J. Phys. Chem. C 2007, 111, 13905-13911
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Voltammetric Sizing and Locating of Spherical Particles via Cyclic Voltammetry Nicole Fietkau,† GuoQing Du,‡ Sine´ ad M. Matthews,‡ Michael L. Johns,‡ Adrian C. Fisher,‡ and Richard G. Compton*,† Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford OX1 3QZ, U.K., and Department of Chemical Engineering, UniVersity of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA, U.K. ReceiVed: May 2, 2007; In Final Form: July 13, 2007
The size of a single spherical particle and its distance from the center of a microdisk electrode are determined by using a simple voltammetric procedure. A single spherical glass particle is positioned at an arbitrary distance from a microdisk electrode. Cyclic voltammograms for the oxidation of a model redox system are then recorded as a function of scan rate and compared to lattice Boltzmann simulations to deduce with good precision the radius of the spherical particle and the distance from the center of the electrode to the center of the particle. The radius of the sphere and its center-to-center distance are independently confirmed by microscopic images recorded at the time of the measurements. Excellent agreement between experiment and theory and independent direct observation are observed.
1. Introduction We have recently shown that simple electrochemical methodologies can provide sensitive measurements for sizing and shaping particles.1-4 Initially, the average size of inert particles was determined by using a simple procedure in which the particles were deposited on a macroelectrode and followed by cyclic voltammetric measurements as a function of the voltage scan rate. Knowing the total mass of the deposited particles, together with their density, accurate sizing was possible.1 We then reported that cyclic voltammetric measurements as a function of voltage scan rate can offer a sensitive measurement of the radius of a spherical particle located in the center of a microdisk electrode.2 Furthermore, the size and shape of hemispherical microdroplets of dodecane immobilized on a regular array of hydrophobic polymer blocks of a partially blocked electrode was verified by comparison of a simple Cottrellian-like potential-step experiment with simulations.3 Cottrellian-like potential-step experiments were also used to determine the precise size of a spherical particle, its distance from the center of a microdisk electrode, and its exact location within an array of three noncollinear microdisk electrodes. The obtained transients were than compared to lattice Boltzmann simulations.4 These approaches extend the application of electrochemical methods for the determination of shape and size, building on the ideas implicit in scanning electrochemical microscopy and those developed for the measurement of thin films.5,6 In the present article, we employ a voltammetric procedure based on current measurements as the voltage is cyclically scanned to size a single particle of known shape (sphere) positioned at different center-to-center distances, dc-c, between the center of the electrode of radius re and the center of the sphere of radius rs (Figure 1). The voltammetric measurements * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: +44 (0) 1865 275 413. Fax: +44 (0) 1865 275 410. † University of Oxford. ‡ University of Cambridge.
Figure 1. Schematic illustrating the experimental setup, in which re is the radius of the microdisk electrode, rs is the radius of the inert sphere, and dc-c is the distance from the center of the electrode to the center of the sphere.
are then compared to lattice Boltzmann simulations to elucidate the size of a spherical particle and its distance from the electrode center, which are confirmed independently by microscopic measurements of the sphere radius and its distance from the center of the electrode. 2. Theory 2.1. Lattice Boltzmann Theory. A 3D-lattice Boltzmann model based on a single relaxation parameter BhatnagarGross-Krook method is used in this work to study the oneelectron electrolysis of species A at a microdisk electrode
A ( e- f B In each simulation, an applied potential is varied linearly with time at a scan rate of V in V s-1. The potential at a given time is determined by
E ) Estart + νt
(1)
We assume that sufficient background electrolyte is present in order not to consider the transport effects induced by migration. The lattice Boltzmann method tracks the time evolution of particle distribution functions, fR,s(x, t), which moves with
10.1021/jp073364n CCC: $37.00 © 2007 American Chemical Society Published on Web 08/25/2007
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Figure 2. Schematic of the halfway bounce back boundary condition.
Figure 4. (a) Microscope image of the naked electrode. (b) Microscope image of the electrode with a glass sphere of 125 µm radius positioned at a center-to-center distances of 180 µm. Note that the electrode radius is slightly distorted due to the different diffraction indices of water and glass.
Figure 3. Schematic of the curved surface boundary condition.
discrete velocities during discrete advances in time, t, from node to node on an ordered lattice in the direction R (R ) 0, 1, ..., b). The distribution function involves collision and streaming steps which are governed by the LB equation
fR,s(x + ∆x, t + ∆t) 1 (eq) fR,s(x, t) ) - (fR,s(x, t) - f R,s (x, t)) ks (R ) 0, 1, 2, 3, 4, 5, 6) (2) (eq) in which fR,s (x, t) is the equilibrium distribution function, and ∆x and ∆t are the lattice constant and time step size, respectively; ks is the collision relaxation parameter and is related to diffusivity for mass-transfer simulations. (eq) A simpler and more efficient form of fR,s (x, t),7 for each species s, is employed in this work without any real loss of accuracy8
(eq) (x, t) ) F(JR,s + KR,seRu) f R,s
(3)
in which eR represents each of the seven discrete velocities
(
(0,0,0) R ) 0 rest channel eR ) ((1,0,0), (0,(1,0),(0,0,(1) R ) 1, 2, 3, 4, 5, 6 nonrest channel
)
(4)
in which JR,s and Knonrestchannel,s are particularly chosen constants; Knonrestchannel,s is equal to 1/2 for this work, and JR,s is determined by considering the following equation (eq) (x, t) ∑R fR,s(x, t) ) ∑R f R,s
(5)
Figure 5. Cyclic voltammetric measurements for the oxidation of 3 mM Fe(CN)64- in a 0.1 M KCl aqueous solution recorded at scan rates ranging from 0.02 to 1.0 V s-1 for the electrode modified with a sphere corresponding to Figure 4b.
As the mass-transfer simulations use lattices of only one speed, all of the nonrest channels share the same coefficient value. The variables F, the particle density, u, the local fluid velocity, and the particle momentum, Fu, are related to fR,s(x, t) as follows
∑R fR,s(x, t) ) F
(6)
∑R fR,s(x, t)eR ) Fu
(7)
(eq) The fR,s(x, t) is to tend toward f R,s (x, t) by the effect of collision. Therefore, the right-hand side of eq 2 becomes zero, leading to a steady state. The real diffusivity is related to the lattice diffusivity by scaling the lattice constant ∆x and time step size ∆x as below
Dreal )
Dlattice∆x2 ∆t
(8)
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Figure 6. Comparison between the voltammetric response of a naked electrode and an electrode modified with a sphere of 125 µm radius at a center-to-center distance of 180 µm for six different scan rates: (a) 0.005, (b) 0.01, (c) 0.05, (d) 0.1, (e) 0.5, and (f) 0.9 V s-1.
Dlattice is related to ks by
(
Dlattice ) Ca (1 - Jrestchannel) ks -
1 2
)
(9)
in which Ca is a lattice-dependent constant and equal to 1/3 for 3D simulation in this work. Jrestchannel is freely chosen between 0 and 1; in this work, a value of 1/4 is used. 2.2. Definition of Boundary Conditions. There are two boundary conditions used to deal with the interaction between the fluid and solid nodes. One is the physical boundary which sits exactly on the mid-grid (Figure 2), and the other is the surface of the spherical obstruction where the physical boundary is a curved surface (Figure 3). The ways to apply them have been described in the previous paper.4
On the electrode surface, the electrochemical measurement boundary condition has been utilized. For the linear sweep voltammetric techniques used in this work, the concentration at the electrode surface is dependent on the applied potential difference. Using the Nernst equation, the relationship between the two can be described as follows
E - Eθ )
RT CA ln nF CB
(10)
in which Eθ is the standard potential, R the universal gas constant, T temperature, n the number of electrons transferred per reaction, F the Faraday constant, and C the concentration of species A and B. It is assumed that the diffusion coefficients
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Figure 7. Experimental and simulated voltammetric response of the electrode with the glass sphere positioned at a center-to-center distance of 180 µm at six different scan rates: (a) 0.005, (b) 0.01, (c) 0.05, (d) 0.1, (e) 0.5, and (f) 0.9 V s-1. For the simulation, the following parameters were used: D ) 0.63×10-5 cm2 s-1, ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, and NZ ) 100.
of the reactant and product are identical, and the presence of the reactant is inferred by mass conservation. The electrochemical measurement boundary condition is therefore defined as
Windows platform. The codes were written in Java and compiled by Java SE Development Kit 6. 3. Experimental Section
1 CB ) 1 + exp(θ)
(11)
in which θ ) (E - Eθ)(nF/RT). The electrolysis current, ie, is determined using eq 12
ie ) - nFDrealAJe
(12)
in which A is the electrode area and Je is the concentration flux at the surface of the electrode. Simulations were conducted using a PC with an AMD processor with clock speeds of 2.16 GHz and a physical memory of 2.56 GB. They were running between 24 and 96 h on a
3.1. Chemical Reactants and Instrumentation. All chemicals employed were of analytical grade and used as received without any further purification. Potassium hexacyanoferrate(II)trihydrate (K4Fe(CN)6) was purchased from Lancaster Synthesis, and potassium chloride (KCl) was supplied by Riedel-de-Hae¨n. All solutions were prepared with deionized water with a resistivity of no less than 18.2 MΩ cm at 25 °C (Millipore water systems, U.K.). The acid-washed glass beads were between 212 and 300 µm in diameter and obtained from Sigma-Aldrich. Electrochemical measurements were carried out by using a µ-Autolab II (ECO-Chemie, Utrecht, Netherlands) potentiostat interfaced to a PC by using GPES (version 4.9) software for Windows. All measurements were conducted by using a three-
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Figure 8. Comparison between experimental and theoretical results in a logarithmic plot of the peak current versus the scan rate for a centerto-center distance of 180 µm. The parameters used for the simulation were as follows: D ) 0.63 × 10-5 cm2 s-1, ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, and NZ ) 100.
electrode cell, in which the working electrode was a homemade platinum microdisk electrode. The counter electrode was a bright platinum wire, and a saturated calomel electrode (SCE) was employed as a reference electrode. All experiments were carried out at 298 ( 2 K. Before measurements were taken, the working electrode was polished with decreasing sizes of alumina slurry (1-0.3 µm) on soft lapping polishing pads and then thoroughly rinsed with pure water. Optical microscope images were recorded by using a Digital Instruments OMV-PAR microscope based on a Sony XC-999P CCD camera, which has a maximum resolution of 752 × 582 pixels over an area of 540 × 400 µm. 3.2. Fabrication of the Microdisk Electrodes and Positioning of the Glass Sphere. The microelectrode consisted of a platinum wire sealed into an epoxy polymer. The fabrication procedure was as follows. Expoxy resin (Epotek H77A, Promatech Ltd, Cirenchester, U.K.) and hardener (Epotek H77B, Promatech Ltd, Cirenchester, U.K.) were mixed manually in the ratio of resin/hardener 20:3 with a spatula. A plate of epoxy with dimensions of 1.0 cm × 1.0 cm × 0.5 cm was fabricated, and the 1.0 cm × 1.0 cm surface was polished to a flat surface with abrasive paper. Half of the polished surface was covered with a thin layer of epoxy, in which a 50 µm platinum microwire (99.99%, Goodfellows, Cambridge, U.K.) was placed and dried at 80 °C for several hours. Before the metal wire was covered with a second layer of epoxy polymer of 0.5 cm thickness, the electrical connection was made. The electrode was cured at 80 °C in the oven; then, its working face was ground down and polished by first using abrasive paper and then decreasing the size of the alumina slurry (25-0.3 µm).9 The electrode radius was then calibrated electrochemically by using a 3 mM K4Fe(CN)6/0.1 M KCl aqueous solution and by simulating the resulting current responses for various scan rates with a microdisk simulation program with a diffusion coefficient of 0.63 × 10-5 cm2 s-1 and a standard heterogeneous rate constant of 0.05 cm s-1 at a temperature of 298 K, which are in good agreement with the reported values.10,11 The glass spheres were immersed into a 3 mM K4Fe(CN)6/ 0.1 M KCl aqueous solution, deposited with a glass pipet onto the electrode surface, and placed in the center of the electrode by using the tip of a very thin glass pipet mounted on an x-y-z micropositioning device.
Figure 9. Comparison between experimental and theoretical results in a logarithmic plot of the peak current versus the scan rate for a centerto-center distance of (a) 60 and (b) 140 µm. The parameters used for the simulation were as follows: D ) 0.63 × 10-5 cm2 s-1, ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, and NZ ) 100.
4. Results and Discussion Figure 4a and b shows microscopic images of a naked microdisk electrode of radius re and the same microdisk electrode modified with a spherical glass particle of radius rs positioned at a certain distance, dc-c, between the center of the electrode and the center of the particle, respectively. Analysis of the microscopic image resulted in an electrode radius of 60 ( 2 µm, a sphere radius of 125 ( 2 µm, and a center-to-center distance of 180 ( 2 µm. Cyclic voltammetric measurements for the oxidation of 3 mM ferrocyanide/0.1 M KCl in aqueous solution were recorded as a function of scan rate (Figure 5). Figure 6 shows the comparison of the voltammetric responses of a naked electrode for a range of scan rates to those of the same electrode with a spherical particle positioned at a distance of 180 µm. The difference in current between the naked and the modified electrode is greater at slower scan rates (at long timescales, t, of the electrochemical experiment) than that at faster ones. The reason for this is that at longer timescales, t, the size of the diffusion layer, approximately given by x2Dt is on the same order of magnitude as the distance between the two centers. On other hand, at shorter times, the size of the diffusion layer does not stretch far enough into the bulk solution for the electrode to “see” the spherical particle; hence, the voltammetric response resembles more that of a naked electrode. Lattice Boltzmann simulations were then used to simulate the current response for a sphere of 125 µm radius located at a center-to-center distance of 180 µm for a range of scan rates
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Figure 11. Comparison of experimental results for different centerto-center distances, dc-c.
Figure 10. Comparison of cyclic voltammograms for the oxidation of 3 mM Fe(CN)64- in a 0.1 M KCl aqueous solution recorded for the naked electrode and the same electrode modified with a sphere of 125 µm radius at different center-to-center distances, dc-c, at (a) 0.002 and (b) 0.7 V s-1.
(Figure 7). For all scan rates, a good fit between experiment and theory was observed. Note that due to capacitative currents that are difficult to account for and the change of diffusion coefficient for the reduction of ferricyanide on the reverse scan (ferrocyanide and ferricyanide have different diffusion coefficients,10 D(Fe(CN)64-) ) 0.63 × 10-5 cm2 s-1 and D(Fe(CN)63-) ) 0.63 × 10-5 cm2 s-1, which has a major effect on the peak currents at the considered scan rates), only the forward scan of each voltammogram was simulated. Figure 8 illustrates the analysis of the peak currents versus the scan rates, which show an excellent agreement between experiment and theory. Similar voltammetric measurements of the same sphere at center-to-center distances of 60 and 140 µm were then recorded and compared to lattice Boltzmann simulations (Figure 9) For both distances, a good fit between theory and experiment is observed. Figure 10a and b shows a comparison of cyclic voltammograms for different distances for 0.002 and 0.7 V s-1, respectively. The decrease in current caused by the presence of the spherical particle is a lot more pronounced at 0.002 V s-1 than that at 0.7 V s-1 due to the different size of the diffusion layer at these scan rates. This effect is nicely illustrated in the logarithmic plot of the peak current versus the scan rate presented in Figure 11. At slow scan rates, the currents of the
Figure 12. Influence of sphere radius on the peak current. For the simulation, the following parameters were used: D ) 0.63 × 10-5 cm2 s-1, re ) 40 µm, dc-c ) 70 µm, V ) 0.05 V s-1, c ) 3 mM, ∆x ) 10-5 m, ∆t ) 0.0005 s, NX ) 100, NY ) 80, and NZ ) 100.
electrode modified with a sphere at different center-to-center distances deviate more from the current response of the naked electrode than those at faster scan rates. At fast scan rates, the currents of the modified electrode tend toward the current of the naked electrode. The exact scan rate at which this occurs depends on the distance of the sphere from the electrode. Note that the data for the centered spherical particle in Figures 10 and 11 were reported in a previous publication and are shown here only for comparison.2 Figure 12 shows the influence of the sphere radius on the peak current for a fixed center-to-center distance and a fixed scan rate. An increase in sensitivity is observed for increasing sphere radii; if the sphere radius is around 40 µm, the peak current is similar to that of the electrode without the sphere. To increase the sensitivity of the measurement, the center-to-center distance and/or the scan rate would have to be decreased. To conclude, the excellent agreement between theory and experiment confirms the values measured independently with the help of the microscope and provides physical support for the use of voltammetric measurements to size and shape particles adjacent to a microdisk electrode. The use of a cyclic voltammetric methodology to size and shape spherical particles
Voltammetric Sizing and Locating of Spherical Particles compared to the chronoamperometric methodology, which we recently reported in this journal, seems to provide a more accurate measurement of the particles. However, the recording of cyclic voltammograms of variable scan rates is more timeconsuming than the recording of a single transient so that the appropriate choice of methodology has to based on whether the emphasis of the measurement is on the accuracy of the measurement or the time necessary to record the measurement. To extend this concept to particles of unknown size or shape, the system would require a calibration for known sizes and geometries. 5. Conclusion We have shown that cyclic voltammetric measurements can be used to determine the size of a spherical particle positioned at different distances from the center of a microdisk electrode. We are confident that this approach can be extended to particles in the nanometer range if the center-to-center distance and the microdisk radius are adjusted accordingly. Work to extend this concept to measure particles of different shapes is currently under way in our labs, as well as the tracking of moving particles.
J. Phys. Chem. C, Vol. 111, No. 37, 2007 13911 Acknowledgment. N.F. thanks the EPSRC for a project studentship. References and Notes (1) Davies, T. J.; Lowe, E. R.; Wilkins, S. J.; Compton, R. G. ChemPhysChem 2005, 6, 1340. (2) Fietkau, N.; Chevallier, F. G.; Li, J.; Jones, T. G. J.; Compton, R. G. ChemPhysChem 2006, 7, 2162. (3) Barnes, A. S.; Fietkau, N.; Chevallier, F. G.; del Campo, J.; Mas, R.; Mun˜oz, F. X.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2006, 602, 1. (4) Fietkau, N.; Du, G.; Matthews, S. M.; Johns, M. L.; Fisher, A. C.; Compton, R. G. J. Phys. Chem. C 2007, 111, 8496.. (5) Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61, 1794. (6) Arkoub, I. A.; Amatore, C.; Sella, C.; Thouin, L.; Warkocz, J.-S. J. Phys. Chem. B 2001, 105, 8694. (7) Flekkøy, E. G. Phys. ReV. E: Stat., Nonlinear, Soft Matter Phys. 1993, E47, 4247. (8) Zeiser, T.; Lammers, P.; Klemm, E.; Li, Y. W.; Bernsdorf, J.; Brenner, G. Chem. Eng. Sci. 2001, 56, 1697. (9) Maisonhaute, E.; del Campo, F. J.; Compton, R. G. Ultrason. Sonochem. 2002, 9, 275. (10) Stackelberg, M. V.; Pilgram, M.; Toome, V. Z. Elektrochem. Angew. Phys. Chem. 1953, 57, 342. (11) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; John Wiley and Sons: New York, 2001.
