J. Phys. Chem. C 2010, 114, 8309–8319
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Transient Voltammetry at Electrodes Modified with a Random Array of Spherical Nanoparticles: Theory Stephen R. Belding and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: February 20, 2010; ReVised Manuscript ReceiVed: March 22, 2010
The simple one-electron reduction, A + e- h B, is studied at a random array of spherical nanoparticles attached to a planar electrode. Electron transfer is assumed to occur exclusively on the surface of the nanoparticles; the electrode simply acts as a conductive support. Voltammetry is simulated using the alternating direction implicit (ADI) variant of the finite difference method. The diffusion-controlled chronoamperometric response is studied as a function of nanoparticle surface coverage, and the cyclic voltammetry is studied as a function of electrochemical rate constant, voltage scan rate, and nanoparticle surface coverage. Also considered is the extent to which a random array of spherical nanoparticles can usefully be approximated as an array of discs with an equal surface coverage. TABLE 1: Dimensional Parameters
1. Introduction The search for improved amperometric sensors and for electrodes capable of resolving fast kinetics for processes of importance in fuel cells, batteries, and other energy conversion devices has led to the increasingly wide study of electrodes modified with small quantities of electrocatalytically active nanoparticles.1–3 The nanoparticles are supported on a conductive substrate with the added expectation that electrolysis occurs solely on the surface of the nanoparticles and not the underlying substrate. It is established4 that, for appropriate surface coverages, such an array can show a current response corresponding to transport-controlled electrolysis at the geometric area of the substrate electrode, despite the latter being partially covered by nanoparticles. In this way, minimal amounts of expensive catalyst, such as platinum or gold, can be used to offer a maximal electrochemical response. In addition, the use of nanoparticles can produce electrochemical behavior at the nanoscale which is different from that seen on either the micro- or macroscales. For example, we have shown that underpotential deposition adsorption of thallium onto gold nanoparticles switches off for gold particles of diameter smaller than ∼50 nm,5 while the kinetics of the hydrogen evolution reaction at silver are significantly altered between the macro- and nanoscales.6 The reduction of hydrogen peroxide at silver nanoparticles shows a mechanism distinct to that on the macroscale.7 Such differences in behavior can occur either through a change in the electronic properties of the particles from the macro to the nano range, influencing, for example, the work function of the metal, or also through the altered surface crystallography at the nanoscale, where high index planes are more likely to occur and where the sheer size limits the extent to which large terraces of a low index plane can exist. In order to explore the electrochemistry of nanoparticles and to discover the kinetics and mechanism of electrocatalytic processes, it is necessary to develop a theory of voltammetry at substrates partially covered with nanoparticles. In previous * To whom correspondence should be addressed. Fax: +44 (0) 1865 275410. Tel: +44 (0) 1865 275413. E-mail: richard.compton@ chem.ox.ac.uk.
dimensional parameter R β cX cX,0 cX* DX N Θ φsph rcyl rd rsph rnp V Vcell Vrandom zcyl
definition transfer coefficient for reduction/unitless transfer coefficient for reduction/unitless concentration of species X/mol m-3 concentration of species X at the electrode surface/mol m-3 concentration of species X in bulk solution/mol m-3 diffusion coefficient of species X/m2 s-1 number of nanoparticles in the array fractional surface coverage/unitless azimuthal spherical coordinate/rad radial cylindrical coordinate/m diffusion domain radius/m radial spherical coordinate/m radius of the nanoparticle/m scan rate/V s-1 voltammogram of a single Voronoi cell/A voltammogram of a random array/A axial cylindrical coordinate/m
papers, we have calculated the steady-state voltammetry of isolated nanoparticles of various shapes8 supported on an inert planar conducting electrode. The predictions were confirmed experimentally.9 More recently, we have attempted a description of voltammetry at an electrode covered with an array of nanoparticles assumed to be flat disks.10 In the present paper, we examine the more realistic case in which the nanoparticles are spherical. The approximation of spheres by discs is shown to be justified in most, but not all, experimental situations. 2. Theory We consider a one-election reduction of the form
A + e- h B
(1)
in which the diffusion coefficients of species A and B are equal (DA ) DB) (see Table 1 for the definitions of dimensional
10.1021/jp1015388 2010 American Chemical Society Published on Web 04/19/2010
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Belding and Compton
Figure 1. Categorization of the diffusional behavior.
Figure 2. (a) Top-down view of a section of a random array of nanoparticles. (b) Top-down view shown in (a) divided into Voronoi cells.
Figure 4. Size distribution function for a random array.
Figure 3. (a) Arbitrary Voronoi cell. (b) Circular cell derived from (a) using the diffusion domain approximation.
parameters used throughout). The reaction occurs on an infinitely large, random array of nanoparticles supported on a planar electrode. The nanoparticles are perfect spheres of equal size that catalyze the reaction shown in eq 1. This reaction is assumed to occur exclusively on the surface of the nanoparticles, with negligible adsorption effects so that the voltammetry is exclusively diffusional in character. 2.1. Categorization of the Diffusional Behavior. Mass transport to a random array of spherical nanoparticles can be understood in terms analogous, but complementry, to those for partially blocked electrodes11,12 and microdisk electrodes.13,14 The interaction between the diffusional zone of each nanoparticle varies as a function of the experimental time scale and of the nanoparticle surface coverage. As the experimental time scale increases and the nanoparticle surface coverage decreases, the diffusional interaction changes from category 1 through to category 4, as shown in Figure 1. In category 1, the diffusion layers are small, and each nanoparticle is diffusionally isolated; mass transport is onedimensional and linear. This behavior leads to peak-shaped cyclic voltammograms. In category 2, the diffusion layers are large and approximately hemispherical, but diffusional isolation between nanoparticles is maintained. This behavior leads to steady-state cyclic voltammograms. In category 3, the depletion layers overlap and are relatively small compared to the nanoparticle separation. The cyclic voltammetry is intermediate between categories 2 and
4. In category 4, the diffusion layers overlap but are relatively large compared to the nanoparticle separation. Mass transport is effectively one-dimensional and linear over the entire array. The current response is equal to that obtained had the substrate been electroactive.Thisbehaviorleadstopeak-shapedcyclicvoltammograms. 2.2. Voltammetry for a Random Array of Nanoparticles. Nanoparticles of equal size are randomly distributed over the surface of an infinitely large supporting electrode, a section of which is shown in Figure 2a. This assembly is divided into hypothetical units called Voronoi cells, as shown in Figure 2b. Each “wall” is situated midway between adjacent nanoparticles. Consequently, to a good approximation, the walls can be considered no-flux boundaries. The Voronoi cells are therefore diffusionally independent and can be simulated separately. The final voltammogram is constructed by summing the voltammograms derived from each Voronoi cell, weighted by the size distribution of the Voronoi cells. An arbitrary Voronoi cell is shown in Figure 3a, the simulation of which is a computationally intractable three-dimensional problem. The problem is computationally simplified by applying the diffusion domain approximation proposed by Gileadi and Amatore.15,16 The irregular Voronoi cell shown in Figure 3a is transformed into the circular diffusion domain shown in Figure 3b. The diffusion domain is of equal area to the original cell and is concentric with respect to the nanoparticle. Since the diffusion domain corresponds to a cylindrical simulation space, the problem is rendered two-dimensional and can be solved numerically using a desktop computer. This technique is well-established and has been used to simulate a range of experiments.1,6,7 The validity of the diffusion domain approximation has been studied in detail by Godino et al.17 In general, a high degree of concordance was shown to occur between the experimental and simulated data. Significant
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Vcell(rd) )
Figure 5. Spherical nanoparticle defined with respect to a system of spherical coordinates (rsph and φsph) and to a system of cylindrical coordinates (rcyl and zcyl).
discrepancies occurred for small arrays. In these cases, a large proportion of the nanoparticles were situated at peripheral points in the array. Over long time scales, diffusion toward these peripheral points become important, and mass transport was convergent toward the entire array. This contrasts with the case of an infinitely large array in which, over long time scales, mass transport is linear toward the entire array (category 4). The diffusion domain approach is best suited for large arrays with significantly more microelectrodes inside than in perimetric positions. The size distribution of the Voronoi cells is given by the function f(rd/〈rd〉), where rd is radius of a given diffusion domain and 〈rd〉 is the mean radius of the Voronoi cells in the array (related to surface coverage, as considered in section 2.3). Although an exact analytical expression for this function does not exist, Ferenc et al.26 have numerically derived a result using Monte Carlo-type simulation methods. The distribution is given by eq 2 and is shown in Figure 4
( )
rd 343 f ) 〈rd〉 15
( ) ( ( ))
7 rd 2π 〈rd〉
5
7 rd exp 2 〈rd〉
Vrandom(〈rd〉) )
( )
rd ∞ f V (r ) dr 0 〈rd〉 cell d d
∫
2 ∂cA ∂2cA ∂cA 1 2 ∂cA 1 ∂ cA + 2 + 2 ) + 2 ∂t rsph ∂rsph rsph rsph tan φ ∂φ rsph ∂φ2
(5) with respect to the following boundary conditions
t)0
0eφeπ
∂cA ) k0cA exp ∂rsph F F -R (E - E0′) - k0(1 - cA) exp (1 - R) (E RT RT
t>0
0eφeπ
[ ( )
rsph g rnp
]
[
]
0 e φ < tan-1 rcyl max
rsph ) 1 cos φ
t>0
tan-1 rcyl max e φ
rnp
rpd ) (1 + 10-3+0.01p)rnp p) 100, 101, 102, ..., 598, 599, 600
t>0
)
( )
zcyl max π π + e φ < tan-1 2 rcyl max 2 rcyl max
(
cos φ -
t>0
tan-1
∂cA )0 ∂rsph
rsph )
(
p) 0, 1, 2, ..., 97, 98, 99
The logarithmic form of this discretization allows it to be used over a range of 〈rd〉 values and is converged for those considered in this paper (1 e 〈rd〉 e 50). When rdp < rnp, corresponding to nanoparticle overlap, diffusion is approximately linear, and the voltammetry can be approximated as18
π 2
rcyl max π cos - φ 2
The diffusion domain radius, rd, is discretized as follows
rpd ) 0.01(p + 1)rnp
( )
E0′)
(3)
rpd e rnp
cA ) c*A
rsph g rnp
(2)
The voltammetric response for the random array (Vrandom) is constructed by summing the voltammograms derived from each Voronoi cell (Vcell). Each voltammogram in this sum is weighted by the function f(rd/〈rd〉) in order to account for the mean diffusion domain radius, 〈rd〉
(4)
The geometry of the nanoparticle with respect to the diffusion domain can be equivalently defined with respect any twodimensional coordinate system. The spatial boundary condition at the surface of the nanoparticle is most easily given in terms of spherical coordinates (rsph and φsph), while the remaining spatial boundary conditions are most easily given in terms of cylindrical coordinates (rcyl and zcyl). These coordinate systems are shown in Figure 5. As explained in section 2.4, the problem is solved more efficiently if spherical coordinates are used. A chronoamperogram or voltammogram is a solution to Fick’s second law of diffusion
t>0
2
( )
rd 2 V (rp)99) rnp cell d
∂cA )0 ∂rsph
rsph ) π 2
)
∂cA )0 ∂rsph
( )
zcyl max π + eφ
