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Diffusional Cyclic Voltammetry at Electrodes Modified with Random Distributions of Electrocatalytic Nanoparticles: Theory Stephen R. Belding, Edmund J. F. Dickinson, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: February 23, 2009; ReVised Manuscript ReceiVed: April 27, 2009
Electrodes modified by electrocatalytic nanoparticles find increasing application in electroanalysis and energy conversion. With the correct choice of nanoparticle material, oxidation or reduction can be brought about via an electrocatalytic process at the nanoparticle. Cyclic voltammetry is studied at such electrodes for the case of a simple homogeneous one-electron reduction involving solution-phase species (A + e- h B). Numerical simulations are used to identify how the voltammetric features are influenced by the electrochemical rate constant of the electron transfer between the A/B couple and the nanoparticle, the surface coverage of nanoparticles, and the voltage scan rate. Data are included in the Supporting Information to enable determination of the surface coverage from experimental data and to discern the onset of diffusion layer overlap for a range of experimental situations. Lastly, the minimum extent of surface modification required to confer upon the electrode a bulk geometric response corresponding to that of the electrocatalytic process is estimated. 1. Introduction Nanoparticles are of increasing importance to several scientific disciplines, notably, biology and catalysis.1,2,5,6 Their widespread use is derived mainly from the expression of characteristics absent in bulk material. Salient features include enhanced mass transport, high surface area, and unique crystal planes.2,7,8 In electroanalysis, an electrode surface can be modified with an array of nanoparticles that mediate electron transfer to an analytical target. For example, silver nanoparticles have been used on a carbon substrate for the sensitive detection of hydrogen peroxide.2 Detection was possible at potentials much lower than those for the analogous unmodified electrode. Such electrodes also have important applications in energy conversion.9 While the theory of voltammetry at flat planar electrodes is well advanced, that for nanoparticle-modified electrodes is in its infancy. The aim of the present paper is to explore the nature of the cyclic voltammetric response derived from nanoparticle-modified electrodes. In a typical experiment, nanoparticles are attached to an electrode surface forming a random array. For well-separated nanoparticles over short times, mass transport occurs by radial diffusion as opposed to linear diffusion, which is prevalent at a planar surface. The former leads to high sensitivity and, in principle, sustained catalytic activity in cases where the surface is likely to become fouled with the products of the homogeneous process. The mode of diffusion over the array varies with time and has been divided into four cases10,11 as shown in Figure 1. Over short time scales, the nanoparticles are isolated; diffusion is initially linear (case 1) and subsequently radial (case 2). Over longer time scales, the diffusion zones for adjacent nanoparticles overlap (“shielding”), leading to less efficient mass transport (case 3). At long times, diffusion occurs linearly over the entire assembly (case 4). Random assemblies are difficult to interpret * To whom correspondence should be addressed. Fax: +44 (0) 1865 275410. Tel: +44 (0) 1865 275413. E-mail: richard.compton@ chem.ox.ac.uk.
Figure 1. (1) Linear diffusion to each nanoparticle. (2) Radial diffusion to each nanoparticle. (3) Overlapping diffusion zones (“shielding”). (4) Overall linear diffusion.
quantitatively because diffusion over the entire array at a given time is a convolution of these four cases. The time scale over which case 4 onsets is expected to decrease with increasing surface coverage. At high surface coverages and low scan rates, the voltammetry is predicted to deviate minimally from that derived from a planar surface.10,12 This is of practical importance since nanoparticles are often fabricated using expensive metals such as gold, palladium, and platinum.7,13,14 Use of these metals can be economized since macroelectrode behavior is observable with partial surface coverages. The purpose of this paper is two-fold; first, we present a quantitative study of cyclic voltammetry at random nanoparticle assemblies for various scan rates. Second, we investigate the time scale over which voltammetry at nanoparticle assemblies can be modeled (or not) by considering purely planar diffusion. Numerical methods are used in each case and are based on the well-known diffusion domain approximation introduced by Gileadi and Amatore.15-20 2. Theory Section The complete simulation of random nanoparticle assemblies is presently an intractable three-dimensional problem which must be simplified in order to be solved. The diffusion layers are
10.1021/jp901664p CCC: $40.75 2009 American Chemical Society Published on Web 06/02/2009
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Z)0
R ) Rmax
∂a )0 ∂R
∂b )0 ∂R
Z ) Zmax
∂a )0 ∂R
∂b )0 ∂R
and
J ) K0a0 exp[-Rθ] -
Re1
K0b0 exp[(1 - R)θ] Z)0
Figure 2. (a) Random microdisc assembly. (b) Random microdisc assembly split into Voronoi cells. (c) Diffusion domain approximation. (d) Cylindrical axis system.
Figure 3. Simulation space.
assumed to be much larger than the nanoparticles; migration may be neglected and diffusion-only conditions assumed. The reaction was assumed to occur exclusively on the surface of the nanoparticles, albeit with negligible observable adsorption effects. The nanoparticles are taken to be of equal size and to exhibit the diffusional behavior of flat microdiscs of the same radius. In addition, the Butler-Volmer model and the diffusion domain approach are used. These approximations have been reviewed3 and experimentally examined1,2 by considering hydrogen peroxide reduction at silver nanoparticles and hydrogen evolution at palladium nanoparticles. More complicated mechanisms are expected to deviate from this model, in particular in cases where adsorption effects are significant (e.g., methanol oxidation at Pt nanoparticles4). As well as diffusion to a random array, diffusion to a planar surface is considered. Such one-dimensional simulation methods are standard21 and will not be further explained here. 2.1. Diffusion to a Single Nanoparticle. Cyclic voltammetry at a nanoparticle is considered for a simple one-electron reduction
A + e- h B
(1)
Depending on the experimental time scale, diffusion toward a microelectrode is either linear or radial. The problem is twodimensional; the response of the entire electrode can be derived from consideration of a single slice as in Figure 2d. This necessitates the use of cylindrical coordinates; the simulation space that results is shown in Figure 3. The boundary conditions are22
and
R)0
∂b )0 ∂R
∂a )0 ∂Z
In order to simulate cyclic voltammetry, θ is swept from an initial value (θi ) 20) to a more reducing potential (θf ) -20) followed by the corresponding reverse sweep.23 The value of θ is calculated as a function of time using24
θ ) θi - στforward
(2)
θ ) 2θf + θi + στreverse
(3)
The problem is solved numerically using the alternating direct implicit method (ADI).25,26 The spatial grid is identical to that recommended by Gavaghan.26-28 In Figure 2d, the maximum of the Z value is taken to be equal to 6(τ)1/2;29 the maximum value of R varies with nanoparticle separation and is considered below. There were 200 temporal increments per unit θ. The system was converged by ensuring the peak currents scaled with (σ)1/2 in the reversible limit for an isolated disc, as quantified by the Randles-Sˇevcˇ´ık equation.30,31 2.2. Diffusion to a Random Nanoparticle Assembly. An assembly is treated as a random array of flat microdiscs wired in parallel. The assembly is divided into a collection of Voronoi cells as shown in Figure 2b; each cell contains one nanoparticle. Each wall lies halfway between adjacent nanoparticles and corresponds to a no-flux boundary. The Voronoi cells are diffusionally independent. The base of each cell is transformed into a circle of the same area. This is called the diffusion domain approximation15,19 and is illustrated in Figure 2c. The flux drawn through the entire assembly is the sum of the fluxes for every possible diffusion domain radius (Rd) weighted by the frequency distribution for a random assembly. The appropriate weighting depends on the average diffusion domain radius (〈Rd〉) for the assembly and is given by32
f(y) )
343 15
2π7 y
5/2
7 exp - y 2
( )
(4)
where
y)
( ) Rd 〈Rd〉
2
(5)
The current through the entire assembly is discretized using pmax
∂a )0 ∂R
∂a )0 ∂Z
R>1
j≈
∑ jp ∫R
p)1
Rp p-1
f(y)dy
(6)
Diffusional Cyclic Voltammetry at Electrodes: Theory
∫RR
p
p-1
f(y)dy ≈
f(yp) + f(yp-1) (yp - yp-1) 2
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(7)
Discretization leads to reliable results provided sufficient Rp values are considered. The converged distribution is
Rp > 1
Rp ) 1 + 10cp
cp ) a + pb
(8) Rp ) 0.01, 0.02, ...0.99, 1.00
Rp e 1
(9)
where a ) -2 and b ) 0.01. When Rd < 1, implying nanoparticle overlap, jp ≈ R2pj(Rp)0).33 The region considered in this paper is 〈Rd〉 ) 1 - 100. This requires voltammograms to be simulated from Rd ) 1.01-1001. Once the simulation is run across a 2D spatial segment of Figure 2d, the flux over the entire electrode is calculated using eq 10 This integral is solved numerically using the trapezium rule.34
Jtotal ) π
∫01 JsegmentRdR
Figure 4. Comparison between the surface coverage calculated using (A) an analytical solution based on a regular array and (B) numerical calculations based on a random array.
(10)
3. Computation Section All programs were written in C++ and compiled using a Borland compiler. The simulations were run on desktop PCs with processor speeds of ∼3 GHz. Approximately 5 min of CPU time was required to simulate a single voltammogram. The study of the random assemblies reported in this paper required simulation of ∼50000 voltammograms. 4. Surface Coverage For a regular array, eq 11 is valid for all surface coverages (Θ)
Θ)
1 〈Rd〉2
(11)
Figure 5. Jpeak as a function of σ and 〈Rd〉 for which R ) 0.5, DA/DB ) 1, and K0 ) 1000.
This formula is not generally valid for random arrays. It is necessary to consider the distribution function in eq 4 to yield
∫0
1
Θ)
f
( ) R2d
〈Rd〉2
R2ddRd +
∫0
∞
f
( ) R2d
∫1
∞
f
( ) R2d
〈Rd〉2
dRd
(12)
R2ddRd 2
〈Rd〉
This equation was solved numerically, and the data are presented in the Supporting Information. Figure 4 plots the comparison between the regular and random cases. It is seen that the random surface coverage is quantitatively described by eq 11 below a coverage of ∼20%. 5. Results and Discussion In the following discussion, cases 1, 2, 3, and 4 can be understood with reference to Figure 1 and the explanation given in the Introduction. As a preliminary, diffusion domain simulations were conducted to explore the effect of 〈Rd〉 for voltammetry in the reversible limit. Figure 5 shows, for a fixed
Figure 6. ∆θpeak-to-peak as a function of σ and 〈Rd〉 for which R ) 0.5, DA/DB ) 1, and K0 ) 1000.
dimensionless electrochemical rate constant (K0 ) 103), the dependence of the peak flux, Jpeak, on 〈Rd〉 and the dimensionless scan rate (σ). Widely separated nanoparticles (large 〈Rd〉) exhibit
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Figure 7. Jpeak as a function of log10(K0) and log10(σ) for several random surface coverages, 1, 3, 10, and 30%.
Figure 8. ∆θpeak-to-peak as a function of log10(K0) and log10(σ) for several random surface coverages, 1, 3, 10, and 30%.
a near-square-root dependence of Jpeak on σ, as predicted by the Randles-Sˇevcˇ´ık equation30,31 for large values of σ corre-
sponding to linear diffusion. At smaller values of σ, a nearconstant flux is seen corresponding to transport under steady-
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Figure 9. Various voltammograms for 3% surface coverage.
TABLE 1: Dimensional Parameters dimensional parameter A [A]0 [B]0 [A]bulk [B]bulk R k0 E Efθ DA DB j V Θ rd
definition 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) scan rate (V s-1) fractional surface coverage diffusion domain radius
state conditions (convergent diffusion), as predicted by the Saito equation.35,36 As the diffusion domain size is decreased from 〈Rd〉 ) 100 to
