode 1: solid line and electrode 2: broken line) and phase difference for two iron electrodes at intermediate added negative coupling strengthsat Vapp = −202 mV. (a) Current for Rcomp = −30� and Rp = −2Rcomp = 60�; regions of different entrainment are also marked. (b) Phase difference �11 (top panel), �12

(middle panel), and �21 (bottom panel) as a function of time for current in (a). (c) Current for Rcomp = −40� and Rp = −2Rcomp = 80�. (d) Phase difference �11 (top panel),�12 (middle panel), and �21 (bottom panel) as a function of time for current in (c).

For Rcomp less than Rcomp = −20� no 2:1 or 1:2 entrainments areseen; therefore, the proportion for these in the current time seriesis 0.

At higher coupling strength 20� ≤ |Rcomp| ≤ 65� all three formsof entrainment are seen. The percentage of 1:1 entrainment falls toa minimum at Rcomp = −40� and then rises again to a value of 1 atRcomp = −75�, where we see anti-phase oscillations (Fig. 6). How-ever, the decrease is not monotonous and there is a local maximaat Rcomp = −25�.

Only two instances of 1:2 entrainment were seen, at Rcomp =−20� and Rcomp = −40�. The percentage of number of 2:1entrained oscillations in a time series rise from zero at Rcomp =−25� to a maxima and then drops back to zero at Rcomp = −75�.

The quantitative analysis of the entrainment regions confirmedthe qualitative observation that at intermediate negative couplingstrengths 1:1, 1:2, and 2:1 entrainment coexist. In particular, atabout Rcomp = 40� the three entrainment regions exist for aboutthe same overall time intervals.

4.2. Simulations

4.2.1. Bistable regionThe existence of stationary patterns due to IR compensation

is modeled with numerical integration of a general electrochem-ical model with negative differential resistance described with Eqs.(1–(4). The system is monostable for small added external resis-

Fig. 6. (a) Current time series (electrode 1: solid line and electrode 2: broken line) for a strong added negative coupling strength; Vapp = −202 mV and Rcomp = −75�,Rp =−2Rcomp = 150�. (b) Phase difference, �11, as a function of time for current in (a).



Fig. 7. Amplitude death in the experiments. Current (electrode 1: solid line and electrode 2: broken line) as a function of time at Vapp = −202 mV. (a) Oscillations atRcomp = −75�, Rp = −2Rcomp = 150� (same as Fig. 6(a)). (b) Coexisting anti-symmetric steady state with oscillations in (a). (c) Anti-symmetric steady states at higher couplingstrength; Rcomp = −200�, Rp = −2Rcomp = 400�. (d) Anti-symmetric steady states after applying the perturbation to time series in (c).

Fig. 8. Percentage of different entrainment ratios in the time series ((©) 1:1 entrain-ment, (�) 2:1, (�) 1:2) as a function of Rcomp .

tances; at Req = Rp = 0.1 and Rc = 0 the polarization curve is shownin Fig. 9(a). When Req = Rp is increased to a value of 10, bistabilityoccurs as shown in Fig. 9(b). With the addition of IR compen-sation Rc = Rcomp = −4.95 to the system (Req = 0.1, Rp = 10) the

bistability is replaced with pattern formation with one electrodeexhibiting larger, the other lower current, than the expected valuesfrom the negative slope of the polarization curve (see Fig. 9(c)). Theresult qualitatively reproduces the experimental findings shown inFig. 2.

The disappearance of the uniform steady states can be explainedthrough the linear stability analysis of the model. (See Appendixfor a detailed analysis of the results concerning this section.) For asingle electrode, without any external compensation, steady statesare stable if the condition −1/Req − diF (e)/de < 0 holds [28]. Forthe two-electrode model the uniform steady states will be sta-ble when −1/Req − diF (e)/de − K < 0, where K is the strength ofelectric coupling between the electrodes. This implies that IR com-pensation contributes to the stability of the cell by reducing thetotal cell resistance from the original R0 = Req to Req = R0 + 2Rcomp

but causes cell instability by introducing negative coupling K < 0. Inthe simple configuration used in the experiment we can evaluatethe overall effect of the IR compensation: −1/Req − K = −1/(R0),where R0 is the equivalent resistance where bistability can beobserved without IR compensation. Apparently, the stabilizingand destabilizing effects of IR compensation cancel each other;

Fig. 9. Polarization scan curves (forward scan rate: s = 0.001 and backward scan rate: s = −0.001) obtained through numerical simulations of the model (Eqs. (1)–(4)).A = 0.05. Results are shown for different values of added parallel resistances Rp and IR compensations Rc . (a) Very small added external resistance; Rp = 0.01, Rc = 0. (b) Largeadded external resistance with no compensation; Rp = 10, Rc = 0. (d) Large added external resistance with almost full compensation. Rp = 10, Rc = −4.95.



Fig. 10. Numerical simulations of the model (Eqs. (5)–(10)). Current (electrode 1: solid line and electrode 2: broken line) at different coupling strengths. Other parameters arereq = 0.085, � = 0.7, v = 30, Ksol = 16, ˛ = 0.2, q = 100, hb = 1, ko = 0.01, ko

2/ko−2 = 0.1eo

1 = 20, and eo2 = 20. (a) Positive coupling: � = 0.25. (b) Uncoupled electrodes: � = 0

(frequency difference = 28 %). (c) Weak negative coupling: � = −0.14. (d) Strong negative coupling: � = −1.

the stability does not depend on the extent of IR compensation,Rcomp.

4.2.2. Oscillatory regionWe investigate the effect IR compensation in a two-electrode

model for active–passive oscillations. During the experiments thetotal cell resistance was kept constant. Increasing the IR compen-

sation makes the coupling strength more negative as seen in Eqs.(5)–(12).

Without IR compensation there is positive coupling due to thepotential drop in the electrolyte. This coupling can be simulatedwith a small positive value for �. At � = 0.04 the oscillators arein-phase synchronized (Fig. 10(a)). As the coupling strength wasdecreased to � = 0 the oscillators are uncoupled (Fig. 10(b)). At a

Fig. 11. Amplitude death in the model. Current (electrode 1: solid line and electrode 2: broken line) as a function of time. (a) Oscillations at � = −4.85. (b) Coexistinganti-symmetric steady states with the time series in (a). (c) Steady state at � = −7. (d) Second anti-symmetric steady state at � = −7.



Fig. 12. Linear stability analysis of the model (Eqs. (5)–(10)) as the parameter � is decreased; other parameters are same as in Fig. 10. (a) Bifurcation diagram: fixed pointvalues of e1 vs. coupling strength −�. Solid line: stable fixed points; broken line: unstable fixed point. Supercritical Hopf bifurcation point is marked with a (*). Regions foroscillations (OSC), Coexistence of steady states with oscillations (CE), and Multiple steady states (MS) are also show. (b) Bifurcation diagram of e2. (c) Limit cycles of the twoelectrodes (electrode 1:dark black line and electrode 2: broken gray line) and unstable fixed points (×) before the Hopf bifurcation. � = −4. (b) Limit cycles and stable fixedpoints (×) after Hopf bifurcation. � = −4.85.

small negative coupling strength, � = − − 0.14, the oscillators are3:4 entrained as shown in Fig. 10(c). As the coupling strength wasmade more negative from this point various entrainment ratiosclose to 1:1 were observed, e.g., 4:5, 9:11, 10:12. Eventually, 1:1entrained, anti-phase oscillations were obtained at large negativecoupling strengths; an example at � = −1 is shown in (Fig. 10(d)).

The oscillator were 1:1 synchronized at more negative couplingstrengths until the point of amplitude death, � ≈ −4.8. Coexis-tence of anti-phase oscillations (Fig. 11(a)) and anti-symmetricsteady states (Fig. 11(b)) was seen at � = −4.85. At higher cou-pling strengths multiple anti-symmetric steady states similar tothe experiments were observed; for example, the anti-symmetricstationary states at � = −7 are shown in Fig. 11(c) and (d).

4.2.2.1. Stability analysis. The appearance of anti-symmetric steadystates, as the parameter K is decreased, can be described with anumerical linear stability analysis of the ordinary differential Eqs.(5)–(10). (The numerical analysis was done using the program AUTOin software XPP.) The obtained bifurcation diagrams showing thesteady states and their stability as a function of coupling strengthare shown in Fig. 12(a) and (b) for electrode 1 and electrode 2,respectively.

For each electrode, the unstable and symmetric fixed point closeto the center of the limit cycle (at around e = 20) goes through apitchfork bifurcation after which unstable, symmetry broken steadystates are created. After a saddle-node bifurcation these symmetrybroken states become stable for � < −4.8 through a Hopf bifurca-tion. The three regions marked as OSC, CE, MS correspond to thevalues of the parameter K for which oscillations, coexistence ofoscillations and steady state, and multiple steady states are seen. A3D plot of the limit cycle and fixed points of the oscillators justbelow and just above the Hopf bifurcation points are shown inFig. 12(c) and (d), respectively. The fixed points (marked with ∗).for the two electrodes are created on the opposite sides of the limitcycles.

These numerical simulations and linear stability analysis clarifythe experimentally observed transition from in-phase entrainmentthrough anti-phase entrainment to amplitude death with increas-ing the (negative) coupling strength between the oscillators. Themodel does show higher mode entrainment between the in-phaseand anti-phase behavior, however, transitions among differententrainment regions similar to the experiments were not observedin the simulations.

5. Conclusions

Stationary and oscillatory patterns were observed due to neg-ative coupling induced by IR compensation in dual-electrodeelectrochemical systems exhibiting negative differential resistance(nickel and iron electrodissolution).

Traditionally, IR compensation has been used to remove thebistability in single electrode systems [1]. Our results show thatwhen IR compensation is applied (globally) to two nickel elec-trodes, the uniform state corresponding to the negative slope onthe polarization curve cannot be stabilized. The stabilizing effect ofIR compensation on bistability is counteracted by the destabilizingeffect due to negative coupling. The negative coupling induced asymmetry breaking bifurcation resulting in new active (high cur-rent) and passive (low current) states that are not seen in thedynamics of the single electrode without interactions. When inter-actions are considered (e.g., reference electrode placed close theelectrode surface), even a single electrode can concurrently exhibitthese active and passive states [19,29]. The formation of thesepatterns cannot be neglected in traditional electrochemical inves-tigations; for example, the patterns strongly affect the impedancespectrum of the system [30].

IR compensation also affected the oscillatory electrodissolution.Without negative coupling in-phase, with strong negative couplinganti-phase synchronization was observed. These results are thesame as those found by inducing negative coupling through refer-



ence electrode placement [31] very close to the working electrode.However, with IR compensation used in our study a fine tuningof negative coupling was possible with which the transition fromin-phase to anti-phase synchrony could be investigated withoutchanging the inherent dynamics of the oscillators. We found thatat negative coupling strengths in between those correspondingto in- and anti-phase synchrony, phase-drift, out-of-phase syn-chrony, and alternating 1:1, 1:2, and 2:1 synchrony patterns canbe observed. Although 1:2 phase locking had been reported foran iron disk electrode, [25], the switching among entrainmentstates indicates the existence of complicated phase dynamics. Therehave been reports of similar behavior in the cardio-respiratorysystem [32]. The signals from the cardiac system and the respi-ratory system stay synchronized with a certain entrainment ratiofor some time and then change the entrainment ratio. Such con-tinuous transitional dynamics in cardiorespiratory system wasattributed to the presence of long-term correlated noise, super-imposed by the activity of lower frequency oscillatory component[33]. The relationship between the dynamics of electrochemicaland the cardiorespiratory system is not clear, however, the fairlylarge frequency difference among the oscillators probably playsan important role. This observation is also supported by the factthat with oscillatory nickel electrodissolution, where the frequencydifference between the oscillators is small (< 10%), such switch-ing entrainment ratios were not observed with negative coupling[5].

With IR compensation it was possible to investigate the effectof extremely strong negative coupling; such very strong negativecoupling could be imposed with reference electrode placementsvery close to the working electrode. We observed amplitude death:because of the strong negative coupling the two oscillators repeleach other to such an extent that the oscillations cease to exist.Amplitude death has been reported in chemical (BZ reactions),biological and electrochemical systems [34–41]. The amplitudedeath reported in this work is different than those observed withpositively coupled oscillations close to Hopf bifurcation in nickelelectrodissolution [41]. In contrast to the symmetric steady statesarising close to Hopf bifurcations with the nickel electrodes, thesteady states were strongly asymmetric. The amplitude death sim-ilar to the results in this work was reported in BZ reactions [35]. Inthis mechanism, the steady states are created on the opposite sidesof the limit cycles of the oscillators.

We developed a model for a dual-electrode iron electrodisso-lution to investigate the effects of IR compensation. The modelshowed that when negative coupling was added to 1:1 in-phasesynchronized oscillators the behavior changed to the anti-phaseamplitude oscillations, similar to what is seen in the experiments.It also explained the transition to a region of coexistence of oscil-lations and anti-symmetric steady states and finally a multipleanti-symmetric steady states as the coupling strength was mademore negative. This transition occurs through a series of pitchfork,saddle-node, and supercritical Hopf bifurcations of the individualsteady states. However, the model fails to describe the switchingbetween several entrainment ratios. There are (at least) two possi-ble qualitative explanations for the switching behavior: either theoscillators itinerate among the basin of attractions of the stableentrainment states due to noise, or the oscillators travel along hete-roclinic orbits of the unstable entrainment states. Either way, noiseeffects are likely to play a major role. Further development of themodel is required to consider external noise (e.g., lack of perfectpotential control) or internal noise due to corrosion that can explainthe switching behavior.

The results presented in this study show that when IR com-pensation is applied to a multi-electrode system stationary andoscillatory patterns can develop; in these cases electrochemicalmethods with spatial and temporal resolution should be applied

to ensure that proper kinetic information is extracted from theexperiments.

Acknowledgements

We thank the National Science Foundation for financial supportunder grant CBET-0730597. We also thank Ms. Carolyn Vallas andNSF REU program for their support.

Appendix A

A.1. Model on bistablity

A simple model for a bistable electrochemical system consistsof a single equation for the charge balance:

Cde

dt= V − e

ReqA− jF (e) (15)

where C is double layer capacitance, V is circuit potential, e is theelectrode potential, R is the series resistance for a single electrodesystem, A is the surface area of the electrode, and jF (e) is the Faradaiccurrent density. For simplicity A = 1, C = 1. The steady states of Eq.(15) are stable [27] if −1/Req − djF (e)/de < 0. For a system of twoelectrodes [23,24,9]:

de1

dt= V − e1

Req− jF (e1) + K

2(e2 − e1) (16)

de2

dt= V − e2

Req− jF (e2) + K

2(e1 − e2) (17)

In these equations the electrical coupling strength is definedthe same manner as in Eq. (4). Around a uniform steady state e1∗ =e2∗ = e∗ where the right hand side of Eqs. (16) and (17) vanish

d(e1 − e∗)dt

= −1Req

(e1 − e∗) − djF (e∗)de

(e1 − e∗) + K

2(e2 − e1) (18)

d(e2 − e∗)dt

= −1Req

(e2 − e∗) − djF (e∗)de

(e2 − e∗) + K

2(e1 − e2) (19)

For e = e2 − e1,

de

dt= −e

Req− djF (e∗)

dee − Ke

thus we get the stability condition −1/Req − djF (e)/de − K < 0.Apparently, large negative coupling can destabilize any uniformsteady state as long as −1/Req − djF (e)/de < K

A.2. IR compensation: theory

Assume we apply a parallel resistance Req = R0 in Eq. (15) withwhich bistability is obtained with a single electrode. Therefore,−1/R0 − djF (e)/de < 0. In the two electrode setup without IR com-pensation (K = 0) with the same Rp = R0 we again obtain unstableuniform steady state since the −1/Req − djF (e)/de < K condition issatisfied.

During IR compensation, a negative resistance is connected inseries with cell, (Rc = Rcoll < 0). Thus, by IR compensation the orig-inal cell resistance R0 is decreased to R0 + 2Rc. This makes theuniform steady state of the cell more stable. However, at the sametime, negative coupling is introduced. In Eq. (16)

K = 2Rc

Rp

1Req

= 2Rc

R0

1R0 + 2Rc

We can see, that from the stability condition −1/Req − K is animportant quantity that will eventually determine the effect of IR



compensation (djF (e)/de is IR compensation independent). How-ever, this quantity is constant:

− 1Req

− K = − 1R0 + 2Rc

− 2Rc

R0

1R0 + 2Rc

= − 1R0

= const

The result implies that in the considered situation IR compen-sation, in contrast to expectation, does not change the stabilitycondition of the steady state.

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