J. Phys. Chem. C 2007, 111, 18049-18054
18049
Diffusion-Limited Currents to Nanoparticles of Various Shapes Supported on an Electrode; Spheres, Hemispheres, and Distorted Spheres and Hemispheres Ian Streeter and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford, OX1 3QZ United Kingdom ReceiVed: August 16, 2007; In Final Form: September 20, 2007
The finite difference method is used to simulate the current at a spherical nanoparticle sitting upon a planar electrode, assuming electrolysis occurs only at the nanoparticle surface. The accuracy of the simulation technique is confirmed by using the same approach to simulate a hemispherical electrode. Nanoparticles with the geometry of a distorted sphere or hemisphere are also considered. Equations are given for the corresponding masstransport-limiting current. For the spherical nanoparticle, this expression is ilim ) -8.71nFDrs[A]bulk, where D is the diffusion coefficient, rs is the radius of the particle, [A]bulk is the bulk concentration of the electroactive species, and n is the number of electrons transferred.
1. Introduction Nanoparticle-modified electrodes play an important role in electroanalysis.1 Metal particles of nanometer dimensions deposited on the surface of an electrode can catalyze heterogeneous electron transfers. Gold, silver, and platinum metals are the most commonly used nanoparticles, although many other transition metals have also been used. The nanoparticles often exhibit chemical and physical properties that differ significantly from the bulk material. Voltammetric studies on nanoparticlemodified electrodes can highlight features of the electron transfer that could not be measured using a macroelectrode of the same material.2-4 For example, Raj et al. showed that the catalytic effect of gold nanoparticles upon dopamine oxidation meant that its potentials could be distinguished from those of ascorbic acid.3 Furthermore, convergent diffusion to the nanoparticles effects a low limit of detection of the target analyte.5,6 The high sensitivity and chemical selectivity of nanoparticle-modified electrodes make them suitable candidates for cheap, disposable sensors of heavy metal ions such as arsenic in water.7-9 A quantitative description of diffusion to a nanoparticlemodified electrode has not yet been tackled in the electrochemical literature. Many studies using these electrodes therefore lack any rigorous interpretation of the experimentally measured current. As nanoparticle-modified electrodes become increasingly more popular with electrochemists, it has become necessary to provide a more complete description of the nature of the diffusional current. The first step toward achieving this target is to use numerical methods to model the mass transport to a diffusionally independent nanoparticle. Figure 1 shows the system under consideration in this paper, which consists of a planar electrode modified with a surface distribution of nanoparticles. The planar electrode is conducting such that there is a point of electrical contact to the particles. However, the planar electrode is electrochemically inactive; the electron transfer is only kinetically feasible on the surface of * Corresponding author. Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, United Kingdom OX1 3QZ. Fax: +44 (0) 1865 275410. Tel: +44 (0) 1865 275413. Email:
[email protected].
Figure 1. Well-separated nanoparticles on an electrode: (a) spherical, (b) hemispherical.
Figure 2. Schematic diagram of (a) a spherical particle and (b) a hemispherical particle sitting upon a supporting planar surface.
the nanoparticle. We assume that the particles are well separated such that they remain diffusionally independent on the experimental time scale. Numerical modeling of diffusion to a nanoparticle is made more complicated by its curved surface. The most favored approach to problems like this is to work in a coordinate system that negates the curvature. For example, one of the simplest mass transfer problems to solve in electrochemistry is that of diffusion to a hemispherical microelectrode set on an insulating plane, or a spherical electrode isolated in solution. In a spherical coordinate system, these systems have an angular isotropy that reduces the mass-transport equations to a function of just the radial coordinate.10,11 A more complicated treatment is required when there is no easy coordinate transformation that will render the curved particle surface “flat”.
10.1021/jp076593i CCC: $37.00 © 2007 American Chemical Society Published on Web 11/10/2007
18050 J. Phys. Chem. C, Vol. 111, No. 49, 2007
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TABLE 1: Dimensionless Parameters Used for Numerical Simulation of the Sphere and Hemisphere parameter
expression
radial coordinate normal coordinate time scan rate potential concentration of species A concentration of species B electrode flux
R ) r/rs Z ) z/rs T ) Dt/rs2 σ ) (F/RT)(νrs2/D) θ ) (F/RT)(E - EQf ) a ) [A]/[A]bulk b ) [B]/[A]bulk j ) -i/(nFD[A]bulkrs)
TABLE 2: Boundary Conditions for Equation 2 boundary
condition
initial conditions axis of symmetry supporting planar surface bulk solution particle surface
a)1 ∂a/∂R ) 0 ∂a/∂Z ) 0 a)1 (i) a ) 0 (ii) a/b ) eθ
In a recent series of papers, the mass-transport equations have been solved for an electron transfer at a planar electrode in the vicinity of a curved object. For example, a partially blocked electrode where the blocking species is a hemispherical droplet12,13 and a microdisc electrode with a sphere at its center as an inert blocking species.14 In these examples, the diffusion equations were solved over two dimensions by finite difference methods, using a cylindrical coordinate system and taking advantage of the axis of symmetry normal to the electrode surface. The rectangular simulation grids used a high mesh density in the region of the sphere to mimic its curvature. In all of these simulations, the electrode itself was flat in the simulation space. In this work, the current at a spherical nanoparticle is simulated numerically by solving the mass-transport equations in cylindrical polar coordinates. Using this coordinate system, the electron transfer occurs at a curved surface in the simulation space. The simulation approach is similar to that used for the inert sphere sitting upon a planar electrode:14 the same simulation grid is used, but the boundary conditions applied at the sphere and plane surfaces are different. We note that an alternative route to modeling this system is to solve the diffusion equations in spherical radial coordinates.15 Using this approach, the particle would be transformed to a flat surface in the simulation space and the supporting planar surface would be represented by a curved surface. However, the spherical radial coordinate system is not appropriate for all curved particles. We use the cylindrical radial coordinate approach here to illustrate its applicability to any curved particle providing it has an axis of symmetry. To validate our approach of modeling a spherical particle using a two-dimensional rectangular grid, we use the same approach to simulate an electron transfer at a hemispherical particle. The voltammetric response of a hemispherical electrode on a plane is well understood, and we would expect simulations to agree with the analytical expressions. As a further extension to our work, we distort the particle geometry to be spheroidal or hemispheroidal in shape. 2. Mathematical Model and Simulation Procedure 2.1. Normalization of the Model. Figure 2 shows the geometries of the systems under consideration, which consist of a spherical or a hemispherical particle of radius rs sitting on
Figure 3. Schematic of the grid used for discretisation of eq 2 for simulations of (a) the sphere and (b) the hemisphere.
an infinite plane. The axial coordinate, z, is defined normal to the plane, and the cylindrical radial coordinate, r, is defined as the distance from the axis of symmetry that runs through the center of the particle. The cylindrical radial coordinate should not be confused for a spherical radial coordinate, which would give a measure of distance from the center of the sphere itself. The model is normalized using the dimensionless parameters in Table 1. The (R,Z)-coordinate system is defined relative to the sphere radius, and dimensionless time, T, is defined in terms of rs and the diffusion coefficient, D. In the normalized space, the particle is bound by the surface R 2 + Z 2 ) 1 for the hemisphere, and R 2 + (Z - 1)2 ) 1 for the sphere. 2.2. Mass-Transport Equations and Boundary Conditions. Equation 1 shows the electron transfer considered in these numerical simulations, where n is the number of electrons transferred. Both species A and B are soluble, but only species A is present in bulk solution.
A + ne- h B
(1)
Species A and B are described in terms of their normalized concentrations, a and b, which are defined in Table 1. The mass transport of species A is described by Fick’s second law of diffusion, which is presented in eq 2 using the normalized parameters from Table 1.
∂a ∂ 2a 1 ∂a ∂ 2a ) + + ∂T ∂R2 R ∂R ∂Z 2
(2)
The boundary conditions for eq 2 are summarized in Table 2. There is a no-flux condition at the planar electrode surface (Z ) 0) and at the axis of symmetry (R ) 0). The bulk solution condition is implemented at a distance 6T (1/2) from the particle
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Figure 4. Various shaped particles on a supporting planar surface: (a) oblate spheroid, (b) prolate spheroid, (c) oblate hemispheroid, (d) prolate hemispheroid. Figure 5. Simulated concentration profile at the spherical particle under diffusion-limiting conditions.
TABLE 3: Dimensionless Parameters Used for Numerical Simulation of Spheroids and Hemispheroids parameter
expression
radial coordinate normal coordinate time electrode flux
R ) r/a Z ) z/b T ) Dt/(a2 + b2) j ) (-ix2)/(nFD[A]bulkxa2+b2)
the expanding grid the spacing of mesh lines is given by eqs 4 and 5
boundary, beyond which the effects of diffusion are not important on the experimental time scale.16 Table 2 shows two different boundary conditions for the particle surface. The first one is used to find the mass-transportlimiting current; the simulation is run until the calculated flux reaches its asymptotic value, to within acceptable error as discussed in Section 3. The second boundary condition at the particle surface is used for simulations of a linear sweep experiment. The expression used comes from the Nernst equation, which is appropriate when the electron transfer in eq 1 is fast and reversible. It is written in terms of the dimensionless potential, θ, which is defined in Table 1. In a simulation of a linear sweep experiment, the value of θ is given at any time by eq 3
θ ) θ0 + σT
hi ) fR hi-1
(4)
kj ) fZ kj-1
(5)
where the parameters h0, k0, fR, and fZ can be chosen for an appropriate level of accuracy. 2.4. Calculating the Flux. The flux of species A through the nanoparticle is related to the concentration gradient in the direction normal to the particle surface. Using the rectangular simulation grid, this flux must be found from the sum of its components in the R and Z directions. Equation 6 shows the expression for the dimensionless diffusional flux density, J
J)
∂a ∂a sin φ + cos φ ∂Z ∂R
where φ is the angle shown in Figure 2. The total dimensionless diffusional flux, j, is found from an integration of J over the electrode surface:
(3)
where θ0 is its initial value and σ is the dimensionless potential sweep rate, also defined in Table 1. When implementing this second boundary condition at the particle surface, it must be assumed that the two species in eq 1 have equal diffusion coefficients such that their concentrations at any point in solution satisfy the relationship a + b ) 1, and the concentration profile of species A may be simulated independently from species B.17 2.3. Simulation Procedure. Equation 2 and its accompanying boundary conditions are discretized and solved by the alternating direction implicit finite difference method in conjunction with the Thomas algorithm. Because the model is axisymmetric, it is sufficient to solve the mass-transport equations in twodimensional space in the region R g 0 and Z g 0. Figure 3 shows the simulation grids, which are based on the grids used in previous simulations of spherical and hemispherical objects.12,14 A high mesh density is required in the region of the particle to maintain a high precision description of the curved boundary surface. The grid then expands moving away from the particle, where not such high precision is required. At any grid point, (Ri ,Zj), the spacing of mesh lines is hi in the R -direction and kj in the Z direction, where i ) 0,1,2, ..., n and j ) 0,1,2, ..., m. In the region of the particle, the spacing of mesh lines is given by hi ) h0 and kj ) k0, and in the region of
(6)
j ) 2π
∫ J cos φ dφ
(7)
The integral in eq 7 is over the limits -(π/2) e φ e (π/2) for the sphere, or 0 e φ e (π/2) for the hemisphere. The current measured in amps, i, is related to the dimensionless flux, j, by the expression in Table 1. 2.5. Spheroids and Hemispheroids. The simulation procedure described can be readily adapted to model diffusion to a spheroid or hemispheroid-shaped nanoparticles on a supporting planar surface. Examples of these shapes are shown in Figure 4. They are axisymmetric about the line r ) 0 and have a cross section the shape of an ellipse. The particle dimensions are described by the parameters a and b, which are defined in Figure 4. The model is normalized using the dimensionless parameters in Table 3. Note that in the special case of a ) b, this is equivalent to the normalization used for the sphere and hemisphere described in Section 2.2. The particle shape is described by a dimensionless parameter B, which is the ratio of the lengths b and a. Under this space transformation, the particles become spherical or hemispherical in shape with a radius of 1. Numerical simulation can therefore be performed using the same simulation grids described in Section 2.3. The mass-transport equation is modified to eq 8, and the flux density
18052 J. Phys. Chem. C, Vol. 111, No. 49, 2007
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Figure 6. Simulated voltammetry for a reversible electrode transfer at the spherical particle. The following scan rates are used: (a) σ ) 10-3, (b) σ ) 1, (c) σ ) 1000.
Figure 7. Simulated concentration profiles at the spherical particle: (a) σ ) 0.1, (b) σ ) 1, (c) σ ) 10, (d) σ ) 100, (e) σ ) 1000.
and total flux are calculated using eqs 9 and 10, respectively, but the rest of the simulation procedure remains the same.
∂a ∂ 2a 1 ∂a ∂ 2a ) (1 + B2) 2 + (1 + B2) + (1 + B-2) ∂T R ∂R ∂R ∂Z 2 J)
∂a sin φ ∂a + B cos φ ∂Z B ∂R
j ) 2π
x2
x1 + B2
∫ J cos φ dφ
(8) (9) (10)
3. Simulated Results 3.1. Simulations at a Hemisphere. The analytical expression for the diffusion-limiting current at a hemispherical electrode is given by eq 11:
ilim ) - 2π nFDrs[A]bulk
(11)
In terms of the normalized parameters, the limiting flux is expected to be j ) 2π, and the flux density is expected to have the uniform value of J ) 1 at all points on the hemisphere surface. The limiting flux was simulated by the numerical approach described in Section 2 using the boundary condition a ) 0 at the surface of the particle. Various values of the grid parameters were used to test the requirements for simulation
convergence. To generate a smooth flux profile for which the surface flux density, J, is correct to within 0.5% error for all values of the angle φ, the following grid parameters are needed: h0 ) k0 ) 2 × 10-4, fR ) fZ ) 1.1. Using these values, the simulated overall flux, j, is correct to within 0.1%. Linear sweep voltammetry was simulated at the hemispherical electrode using the two-dimensional numerical approach. It was compared to voltammetry simulated using the commercial program Digisim,18 which uses the more conventional onedimensional approach. Simulations were performed for scan rates in the range -3 e log σ e 3, which includes both nearsteady-state and near-planar diffusion. Using the same grid parameters as before (h0 ) k0 ) 2 × 10-4, fR ) fZ ) 1.1), there was excellent agreement between the two simulation approaches, and the predicted peak currents agreed to within 0.2% for all scan rates. These values of the grid parameters will be used in Section 3.2 to simulate voltammetry and to find the diffusion-limiting flux at the spherical nanoparticle, for which there is no known analytical expression. 3.2. Simulations at a Sphere. Figure 5 shows the concentration profile simulated at the spherical nanoparticle under diffusion-limiting conditions. It can be seen that there is significant depletion of the electroactive species where the sphere meets the supporting plane. At large distances from the particle, the isoconcentration surfaces are approximately hemi-
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J. Phys. Chem. C, Vol. 111, No. 49, 2007 18053
Figure 8. Simulated flux profiles at the spherical particle: (a) diffusion-limiting profile, (b) σ ) 1, (c) σ ) 1000.
3.3. Simulations at Other Shapes. The mass-transportlimiting flux is simulated at spheroids and hemispheroids of various geometries. The mass-transport-limiting current is then given by eq 13
x
ilim ) - jnFD[A]bulk
Figure 9. Comparing the diffusion-limiting flux for spheroids and hemispheroids of equal surface area.
TABLE 4: Mass-Transport-Limiting Flux B 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
hemispheroid
spheroid
j ) 6.13 j ) 6.49 j ) 6.57 j ) 6.47 j ) 2π j ) 6.06 j ) 5.84 j ) 5.61 j ) 5.41 j ) 5.22
j ) 7.18 j ) 8.14 j ) 8.60 j ) 8.75 j ) 8.71 j ) 8.57 j ) 8.38 j ) 8.17 j ) 7.96 j ) 7.76
(13)
where the simulated value of j is given in Table 4. Figure 9 presents a comparison of the mass-transport-limiting current at particles of different shapes but equal surface areas. The value plotted is ilim for a particle of unit area, normalized by dividing by ilim for the hemispherical particle. Although a wide range of particle geometries are considered, the masstransport-limiting current is fairly consistent for the particles considered here. In particular, a hemisphere and a sphere of equal surface area differ in their diffusion-limiting current by less than 2%. Also marked in Figure 9 is the expected limiting value of ilim as B tends toward zero. This value is found by noting that at low values of B the spheroid and hemispheroid tend toward the geometry of a microdisc, for which the analytical expression for the diffusion-limiting current is well known.19 4. Conclusions
spherical in shape. The diffusion-limiting flux is found to be j ) 8.71 ( 0.01, which leads to the following expression for the diffusion-limiting current at a spherical particle:
ilim ) -8.71nFDrs[A]bulk
a2 + b 2 2
(12)
Linear sweep voltammetry was simulated at the spherical particle at a range of scan rates. Figure 6 shows some sample voltammograms, and Figure 7 shows some concentration profiles taken at the voltammograms’ peak potentials. The transition from near-steady-state to near-planar diffusional behavior can be seen as the experimental time scale decreases. Figure 8 shows the variation of the flux density, J, across the surface of the sphere under various diffusion regimes. The angle φ was defined in Figure 2; a value of φ ) -(π/2) refers to the point where the sphere meets the planar surface, and a value of φ ) (π/2) refers to the “top” of the sphere. Moving from slow to fast experimental timescales, the flux density becomes more uniform over the sphere surface. This is because the presence of the electroinactive plane has less effect when the diffusion-layer thickness is small compared to the sphere radius.
This work describes the first steps toward a complete understanding of the diffusional current measured at a nanoparticle-modified electrode. We aim to further develop the theory of diffusion to these electrodes so that researchers can in turn provide a more rigorous analysis of their experimental results. We have provided eqs 12 and 13, which give simple expressions for the mass-transport-limiting current at an isolated nanoparticle. These expressions are appropriate for an electrode modified with a sparse distribution of nanoparticles. We have also described the reversible current at isolated nanoparticles in a linear sweep experiment. The next step in developing the theory of nanoparticlemodified electrodes is to use the numerical methods developed in this paper to model higher surface densities of nanoparticles, for which the approximation of diffusional independence cannot be made.20 The techniques described here can also be applied to many other related systems, including quasi reversible electron transfers, mechanisms involving a homogeneous step, or particles with a more complicated shape. Acknowledgment. I.S. thanks the EPSRC for studentship. References and Notes (1) Welch, C. M.; Compton, R. G. Anal. Bioanal. Chem. 2006, 384, 601-619.
18054 J. Phys. Chem. C, Vol. 111, No. 49, 2007 (2) Miscoria, S. A.; Barrera, G. D.; Rivas, G. A. Electroanalysis 2005, 17, 1578-1582. (3) Raj, C. R.; Okajima, T.; Ohsaka, T. J. Electroanal. Chem. 2003, 543, 127-133. (4) Yu, A.; Liang, Z.; Cho, J.; Caruso, F. Nano Lett. 2003, 3, 12031207. (5) Simm, A. O.; Ward-Jones, S.; Banks, S. E.; Compton, R. G. Anal. Sci. 2005, 21, 667-671. (6) Welch, C. M.; Banks, S. E.; Simm, A. O.; Compton, R. G. Anal. Bioanal. Chem. 2005, 382, 12-21. (7) Dai, X.; Nekrassova, O.; Hyde, M. E.; Compton, R. G. Anal. Chem. 2004, 76, 5924-5929. (8) Dai, X.; Compton, R. G. Anal. Sci. 2006, 22, 567-570. (9) Dai, X.; Compton, R. G. Electroanalysis 2005, 17, 1325-1330. (10) Delmastro, J. R.; Smith, D. E. J. Phys. Chem. 1967, 71, 21382149. (11) Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1983, 158, 193215.
Streeter and Compton (12) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 580, 265-274. (13) Barnes, A. S.; Fietkau, N.; Chevallier, F. G.; del Campo, J.; Mas, R.; Munoz, F. X.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2007, 602, 1-7. (14) Fietkau, N.; Chevallier, F. G.; Jiang, L.; Jones, T. G. J.; Compton, R. G. Chem. Phys. Chem. 2006, 7, 2162-2167. (15) Svir, I. B. Analyst 2001, 126, 1888-1891. (16) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; John Wiley and Sons: New York, 2001. (17) Streeter, I.; Compton, R. G. Phys. Chem. Chem. Phys. 2007, 9, 862-870. (18) http://www.bioanalytical.com/products/ec/digisim/. (19) Saito, Y. ReV. Polarogr. (Jpn.) 1968, 15, 177-187. (20) Streeter, I.; Baron, R.; Compton, R. G. J. Phys. Chem. C, in press.
