Unidirectional Negative Coupling Induced Dynamical Patterns in an Epoxy-Based Dual-Electrode Microchip Flow Cell

We explore the spatiotemporal pattern formation of oscillatory electrodissolution of nickel in sulfuric acid with two micro-wires in an on-chip integrated microﬂuidic ﬂow cell. In the ﬂow channel, the reference and counter electrodes are placed at the opposite end of the ﬂow channel (contralateral placement). A theory based on modeling the interactions between the reactions indicates that there is a unidirectional negative coupling through the electrode potential from the upstream to the downstream electrodes. Experiments and numerical simulations with a kinetic model show that such unidirectional negative coupling could induce complex dynamics. At small cell resistances the uniform stationary state loses stability and a non-uniform stationary state, or a coupling induced oscillatory state can occur. At large cell resistance the oscillations can produce anti-phase synchrony and complex waveforms. The nature of the unidirectional coupling is conﬁrmed in independent experiments with synchrony analysis. The ﬁndings indicate that contralateral placement should be used with great care in electroanalytical application with reactions that exhibit negative differential resistance (NDR) region, e.g., due to passivation or adsorption of an inhibitor. © The Author(s) 2018. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/2.0381807jes]

6][17] When the electrode potential at two reaction sites are different, a cross-current can occur through a resistive circuit element (e.g., electrolyte), which can induce processes that diminish or enlarge the difference between the electrode potentials, resulting in positive or negative coupling, respectively.When the coupling strengths between the sites are the same, global coupling occurs.Potential drop in the electrolyte typically generates coupling strengths that decreases with increasing the distance between the sites: such coupling is called long-range. 2 For globally coupled electrodes, negative global coupling was studied previously using a close reference electrode (RE) [18][19][20] that lead to a variety of nonlinear phenomena including stationary spatial patterns, 21,22 remote triggering of waves, 23 oscillatory cluster formation, 3 and rotation of a complex breathing active region. 19These patterns with negative coupling typically arise at the negative differential resistance region in which current decreases with the increase of the potential in anodic reactions.Alternatively, the negative impedance device commonly used in electrochemistry to compensate the electrolyte resistance (IR compensation) can also induce negative global coupling that can produce stationary patterns through a spatial symmetry breaking mechanism. 24With electrode arrays, 25 electrodissolution reactions exhibited symmetry breaking bifurcation, anti-phase synchronization, and amplitude death due to the bidirectional negative coupling imposed by IR compensation.
Lab-on-chip devices with microfluidic flow cells gain interest in practical applications such as cellular systems analysis 26,27 and electrophoretic separations. 280][31] It was shown that the microcell design typically used in electroanalytical chemistry produces large ohmic drops that can induce spontaneous oscillations that are seen in macrocell settings only with external resistors, 29,30 and these ohmic drops will bring the current oscillations of the two electrodes into synchrony. 31In previous research 31 with two nickel electrodes, the microcell was placed at ipsilateral (traditional) placement configuration where the reference and counter electrodes are at the same downstream end of the flow channel.In this configuration, the oscillatory metal dissolution reactions exhibited in-phase synchrony; the synchronization was attributed to the intensified bidirectional positive electrical coupling among the electrodes that exists due to the directed current flow between the electrodes. 30,31However, in many microscale electroanalytical applications with multi-working electrode, 32,33 the reference and counter electrodes were placed at the opposite end of the flow channel to optimize the performance.The rearrangement of the cell geometry is expected to change the nature of coupling within the system and thus the overall dynamical features of the reactions. 34n this paper, the dynamical features of electrochemical reactions in dual-electrode microfluidic cell are investigated with contralateral placement of reference/counter electrodes.In contralateral placement configuration, the reference and counter electrodes are placed at the opposite end of the flow channel where two working electrodes reside.A theory is formulated based on the cell topology to reveal the type and strength of interactions between the two electrodes.The effect of the predicted unidirectional negative coupling through the electrode potential is studied by numerical simulations that were carried out with a kinetic model for dual-electrode Ni electrodissolution reaction.The dynamical features are studied at both stationary (e.g., pattern formation and induced oscillations at negative differential resistance region) and non-stationary states (e.g., current oscillations, synchronization patterns).The theoretical predictions are tested in experiments with nickel electrodissolution reaction that takes place on the surface of two electrodes in on-chip integrated electrochemical cell.The relevance of our findings are discussed for design of efficient generator-collector type of electrochemical detector systems.

Materials and Methods
Schematic.-Aschematic of the contralateral placement configuration of dual-electrode microfluidic flow cell set-up is shown in Fig. 1a.The overall cell design follows that used in electroanalytical applications.The 2 M sulfuric acid/0.01M NiSO 4 electrolyte is pumped in a 200 µm (width) × 100 µm (height) flow channel at flow rates Q = 1.5 µL/min that corresponds to a linear velocity of 0.125 cm/s.The electrolyte flows over the reference electrode, front (upstream) and rear (downstream) working electrodes to the reservoir.In most of the experiments two 100 µm diameter nickel wires (Puratronic, 99.994%, Alfa Aesar) are placed approximately to the middle of the flow channel as working electrodes.Each working electrode is connected to a potentiostat (Gamry Reference 600, in bipotentiostat configuration) through an external resistor.A 2 mm diameter nickel wire is placed at the most upstream end of flow channel, this electrode is used as a quasi-reference electrode for the electrochemical cell.At Electrode fabrication and fluidics.-Afive-electrode array of nickel was made in an epoxy (Armstrong C7 with activator A, Ellsworth Adhesives) mold.Four Ni wires that have diameter 100 µm are equally spaced with 5 mm distance; in addition a 2 mm diameter nickel rod was placed at one end of the array.The centers of each electrodes are approximately aligned.The electrodes were polished so that only the ends are exposed.Depending on the types of experiment, no more than two working electrodes of the array were connected during the experiments.(The four-electrode array gave flexibility in choosing a properly connected/positioned working electrode.)The PDMS (polydimethylsiloxane) chip with 200 µm (width) × 100 µm (height) fluidic channel was created by standard soft photolithography using a negative photoresist.The cell assembly follows the same methods and procedures used before with traditional placement of dual-electrode experiment.Further details about cell assembly are given in previous publications. 29,31efore the experiments, the series resistances of the upstream electrode (electrode 1) R 1 and downstream electrode (electrode 2) R 2 were determined with impedance spectroscopy.First, an auxiliary Ag/AgCl/3M NaCl reference electrode was placed in the reservoir containing the counter electrode.The solution resistance was determined when only electrode 1 or electrode 2 was connected, giving solution resistances R 1 and R 2 , respectively.By definition, the solution resistance between the two working electrodes Finally, the auxiliary reference electrode was disconnected, and the upstream nickel quasi-reference electrode was used throughout the measurements.

Results and Discussions
The interactions between the electrodes in the flow channel were first approximated by a theory through developing a kinetic model for dual-electrode Ni electrodissolution reaction.The model was used to perform numerical simulations to predict the expected dynamical behavior of the reaction.Finally, experiments were carried out to test the predictions from theory and numerical simulations.
Theory.-The kinetic model of a two-electrode system is developed through mass and charge balance equations.For charge balance, the system is modeled with the use of a Randles equivalent circuits 35 for the individual electrodes (Figure 1b).The two electrodes having capacitance / surface area (C d ) are connected to the potentiostat through individual resistors (R ind,1 and R ind, 2 ).The electrodes, that have surface area A, are connected to each other through a solution resistance R 12 .The current of the upstream electrode (I 1 ) flows in the channel to the downstream electrode (electrode 2) through resistive circuit element R 12 .The currents generated by each electrodes are obtained from double layer charging and charge transfer processes (with Faradaic current densities J F, 1 and J F, 2 ): where E 1 and E 2 are the electrode potentials and t is the time.The potentiostat maintains constant circuit potential V, therefore: The equations can be greatly simplified by setting R ind,2 = R ind,1 +R 12 , i.e., the individual resistor attached to the downstream electrode is always increased by R 12 relative to that of the upstream electrode.Under this condition, the electrodes have the same total effective resistance, R 0 = R ind, 1 ; this total resistance will express the resistance at which the behavior of the coupled system is comparable to the individual (uncoupled) electrode behavior.By combining Equations 1-4 we obtain differential equations for the dynamical evolution of the electrode potentials: From a coupling perspective, Equations 5-6 can be written as Thus we can see that the flow channel geometry imposes an electrical coupling between the electrodes with coupling strengths: [10] Equations 9-10 indicate there is no coupling from downstream electrode to the upstream electrode while the coupling from upstream electrode to the downstream electrode K 1→2 is a negative value that depends on the distance between two working electrodes, the electrode surface area A, and the total cell resistance R 0 .
Effect of total cell resistance, R 0 .-Equations 10 shows that the coupling strength from the upstream electrode to the downstream electrode K 1→2 decreases with increase of total resistance.The coupling disappears (K 1→2 → 0) in the limit of infinite resistance (R 0 → ∞).
Numerical simulations.-Modelformulation.-Numericalsimulations were carried out to investigate the dynamical behavior of oscillatory Ni electrodissolution in a dual electrode flow cell with unidirectional negative coupling topology defined by Equations 7-10.Linear sweep scans at both stationary state as well as oscillatory state are performed.The effects of changing the total resistance (R 0 ) on the dynamical behaviors were also studied.
An ordinary differential equation model is developed that relies on the kinetic model of nickel dissolution in sulfuric acid by Haim et al. 5,8,36 The model has two dimensionless variables for each oscillator l = 1 and 2: the double layer electrode potential (e l ) and the total surface coverage of oxide species (θ l ).The kinetic model is combined with the charge balance in Equations 7-8 to give a dimensionless set of equations as follows where τ is dimensionless time, v is dimensionless circuit potential, a is dimensionless electrode surface area, r 0 is dimensionless total resistance, c l (l = 1,2) is the coupling term κ l is the dimensionless coupling strength: Kinetic parameters C h = 1600, α = 0.3, β = 6 × 10 −5 , γ = 0.001 were chosen to produce smooth (nearly sinusoidal waveform) limit cycle oscillations close to supercritical Hopf bifurcation. 1 and 2 parameters, which correspond to surface molar capacities, were set to slightly different values ( 1 = 0.01 and 2 = 0.0102) to simulate surface heterogeneities that produce the experimentally observed different natural frequencies of the two oscillators 5,8 .The ordinary differential equations were solved for a wide range of r 12 , r 0 , and a values with MATLAB using a variable stepsize fourthorder Runge-Kutta algorithm (ode45).
The effect of coupling on the dynamics of the system is analyzed by reconstructing the phase of the oscillators; the Hilbert transform of the time series of the electrode potential is used in defining the phase 37 using the Hilbert transform approach PV in Equation 15implies that the integral should be evaluated in the sense of Cauchy principal value.<e> is the temporal average of the e time series.The phase difference between the oscillators φ = φ 2 −φ 1 is unwrapped: When the angle value crosses the integer multiple of 2π in an increasing manner the phase value is increased by 2π (i.e., the phase value is not bounded by 2π).The frequency of the oscillators (ω l ) is obtained from the slope of the linear fit to the phase vs. time plots each of the oscillators.
Dynamics with small total cell resistance: stationary state and induced oscillations.-Patternformation.-Thedynamics at stationary state were investigated in the numerical simulation by changing the dimensionless circuit potential (v) in Equation 11.The cyclic voltammograms with very small external resistance (r 12 = 4, a = 1, r 0 = 0.05, scan rate s = 0.0005) are shown in Figures 2a-2b.When two working electrodes are connected individually (without coupling), the cyclic voltammograms for each individual electrode almost perfectly overlay as an N-shaped curve (Figure 2a).When two electrodes are coupled, as v is increased, in the negative differential resistance (NDR) region the downstream electrode develops a new stationary state from v = −0.7 to 0.1 (Figure 2b), which have current higher than upstream electrode.At backward scan, similarly the downstream electrode develops another state v = −0.4 to −1.1, which has lower current than the upstream electrode.The upstream electrode (electrode 1) remain the same behavior as without coupling.The new stationary states obtained here through the symmetry breaking with unidirectional negative coupling are similar to those obtained with two-electrode system in the presence of IR compensation. 25However, while bidirectional coupling affects both electrodes and the symmetry broken pattern displays one electrode with elevated, and the other electrode with lowered activity, the unidirectional coupling creates new (increased or lowered activity depending on the direction of the scan) only for the downstream electrode.Induced oscillation.-Thelinear sweep voltammetry at larger total resistance (r 0 = 4) of the two electrodes are shown in Figures 2c-2d.When electrodes are uncoupled (Figure 2c), the linear sweep voltammograms for each electrode (nearly) overlay as the expected N-shaped curve without any oscillations.When two electrodes are coupled (Figure 2d), as the circuit potential is increased, electrode 2 (downstream electrode) shows current oscillations from v = 2.9 to 5.0.In the meantime the electrode 1 still remains at stationary state, follow the same trend as uncoupled electrodes (see Figure 2c).The numerical simulation findings thus show that negative coupling can induce oscillations of the downstream electrode.Negative coupling has been shown to shift the onset of Hopf-bifurcation in the coupled Stuart-Landau equation. 38The numerical results show that such shift can also occur with unidirectional coupling.
Dynamics with large total cell resistance at oscillatory state: synchronization and complex behaviors.-Whenthe cell has very large resistance (r 0 ), the system can exhibit oscillatory behavior. 5The oscillatory dynamics was investigated by a slow linear sweep of the dimensionless circuit potential (v), as summarized in Figures 3a-3c (r 0 = 10, r 12 = 4, a = 1, s = 0.014).Without coupling, the linear sweep voltammograms for upstream electrode (electrode 1, Figure 3a) and downstream electrode (electrode 2, Figure 3b) behavior are the same: there is an oscillatory range between v = 8.7 and 13.3.When the two electrodes are coupled (Figure 3c), the downstream electrode (electrode 2) starts to oscillate at lower potential v = 8.3 compared to uncoupled behavior (Figure 3b).The earlier onset of oscillations of the downstream electrode is consistent with the finding at low total resistances; in both simulations there are potential regions where the downstream electrode generates oscillatory current while the upstream electrode has stationary behavior.As the potential is increased, at the onset of oscillations of the upstream electrode in Figure 3c, there is a phase shift between downstream electrode and upstream electrode; at large potential this phase shift disappears.
The simulations thus show that in either stationary state or nonstationary state, the unidirectional negative coupling could change the dynamical behavior of downstream electrode.Steady state patterns between dual working electrodes or current oscillation of downstream electrode can be obtained at stationary state, earlier oscillation onset can be observed at non-stationary state.Since the coupling is  unidirectional, the dynamical behaviors of the upstream electrode will not be affected.Effects of total resistance (r 0 ) on synchronization.-Equation14 indicates that, at a fixed distance between two electrodes, the coupling strength decreases with increasing of total resistance.We performed a series of simulations at different total resistances r 0 = 20, 30 and 100.The dimensionless circuit potential v were set at just above the onset of oscillations in a manner by which the temporal average <e> of the oscillation remained the same at different r 0 .The oscillations are thus compared at same mean electrode potential.With low resistance (r 12 = 1.29, a = 1, r 0 = 20, and v = 15.0) the unidirectional coupling is relatively strong κ 2 = − 0.0030; the coupling results in a nearly antiphase synchronized current oscillations (Figure 4a) at constant phase difference (phase locking) of about φ = 2.6 rad (Figure 4b, lowest curve).The phase differences between the oscillators are shown in Figure 4b for two other total resistances and coupling strengths.At medium total resistance (r 0 = 30, κ 2 = −0.0014), the oscillations synchronize intermittently and the synchronized time intervals are separated by quick phase slips (i.e., phase slipping occurs).At large total resistance (r 0 = 100, κ 2 = −1.3× 10 −4 ), the phases of the two oscillators exhibit phase drift behavior (i.e., the phase difference increases nearly linearly with time).The numerical simulations thus show that the increase of the total cell resistance r 0 has an effect of weakening the synchronization between two electrodes by decreasing the coupling strength between the oscillators.Complex behavior with strong coupling.-Withstrong coupling large changes of the oscillation waveforms can be observed.Two examples are shown in Figures 4c-4d.In Figure 4c the downstream electrode exhibits a high-order periodicity (at r 12 = 25, r 0 = 5, a = 1, v = 3.0); the current oscillations repeat after about five large amplitude cycle (which is followed by a very small peak, or 'shoulder').In Figure 4d (r 12 = 5, r 0 = 7, a = 1, v = 6.0,) we can see the dramatic change in the oscillator waveform from nearly smooth (see the corresponding waveform of the current oscillations of the upstream electrode, which would also be expected for the downstream electrode without coupling) to strongly anharmonic shape.Such diverse complex behaviors were not observed in the ipsilateral (traditional) electrode placement configuration that imposed bidirectional positive coupling. 29perimental results .-Weperformed experiments in a dual electrode setup to confirm the unidirectional negative coupling induced spatiotemporal patterns predicted by the theoretical and numerical results.
Dynamics with small total cell resistance: stationary patterns and induced oscillations.-Patternformation.-Theexperiments were performed with a linear sweep voltammetry with the individual electrodes in the flow channel (uncoupled behavior) and the two electrodes placed in the flow channel (coupled behavior).In these sets of experiment, the distance from electrode 2 (downstream) to the reservoir was always set to L 2c = 1.5mm, which yields a collective resistance R c = 1.3 k ; the distance between two working electrodes L 12 = 5.6 mm, which results in solution resistance R 12 = 4.3 k .The diameter of the two electrodes is D 1 = D 2 = 100 µm, with geometric surface area A = 7.85 × 10 −3 mm 2 .To characterize comparable dynamical behavior between the coupled and the uncoupled electrodes, the added external resistance is adjusted in a manner by which the total cell resistance at the coupled state is equal to the total series resistance in the uncoupled state.The linear sweep voltammograms (with relatively slow scan rate of 10 mV/s) of two working electrodes at uncoupled or coupled states are illustrated in Figures 5a-5b and Figure 5c, respectively.The upstream and downstream electrodes (with R 0 = 10 k ) both showed the expected, nearly identical N-shaped curves (Figures 5a-5b) when they were alone in the flow channel.(Note that at high potential it is not possible to obtain reproducible results because of the oxygen bubbles from water electrolysis block the current flow in the flow channel.)When both electrodes resided in the flow channel, nearly the same currents were maintained until the peak point.After the peak potential the current of the downstream electrode maintained a high  value (about 14 µA) while the current of the upstream electrode followed the expected decrease due to the expected negative differential resistance.The observed spatial pattern is similar to the simulation results in Figure 2b, which confirms the numerical prediction of the coupling induced generation of a high-current stationary state that can be achieved during the forward scan.Unfortunately, reproducible experiments are not possible to obtain at higher potential for tests of the presence of the low stationary state in the backward scan.The rest of the experiments were performed at large resistance and lower potential, where the oxygen evolution interference is avoided.Induced oscillation.-Atanother set of experiments, the distance between the two electrodes were increased to 16.0 mm (R 12 = 12 k ).Under these conditions, no oscillation were observed for upstream (Figure 6a) or downstream electrode (Figure 6b) when only one of the electrodes were used with R 0 = 1 k .When both of the electrodes were present (Figure 6c), downstream electrode shows current oscillations between 1.58 V and 1.70 V while the upstream electrode still remains at a stationary state that would be expected from the single electrode behavior.Similar experiments were also performed in another set of experiments with R 0 = 20 k .When we applied a constant potential (V = 1.72 V), without coupling the two electrodes generate stationary current of about 10.0 µA (see Figure 6d).With both electrodes in the flow channel (Figures 6e), the downstream electrode exhibits current oscillations with frequency of 0.309 Hz and amplitude of 1.2 µA.The upstream electrode, however, exhibits a nearly stationary state behavior with very small oscillations amplitude (0.1 µA).The mean currents of the both electrodes exhibit nearly the same value (10.2 and 10.3 µA) that was observed without coupling.The experiments thus confirm the presence of coupling induced oscillations in the experimental system.In the experiments, there is a very small amplitude oscillation observed in the time series of the upstream electrode; this small amplitude could be related to the representation of the electrochemical cell as a one-dimensional resistance circuitry.In reality, the potential drop in the electrolyte is described by the Poisson equation, 36 and thus additional weak coupling contributions may exist in the system.Nonetheless, the ordinary differential equation model successfully describes the main dynamical features of the coupling induced symmetry breaking and oscillation inducement phenomena.
Dynamics with large total cell resistance at oscillatory state: synchronization and complex behaviors.-Experimentswith the two electrodes were also performed with large total cell resistance; under these conditions single electrodes produce current oscillations. 27In these experiments the distance from the downstream electrode to the reservoir is L 2c = 1.5 mm (R c = 1.3 k ) and the distance between the electrodes is approximately 6.0 mm, which results in R 12 = 4.5 k .The linear sweep voltammetry scans of the single upstream electrode and downstream electrodes with total cell resistance R 0 = 20 k are shown in Figures 7a and 7b, respectively.The currents of both of the electrodes start to oscillate at about 1.77V and the variation of frequency and amplitude with potential are very similar.When both electrodes are in the flow channel (coupled electrodes in Figure 7c, R 0 = 20 k ), both the onset of current oscillations of downstream electrode (V = 1.73 V) is at a lower potential than that of the upstream electrode (V = 1.74 V).As the potential is increased, we saw a transition behavior from phase drifting between two working electrodes to anti-phase synchrony and then back to phase drifting behavior.At very large potential (V = 1.92 V), the current of downstream electrode start to drop very quickly, the oscillations are lost.8a) with same frequency, ω 1 = ω 2 = 0.680 Hz and achieved a phase locked behavior with nearly anti-phase configuration (phase difference φ = −3.58rad in Figure 8b).When the total resistance was increased to medium value of R 0 = 50 k and thus the coupling weakened to K = −0.22k −1 mm −2 , the two oscillators are intermittently synchronized (Figure 8c) with a small frequencies difference ω = 9 mHz (ω 1 = 0.536 Hz, ω 2 = 0.527 Hz) and achieved phase slipping behavior with one phase jump event in the 300 s experiment (see Figure 8d).At large R 0 = 100 k and thus weak coupling, K = −0.056k −1 mm −2 , the two oscillators exhibited phase drifting behavior (see Figures 8e-8f) with ω = 12 mHz (ω 1 = 0.543 Hz, ω 2 = 0.531 Hz); the phase drifting behavior confirms the absence of phase synchronization expected for weakly coupled oscillations.The experiments thus show that by increasing the total resistance and weakening the coupling strength between the oscillators the synchrony is progressively lost through a route of anti-phase-synchrony → phase slipping with rare phase slips → phase drift.This trend confirms the numerical simulations showing that by increase of total resistance the synchrony is progressively lost.Evidence for unidirectional coupling.-Inorder to confirm the existence of unidirectional coupling, we choose the phase slip behavior shown Figure 8d and carefully analyze the phase changes of the current oscillations of each electrode.For the purpose to compare the phase dynamics, we remove the linear trends of phase changes of each oscillator by the equation where φ l , l = 1,2, is the phase of each oscillator, φ detrend,l is the phase after detrending, and ω 1 is the frequency of electrode 1.
The detrended phases and phase difference between two oscillators as a function of time are shown in Figure 9a.During the 100s interval, the phase of electrode 2 shows a slipping behavior while the phase of electrode 1 remains almost constant.These variations indicate the overall phase slip behavior of the phase difference (thick solid line) is due to the phase changes of downstream electrode only; therefore the coupling between two electrodes is unidirectional in direction from the upstream to the downstream electrode.
A method of detection of coupling directionality from phase dynamics can also be used to characterize the coupling directionality topologies.The method is based on testing whether the time evolution of the phase of one oscillator is influenced by the phases of the other oscillators. 37,38The dependence of rate of phase change on the phase difference between two oscillators are approximated in the form [18]   where φ k are the phases, ω k are natural frequencies (in rad/s) of the oscillations, and H k are the interaction functions in a Fourier expanded form The interaction function can be determined from an experiment close to the onset of phase synchronization when the phase difference displays phase slipping behavior by plotting dφ k (t)/dt − ω k as a function of the phase difference φ(t).To extract phase φ k from experimental data, the linear interpolation method is used 38 in which the phase φ k is increased by 2π at every local maximum of the current; between the local maxima the phase grows linearly with time.Because the natural frequency of the oscillations could drift with time, we approximated the natural frequencies with the assumption H(0) = 0; this assumption is due to the difference nature of the coupling (i.e.,   at zero phase difference the electrodes have nearly identical electrode potential and thus do not affect each other) and was validated in macrocell studies with nickel electrodissolution. 38 measure of the interaction strength is the amplitude (c k ) of the interaction function defined as [20]   where the Fourier coefficients are taken from the corresponding interaction functions.
Based on the amplitudes of the interaction functions, a directionality index (d) can defined as [21]   This index d = +1 if electrode 1 predominantly drives electrode 2, whereas it is d = −1 in the opposite case.When d = 0 is the coupling is bidirectional.Such quantitative measure gives direct information on the nature of coupling topology.For example, in contrast to the synchrony index (σ), directionality index (d) does not depend on the natural frequencies of the oscillators.
As shown in previous macrocell results for globally coupled oscillators, the phase interaction function is a positive sinusoidal waveform. 38For contralateral placement configuration, at intermediate coupling strength, the interaction function (Figures 9b-9c) is a negative sinusoidal function with amplitude c 2 = 0.23 for the downstream electrode and a nearly straight line with negligible amplitude c 1 = 0.01 for the upstream electrode.The directionality index d = 0.9 indicates the presence of strong unidirectional coupling from the upstream to the downstream electrode.
Further evidence for the existence of unidirectional coupling can be obtained by comparing the frequencies of oscillations before and after coupling.The frequencies of each oscillations as a function of the absolute value of coupling strength are shown in Figure 10.First we consider a scenario when the natural (uncoupled) frequency of electrode 1 larger than electrode 2 (ω 1 = 0.565 Hz, ω 2 = 0.527 Hz, Figure 10a).When we coupled the two electrodes with coupling strength |K| = 2.58 k −1 mm −2 , two oscillators are strongly synchronized with same frequency ω 1 = ω 2 = 0.573 Hz.Note that the frequencies in the synchronized state are close to the natural frequency of the electrode 1; the oscillators speed up during the synchronization process.Now we consider the opposite scenario with the natural (uncoupled) frequency of electrode 1 smaller than electrode 2 (ω 1 = 0.688 Hz, ω 2 = 0.705 Hz, Figure 10b).When we coupled two electrodes with coupling strength |K| = 1.25 k −1 mm −2 , two oscillators are strongly synchronized with near same frequency, ω 1 = 0.682 Hz, ω 2 = 0.683 Hz.This time oscillations slow down during the synchronization process, however, the synchronized frequency is again close to the frequency of the expected driver electrode positioned upstream (electrode 1).
The phase and frequency analyses thus all confirm that the coupling is from the upstream to the downstream electrode, therefore, the coupling is unidirectional.Complex waveform oscillations.-Diverseoscillation waveforms were observed when two the electrodes were strongly coupled (e.g., K = −1.2k −1 mm −2 ).In these set of experiments we kept the distance from electrode 2 to the reservoir L 2c = 3 mm (R c = 2.6 k ), and the distance between the electrodes L 12 = 5.8 mm (R 12 = 4.8 k ) with R 0 = 20 k ; the current oscillations at different circuit potentials are shown in Figure 11.At relatively low potential V = 1.75 V, smooth anti-phase current oscillations are observed (Figure 11a) with nearly the same frequency ω 1 = 0.594 Hz, ω 2 = 0.593 Hz.At larger potentials, the waveform of downstream electrode started to exhibit small secondary peaks in between the expected large amplitude peaks.As the potential increased from 1.80V (Figure 11b), 1.81V (Figure 11c), and 1.90 V (Figure 11d), the frequencies of the oscillations became larger and a transition from anti-phase to out of phase synchronization was observed.Note that only the waveform of current oscillations of the downstream electrodes were altered to display secondary peaks (due to the coupling) while the oscillations of the upstream electrodes remained unaffected.
When the two electrodes were placed even further from each other (L 12 = 12.5 mm, R 12 = 10.4 k ), the oscillations became even more complex.For instance, the secondary peak during the anti-phase synchronization at low potential (Figure 12a, V = 1.88 V) became more prominent.In other experiments, the oscillations of the two electrodes had large frequency difference (110 mHz with ω 1 = 0.590 Hz, ω 2 = 0.700 Hz) a nearly 6:7 entrainment occurred: 6 simple oscillations of the currents of the upstream electrode were accompanied with 7 complex (amplitude modulated) oscillations of the currents of the downstream electrode (Figure 12b, V = 1.80 V).The oscillations of the downstream electrode thus exhibited higher order periodicity.Such entrainment is a stable behavior that could exist for 300 oscillatory cycles (200s).(The Figure 12b shows only 25s for clarity.) In some experiments, often at large potentials, (Figure 12c, V = 1.80 V) we observed that mean current of the oscillations of the downstream electrode alternated between a low value (where the oscillations are nearly in-phase with the upstream electrode) and a high value (where the oscillations have complex waveform and dominant anti-phase relationship with the upstream electrode).The dynamics is thus affected by a large timescale process; we note that such oscillations were not observed in the numerical simulations and thus could be attributed to factors not included in the model (e.g., one possibility is the periodic removal of oxygen bubbles from the electrode surface that results in partial blocking of the flow channel.)

Conclusions
We showed that nickel electrodissolution reaction can display a wide range of complex dynamical behaviors with two electrodes in a flow channel with contralateral placement of reference/counter electrodes.The dynamical behaviors (e.g., formation of stationary patterns, induced oscillations, anti-phase synchronization of smooth oscillations) were interpreted with the presence of unidirectional negative coupling between the electrodes.The sign and the strong asymmetry of the coupling was attributed to asymmetrical potential drops in the flow channel: The current flow is away from the reference electrode and only affects the behavior of the downstream electrode.This asymmetrical coupling is very different in nature from the asymmetrical coupling found before with electrode size disparity, 39 where the large current generated by the large electrode drives the behavior of the small electrode.Here, asymmetrical coupling can be observed even between electrodes of the same size.
Numerical simulations and experiments were carried out to show the complex dynamical behaviors due to the unidirectional negative coupling of electrochemical reactions with hidden negative differential resistance characteristics: at low resistance where the individual electrodes exhibit a single stationary state, a symmetry broken spatial pattern or induced current oscillation of the downstream electrode can be observed; because the coupling is from the upstream to the downstream electrode, the behavior of the upstream electrode remains unchanged due to the presence of the downstream electrode.At large resistance where the individual electrodes exhibit oscillatory behavior, anti-phase synchrony was observed where the upstream electrode drove the current oscillations of the downstream electrode.Because of the strong negative coupling, complex oscillation waveforms, often accompanied with high order entrainment were observed.Previous research with traditional placement configuration of dual-electrode microcell 29 only showed in-phase synchrony due to the positive bidirectional coupling.The change of the position of the reference electrode from ipsilateral to contralateral placements gives rise to a richer set of dynamics with not only more complex synchronization dynamics, but also the stationary pattern formation and induced oscillatory dynamics.
The contralateral placement of reference electrode is often used in electroanalytical applications to minimize the ohmic drop of the upstream electrode. 30,31The findings suggest that for chemical reactions that display negative difference resistance (e.g., due to passivation, adsorption of an inhibitor, or desorption of a catalyst) 2 great care should be taken to avoid the interference between the electrodes due to negative coupling.The coupling can be weakened by increasing the cell total resistance (e.g., by adding individual resistance to the electrodes) and by placing the electrodes close to each other.

Figure 1 .
Figure 1.Experimental setup: dual electrodes with reference and counter electrodes opposite placement in a flow cell.(a) Schematic diagram of dual electrode cell.WE 1, 2 : Ni electrodes embedded in epoxy, RE: Ni reference electrode, CE: Pt counter electrode.(b) The equivalent circuit of a dual-electrode electrochemical cell, E 1, 2 : electrode potential, I F1, 2 : Faradaic current, C d : double-layer capacitance, R ind : individual resistors, R 12 : solution resistance between two working electrodes, I 1 and I 2 : current, V: circuit potential.

Figure 2 .
Figure 2. Numerical simulations: Dynamical behaviors of current oscillations with small total cell resistance.Thin line: downstream electrode.Thick line: upstream electrode.(r 12 = 4, a = 1, scan rate s = 0.0005).(a and b) Cyclic voltammetry scans with total cell resistance r 0 = 0.05.(a) Individual scan.(The forward and backward scans of upstream and downstream electrodes are overlaid.)(b) Coupled electrodes.(c and d) linear sweep voltammetry scans with r 0 = 4. (c) Individual electrodes (the scans of upstream and downstream electrodes are overlaid).(d)Coupled electrodes.

Figure 7 .
Figure 7. Experiments: Effects of negative coupling of on synchronization of current oscillations at R 0 = 20 k .(a) Individual scan of downstream electrode, (b) Individual scan of upstream electrode (c) Coupled electrodes.Thin line: downstream electrode, thick line -upstream electrode.D = 100 µm, A = 7.8 × 10 −3 mm 2 , L 2c = 1.5mm,L 12 = 6.0 mm.The experiments thus confirm the numerical simulations indicating that the strong negative coupling can advance the onset of current oscillations.Effects of total resistance (R 0 ) on synchronization.-Experimentswere carried out with smooth electrochemical oscillations (nearly sinusoidal waveform close to the Hopf bifurcation point) obtained at circuit potentials with 30 mV above the Hopf bifurcation.In these sets of experiments the distance from electrode 2 (downstream) to the reservoir was always set to L 2c = 3.0 mm, which yielded a collective resistance R c = 3.4 k ; the distance between two working electrode L 12 = 6.0 mm (R 12 = 4.6 k ).The time series of current oscillations and the corresponding phase differences at total resistances (R 0 ) 20 k , 50 k , and 100 k are shown in Figure 8.With small total resistance R 0 = 20 k (coupling strength K = −1.2k −1 mm −2 ) the smooth current oscillators are synchronized (Figure8a) with same frequency, ω 1 = ω 2 = 0.680 Hz and achieved a phase locked behavior with nearly anti-phase configuration (phase difference φ = −3.58rad in Figure8b).When the total resistance was increased to medium value of R 0 = 50 k and thus the coupling weakened to K = −0.22k −1 mm −2 , the two oscillators are intermittently synchronized (Figure8c) with a small frequencies difference ω = 9 mHz (ω 1 = 0.536 Hz, ω 2 = 0.527 Hz) and achieved phase slipping behavior with one phase jump event in the 300 s experiment (see Figure8d).At large R 0 = 100 k and thus weak coupling, K = −0.056k −1 mm −2 , the two oscillators exhibited phase drifting behavior (see Figures8e-8f) with ω = 12 mHz (ω 1 = 0.543 Hz, ω 2 = 0.531 Hz); the phase drifting behavior confirms the absence of phase synchronization expected for weakly coupled oscillations.The experiments thus show that by increasing the total resistance and weakening the coupling strength between the oscillators the synchrony is progressively lost through a route of anti-phase-synchrony → phase slipping with rare phase slips → phase drift.This trend confirms the numerical simulations showing that by increase of total resistance the synchrony is progressively lost.Evidence for unidirectional coupling.-Inorder to confirm the existence of unidirectional coupling, we choose the phase slip behavior shown Figure8dand carefully analyze the phase changes of the current oscillations of each electrode.For the purpose to compare the phase dynamics, we remove the linear trends of phase changes of

Figure 9 .
Figure 9. Experimental evidence for unidirectional coupling: Phase dynamics.(a) Detrended phase vs. time plot.Thick dashed line: upstream electrode, thin solid line: downstream electrode, thick solid line: phase difference between two electrodes.Phase interaction functions for (b) upstream electrode and (c) downstream electrodes.The experimental parameters are given in Figures 8c-8d.