Influence of Surface Tension-Driven Convection on Cyclic

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Influence of Surface Tension-Driven Convection on Cyclic Voltammograms of Langmuir Films of Redox-Active Amphiphiles Guiyu Bai, Nicholas L. Abbott,*,† and Michael D. Graham*,‡ Department of Chemical Engineering, University of WisconsinsMadison, Madison, Wisconsin 53706 Received May 17, 2002. In Final Form: September 20, 2002 Simulations were conducted to elucidate the effects of Marangoni convection on the cyclic voltammograms of Langmuir films of redox-active amphiphiles. Oxidation and reduction of the amphiphiles occur at a line electrode that contacts the Langmuir film, and oxidation is assumed to introduce an electrostatic contribution to the surface equation of state of the film. A boundary element method is used to compute the bulk fluid velocities that are induced by the corresponding surface tension gradients and a finite element method used to compute the surface transport of the surfactants. When compared with the case of purely diffusional transport processes, the simulation results reveal that when Marangoni convection is included, the cyclic voltammograms become asymmetric. The anodic peaks tend to be larger and sharper than the cathodic and the ratio of cathodic peak to anodic peak is about 0.808. The results of the simulation are compared to past reports of cyclic voltammograms of Langmuir monolayers of a ferrocenyl amphiphile, and good agreement is found.

1. Introduction Marangoni flow is caused by tangential stress exerted at a fluid-fluid interface because of a surface tension gradient. Often, these gradients are due to concentration gradients or temperature gradients. It occurs in many settings, such as crystal growth, alloy melting, film wetting, and the condensation of multicomponent mixtures.1 Recent years have seen a resurgence in interest in Marangoni flows because of their potential importance for control of fluid motions at the scales characteristic of microfluidic devices.2,3 In particular, the present study is motivated in part by recent demonstrations that electrochemical methods can be used to transform a redox-active surfactant (e.g., Fc(CH2)11-N+(CH3)3Br-) between states that differ in surface activity and thereby achieve active control of a variety of interfacial phenomena, including Marangoni flows.4-7 In this paper, we report a modeling study of the effects of electrochemically driven Marangoni flows in a simple context, cyclic voltammetry of insoluble redox-active amphiphiles assembled at the surface of aqueous solutions of electrolyte. The properties of Langmuir monolayers of waterinsoluble redox-active surfactants at the air/water interface have been experimentally investigated using cyclic voltammetry by Majda and co-workers.8,9 A key element * Corresponding authors. † E-mail: [email protected]. ‡ E-mail: [email protected]. (1) Pline, A.; Zurawski, R.; Jacobson, T.; Kamotani, Y.; Ostrach, S. Hardware and Performance Summary of the Surface Tension Driven Convection Experiment-2 aboard the USML-2 Spacelab Mission. In 47th IAF Congress, Beijing; 1996, p 47. (2) Kataoka, D. E.; Troian, S. M. Nature (London) 1999, 402, 794797. (3) Stone, H. A.; Kim, S. AIChE J. 2001, 47, 1250-1254. (4) Bennett, D. E.; Gallardo, B. S.; Abbott, N. L. J. Am. Chem. Soc. 1996, 118, 6499-6505. (5) Gallardo, B. S.; Hwa, M. J.; Abbott, N. L. Langmuir 1997, 11, 44209-4212. (6) Aydogan, N.; Gallardo, B. S.; Abbott, N. L. Langmuir 1999, 15, 722-730. (7) Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig, V. S.; Shah, R. R.; Abbott, N. L. Science 1999, 283, 57-60.

Figure 1. (a) Molecular structure of n-octadecyl ferrocene carboxamide (C18Fc) in its reduced (left)/oxidized (right) state. (b) Cyclic voltammogram recorded by Majda and co-workers9 at a gold microband electrode positioned at a air/water interface covered with C18Fc. The dashed line is the background recorded at the same electrode before the C18Fc monolayer was spread at the interface. Temperature (T) ) 27 °C, electrode width ) 0.158 cm, and scan rate ) 200 mV/s.

of those studies was a one-dimensional gold microband electrode that was positioned in the plane of the water surface. The surfactant C18Fc (structure shown in Figure 1a) can freely diffuse on the water surface. The cyclic voltammograms (CVs) of this process were recorded and interpreted to result from purely diffusional processes (Figure 1bsnote that here and in the simulated CVs presented below, an oxidizing (positive) potential is on the left, and reducing (negative) is on the right). Nevertheless, inspection of Figure 1b reveals that the anodic peak is higher than the cathodic peak, which, as we see below, is not expected for a purely diffusional process. (8) Charych, D. H.; Goss, C. A.; Majda, M. J. Electroanal. Chem. 1992, 323, 339-345. (9) Charych, D. H.; Landau, E. M.; Majda, M. J. Am. Chem. Soc. 1991, 113, 3340-3346.

10.1021/la0259611 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/09/2002

Surface Tension-Driven Convection

Figure 2. Schematic illustration of the simulation domain. The domain is 0.04 cm wide and 0.02 cm high. On the water surface, the surfactants can freely diffuse; redox reactions occur only at the electrode.

In the present work, we explore the proposition that this asymmetry results from Marangoni convection induced by gradients in surface tension that accompany the electrode redox processes. We report simulations of cyclic voltammograms of insoluble redox-active surfactants that include the effects of surface electrochemical reaction, surface transport and Marangoni flow. The surface tension gradients result from oxidation-induced changes in the surface pressure due to the electrostatic interaction among the oxidized amphiphiles. The simulation results, which are in good agreement with experiment, reveal that Marangoni flows can introduce asymmetry into cyclic voltammograms.

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Figure 3. Electrostatic contribution of the oxidized species to the surface pressure, calculated from eq 4. oxidized species is10

∆gel ) -

Ψ0

0

σ dΨ

(1)

where Ψ is surface potential corresponding to the surface charge density σ and Ψ0 is the potential of the monolayer. The relation between surfactant charge density and surface potential is

σ)

-4CeNAe eΨ sinh K 2kT

( )

(2)

where K is the Debye-Hu¨ckel parameter

K)

2. Models and Methods The system considered is shown in Figure 2. Inert electrolyte fills the fluid domain with a concentration high enough to eliminate migration effects (i.e. the bulk electrostatic potential is constant). The domain has a rectangular cross-section that is 0.04 cm long and 0.02 cm high, bounded by fixed walls where the velocities vanish and a symmetry plane on which there is no horizontal velocity or shear stress. The system is assumed to be infinite and uniform in the z-direction, allowing a two-dimensional description of the flow field. The liquid-gas surface is assumed to be flat (we justify this assumption below); insoluble surfactant molecules can freely move on the surface by diffusion and fluid flow. Reduction and oxidation occur only at the corner of the electrodesthe three-phase contact line. The viscosity of the electrolyte solution is 1.0 cP at room temperature. The thickness of the electrode is half of the length of the domain (0.02 cm). All simulations are run at a scan rate of 0.2 V/s with the potential oscillating in a triangular waveform between +0.5 and -0.5 V. Although in the simulation the electrode is infinite and uniform in the z-direction, to calculate the current and compare to experiment, we pick a fixed electrode width in the z-direction (W ) 0.158 cm). Unless otherwise noted, the diffusion coefficients (D) of both the reduced and oxidized surfactants are taken to be 2.4 × 10-6 cm2/s, as found in the experiments of Majda and coworkers.10 2.1. Electrostatic Contribution to Surface Pressure. To permit comparison to past experiments, we assume an average surface area per molecule of 120 Å2. We assume that the surface pressure of the reduced surfactant is negligible at this surface concentration. This assumption is consistent with previous work with insoluble10 and soluble (ferrocenyl) surfactants.4,6,7 We further assume that electrostatic effects dominate the change in surface pressure with oxidation state. The electrostatic contribution to the excess Gibbs free energy of a surface containing the

∫

(

)

2e2N0Ce kT

1/2

(3)

Ce is the electrolyte concentration, T is the absolute temperature, e is electronic charge, NA is Avogadro’s constant, k is Boltzmann’s constant, and  is the permittivity of solution (7.08 × 10-10 C2 J-1 m-1). For 0.05 M HClO4, K ) 0.735 nm-1. With Co denoting the surface concentration of oxidized species, the change in surface pressure upon oxidization is thus given by

∆(γ0 - γ) )

( ( ) )

eΨ0 8CeNAkT cosh -1 K 2kT

(4)

where

Ψ0 )

( )

-CoK 2kT sinh-1 e 4Ce

(5)

and γ0 is the surface tension of a surfactant-free interface. Figure 3 shows the surface pressure as a function of area per molecule of oxidized surfactant. At the simulation conditions (120 Å2/ (oxidized molecule)), the surface pressure is about 5.0 mN/m. 2.2. Surfactant Transport. The conservation equation for transport of insoluble surfactant on a deforming interface is11

∂CR + ∇s‚(CRus) + CR(∇s‚n)(us‚n) ) DR∇s2CR ∂t

(6)

In this equation, CR is the concentration of reduced (R ) r) or oxidized (R ) o) surfactant on the surface and us is the fluid velocity at the surface. Electric-field-driven migration and (10) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997. (11) Stone, H. A. Phys. Fluids A 1990, 2, 111-112.

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electron hopping between oxidized and reduced species are neglected because of the assumed presence of excess inert electrolyte and the relatively slow redox kinetics on the surface. For most simulations, we assume that the diffusivities of the oxidized and reduced species are the same: Do ) Dr ) D. The third term on the left-hand side results from a local change of surface area due to the deformation of the surface. Under the assumption that the surface does not deform (us‚n ) 0), we find below that the velocities at the surface are around V ) 0.03 cm/s. Balancing the viscous and capillary stresses, we can estimate the radius of curvature R of the interface. The viscous stress is given by τv ∼ µV/L and the capillary stress by τc ∼ γ/R. Equating these, using characteristic parameter values F ) 1 g/cm3, µ ) 1 cP, L ) 0.02 cm, and γ ) 70 mN/m, gives R ≈ 5000 cm, which is much larger than the characteristic length of the flow domain. Therefore, interface deformation is unimportant, the flat-surface assumption is justified and the us‚n ) 0 term in eq 4 is negligible. The current is determined by two factors: the kinetics of the electron transfer between electrode and ferrocene surfactants and the rate of surfactant transport (diffusion and convection) to the electrode. Here we assume that the latter effect is ratelimiting. Accordingly, the Nernst equation is applied at the electrode (x ) 0.02)

δE )

()

Co kT ln , e Cr

(7)

where δE is the difference between the applied potential and standard potential. At the other contact line (x ) 0), the surfactant concentrations satisfy a no flux boundary condition. To compute the current passed at the electrode, we evaluate

dMo , dt

i(t) ) F

(8)

where Mo is the total moles of oxidized species on the surface and F is Faraday’s constant. The bulk and interfacial transport processes are coupled through the tangential stress boundary condition on the surfactant covered surface

Figure 4. Evolution of current when Marangoni convection is neglected and the surface is covered with almost pure reduced species. The inserts are expanded views of the cathodic and anodic peaks. sized elements on the bottom boundary, 20 on each side and on the bottom surface of the electrode, and 40 on the free surface. To compute the evolution of the surfactant concentrations, a finite element method is used for spatial discretization and timeintegration is performed with a forward Euler method. Because convective transport dominates on much of the interface (the Peclet number VL/D is large), the streamline upwind PetrovGalerkin (SUPG) method15 is implemented to improve the numerical stability of the computations.

3. Results and Discussion

where p is the pressure and u the fluid velocity. On the solid boundaries, ux ) uy ) 0; on the symmetry boundary, ux ) 0,∂uy/∂x ) 0; on the free surface, u ) us, with uy ) 0 and the shear stress is as given above. 2.3. Solution Method. At every time step, the surface tension gradient at the surface must be computed from the oxidizedsurfactant concentration profile Co, and then the velocity field was computed. Using this velocity field, the concentration profiles of surfactants are updated. We solve the flow problem at each time step with a boundary element method (BEM). In this approach, the Stokes flow problem is recast as an integral equation for the stresses and velocities on the boundary of the domain, and then the boundary is discretized into elements.12 We approximate the shear stress and velocities on each element using linear interpolation functions, a very simple method that nevertheless yields satisfactory results.11-14 There are 40 equal-

3.1. Cyclic Voltammograms Neglecting Convection. Figure 4 shows the evolution of the cyclic voltammograms when Marangoni convection is neglected; the fluid velocity is set to zero so surface diffusion is the only means of surfactant transport. In this simulation, the surface is initially covered with largely reduced species (corresponding to an electrode potential of -0.5 V). At the beginning of the simulation, the integrated anodic current is larger than the cathodic: more oxidized species is produced than consumed. After the first cycle, the anodic peak shrinks while the cathodic peak increases, and after several cycles the CV reaches a steady state oscillation in which the anodic and cathodic peaks are equal in size. The steady state can also be viewed by tracking the total amount of surfactant on the surface (Figure 5). After about 17 cycles, the steady state is reached and the amounts (moles) of oxidized and reduced species oscillate around the same average value. Inspection of Figure 5 also reveals that at steady state, the cyclic voltammograms are totally symmetric; i.e., they are invariant under a rotation of 180° around the origin. Figure 6 shows the concentration profiles of reduced and oxidized species along the free surface at various locations on the cyclic voltammogram. At the end away from the electrode (x ) 0), the concentrations remain constant, while they oscillate near the electrode according to the Nernst equation and diffusive transport. The concentration profiles are identical if we exchange the reduced and oxidized labels and shift half a cycle in time. Finally, we have studied the effect of letting the two diffusivities be different. A 20% change in the

(12) Pozrikidis, C. Boundary integral and singularity methods for linearized viscous flow; Cambridge University Press: Cambridge, England, 1991. (13) Higdon, J. J. L. Fluid Mech. 1985, 159, 195-226.

(14) Milliken, W. J.; Stone, H. A.; Leal, L. G. Phys. Fluids A 1993, 5, 69-79. (15) Brooks, A. N.; Hughes, T. J. R. Comp. Methods Appl. Mech. Eng. 1982, 32, 199-259.

∂ux ) ∇γ, ∂y

τyx ) µ

(9)

where τyx is the shear stress at the interface. The quantity ∇γ is the Marangoni stress. Finally, we briefly describe the equations governing the bulk fluid behavior. Assuming that the Reynolds number Re () FVL/ µ) is much less that unity (which we have verified a posteriori), the fluid motion is governed by Stokes equations:

- ∇p + µ∇2u ) 0

(10)

∇‚u ) 0,

(11)

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Figure 5. Evolution of the total amounts of reduced and oxidized species on the surface. All the parameters and conditions used here are the same as for Figure 4.

Figure 7. Evolution of current in the presence of Marangoni convection. The surface is initially covered with almost pure reduced surfactant.

Figure 6. Steady-state cyclic voltammogram without Marangoni flow. The small figures show the concentration profiles of reduced and oxidized species at corresponding points on the CV. Parameters used here the same as for Figure 4.

diffusivity of either species does break the symmetry of the CV, but only by a negligible amount. 3.2. Cyclic Voltammograms Including Convection. According to the surface equation of state used in our simulations, a surface tension gradient tends to drive flow from regions of high to low oxidized species concentration. Figure 7 shows the effect of the Marangoni stress on CVs at a surface initially covered with the reduced species (δE ) - 0.5 V). When compared to CVs calculated in the absence of convection (Figure 4), both the anodic and cathodic peaks are smaller in the first cycle. Oxidized species generated at the electrode lowers the surface tension near the electrode and drives surfactant (both oxidized and reduced) away from it. Figures 8 and 9 show the fluid velocity at the interface and in the bulk during one cycle of the CV. The stress at the interface generates a circulation in the fluid with a maximum velocity of about 0.03 cm/s. At steady state, the cyclic voltammograms are distinctly asymmetric, with the anodic peak sharper and higher than the cathodic one (Figures 8 and 10). The ratio of the cathodic and anodic peak heights is 0.808, which is almost identical to the result 0.80 that we calculated from Figure

Figure 8. Steady-state cyclic voltammogram with convection. The inserted small figures show the velocity profiles on the surface at different points on the CV; positive velocities are directed toward the electrode.

1b.9 We also note that another signature of cyclic voltammograms in the presence of Marangoni convection is a shift of the anodic peak toward a negative potential. For the purely diffusional process (Figure 4), the anodic and cathodic currents peak in the negative and positive potentials, respectively. In the presence of convection, however, the surfactants are driven away from the electrode when the potential is increased. This effect, combined with diffusion, causes the anodic current peak to occur earlier in the cycle than during a purely diffusional process. To illustrate the effects of changing the initial surface preparation, Figure 11 shows the evolution of oxidized

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Figure 11. Evolution of oxidized and reduced species totals when the interface contains initially a 1:1 ratio of reduced and oxidized surfactant. Other parameters used here are the same as in Figure 7.

Figure 9. CV and instantaneous velocity fields corresponding to Figure 8. Each arrow shows the instantaneous magnitude and direction of flow computed at its tail.

Figure 12. Evolution of moles of oxidized surfactant when the surface is initially covered with different concentrations of oxidized species. Marangoni convection is included in this simulation.

Figure 10. CV and instantaneous surface concentrations corresponding to Figure 8.

and reduced species on the surface when the initial state of the surface corresponds to half oxidized and half reduced species (δE ) 0 V). It is interesting to note that the presence of convection influences the steady-state composition of the surface. Specifically, convection leads to an enrichment of the reduced species. In contrast, in the absence of convection the steady-state compositions of reduced and oxidized species are equal (Figure 5). The enrichment of the reduced species on the surface is caused by convective motion toward the electrode when cathodic potentials are applied. The result is that, until steady state is reached,

the electrode produces more reduced species in the anodic half-cycle than it consumes in the cathodic halfcycle. To illustrate that the CVs reach the same steady state independent of initial conditions, we ran simulations where the initial ratios of oxidized and reduced surfactants were varied, but the total surfactant concentration was kept constant. In these simulations, three concentration ratios were picked: pure reduced species, one-third oxidized species, and half-oxidized and half-reduced species. Figure 12 shows the evolution of oxidized species in these cases. All of them reach the same steady state; the averages are the same and the oscillation curves exactly overlap each other. Whereas all the simulations reach the same steady state, the first cycle is very sensitive to the initial condition of the surface. Figure 13 shows the evolution of current when the surface is initially covered by two-thirds reduced species and one-third oxidized. There is initially a large positive current because convection drives the surfactants to the electrode. When compared to the evolution of current that the surface is initially covered with pure reduced species, we see that the presence of oxidized species enhances the cathodic current during the first cycle. These results clearly show that Marangoni convection can have a significant qualitative effect on the dynamics

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proportional to the microbandwidth, ruling out the possibility of diffusion distortion in the direction parallel to the electrode edge.10 It also seems unlikely that the asymmetry arises from different diffusivities of the reduced and oxidized species; our simulations of this case show a negligible change in the CVs. 4. Conclusion

Figure 13. Evolution of the CV when the surface is initially covered with one-third oxidized and two-thirds reduced surfactant. Marangoni convection is considered in this simulation.

of Langmuir monolayers of redox-active amphiphiles during cyclic voltammetry. The most experimentally accessible result of our simulations is of course the CV itself, which shows a distinct asymmetry qualitatively and quantitatively similar to that found experimentally by Majda and co-workers.8,9 It is unlikely that the experimental asymmetry comes from C18Fc adsorption on the microelectrode, because, in their studies, Majda and co-workers periodically detached microband electrodes from the interface and repositioned them on the clean surface, and the obtained background currents were identical with what was seen at a fresh electrode on a clean air/water interface. They also changed the width of the microband and found the currents were linearly

We report the use of numerical simulations to investigate the effects of Marangoni convection on the cyclic voltammograms of Langmuir films of redox-active amphiphiles. When the possibility of Marangoni convection is ignored, the cyclic voltammograms are calculated to be symmetric or very nearly so at steady state. When Marangoni convection is included in the simulation, however, we observe the cyclic voltammograms to possess at steady state an anodic peak that is higher and sharper than the cathodic, consistent with past experimental observations by Majda and co-workers. It is also found that the convection can change the steady-state mass ratio of the reduced and oxidized states of the surfactants within a Langmuir monolayer. Acknowledgment. This work was supported in part by the Camille and Henry Dreyfus (Teacher-Scholar Award), the David and Lucille Packard Foundation, the donors of the Petroleum Research Fund (ACS-PRF 35409AC7), administered by the American Chemical Society, and the National Science Foundation (CTS 9911863 and EEC/BES/CTS 0085560). LA0259611