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Spontaneously Synchronized Electrochemical Micro-oscillators with Nickel Electrodissolution Yanxin Jia and István Z. Kiss* Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, United States S Supporting Information *

ABSTRACT: We investigate the dynamical behavior of oscillatory electrodissolution of two nickel microwires in an on-chip integrated microfluidic flow cell. With equivalent circuit analysis, an equation is derived that shows that the electrical coupling strength between the electrodes can be intensified by increasing the distance from the downstream electrode to the reference/counter electrodes, increasing the electrode size, and decreasing the total resistance within the cell. Experiments and numerical simulations with an ordinary differential equation model show that by intensifying the coupling between the electrodes spontaneously synchronized oscillations occur due to large potential (IR) drop in the small flow channel. Experiments with different distances, electrode diameters, and total resistances confirm a correlation between the statistical entropy based synchrony index and the theoretically derived coupling strength formula. The findings thus show that dynamical features of the chemical reaction that have been previously seen with macroelectrodes with application of external coupling resistances can occur spontaneously with microwires. The delineation of coupling effects in the reaction system could facilitate kinetic and electroanalytical applications in design of microchip-based setups that require collector-generator multielectrode wires: the accuracy of the measurement can be improved by proper cell design that diminishes electrical cross talk. observed on the surface of a single electrode19,20 and assemblies of discrete electrodes.21−31 In traditional macrocells, the geometry of the arrangement of the cathode, anode, and the reference electrode determines the strength and length-scale of the effective coupling. For example, by tuning the distance between working and reference electrodes in H2O2 reduction on Pt32 and iron electrodissolution,29 transitions from asynchronous to synchronous oscillations have been observed in dual electrode configurations. To explore self-organized temporal and spatial features on the microscale, on-chip integrated electrochemical cells have been constructed in various designs.33,34 In formic acid oxidation on Pt electrodes,35−37 the major dynamical features seen at the macroscale have been reproduced in a glass− poly(dimethyl siloxane) (PDMS) on-chip integrated flow cell. In addition, it was demonstrated that far reference-to-working electrode placement could result in synchronized behavior.33 An epoxy-based design was utilized to investigate oscillatory Ni electrodissolution18,23,38,39 of a single microwire.34 It was shown that the microcell design typically used in electroanalytical chemistry produces large ohmic drops that can induce oscillations that are seen in macrocell settings only with external resistors.

1. INTRODUCTION Qualitative features of spatial and temporal aspects of dynamical behavior (e.g., pattern formation or oscillations) of chemical reacting systems are often determined by interplay of reaction kinetics and physical effects (e.g., mass transfer).1 Scaling down the reactor size from traditional macroscale to micro- and nanoscales holds the promise of the introduction of novel chemical (e.g., fluctuation kinetics2−4) and mass transfer (e.g., microfluidic flow5) effects, which could result in emergent behavior not seen at the macroscale. A wide range of structures including Turing patterns,6 inwardly moving spirals, and waves (antipacemakers)7 have been observed in nanoscale droplets of the Belousov−Zhabotinsky reaction; the droplets obtained in water-in-oil microemulsion are used to tune the diffusion coefficient ratio of the activator and inhibitor species in the reaction. Microscale BZ droplets8−11 were fabricated using microfluidic assembly to produce complex synchronization structures. Quorum transition and complex organizing centers were investigated in groups of oscillatory BZ beads.12−14 Surface reaction in catalytic solid−gas systems can form oscillatory micro-15 or nanoscale16 patterns. Electrochemical systems produce a wide range of dynamical behavior17,18 in which the coupling among reacting sites has a profound effect on the self-organized features.19 Investigations have shown that a major form of coupling is the electrical interaction through potential drops in the electrolyte.19,20 Various forms of stationary and oscillatory patterns have been © 2012 American Chemical Society

Received: May 15, 2012 Revised: August 14, 2012 Published: August 15, 2012 19290

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Figure 1. Experimental setup: dual electrode flow cell. (a) Schematic diagram of dual electrode cell. WE1,2, Ni electrodes embedded in epoxy; RE, Ag/ AgCl/3 M NaCl reference electrode; CE, Pt counter-electrode. (b) The equivalent circuit of a dual-electrode electrochemical cell: E1,2:, electrode potential; IF1,2, Faradaic current; Cd, double-layer capacitance; Rind, individual resistors; R1,2, solution resistance between two working electrodes; I1 and I2, current; Rc, solution resistance between downstream (WE2) electrode and the reference electrode; V, circuit potential. (c) Optical microscope image of PDMS flow channel sealed over 50 μm (top), 100 μm (middle), and 500 μm (bottom) diameter nickel wires.

99.995%) are applied so that the flow channel is placed to the middle of the electrode forming an approximately 500 μm × 200 μm active electrode surface (see bottom panel of Figure 1c). Each working electrode is connected to a potentiostat (Gamry Reference 600) through an external resistor. At the end of the flow channel there is an approximately 6 mm diameter reservoir; the Ag/AgCl/3 M NaCl reference electrode (exhibiting 209 mV electrode potential vs standard hydrogen electrode) and a 0.5 mm thick Pt wire counter electrode is placed into the center of the reservoir. All the given potentials are with reference to the Ag/AgCl/3 M NaCl reference electrode. 2.2. Electrode Fabrication and Fluidics. A four-electrode array of nickel wires (either 50, 100, or 500 μm diameter) was made by casting the electrodes with 5 mm spacing in an epoxy (Armstrong C7 with activator A, Ellsworth Adhesives) mold. The electrodes were polished so that only the ends are exposed. Two electrodes with proper spacing were selected from the fourelectrode array for the experiments. (The four-electrode array gave flexibility in choosing properly connected/positioned working electrodes.) The PDMS chip with 200 μm (width) × 100 μm (height) fluidic channel was created by standard soft photolithography using a negative photoresist.42,43 To create the fluidic microchip from the silicon master a mixture of 20:1 elastomer base and curing agent was used (Sylgard 184, Ellsworth Adhesives). Further details about construction of epoxy embedded arrays of electrodes are given in previous publications.34,44 2.3. Cell Assembly and Procedures. The electrode was wet polished with series of sandpapers (P180−P4000) with a Buehler Metaserv 3000 polisher. The electrode and the PDMS chip were then cleaned with isopropyl alcohol and dried with N2. The PDMS chip was reversibly sealed over the electrode such that the electrode is at a specified distance to the outlet reservoir. Proper placement of the electrode and measurement of the distance to the reservoir were made with an Olympus SZX-12 microscope equipped with Q Imaging QICAM Fast 1394 camera. Unless otherwise noted, the distance to the reservoir was

In this paper, the dynamical features of microscale oscillatory Ni dissolution in sulfuric acid are investigated in a dual-electrode, on-chip integrated setup. Equivalent circuit analysis is applied to formulate a theory for the electrical coupling strength between the electrodes as a function of the cell parameters: electrode surface area, length of the flow channel, and total cell resistance. Numerical simulations are carried out with an ordinary differential equation model for dual electrode Ni electrodissolution to investigate the effects of cell geometry on the synchronization of the current oscillations by tuning the coupling strength between the electrodes. The predictions are tested in experiments with an on-chip integrated electrochemical cell that uses an epoxy-based substrate for two embedded metal wires with the PDMS fluidics. A correlation is tested between a statistical entropy based synchrony index (obtained from the phase difference between the oscillators) with the theoretically predicted coupling strength for a wide range of experimental conditions with different electrode sizes, cell lengths, and total resistances. Finally, the results are discussed and design rules are formulated for tuning the dynamical characteristics of the chemical reaction system through intensifying or minimizing the coupling strength between the electrodes.

2. MATERIAL AND METHODS 2.1. Schematic. A schematic of the dual-electrode microfluidic flow cell setup is shown in Figure 1a. The overall cell design follows that used in electroanalytical applications.40,41 The 2 M sulfuric acid/0.01 M NiSO4 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 front (upstream) and rear (downstream) working electrodes to the reservoir. Three different sizes of nickel working electrodes are used. In most of the experiments two 50 or 100 μm diameter nickel wires (Puratronic, 99.994%, Alfa Aesar) are placed approximately to the middle of the flow channel as shown in the top and middle panels of Figure 1c, respectively. In another set of experiments two 500 μm diameter Ni wires (Goodfellow Cambridge Ltd., 19291

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5.0 mm. Similar to the standard glass-based substrate45 the seal is leak-free for the investigated flow rates Q = 1.5 μL/min.34,44 Before the start of the experiment the electrolyte was pumped (Harvard Apparatus microsyringe pump) through the flow channel for 15 min and the open circuit potential (OCP) was monitored during this time. When the OCP reached the expected value of −120 mV to −150 mV, the series resistance of the flow channel was determined with impedance spectroscopy with a Gamry Reference 600 potentiostat. The series resistances of the upstream (electrode 1) and downstream (electrode 2) electrodes are denoted as R1 and R2, respectively. These values are used to calculate the resistance between the electrodes, R1,2 = R1 − R2. We will also refer to R2 as Rc because conceptually it plays a similar role in the dynamical behavior as a collective resistance in our previous studies.24,25 Polarization scan experiments were carried out with the frontpanel control of the Gamry Reference 600 potentiostat, and the potential/converted current was digitized with a National Instrument PCI-6251 data acquisition card and Labview software. Constant potential experiments were carried out by scanning the circuit potential (V) from passivated electrodes (at 500 mV) to the target potential where the current data was saved at a data acquisition rate of 1 kHz after discarding about 3−5 min of transient response. The experiments were carried out at room temperature (21 ± 2 °C).

Cd

dE1 + AJF,1 dt

(1)

I2 = CdA

dE 2 + AJF,2 dt

(2)

(3)

V = I2R ind,2 + (I1 + I2)R C + E2

(4)

dE1 V − E1 = − JF,1 dt A(R ind,1 + R1,2 + 2R C) RC + (E2 − E1) A R ind,2(R ind,1 + R1,2 + 2R C)

dE1 V − E1 = − JF,1 + K (E2 − E1) dt AR 0

(7)

Cd

dE 2 V − E2 = − JF,2 + K (E1 − E2) dt AR 0

(8)

K=

RC AR 0(R 0 − 2R C)

(9)

Equation 9 allows the analysis of the effect of cell geometry on the dynamical behavior of the electrochemical reaction through tuning coupling strength of a coupled ordinary differential equation model. Effect of Total Cell Resistance, R0. Equation 9 shows that the coupling strength decreases with increase of total resistance. This diminishing coupling strength with large total resistance is demonstrated in Supporting Information, Figure S1a with some representative, constant values for electrode surface area A and collective resistance Rc. The coupling disappears (K → 0) in the limit of infinite resistance, R0 → ∞. The smallest total resistance for a given cell geometry can be obtained without any added resistance to the upstream electrode, that is, R0 = R1,2 + 2RC. This gives a theoretical upper limit for coupling strength as Kmax= Rc/ (AR1,2 (R1,2 + 2Rc)). Effect of Distance to the Reservoir, L. The distance of the downstream electrode to the reservoir can be conveniently controlled during the assembly of the cell. This distance determines the collective resistance Rc; it was previously shown34 that there is a linear relationship between the quantities Rc ∝ L, therefore, the effect of Rc on the coupling strength also describes the effect of changing the flow channel length from the downstream electrode to the reservoir. As it is shown in eq 9, an increase of Rc and thus L increases the coupling strength at fixed values of R0. This increase is demonstrated with typical experimental resistance values in Supporting Information, Figure S1b. The coupling can be eliminated (K → 0) with zero distance (L → 0, Rc → 0). In a given cell geometry with a fixed distance between the two electrodes one can obtain the strongest coupling without added individual resistance (Rind,1 = 0) at large distance to the reservoir compared to the distance between the electrodes (Rc ≫ R1,2) giving a maximum coupling strength of Kmax= 1/(2AR1,2). Therefore, the distance between the two electrodes defines a maximum coupling strength (that corresponds to large L and Rc), that can be diminished (down

By combining eqs 1−4 we obtain differential equations for the dynamical evolution of the electrode potentials: Cd

Cd

Thus we can see that the flow channel geometry imposes an electrical coupling between the electrodes with coupling strength K:

where E1 and E2 are the electrode potentials and t is the time. The potentiostat maintains constant circuit potential V, therefore: V = I1(R ind,1 + R1,2) + (I1 + I2)R C + E1

(6)

Note that the equations could be greatly simplified by setting Rind,2 = Rind,1 + R1,2, that is, the individual resistor attached to the downstream electrode is always increased by R1,2 relative to that of the upstream electrode. Under this condition, the same total resistance, R0, acts on each of the electrodes: R0= Rind,1 + R1,2 + 2RC. This total resistance expresses the resistance at which the behavior of the coupled system is comparable to the individual electrode behavior. Now we can further simplify eqs 5−6

3. RESULTS AND DISCUSSIONS 3.1. Theory. A two-electrode system coupled through the potential drop in the electrolyte is modeled with the use of a Randles equivalent circuit46 for the individual electrodes (Figure 1b). The two electrodes (having capacitance/surface area Cd) are connected to the potentiostat through individual resistors (Rind, 1 and Rind, 2). The electrodes, that have surface area A, are connected to each other through a solution resistance R1,2. The current of the upstream electrode (I1) flows in the channel to the downstream electrode (electrode 2) through resistive circuit element R1,2. In the channel below the downstream electrode the total current (I1 + I2) flows through resistance Rc. The current generated by each electrode is obtained from double layer charging and Faradaic current density (JF,1 and JF,2): I1 = CdA

d E2 V − E2 = − JF,2 dt A(R ind,2 + 2R C) RC + (E1 − E2) AR ind,2(R ind,1 + R1,2 + 2R C)

(5) 19292

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Figure 2. Numerical simulations: Effects of collective resistance (rC), electrodes surface area (a) and total resistance (r0) on phase difference between two electrodes. (a−c) phase difference vs time plot at different collective resistance rC = 0, 1.94, 6.67, respectively: r0 = 20, a = 1, potential v = 15. (d−f) phase difference vs time plot at different electrode surface area a = 0.25, 1, 4, respectively: r0 × a = 20, rC = 1.935, v = 15. (g −i), phase difference vs time plot at different total resistance r0 = 10, 20, 50, respectively: a = 1, rC = 1.935. (g) v = 7.89, (h) v = 15, (i) v = 35.4.

⎞ ⎛ C exp(0.5el) de l v − el = −⎜ h + α exp(el)⎟ dτ ar0 ⎠ ⎝ 1 + C h exp(el)

to zero) with close placement of the reservoir to the downstream electrode. Effect of Electrode Surface Area, A. For the comparison of the behavior of electrodes of different sizes we shall consider that equivalent dynamics can be obtained by keeping the parameter ρ = AR0 constant47,48 in eqs 7−9. Therefore, it is worthwhile to rewrite eq 9 in the form K=

RC ρ(ρ /A − 2R C)

(1 − θ l) + κc l Γl

(11)

dθl exp(0.5el) βC h exp(2el)θ l = (1 − θ l) − dτ 1 + C h exp(el) γC h + exp(el) (12)

where τ is dimensionless time, v is dimensionless circuit potential, a is dimensionless electrode surface area, r0 is dimensionless total resistance, cl is the coupling term c1 = −c 2 = e 2 − e1 (13)

(10) 34

In a previous publication it was shown that in typical experimental conditions the collective resistance does not depend on electrode sizes at given distance of the electrode to the reservoir. Therefore, we can conclude that at a fixed distance to the reservoir the coupling strength increases with an increase in electrode size: for small electrodes (A → 0) the coupling diminishes (K → 0) while for a large electrode there exists a maximum surface area Amax = ρ/ (R1,2 + 2Rc) at which the coupling strength is maximal (Kmax = Rc/(R1,2 ρ)). This variation is demonstrated in Supporting Information, Figure S1c where an approximately 46× increase in the electrode surface area A from 2.0 × 10−3 mm2 to 9.2 × 10−2 mm2 resulted in an approximately 112× increase of coupling strength K with parameters typically applied in the experiments. 3.2. Numerical Simulations. Numerical simulations were carried out to investigate the effects of changing the total resistance (R0), distance of reference/counter electrodes to the reservoir (through Rc), and electrode surface area (A) on the dynamical behavior of oscillatory Ni electrodissolution in a dual electrode flow cell. An ordinary differential equation model is developed that relies on the kinetic model of nickel dissolution in sulfuric acid by Haim et al.22,25,49 The model has two dimensionless variables for each oscillator l = 1 and 2: the double layer electrode potential (el) and the total surface coverage of electroactive species (θl). The kinetic model is combined with the charge balance in eq 7 and eq 8 to give a dimensionless set of equations as follows

and κ is the dimensionless coupling strength: κ=

rC 1 a(r0 − 2rC) r0

(14)

and rc is the dimensionless collective resistance. (Detailed definitions of all model parameters are given in Table S1.) Model parameters v = 15, ar0 = 20 and kinetic parameters Ch = 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. Γ1 and Γ2 parameters, which correspond to surface molar capacities, were set to slightly different values (Γ1 = 0.01 and Γ2 = 0.0102) to simulate surface heterogeneities that produce the experimentally observed different natural frequencies of the two oscillators.22,25 The ordinary differential equations were solved for a wide range of rc, r0, and a values with MATLAB using a variable stepsize fourthorder Runge−Kutta algorithm (ode45). The results shall be interpreted in the framework of phase description of oscillators.50,51 With relatively weak coupling strengths that do not alter the amplitude of the oscillators the phase dynamics is indicative of the extent of the coupling strengths. Phase drift occurs with no (or very weak) coupling: the two coupled systems oscillate independently from each other and 19293

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phase slipping) by increasing the coupling strength between the oscillators. 3.2.2. Effects of Electrode Surface Area (a) on Synchronization. The simulation on the effects of electrodes surface area (a) on synchronization properties was carried out by changing the surface area parameter (a) in the coupling strength κ in eq 14 while keeping the quantity ρ = ar0 = 20 constant. The results are summarized in Figure 2d−f (middle column) in which the phase difference is shown as a function of time for data obtained by numerical integration of eqs 11−12. Results show that at relative small surface area a = 0.25, the two oscillators are weakly coupled with phase drift behavior (approximately linear increase of phase difference with time, see Figure 2d). When the electrode surface areas are set to a medium value a = 1, the two oscillators intermittently synchronized with a phase slip behavior (Figure 2e). With an increase the surface area to a = 4, phase locked behavior with a constant phase difference of about Δϕ = 0.02 rad (figure 2f) was achieved. Therefore, as the coupling between two oscillators becomes stronger with the increase of the electrode surface area a, in-phase synchronized state develops through phase slipping behavior. 3.2.3. Effects of Total Resistance (r0) on Synchronization. Change of the total resistance (r0) could affect dynamical properties in two aspects: the change of inherent oscillator properties (e.g., period, amplitude, phase response) through eq 11 and the change of coupling strength in eq 14. To explore the relative effects of these two influences, simulations were carried out by changing the r0 parameter in eqs 11−14. Previous studies have shown that by increase the total resistance in the system, the region of oscillation shifts to the more anodic potentials.19 To characterize comparable dynamical behavior at different total resistance, the dimensionless circuit potential (v) adjusted in a manner by which the temporal average ⟨e⟩ of the oscillation remained the same (⟨e⟩ = −3.42) at different r0. The oscillations are thus compared at the same mean electrode potentials. The phase differences of two oscillators with different r0 are shown in right column of Figure 2. At relatively small total resistance r0 = 10 (strong coupling strength, κ = 0.032) phase locked behavior with a constant phase difference of about Δϕ = 0.15 rad (Figure 2g) was achieved. At medium value r0 = 20 (intermediate coupling strength, κ = 0.0060) the phase difference exhibited phase slipping (Figure 2h). Finally, at large total resistance r0 = 50, (weak coupling, κ = 8.4 × 10−4) the phases of the two oscillators exhibit phase drift behavior (Figure 2i). For detailed description of the change of extent of synchrony as a result of change in coupling strength, we use an information entropy based synchrony index (order parameter) to quantify the deviation of the cyclic phase difference distribution from a uniform distribution. Synchrony index based on the Shannon entropy is defined as52

the phase difference increases linearly with time. At intermediate coupling strength phase slipping takes place (intermittent synchronization) where phase locked and phase drifting periods alternate. At strong coupling the two oscillators exhibit phase locking where the phase difference remains constant. The critical coupling strength at which phase locking takes place is proportional to the natural frequency difference between the oscillators, and, for bidirectional coupling, inversely proportional to the amplitude of the odd component of a phase-interaction function that describes the rate of phase advance due to coupling as a function of the phase difference between the oscillators. Because of the efficiency of phase model description, the effect of coupling on the dynamics of the oscillatory system (eqs 11−14) is analyzed by reconstructing the phase of the oscillators; the Hilbert transform of the time series of the electrode potential H (t ) =

1 PV π

∞

∫−∞

e(τ ) − ⟨e⟩ dτ t−τ

(15)

51

is used in defining the phase ϕ(t ) = arctan

H(e(t )) e(t ) − ⟨e⟩

(16)

PV in eq 15 implies that the integral should be evaluated in the sense of Cauchy principal value. ⟨e⟩ is the temporal average of the time series e(t). The phase difference between the oscillators Δϕ = ϕ2 − ϕ1 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 (ωl) is obtained from the slope of the linear fit to the phase versus time plots each of the oscillators. 3.2.1. Effects of Distance to the Reservoir (Collective Resistance, rc) on Synchronization. The distance to the reservoir primarily affects the collective resistance in the cell. The effect of collective resistance (rc) on synchronization properties in the numerical simulation was tested by changing the rc parameters in eq 11−14. At close placement of the reservoir rc = 0 and thus κ = 0. The time series of e and the phase difference of two oscillators are shown in Supporting Information, Figure S2a. Because of the lack of coupling (rC = 0, κ = 0) and small (2%) mismatch in Γ parameters of the oscillators the natural frequencies of the oscillators are slightly different, ω1 = 0.0722 and ω2 = 0.0712 giving a frequency difference of 10 × 10−4. The linear increase of phase difference over time shown in Figure 2a confirms the absence of synchrony. With the increase of distance of the electrode to the reservoir rc is increased; at rC = 1.94 (κ = 0.0060) the two oscillators are intermittently synchronized (Supporting Information, Figure S2b) with a somewhat smaller frequency difference (Δω =7 × 10−4, ω1 = 0.0729, ω2 = 0.0722) than that for the uncoupled case and the phases exhibit “slipping” behavior (Figure 2b) in which phase locked states (Δϕ ≈ 0) alternate with a relatively quick drift that correspond to an oscillatory cycle (2π). Such phase slipping behavior occurs at intermediate coupling strengths in theory of weakly coupled oscillators.51 When the distance to the reservoir is increased even further by setting rC = 6.67 (κ = 0.050) the two oscillators became synchronized (Supporting Information, Figure S2c) with a common frequency of ω1 = ω2 = 0.0717 and achieved a constant phase difference of about Δϕ = 0.17 rad (Figure 2c) indicating a nearly in-phase synchronized state. The simulations thus show that the increase of distance to the reservoir as modeled by increase of the collective resistance rC has an effect of inducing in-phase synchronized state (through

σ=

Smax − S Smax

(17)

where N

S = − ∑ pk ln pk k=1

(18)

is the entropy of the distribution (pk) of cyclic phase difference ψ = Δϕ mod 2π and Smax = lnN is the maximum entropy related to the uniform distribution of phases in N bins. For perfect phase locking, the cyclic phase difference distribution is a Dirac-delta function (single peak at the phase locked state), therefore, the 19294

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results show that by increase of r0 the critical coupling strength at which synchrony is induced, κcrit, is increased, that is, oscillators at larger total resistance require larger coupling strength to synchronize. The effect of r0 on dynamical behavior shown in Figures 2g−i can now be interpreted. First, the increase of r0 decreases the inherent coupling strength between the oscillators (eq 14). Second, at an increased value of r0 larger coupling strength is required to achieve synchrony. Therefore, we can see that both effects of total resistance on the dynamics indicate that an increase of total resistance is expected to diminish the synchrony between the two oscillators. The numerical simulations thus show that the classical phase synchronization phenomena (phase drifting, phase slipping, phase locking) that previously had been reported for two electrochemical oscillator models coupled through external resistances22,48 could also exist in the microfluidic device albeit with coupling inherent in the system through potential drops in the flow channel. The intensified electrical coupling originates from the cell design that utilizes narrow flow channel. The simulations also reveal that in the given electrochemical model system, an increase of total resistance increases the critical coupling strength for synchrony. To explain this dependency we have determined the natural frequency difference and the amplitude of the odd part of the phase interaction function using the computer software XPP53 at the three values of r0 equal to 10, 20, and 50 (same as in Figure 2 g−i). With respect to the r0 = 10 case, the natural frequency difference (0.960 × 10−3 ) increased by 12% (r0 = 20) and 23% (r0 = 50), while the amplitude of the odd part of the interaction function decreased by 18% (r0 = 20)

entropy is 0; for phase drifting the cyclic phase difference distribution is flat, giving maximal entropy. The synchrony index is a rescaled entropy with values of 0 ≤ σ ≤ 1 where σ = 0 corresponds to a uniform distribution (no synchronization) and σ =1 corresponds to a Dirac-delta function (perfect synchronization with a phase locked state). The numerical simulation on effects of coupling strength (κ) on extent synchrony measured by synchrony index (σ) at different fixed total resistance (r0) is shown in Figure 3. At r0 = 10,

Figure 3. Numerical simulations: Effect of coupling strength (κ) on extent synchrony measured by synchrony index (σ) at different total resistance (r0). Circle, r0 = 10; square, r0 = 20; triangle, r0 = 50.

the critical coupling strength for large change of synchrony behavior (from phase drift to phase lock) occurred at κcrit = 0.0055. Similarly, the critical coupling strengths at at r0 = 20 and 50 were found to be at κcrit = 0.0085 and 0.0155, respectively. The

Figure 4. Experiments: Effects of total resistance (R0) on synchronization of current oscillations. (Left column) Current vs time plot; (right column) phase difference vs time plot. (a and b) With very small total resistance, very strong coupling results in phase locked behavior (R0 = 20 kΩ, V = 1.65 V, K = 1.3 kΩ−1 mm−2). (c and d) With small total resistance, strong coupling results phase locked behavior (R0 = 30 kΩ, V = 1.75 V, K = 0.78 kΩ−1 mm−2). (e and f) With medium total resistance, intermediate coupling results in rare phase slip behavior (R0 = 40 kΩ, V = 1.76 V, K = 0.27 kΩ−1 mm−2). (g and h) With large total resistance, weak coupling results in frequent phase slip behavior (R0 = 50 kΩ, V = 1.85 V, K = 0.17 kΩ−1 mm−2). (i and j) With very large total resistance, very weak synchrony results phase drift behavior (R0 = 100 kΩ, V = 2.3 V, K = 4.0 × 10−2 kΩ−1 mm−2). Distance from electrode 2 to the reservoir, L = 5 mm; electrode diameter, D = 100 μm; A = 7.85 × 10−3 mm2; R1,2 = 1.5 kΩ. 19295

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Figure 5. Experiments: Effects of collective resistance (RC) on synchronization of current oscillations. (Left column) current vs time plot; (right column) phase difference vs time plot. (a and b) With close placement, very weak synchrony results in phase drift behavior (L = 1 mm, V = 1.85 V, K = 3.9 × 10−2 kΩ−1 mm−2). (c and d) With medium placement, intermediate synchrony results phase slip behavior (L = 5 mm, V = 1.85 V, K = 0.17 kΩ−1 mm−2). (e and f) With distant placement, strong synchrony results phase locked behavior (L = 16 mm, V = 1.7 V, K = 1.4 kΩ−1 mm−2). D = 100 μm, A = 7.85 × 10−3 mm2, R0 = 50 kΩ, R1,2 = 2 kΩ.

synchronized (Figure 4e) with a small frequencies difference Δω = 0.008 Hz (ω1 = 0.361, ω2 = 0.353 Hz) and achieved phase slip behavior with two phase jump events in the 300 s experiment (see Figure 4f). When the total resistance increased to relative large value of R0 = 50 kΩ (K = 0.17 kΩ−1 mm−2), the two oscillators were still intermittently synchronized (see Figure 4g), however, with a larger frequency difference of Δω = 0.016 Hz (ω1 = 0.374 Hz, ω2 = 0.390 Hz), and achieved the phase slip behavior with five phase jump events in 300 s (see Figure 4h). At even larger R0 = 100 kΩ (K = 0.040 kΩ−1 mm−2), two oscillators were only weakly coupled (see Figure 4i) with a large frequencies difference Δω1 = 0.026 Hz (ω1 = 0.311 Hz, ω2 = 0.337 Hz), and achieved the phase drift behavior (see Figure 4j), which confirms the absence of phase synchronization in weakly coupled oscillators. 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 strong in-phase-synchrony → in-phase-synchrony with a small phase difference → phase slipping with rare phase slips → phase slipping with frequent phase slips → phase drift. This trend confirms the numerical simulations presented in section 3.2.3 showing that with an increase of total resistance the synchrony is progressively lost. The observed synchronization structures (phase drift/slip/ lock) are similar to those seen with macroelectrodes coupled with external resistors.22,48 However, with strong contrast to the macroelectrode studies, the phase slipping and phase locking behaviors in the on-chip integrated device with microlelectrodes occur without any cross resistance, i.e., the electrical coupling is inherent in the setup. We note that in general the dynamical behavior also depends on the natural frequency difference between the oscillators (Δω0) because the critical coupling strength for synchrony is proportional to Δω0.50,51 We observed that the typical natural frequency difference between the oscillators lies in the range of 10−35 mHz in the experiments. If the natural frequency difference was larger than the given value,

and 41% (r0 = 50). Therefore, both phase-interaction function and frequency difference variations point to a critical coupling strength increase; in addition, the phase interaction function amplitude decrease plays a somewhat more important role than the changes of natural frequencies (by a factor of 1.5−1.8) in increasing the critical coupling strength. 3.3. Experimental Results. We have performed experiments in a dual electrode setup to confirm the theoretical and numerical findings that increasing the distance to the reservoir, increasing the electrode area, and decreasing the total resistance intensifies the electrical coupling between the oscillators and thus results in in-phase synchronized oscillations. 3.3.1. Effects of Total Resistance (R0) on Synchronization. The experiments were carried out with smooth electrochemical oscillations (nearly sinusoidal waveform close to the Hopf bifurcation point) obtained at circuit potentials 30 mV above Hopf bifurcation. In these sets of experiment the distance from electrode 2 (downstream) to the reservoir was always set to L = 5.0 mm, which yielded a collective resistance Rc = 3.5 kΩ. The diameter of the two electrodes are D1 = D2 = 100 μm, with surface area A = 7.85 × 10−3 mm2. The time series of current and the phase difference of two oscillators at different total resistances (R0) from 20 kΩ to 100 kΩ are shown in Figure 4 in the left and right columns, respectively. (The total resistance was varied by adjusting the individual resistors connected to the electrodes.) With very small total resistance R0 = 20 kΩ (coupling strength K = 1.3 kΩ−1 mm−2) the smooth current oscillators are synchronized (Figure 4a) with nearly the same frequency, ω1 = 0.511 Hz, ω2 = 0.510 Hz, and achieved a phase locked behavior with small phase difference Δϕ = 0.18 rad (figure 4b). At small R0 = 30 kΩ (K = 0.78 kΩ−1 mm−2), the two oscillators are still synchronized (see Figure 4c, ω1 = ω2 = 0.498 Hz); however, they achieved a phase locked behavior with a slightly increased phase difference, Δϕ = 0.30 rad (see Figure 4d). When the total resistance was increased to medium value of R0 = 40 kΩ (K = 0.27 kΩ−1 mm−2), the two oscillators are intermittently 19296

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exhibit phase slipping. At intermediate placement (L = 5 mm) the coupling strength falls into the critical region when the total resistance (R0) can effectively tune the synchrony between the oscillators. 3.3.3. Effects of Electrode Surface Area (A) on Synchronization. Experiments with different electrode sizes give comparable dynamics when the AR0 quantity is constant.47,48 As a reference point, we consider results presented in section 3.3.1 with medium size electrodes (D = 100 μm, A = 7.85 × 10−3 mm2) that exhibit a high level of synchrony at R0 = 20 kΩ, a low level of synchrony with R0 = 100 kΩ, and tunable synchrony with R0 = 50 kΩ, where experimental conditions (distance to the reservoir) strongly affect the observed behavior. The results of the experiments with two small electrodes (D = 50 μm, A = 2.0 × 10−3 mm2) at various total resistances (R0) are shown in Figures 6a−f. In all these experiments (that correspond to AR0 values for the medium size electrode in the investigated range of 20−100 kΩ) the phase of the oscillators exhibited drift behavior that indicates lack of coupling between the electrodes. In contrast, with large electrodes (D = 500 μm, A = 0.0900 mm2), the oscillations were always strongly synchronized (Figures 6g,i) with phase locked behavior (Figures 6h,j). (The total resistances were R0 = 4.4 kΩ, and R0 = 10.9 kΩ that compares to the intermediate or weakly coupled medium size electrode resistances 50 kΩ and 125 kΩ, respectively). In summary, the experiments confirm the theoretical and numerical prediction that with small electrodes the weak coupling cannot induce synchrony while with large electrodes the strong coupling results in highly synchronized oscillations. 3.3.4. Effect of Coupling Strength (K) on Synchrony Measured by the Synchrony Index (σ). The Experiments in Figures 4−6 and Table 1 with different total resistances, distance to reservoir, and electrode surface areas are summarized in Figure 7 where the synchrony index (σ) is shown as a function of the calculated coupling strength (K). (Because the coupling strength covered a large region, the logarithm of the coupling strength is used.) The results show that when K < 0.1 kΩ−1 mm−2, the system always exhibits phase drift (circles) behavior with the value of σ around zero, indicating weak synchrony between two oscillators. When K > 1 kΩ−1 mm−2, the system exhibits phase locked (triangles) behavior with σ > 0.5 indicating strong synchrony. When 0.1 kΩ−1 mm−2 < K < 1 kΩ−1 mm−2, the system could exhibit phase locked, phase slip (squares), or phase drift behavior, the occurrence of which depends on the parameter R0, Rc, and A and the actual natural frequency difference between the oscillators. The plot thus confirms that the dynamical behavior of the system related to the synchronization of the oscillators can be predicted by the theoretical value of the coupling strength K; in other words, the graph facilitates the design of cell geometry and electrode sizes for strong (K > 1 kΩ−1 mm−2), weak (K < 0.1 kΩ−1 mm−2) or tunable (0.1 kΩ−1 mm−2 < K < 1 kΩ−1 mm−2) coupling strength for the synchrony behavior. The characterization of electrical coupling strength using eq 9 could also facilitate design of electroanalytical applications52,53 with on-chip integrated devices that utilize multiple electrodes (e.g., generator−generator or generator−collector configurations). In such devices there is a need for minimizing the electrical crosstalk between the electrodes. For example, in flow cells with a similar configuration to those in our study (200 μm (width) × 100 μm (height) flow channel with electrode sizes of about 100 μm × 100 μm in a typical background electrolyte of 1 M solution) and analyte concentrations and flow rates such that approximately 10 μA current is generated, it is strongly

then the electrodes were repolished and the cell was reassembled. Note that in the experiments we often tuned the coupling strength to a large extent so that the natural frequency difference did not play a major role in the overall qualitative dynamics; in the experiments above the coupling strength was intensified by a factor of 33 while the total resistance was changed from 20 to 100 kΩ. To further explore our capability of tuning the electrical coupling between the electrodes, we also investigated the effect of distance to reservoir and electrode size on the dynamical behavior. 3.3.2. Effects of Distance to Reservoir through Collective Resistance (Rc) on Synchronization. The distance to the reservoir, L, affects (in a linear manner) the collective resistance RC of the cell.9 We performed experiments in the two-electrode setup by changing the distance L from electrode 2 to the reservoir to change the solution collective resistance Rc while keeping the total resistance R0 and electrode surface area A constant. According to eq 9 the distance (L) through the Rc parameter can effectively tune the coupling strength between the oscillators. The time series of current and the phase difference of two oscillators with R0 = 50 kΩ, D1 = D2 = 100 μm (A = 7.85 × 10−3 mm2) at different collective resistance (Rc) are shown in Figure 5 left and right columns, respectively. At close placement of the downstream electrode with L = 1 mm (Rc = 0.84 kΩ, K = 3.9 × 10−2 kΩ−1 mm−2), the two oscillators had little impact on each other (see Figure 5a): they achieved a phase drift behavior (Figure 5b) with a frequency difference of Δω = 0.009 Hz (ω1 = 0.443 Hz, ω2 = 0.434 Hz). When we increased L to 5 mm (Rc = 3.5 kΩ, K = 0.17 kΩ−1 mm−2), the increased coupling strength resulted in phase slip behavior with intermittent synchrony (Figure 5c,d). At distant placement with L = 16 mm (Rc = 14.7 kΩ, K = 1.4 kΩ−1 mm−2) the strong coupling synchronized the oscillators (Figure 5e) with a common frequency of ω1 = ω2 = 0.055 Hz and achieved a phase locked behavior with Δϕ = 0.31 rad (see Figure 5f). The effects of L were investigated at different R0 values and the corresponding phase dynamics (drifting, slipping, locking) are summarized in Table 1. The results shown in each column Table 1. Experiments: Effects of R0 and L on Synchronization of Dual Electrodesa R0

L = 20 kΩ

L = 50 kΩ

L = 100 kΩ

1 mm 5 mm 15 mm

phase slip phase lock phase lock

phase drift phase slip phase lock

phase drift phase drift phase slip

a

At large distance, phase locked behavior which results from strong synchrony is dominant. At medium distance, tunable synchrony is achieved. At small distance, phase drift behavior which results from weak synchrony is dominant. D = 100 μm, A = 7.85 × 10−3 mm2. Circuit potential is set to about 30 mV above the Hopf bifurcation point.

indicate that the coupling between two oscillators increases with the increasing of distance (collective resistance Rc). The results shown in each row confirm again that coupling strength between two oscillators decreases with an increase of the total resistance resulting in lower level of synchrony. At small distance (1 mm), phase drift behavior, which results from weak synchrony is dominant; only at low R0 (20 kΩ) did we observe phase slipping. At far placement (15 mm) phase locked behavior occurs except at large total resistance (100 kΩ) at which the oscillators tend to 19297

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Figure 6. Experiments: Effects of electrode surface area (A) on synchronization of current oscillations. (Left column) current vs time plot; (right column) phase difference vs time plot. (Top row, a−f) with small electrodes, lack of coupling results in phase drift (D = 50 μm, A = 2.0 × 10−3 mm2, L = 5 mm, R12 = 2 kΩ): (a and b) R0 = 80 kΩ, V = 1.65 V, K = 0.26 kΩ−1 mm−2; (c and d) R0 = 200 kΩ, V = 1.8 V. K = 4.0 × 10−2 kΩ−1 mm−2; (e and f) R0 = 400 kΩ, V = 2 V, K = 9.8 × 10−3 kΩ−1 mm−2. (Bottom row, g−j) with large electrodes, strong coupling results in phase locked (D = 500 μm, A = 0.09 mm2, R1,2 = 2 kΩ): (g and h) R0 = 4.4 kΩ, V = 1.65 V, L = 0.6 mm, K = 0.40 kΩ−1 mm−2; (e and f) R0 = 10.9 kΩ, V = 2 V, L = 5 mm, K = 1.5 kΩ−1 mm−2.

potential drop in the electrolyte is intensified. The microcell exhibits this strong coupling because of the large potential (IR) drop in the small flow channel that has large resistance. The extent of in-phase synchrony is determined by a theoretically derived coupling strength formula (eq 9) that predicts that the coupling strength increases with far placement of counter/ reference electrodes, large electrode sizes, and small total resistance. The distance between the two electrodes affects the maximum coupling strength that can be achieved in the cell. The experimental procedure can be expanded to study the interaction of a population of electrodes within 1D or 2D network configuration. Such networks could provide insights of importance of network topology on dynamical behavior of nonlinear chemical systems. The findings provide valuable cell design information for improvement of the accuracy of the measurements in electroanalytical applications that require collector−generator multielectrode wires.54,55

Figure 7. Experiments: Effect of coupling strength (K) on synchrony measured by synchrony index (σ) at different conditions shown in Figures 4−6 and Table 1. Circle, phase drift; square, phase slip; triangle, phase locked.

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recommended that the reservoir is placed 1 mm or less to the downstream working electrode. However, it would be expected that reactions on electrode sizes of less than 50 μm × 50 μm (and corresponding current levels of 2−3 μA) could be safely investigated with the reservoir placed at a typical 5−10 mm distance to the downstream electrode in the given configuration.

ASSOCIATED CONTENT

S Supporting Information *

Figure S1: The calculated coupling strength vs cell parameters. Figure S2: Numerical simulations showing the effects of collective resistance on synchronization of oscillations. Table S1: Detailed description of model parameters in eqs 11−14. This material is available free of charge via the Internet at http://pubs. acs.org.

4. CONCLUSION The dynamical behavior of oscillatory electrodissolution of two nickel electrodes was characterized in an on-chip integrated microfluidic flow cell as a function of cell geometry (distance to the reference/counter electrodes), electrode size, and total resistance. In contrast to the macrocell experiments where a cross-resistor is needed to induce synchrony between the two oscillators,22,25 in the microcell spontaneously synchronized oscillations were observed when the electrical coupling due to

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +1-314-977-2139. Fax: +1-314-977-2521. E-mail: izkiss@ slu.edu. Notes

The authors declare no competing financial interest. 19298

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant No. CHE-0955555.

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