J. Phys. Chem. C 2008, 112, 9351–9357
9351
Characterisation and Modeling of Conducting Composite Electrodes Hong Zhao and Danny O’Hare* Department of Bioengineering, Imperial College, London, SW7 2AZ, United Kingdom ReceiVed: December 1, 2007; ReVised Manuscript ReceiVed: March 28, 2008
We have carried out a systematic investigation of the surface electrochemistry of glassy carbon-epoxy composite electrodes. Key findings include the following: the distributed resistance in electrodes results in shifts in E1/2, and the overlap of diffusion layers on electrode surface leads to a potential dependent il. Two different types of carbon fiber array electrodes were studied for comparison with glassy carbon-epoxy electrodes. A numerical model was built, and results show shifts in E1/2 and sloping il which are in good qualitative agreement with experiments. Introduction Conducting composite electrodes have been of great interest for many years because of their versatile use in electrochemistry. Pioneering work in this area was done by Adams1 in 1958. Carbon-epoxy composite electrodes play an important role in biomedical applications because of their good biocompatibility, low cost, and ease of construction. They can be bulk modified with stabilized enzymes2,3 to make biofuel cells or used in electro-analysis to detect acids,4 glucose5 and enzymes.6 Despite the wide use of composite electrodes, the systematic investigation of such devices is limited in the literature. More recently, O’Hare et al.7 and Ramirez-Garcia et al.8 have used similar materials to study the properties of these devices. O’Hare et al. showed that graphite epoxy composite electrodes have microscopic conducting features that behave like random assembles of microelectrodes. Ramirez-Garcia et al. showed that the conductivity of similar devices depends on the composite ratios which could be predicted by percolation theory. Overall, such devices display good stability, temporal characteristics, and microelectrode-array-like behavior. However, their low sensitivity to flow is accompanied by high capacitance and bulk resistance, which can blur the voltammetric details and make interpretation difficult. Theoretical work in this area has focused exclusively either onthevoltammetricconsequencesofadispersedelectrode-electrolyte interface or on the relationship between the composition of the composite and its conductivity. Of the former approaches, modeling work by Compton’s group on partially blocked microelectrodes9–11 is particularly noteworthy. Advances in understanding of the conductivity of composites is dominated by percolation theory, first applied to electrochemistry by Navarro-Laboulais et al.12–14 Both models provide useful insights into composite electrodes but fail to provide a comprehensive treatment. Furthermore, the consequences of finite and distributed resistances in the bulk of these electrodes have not been considered hitherto, and the link between the resistance and the diffusion field at the electrode’s surface has not been explored. In this paper, we report numerical and experimental approaches considering the inner structure of composite electrodes and the consequences for the surface electrochemistry, which leads us to consider the clustering network. * Corresponding author. Phone: + 44-20-75945173. Fax: + 44-2075945177. E-mail:
[email protected].
Martin et al.15 extensively studied the clustering of carbon nanotubes blended in resin under different conditions. Some of their key findings were that clustering only occurs after the hardener was added to the mixture, clustering is significantly dependent on the curing temperature, and no clustering occurs at room temperature. In order to study the random network of conductivity with no bias to any specific materials, glassy carbon spherical powder was used as the conducting material, which is relatively monodisperse. The rigidity makes the contact between any two spherical balls a point-to-point contact. This allows for more straightforward and realistic modeling than would be the case for more irregular graphite particles. Carbon fiber (CF) array electrodes were made for comparison with glassy carbon-epoxy electrodes since these devices show dispersed interfaces without the complication of distributed internal resistances. Chemical pretreatment can alter the identity and density of functional groups on the carbon surface16 and may also affect the contact resistance. Consequently, we have compared acid pretreatment with reducing agent pretreatment on glassy carbon spherical powder and report the consequences for functionality using acid-base titration. Any systematic variation in the voltammetry is reported. Although numerical reports exist for two-dimensional (2-D) discs with close random packing17 and for randomly close packed three-dimensional (3-D) monosized spheres,18 3-D modeling of close packing for various sphere sizes still represents a great challenge because of the enormous computational requirements. We have used the concept of conducting pathways to model the composite structure to reduce the amount of numerical work needed. Three-dimensional numerical models based on percolation theory are presented, and these allow the distributed resistances to be embedded in the composite and enable qualitative prediction of the voltammetric properties. Experimental Methods Chemicals. All chemicals were reagent grade and used without further purification. Voltammetric experiments were undertaken using 10 mM Ru(NH3)6Cl3 in 1 M potassium chloride aqueous solution. Instrumentation. All cyclic voltammetry experiments were performed using a CHI 650A potentiostat (CH Instruments). All of the experiments were performed in a Faraday cage to
10.1021/jp711366u CCC: $40.75 2008 American Chemical Society Published on Web 05/31/2008
9352 J. Phys. Chem. C, Vol. 112, No. 25, 2008
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TABLE 1: Modeling Parameters for Conductive Media (V) Section modeling equations and parameters subdomain
boundary
description
-∇ · (σ∇V-Je) ) Qj σ ) 10 S m-1 Je ) 0 A m-2 Qj ) 0 A m-3 V(t0) ) 0 V n · J ) σ(V-Vref)/d V)0 n·J ) 0 σ ) 107 S m-1; σ ) 106 S m-1; σ ) 105 S m-1; σ ) 104 S m-1; σ ) 3 × 103 S m-1; σ ) 103 S m-1; σ ) 3 × 102 S m-1; σ ) 102 S m-1;σ ) 10 S m-1 d ) 0.001 m Vapplied ) (E1-Vt) · (tet1/2) VVapplied )[E2+Vt] (t-t1/2)] · (t>t1/2) V E1 ) 0.2 V E2 ) -0.6 V V ) 0.01 V s-1 t1/2 ) (E1-E2)/V s E0 ) 0.2 V
electrical conductivity external current density current source electric potential
electrical conductivity thickness applied potential upper potential lower potential scan rate switch scan time standard potential
TABLE 2: Modeling Parameters for Species O (c) Diffusion Section modeling equations and parameters subdomain
boundary
δts∂c/∂t+∇ · (-D∇c) ) R δts ) 1 D ) 9.1 × 10-10 m2 s-1 R ) 0 mol m-3 s-1 c(t0) ) 1 mol m-3 -n · N ) N0+kc(cb-c); N ) -D∇c c ) c0 n · N ) 0; N ) -D∇c N0 ) -k0[c · e-Rf(V-E0)-c2 · e(1-R)f(V-E0)] mol m-2 s-1 kc ) 0 m s-1 c0 ) 1 mol m-3 k0 ) 0.001 m s-1 R ) 0.5 f ) F/(R · T) ) 38.9217 V-1
description time-scaling coefficient diffusion coefficient reaction rate initial concentration
inward flux mass transfer coefficient concentration rate constant transfer coefficient
TABLE 3: Modeling Parameters for Species R (c2) Diffusion Section modeling equations and parameters subdomain
boundary
δts∂c/∂t+∇ · (-D∇c2) ) R δts ) 1 D ) 9.1 × 10-10 m2 s-1 R ) 0 mol m-3 s-1 c2(t0) ) 0 mol m-3 -n · N ) N0+kc(cb-c2); N ) -D∇c2 c2 ) c20 n · N ) 0; N ) -D∇c2 N0 ) k0[c · e-Rf(V-E0)-c2 · e(1-R)f(V-E0)] mol m-2 s-1 kc ) 0 m s-1 c20 ) 1 mol m-3 k0 ) 0.001 m s-1 R ) 0.5 f ) F/(R · T) ) 38.9217 V-1
reduce electrical noise. Ag|AgCl reference electrode (CH Instruments) and homemade platinum mesh counter electrode were used in all experiments. A low speed diamond saw (IsoMetTM, Buehler) was used for precise cutting of the electrodes. Glassy Carbon Spherical Powder Preparation. Glassy carbon spherical powder (10-20 µm, Alfa, type 2) was pretreated in two different ways for comparison. 1. Reducing Agent Pretreatment. The untreated glassy carbon spherical powder was washed in 10 mM boiling sodium cyanoborohydride in methanol solution, stirred for 2 h, and then washed with distilled water thoroughly and dried in vacuum over anhydrous silica gel.
description time-scaling coefficient diffusion coefficient reaction rate initial concentration
inward flux mass transfer coefficient concentration rate constant transfer coefficient
2. Acid Pretreatment. The untreated glassy carbon spherical powder was washed in 3 M boiling nitric acid, stirred for 2 h, and then washed in copious volumes of deionized water and dried in a similar condition as stated above. Glassy Carbon-Epoxy Composite Electrode Preparation. The pretreated glassy carbon spherical powder was mixed with epoxy resin and hardener (“Araldite” RX771C/NC and HY1300, Robnor Resins Ltd.) to different final ratios of 60:40, 50:50, and 40:60 (w/w). The ratio between epoxy resin and hardener was fixed to 100:37 (v/v). Glassy carbon spherical powder and epoxy resin was mixed first by hand until a smooth black paste was achieved. Hardener was then added to the paste, which was mixed again for a few minutes before being degassed under
Conducting Composite Electrodes
Figure 1. Cyclic voltammograms of 0.5 mm thick 60:40 glassy carbon-epoxy electrodes at 10 mV s-1 in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
vacuum. Then the whole mixture was packed in a 2 mm plastic tube to form a cylinder shape and degassed again. Curing was achieved at room temperature for 72 h to avoid any clustering. The electrode rods were then cut into slices of different thicknesses (0.5 mm, 1 mm, and 2 mm) using a diamond saw followed by ultrasonic cleaning in ethanol and deionized water. One side of each slice was then attached to silver wires using silver epoxy; the other face was used as the electrode surface. The resulting electrodes were then sealed in plastic micropipette tips using epoxy resin (Torr Seal). Carbon Fiber Array Electrode Preparation. Two different types of carbon fiber array electrodes were fabricated: clustered and isolated. 1. Isolated Carbon Fiber Array Electrodes. Ten single carbon fibers (5 µm and 10 µm, Goodfellow; 30 µm, World Precision Instruments, Inc.) were stuck onto a self-adhesive copper tape (Farnell) separated by a few millimeters. Scores of polymer fibers (17 µm, polyaramid, Goodfellow) were placed between the carbon fibers to ensure isolation at the electrode surface. Silver epoxy was used to improve electrical connection between the fibers and the copper tape. A gold pin (MultiContact) was attached to the copper tape which was rolled to form a cylinder and then sealed in epoxy resin, as above. The surface was then cut and cleaned using the procedures described above. 2. Clustered CF Array Electrodes. A similar fabrication method was used except that hundreds of carbon fibers (7 µm, Goodfellow) were used and no polymer fibers were included. Thus, the clustered CF array electrodes would have an inner random conducting network, while the isolated CF array electrode would have no connection between each fiber except at the end where they were attached to the copper tape. Procedures. Before starting an electrochemical experiment, each electrode was polished with 1 µm, 0.3 µm, and 0.05 µm alumina aqueous slurry successively. The electrode was sonicated in deionized water between grades of alumina. Then the electrode was scanned from 1.5 to -1.5 V for 15 min in 0.1 M sulfuric acid and held at -1.5 V for another 15 min. All solutions were bubbled with nitrogen for 30 min before use to remove dissolved oxygen. Titration. Titration experiments were done on untreated glassy carbon spherical powder and on samples subjected to the two different pretreatments. A 2 g sample was added in 20 mL 0.1 M sodium chloride and titrated with 0.1 M hydrochloric acid until the pH of the solution became stabilized (around pH 2 to 3). The 0.1 M sodium hydroxide solution was then added gradually until the pH stabilized (around pH 11 to 12). The solution of acid pretreated glassy carbon spherical powder has a very low pH, so only base titration was applied.19 Image Processing. A microscope (Leica, model MZ16FA) and a digital camera (Sony, Cybershot DSC 727) were used to
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Figure 2. Cyclic voltammograms of glassy carbon electrodes with different series resistances at 100 mV s-1 in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
Figure 3. Cyclic voltammograms of 0.5 mm glassy carbon-epoxy electrodes with different ratio at 10 mV s-1 in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
Figure 4. Cyclic voltammograms of 60:40 glassy carbon-epoxy electrodes with different electrode thickness at 50 mV s-1 in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
Figure 5. Cyclic voltammograms of 0.5 mm thick 60:40 glassy carbon-epoxy electrodes with different scan rate in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
obtain the pictures of the carbon fiber array electrodes’ surface. The images were then loaded into a data processing program (IgorPro 5.0) and converted to gray scale images. The numbers of fibers were counted, and the exact positions of each fiber were recorded. Calculating the Regularity Index (RI) and Ripley’s K Function. Although the separation and clustering of carbon fiber array electrodes can be observed qualitatively from micrographs,
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Figure 6. Cyclic voltammograms of an acid pretreated GC-epoxy composite electrode (60:40, 0.5 mm) at different scan rate in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
Figure 8. Cyclic voltammograms of (a) isolated carbon fibre array electrodes with different diameter and (b) clustered carbon fibre array electrodes at 10 mV s-1 in 10 mM Ru(NH3)6Cl3 in 1 M aqueous KCl.
Figure 7. Titration using acid/base on untreated GC, acid pretreated GC, and reducing agent pretreated GC.
the regularity index and Ripley’s K function/L function20 were used to describe the separation/clustering more quantitatively. For a number of events in a closed proximity, RI is defined as the fraction of mean and standard deviation of the distance between each event. Ripley’s K function provides more graphical information on both the extent and the length scale by comparing the clustering to a Poisson distribution as a function of distance. The definition of the K function is as follows:
λK(h) ) E (the number of events within distance h of an arbitrary event) where λ is the intensity or mean number of events per unit area. A suitable estimate of K(h) is given by N
K(h) )
N
Ih(dij) 1 2 λ R i)1 j)1,j*i wij
∑ ∑
where R is the area of region; dij is the distance between the ith event and jth event in the region; Ih(dij) is the indicator function which is 1 if dij e h and 0 otherwise; and wij is the conditional probability that an event is observed in the region, given that it is a distance dij from the ith event. Having wij in the equation assumes the region is infinitely large. L(h) is often used as an exploratory tool by comparing K(h) estimated from the observed data with πh2.
L(h) )
K(h) π
Comparing L(h) with Poisson distribution, f(h) ) h, one would be able to tell if the events have a clustering at that scale,
which is above the line, or greater regularity than the Poisson distribution, which is below the line. Numerical Modeling. A numerical 3-D model was developed using the finite element method (Comsol 3.2). Transit diffusion and conductive media DC models were coupled to solve the problem in this module. An electrode array of nine electrodes was combined in the model in a rectangular lattice. The distance between the centers of two nearby electrodes is 20 times the diameter of the electrodes. The conductivity of the solution was kept constant, and the conductivity of each electrode varied. Different resistances on each electrode were implanted to mimic the different conducting pathways. Potential difference on the surface of each electrode was thus generated as a result. Diffusion of a redox couples in the subdomain and Butler-Volmer equation on the electrode surface were combined to model the electrochemical reaction. The Butler-Volmer approach21 of a one electron transfer reaction O + e T R is
i ) FAk0CO(0, t) · e-Rf(E-E ) - CR(0, t) · e(1-R)f(E-E ) 0
0
and the normal flux J is defined by
J)
i nFA
At the beginning, only species O exists in the solution. As the potential on the electrode surface comes close to the formal potential (E0′), the species O near the electrode surface would be reduced to R and change the local concentration. Flux is thus generated by diffusion and integrating the flux over the electrode surface gives the current. When the voltammetric scan direction changes, species R is oxidized to O again. The individual current response at each conducting element as well as overall response of the array were calculated. All of the subdomain and boundary conditions are listed in Table 1, Table 2, and Table 3.
Conducting Composite Electrodes
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Figure 10. Ratio of potential on the electrodes surface (potentialS) and the potential on the electrodes (potentiale) with changing electrical conductivity in conducting pathway.
Figure 9. L(h) for (a) an isolated 5 um carbon fibre array electrode, (b) an isolated 10 um carbon fibre array electrode, (c) an isolated 30 um carbon fibre array electrode, and (d) a clustered 7 um carbon fibre array electrode. The maximum value of h in each figure equals to onetenth of the diameter of the electrode respectively.
Results and Discussion 1. GC-Epoxy Electrodes. A 10 mM sample of Ru(NH3)6Cl3 in 1 M KCl was used in all cyclic voltammetry experiments. Nine different types of GC-epoxy electrodes were studied by varying the composite ratios from 60:40, 50:50, to 40:60 (w/w %) and electrode thicknesses from 0.5 mm, 1 mm, to 2 mm. The cyclic voltammograms of the GC-epoxy electrodes at slow scan rates show currents of comparable magnitude to
macro-electrodes and the sigmoidal voltammograms characteristic of microelectrodes (Figure 1). Two dominant characteristics are evident in these voltammograms: (i) the shift of E1/2 compared with highly active glassy carbon electrodes, and (ii) a sloping diffusion limited current. These are caused by the difference resistance in each conducting pathways. Random packing can lead to conducting pathways of varying length, and thus resistances between the hook-up contact and the electrode surface vary according to their length. The different conducting pathways could have a resistance that may vary by a few orders of magnitude. These phenomena were explored by adding series resistors between a glassy carbon electrode (4 mm, CH Instruments) and the potentiostat for cyclic voltammetry measurements. Experiments showed a linear increase of E1/2 versus resistance and a linear decrease of ip versus the logarithm of resistance (Figure 2). These experiments modeled the effects of different resistance in each conducting pathway. Voltammograms of different composite ratios and different electrode thicknesses were compared. An increasing E1/2 and decreasing current could be observed with decreasing conducting material fraction (Figure 3). Increasing electrode thickness has a similar effect (Figure 4). In both cases, this is due to the fact that fewer conducting pathways exist and more resistance is embedded in each pathway, respectively. Increasing the scan rates from 10 mV s-1 to 1 V s-1 increases the magnitude of the current response and the hysteresis in the voltammograms (Figure 5). For the acid pretreated composite electrodes, the cyclic voltammograms (Figure 6) also have sloping diffusion limited currents. The current responses are much higher than the electrodes pretreated with sodium cyanoborohydride, so are the capacitances. For the same composite material ratio and electrode thickness, these differences in the voltammograms indicate that acid pretreated glassy carbon spherical powders are more electrochemical active than reducing agent pretreated ones. The capacitance for these electrodes (around 100 µF) was also higher compared with the reducing agent pretreated electrodes (around 1 µF), consistent with more oxygen functional groups on the surface associated with increased pseudocapacitance and surface roughness. Titration Experiments. Titration can provide useful information about the surface functional groups since each functional group has its own pH range (carboxyl, 2.5 to 4.5; lactone, 4.5 to 8.0; and phenol, 8.0 to 10.5).22 Figure 7 shows that the titration curves of untreated glassy carbon spherical powders and reducing agent pretreated glassy carbon spherical powders are of similar shape, but the reducing agent pretreated powders have more functional groups on the
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Figure 11. Numerical modeling of microelectrode array with different resistances in each conducting pathways. Iso-surface plot shows the concentration, and streamline shows the electric field.
surface and an increased buffer capacity. The acid pretreated glassy carbon spherical powders have a low pH initially and a much increased buffer capacity. Reducing agent pretreated glassy carbon powders surprisingly contain more carboxyl groups than untreated glassy carbon powders, and acid pretreated glassy carbon powders probably have similar amount of carboxyl to the reducing agent pretreated ones, although they have more lactone and phenolic groups. It is well-known that electrochemical activity is associated with increased oxygen functionality.23 The titration data are consistent with the voltammetric results: acid pretreated GC-epoxy electrodes have more functional groups and activated areas on the surface and bigger current response in cyclic voltammograms; reducing agent pretreated GC-epoxy electrodes have fewer functional groups than the acid pretreated samples but still have more than the untreated glassy carbon. 2. CF Array Electrode. The i-V curves for the isolated CF array electrodes (Figure 8a) have the sigmoidal shape typical of a microelectrode and a diffusion limited current several times higher than a single fiber, as expected. The clustered CF array electrodes (Figure 8b) have a sloping diffusion limited current similar to GC-epoxy electrodes. The only differences between these two types of CF array electrodes are the regularity of carbon fiber clusters on the surface and inner conducting network. The sloping current cannot be caused by the regularity because the clustered CF array electrodes have their fibers closer together, and the overlap of diffusion layer
Figure 12. Numerical modeling cyclic voltammograms of (a) each electrode in the microelectrode array with different resistances and (b) overall microelectrode array.
Conducting Composite Electrodes would form a better defined limited current. Thus, the sloping current must be caused by the resistance between the conducting pathways. The carbon fiber electrodes were examined under a microscope. The isolated ones showed a good separation between each fiber, and the clustered ones showed several clusters as well as some individual fibers. In order to describe the separation quantitatively, the regularity index (RI) and Ripley’s K function/L function were used to check the regularity of carbon fiber array electrodes. Both types of CF array electrodes have a similar RI value around 2 indicating the limited utility of this metric in this application. However, the L function (Figure 9) provides more information. Isolated CF array electrodes showed regularity close to the random Poisson distribution, whereas clustered CF array electrodes clearly showed clustering structure on the millimeter length scale. 3. Modeling. The resistances inside the conducting pathways led to potential loss in the electrodes causing the actual potential on the electrode surface to be smaller than the applied potential. This potential loss was modeled and is shown in Figure 10. As a consequence of the variability in the resistance of the different pathways, different regions of the electrode surface will experience different potentials and, rather less obviously, will also experience different scan rates. Figure 11 shows the concentration profile of species O at certain times on the anodic wave in iso-surface and electric field in streamline. The resistance of each electrode increases from No. 9 (highly conductive) to No. 14 (highly nonconductive). Because of the ohmic drop in each electrode (Nos. 9 to 14), they all develop different concentration gradients. Some electrodes have little electrochemical reaction taking place because of their huge internal ohmic drop, and the potential on the surface could not drive the reaction. Cyclic voltammetric responses of each electrode as well as the overall electrode response were calculated (Figure 12), which were qualitatively similar to the experimental data of clustered carbon fiber array electrodes (Figure 8b). For microelectrodes, the diffusion limited current would always reach the same level for quasi-steady potential perturbations given sufficient potential scan range. However, for a given potential range, some electrodes would reach the diffusion limited current while other electrodes may just reach the E0′ and yet others may not even reach a sufficient potential to support any electro-chemical reaction due to the large internal resistance. The slope of the sloping limited current would depend on the combination of resistance in each conducting pathway. Conclusions We have undertaken a systematic investigation of the effects of composition and formulation on the voltammetric behavior and nonfaradaic properties of conducting composite electrodes. Voltammetric results showed a shift in E1/2 (half-wave potential) and a sloping il (diffusion limiting current). Comparison with carbon fiber arrays enabled separation of the effects of a distributed interface from 3-D variations in the individual conducting pathways. Reducing agent pretreatment and acid pretreatment both created more functional groups and increased the current response of cyclic voltammetric experiments, though the effects of acid pretreatment were greater. A 3-D numerical model coupling the resistance in each conducting pathway with diffusion has been developed. The good qualitative agreement between the numerical model and the experimental data shows
J. Phys. Chem. C, Vol. 112, No. 25, 2008 9357 that the sloping, potential dependent diffusion-limited current commonly observed for these devices can be accounted for by the existence of conducting pathways of varying resistance. Acknowledgment. Useful discussions with members of the Biosensor Group, Imperial College London are gratefully acknowledged. References and Notes (1) Adams, R. N. Carbon Paste Electrodes. Anal. Chem. 1958, 30 (9)), 1576. (2) Lvovich, V.; Scheeline, A. Amperometric sensors for simultaneous superoxide and hydrogen peroxide detection. Anal. Chem. 1997, 69 (3)), 454–462. (3) Khurana, M. K.; Winlove, C. P.; O’Hare, D. Detection mechanism of metallized carbon epoxy oxidase enzyme based sensors. Electroanalysis 2003, 15 (12)), 1023–1030. (4) Rubianes, M. D.; Rivas, G. A. Carbon nanotubes paste electrode. Electrochem. Commun. 2003, 5 (8), 689–694. (5) Li, J.; Chia, L. S.; Goh, N. K. Renewable silica sol-gel derived carbon composite based glucose biosensor. J. Electroanal. Chem. 1999, 460 (1-2), 234–241. (6) Sampath, S.; Lev, O. Inert metal-modified, composite ceramiccarbon, amperometric biosensors: Renewable, controlled reactive layer. Anal. Chem. 1996, 68 (13), 2015–2021. (7) O’Hare, D.; Macpherson, J. V.; Willows, A. On the microelectrode behaviour of graphite-epoxy composite electrodes. Electrochem. Commun. 2002, 4 (3), 245–250. (8) Ramirez-Garcia, S.; Alegret, S.; Cespedes, F. Carbon composite microelectrodes: Charge percolation and electroanalytical performance. Anal. Chem. 2004, 76 (3), 503–512. (9) Kovach, P. M.; Deakin, M. R.; Wightman, R. M. Electrochemistry at Partially Blocked Carbon-Fiber Microcylinder Electrodes. J. Phys. Chem. 1986, 90 (19), 4612–4617. (10) Brookes, B. A.; Davies, T. J.; Fisher, A. C. Computational and experimental study of the cyclic voltammetry response of partially blocked electrodes. Part 1. Nonoverlapping, uniformly distributed blocking systems. J. Phys. Chem. B 2003, 107 (7), 1616–1627. (11) Davies, T. J.; Brookes, B. A.; Fisher, A. C. A computational and experimental study of the cyclic voltammetry response of partially blocked electrodes. Part II: Randomly distributed and overlapping blocking systems. J. Phys. Chem. B 2003, 107 (26), 6431–6444. (12) Navarro-Laboulais, J.; Trijueque, J.; Garcia-Jareno, J. J. Electroc hemical impedance spectroscopy of conductor-insulator composite electrodes: properties in the blocking and diffusive regimes. J. Electroanal. Chem. 1998, 444 (2), 173–186. (13) Navarro-Laboulais, J.; Trijueque, J.; Garcia-Jareno, J. J. Determination of the electroactive area of graphite plus polyethylene composite electrodes. Uncompensated resistance effects and convolution analysis of chronoamperograms. J. Electroanal. Chem. 1998, 443 (1), 41–48. (14) Beaunier, L.; Keddam, M.; Garcia-Jareno, J. J. Surface structure determination by SEM image processing and electrochemical impedance of graphite plus polyethylene composite electrodes. J. Electroanal. Chem. 2004, 566 (1), 159–167. (15) Martin, C. A.; Sandler, J. K. W.; Shaffer, M. S. P. Formation of percolating networks in multi-wall carbon- nanotube-epoxy composites. Comp. Sci. Technol. 2004, 64 (15), 2309–2316. (16) Kinoshita, K. Carbon, electrochemical and physicochemical properties; John Wiley & Sons, Inc.: New York, 1988. (17) Xu, N.; Blawzdziewicz, J.; O’Hern, C. S. Random close packing revisited: Ways to pack frictionless disks. Phys. ReV. E 2005, 71 (6), -061306. (18) Aste, T.; Saadatfar, M.; Senden, T. J. Geometrical structure of disordered sphere packings. Phys. ReV. E 2005, 71 (6), 061302. (19) Boehm, H. P. Some Aspects of the Surface-Chemistry of CarbonBlacks and Other Carbons. Carbon 1994, 32 (5), 759–769. (20) Bailey, T. C.; Gatrell, A. C. InteractiVe spatial data analysis; Wiley: New York, 1995. (21) Bard, A. J.; Faulkner, L. R. Electrochemical methods: fundamentals and application, 2nd ed.; John Wiley & Sons, Inc.: New York, 2001. (22) Barkauskas, J.; Cannon, F. S. Potentiometric titrations: Characterize functional groups and adsorbed species on activated carbon. In Carbon, 2003; An International Conference on Carbon, Oviedo, Spain, 2003. (23) Chen, P. H.; McCreery, R. L. Control of electron transfer kinetics at glassy carbon electrodes by specific surface modification. Anal. Chem. 1996, 68 (22), 3958–3965.
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