J. Phys. Chem. C 2009, 113, 9053–9062
9053
Hydrogen Peroxide Electroreduction at a Silver-Nanoparticle Array: Investigating Nanoparticle Size and Coverage Effects Fallyn W. Campbell, Stephen R. Belding, Ronan Baron, Lei Xiao, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: January 9, 2009; ReVised Manuscript ReceiVed: March 23, 2009
The cathodic reduction of hydrogen peroxide displays altered electrochemical behavior between silver macroand nanoscale electrodes. In acidic media, two parallel reduction mechanisms have been reported: “normal” and “autocatalytic”. The reduction potentials are reported in the literature versus mercury/mercurous sulfate reference electrode. The “normal” reduction of H2O2, in the presence of H+, forms water and the intermediate OH(ads), taking place at 1 ∂Z
R)0
TABLE 2: Dimensional Parameters dimensional parameter
definition
re F R T A [A]0 [B]0 [A]bulk [B]bulk R k0 E Ef0 DA DB j i ν N
electrode radius (m) Faraday constant (C mol-1) universal gas constant (J K-1 mol-1) temperature (K) electrode area (m2) concentration of species A at the electrode surface (mol m-3) concentration of species B at the electrode surface (mol m-3) concentration of species A in bulk solution (mol m-3) concentration of species B in bulk solution (mol m-3) transfer coefficient (unitless) electrochemical rate constant (m s-1) applied potential (V) formal reduction potential (V) diffusion coefficient of species A (m2 s-1) diffusion coefficient of species B (m2 s-1) flux across the electrode (mol · m2 s-1) current (A) i/ilim on a steady-state voltammogram (unitless) scan rate (V s-1)
scaling factors. The dimensionless parameters used in this paper are defined in Tables 1 and 2. The problem amounts to solving Fick’s second law
∂a 1 ∂a ∂ 2a ∂2a ) 2 + + 2 ∂τ R ∂R ∂R ∂Z
(A6)
∂b )0 ∂Z
The problem is solved numerically using the alternating direct implicit method (ADI).31,32 This requires discretization in both space and time. Simulation efficiency is increased by using an expanding spatial mesh. The mesh is chosen so as to leave a high density at the singularities, that is, the edge of the electrode and the symmetry axis. An exponentially expanding mesh is used. The nature of the expansion is identical to that used by Gavaghan.33 Once the simulation is run across a 2D spacial segment of Figure 5A, the flux over the entire electrode is calculated using eq A7.
Jtotal ) π
∫01 JsegmentRdR
(A7)
This integral is solved numerically using the trapezium rule.34 Appendix 2: Simulation of Microelectrode Arrays The simulation procedure for a microelectrode in an array is identical to that described above, except the value of Rmax must be chosen so as to characterize the separation of the electrodes of the array. Electrode separation is calculated assuming a regular square grid. Each microelectrode can be considered as being at the center of a unit cell, as shown in the Figure 5D. Each cell is diffusionally independent. The square base of each cell is taken to be equivalent to a circle of equal area. This is called the diffusion domain approximation and is illustrated in Figure 5C,D. The current over the entire array is calculated by summing together the contribution from each microelectrode. Appendix 3: Computation
(A2)
where, in order to account for the symmetry of the situation, cylindrical coordinates are used. This equation is solved subject to several boundary conditions. The behavior of the system at the interface between electrode and solution is quantified by the Butler-Volmer equation
J ) K0a0 exp[-Rθ] - K0b0 exp[(1 - R)θ]
∂b )0 ∂R ∂b )0 ∂R ∂a )0 ∂Z
(A3)
At very large distances from the electrode surface, a ) 1 and b ) 0. The values of Zmax and Rmax are chosen so as to contain the entire diffusion layer. The expressions used here are30
Zmax ) 6√τtotal
(A4)
Rmax ) 1 + 6√τtotal
(A5)
In addition, there is assumed to be zero flux through the symmetry axis or through the insulating surface surrounding the electrode. The boundary conditions are summarized below
All programs were written in C++ and compiled using a Borland compiler. The simulations were run on a desktop PC with a 3.4 GHz Pentium processor and 2GB of RAM. Approximately 2 min of CPU time was required to simulate a single voltammogram. The integrity of the source code and efficiency of the program were investigated by comparing simulations for an isolated disk at steady state with the Saito equation.35 The program was converged to yield better than 0.1% accuracy. Acknowledgment. F.W.C., R.B., and L.X. thank EPSRC for funding. F.W.C. also thanks Abington Partners for partial funding. References and Notes (1) (a) Sˇljukic´, B.; Baron, R.; Salter, C.; Crossley, A.; Compton, R. G. Anal. Chim. Acta 2007, 590, 67. (b) Genies, L.; Faure, R.; Durand, R. Electrochim. Acta 1998, 44, 1317. (2) Hughes, M. D.; Xu, Y.; Jenkins, P.; McMorn, P.; Landon, P.; Enache, D. I.; Carley, A. F.; Attard, G.; Hutchings, G. J.; King, F.; Stitt, E. H.; Johnston, P.; Griffin, K.; Kiely, C. J. Nature 2005, 437, 1132. (3) Arrigan, D. W. M. Analyst 2004, 129, 1157. (4) Compton, R. G.; Wildgoose, G. G.; Rees, N. V.; Streeter, I.; Baron, R. Chem. Phys. Lett. 2008, 459, 1. (5) Welch, C. M.; Banks, C. E.; Simm, A. O.; Compton, R. G. Anal. Bioanal. Chem. 2005, 382, 12. (6) Welch, C. M.; Compton, R. G. Anal. Bioanal. Chem. 2006, 384, 601.
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