Diffusion at Double Microband Electrodes Operated within a Thin Film

8694

J. Phys. Chem. B 2001, 105, 8694-8703

Diffusion at Double Microband Electrodes Operated within a Thin Film Coating. Theory and Experimental Illustration† Idir Ait Arkoub, Christian Amatore,* Catherine Sella, Laurent Thouin, and Jean-Ste´ phane Warkocz De´ partement de Chimie, Ecole Normale Supe´ rieure, UMR CNRS 8640 “PASTEUR” 24 rue Lhomond, 75231 Paris Cedex 05, France ReceiVed: February 28, 2001; In Final Form: May 29, 2001

The diffusion transport of reversible redox species was investigated at microbands in a thin film coating. The effect of the film thickness onto the diffusion was studied theoretically and by numerical simulations both at single-band and double-band microelectrodes. According to the diffusion length of the species in this restricted volume, the corresponding theory and simulations show that several diffusional regimes are expected which provide a way to estimate both the diffusion coefficient and concentration of the species but also to evaluate at the same time the film thickness. This has been demonstrated experimentally by examining the electrochemical responses of microbands coated by FeIII-loaded Nafion films of different thicknesses. A very good agreement is obtained between theory and experiments which shows that the electrochemical responses of such devices afford a possibility of monitoring the physical and geometrical properties of a polymer film in relation to its chemical environment.

Introduction Electrodes coated by a polymer membrane incorporating electroactive species have been extensively studied because of the high number of applications to electrocatalysis, analysis, sensors, and other related devices.1,2 In several studies, the redox centers incorporated into polymeric structures are used as redox mediators in catalytic processes, the redox couple being then confined within the polymer coating instead of being dispersed in the solution. For example, in the context of electrochemical biosensors, mediators are usually used to shuttle electrons between the immobilized enzyme and the electrode. The intrinsic electrochemical reactivity of redox polymer coatings is generally dependent on the concentration and the apparent diffusion coefficient of the electroactive species in the polymer. Physical processes like swelling and morphological changes of the polymer are often considered as ancillary processes and are normally incorporated into the primary process or even neglected. However, solvent environments can affect mechanisms of electron-transfer reactions and mass transport rates.3 In particular, swelling of polymeric materials is central for the design of artificial noses and tongues based on optical sensors or chemiresistor arrays.4 Indeed physical swelling of polymeric films induced by sorption of chemicals may give rise to a specific response, i.e., change in fluorescence of the incorporated dyes in optical sensors5 or variation of the intrinsic conductivity of polymeric chains in chemiresistor arrays.6 In the present paper, we investigate the possibility of monitoring the swelling of polymeric films through its effects on the mass transport of incorporated redox species. This constitutes an interesting approach in view of the conception of new kinds of sensor devices.7,8 Alteration of the film should lead to a variation of both the diffusion coefficient and †

Part of the special issue “Royce W. Murray Festschrift”. * To whom correspondence should be addressed. Fax: +33-1-44323863. E-mail: [email protected].

concentration of the redox species loaded in (or bound to) the film as well as to a variation of the space in which diffusion occurs. The electrochemical response of the coated electrode is then expected to be a function of these parameters. It should thus afford a way to monitor the physical and geometrical properties of the polymer in relation to its chemical environment, provided that the specific and complex diffusional patterns thus created when the diffusion layers reach their geometrical limits imposed by the film are fully understood and mastered. The purpose of this work is therefore to examine the mass transport of reversible redox species at simple and double-band microelectrodes coated by a thin film, considering the potentiality of such devices in the realization of chemical sensors. The effect of the film thickness onto the diffusion transport at simple and double-band microelectrodes has then been investigated by numerical simulation. Our theoretical expectations are compared experimentally to the electrochemical responses of a doubleband microelectrode assembly modified with Nafion containing FeIII cationic counterions as reversible redox species. This system (viz., Nafion and FeIII/II redox couple) was chosen as a test one because Nafion has been extensively studied.9 Experimental Section Double-Band Microelectrode Assemblies. The double-band assemblies were constructed along the same “sandwich” technique previously reported.8,10 An insulator (mylar film of 2.5 µm nominal thickness, Energy Beam Sciences) was placed between two sheets of platinum foils (Goodfellow, 5 µm thickness) and the whole assembly inserted between two pieces of soft glass (about 3 mm thick and 8 mm width). Each layer of this “sandwich” was sealed with a small amount of epoxy resin (Epon 828 with 10% triethylene tetramine, Aldrich). The whole system was then firmly pressed into position. After hardening of the epoxy, each platinum foil was independently connected to an electrical copper lead with silver epoxy (Elecolit, type 340). This assembly was glued inside a glass tube of 1 cm

10.1021/jp010764g CCC: $20.00 © 2001 American Chemical Society Published on Web 07/25/2001

Diffusion at Double Microband Electrodes internal diameter. The cross section of this assembly was then exposed with a diamond saw and polished by successive steps using five abrasive papers of finer and finer grades (Presi P600, P1000, P2500, and P4000). Each assembly was composed of two paired-microband Pt electrodes whose characteristics were controlled under a binocular microscope (thickness, w ) 5 µm; length, l ≈ 3.5 mm; surface area ≈ 0.0175 mm2; insulating gap, g ) 2.5 µm). The electrical independence of each electrode was checked by controlling that the electrical resistance between them was infinite. Each of the paired bands could be disconnected electrically, the other being then used as a single band. Nafion-Coated Electrodes. A preliminary treatment of the glass surface was necessary to ensure a good adhesion of the Nafion film onto the assembly. The electrode was immersed in a 5% 3-aminopropyltrimethoxysilan solution (Fluka) for one minute, rinsed with acetone, and dried at room temperature during at least 20-30 min. A total of 5-30 µL of a 5 wt % Nafion solution (mixture of lower aliphatic alcohols and 1520% water, prepared from Nafion-117 perfluorinated membrane, Aldrich) was cast on the surface of the assembly, and the solvent was evaporated at room temperature for at least 10 h. Estimated values of film thicknesses were calculated on the basis of a Nafion dry density of 1.98 g cm-3.11 For example, the thickness of a film obtained by casting 10 µL of a 5 wt % Nafion solution onto the surface of the assembly (effective area 0.4 cm2) was estimated to be about 5 µm by this method. After use, Nafion films were removed by cleaning with acetone, and the assembly was polished again. Loading of Electroactive Species. Supporting electrolyte solutions were prepared from deionized water and H2SO4 (97% Fluka). Fe2(SO4)3‚5H2O (Aldrich) was used to prepare 3-10 mM FeIII stock solutions. Electrochemical experiments have been performed in FeIII-free 0.1 M H2SO4 aqueous solution in which the electrochemical responses showed well-shaped voltammograms and for which the porosity of Nafion was maximum.12 In 0.1 M H2SO4 aqueous solution, FeIII exists mainly as cationic complexes such as Fe(SO4)+ and Fe(HSO4)2+. Under these conditions, the hydrated ferric ion Fe(H2O)63+ content is rather low. Fe2+ ion does not form stable sulfate complexes, and its hydrated form Fe(H2O)62+ prevails. Nafioncoated electrodes were immersed in 3-10 mM FeIII, 0.1 M H2SO4 aqueous solution for at least 15 h. A long time of incorporation was used to ensure saturation in encapsulating FeIII cations in Nafion. Indeed, the partitioning of FeIII cations into the membrane-type coating can be described by a classical ion-exchange mechanism involving electrostatic forces. Prolonged exposure to solutions of the cations led to the gradual dehydration of Nafion coatings13 which in turn gives rise to the compression of the hydrophilic channels in the film.14,15 When the FeIII-loaded electrode was placed in a FeIII-free 0.1 M H2SO4 aqueous solution for hours, some FeIII leaked out into the bulk solution. During the short duration (15 s) of the experiments reported here, leakage was negligible as evidenced by the fact that the currents monitored at the assemblies were sufficiently stable and reproducible during all of the measurements reported here. Electrochemical Equipment. All experiments were carried out at room temperature in 0.1 M H2SO4. For electrochemical measurements, the cell (ca. 15 mL) was degassed and placed under an inert atmosphere blanket with argon. It was equipped with a platinum counter-electrode coil of ca. 1 cm2 surface area and a SCE reference electrode (Radiometer analytical). The potential of each microband was controlled separately by an integrated computer-driven bipotentiostat (Autolab PGstat 20;

J. Phys. Chem. B, Vol. 105, No. 37, 2001 8695

Figure 1. Schematic representations of microbands assemblies and of their dynamics. (A) Single-band mode, only one electrode being connected (solid black). (B) Double-band mode showing the crosstalk between generator (solid black) and collector (hatched shade) electrodes. (C and D) Expansion of the diffusion layer at two different times at a coated single band: (C) short time planar diffusion in the near vicinity of the band; (D) long time planar lateral diffusion (see text). h is the thickness of the Nafion film, w is the width of each Pt microband, and g is that of the Mylar insulating gap. Insulating materials (glass, Mylar, and bulk solution) are shown in solid gray, whereas the space available to diffusion is represented in white.

GPES software; Ecochemie). Cyclic voltammetry was performed at a 5 mV s-1 scan rate, and chronoamperometry measurements were recorded during 15 s with a time window of 20 ms per data point. Either in single-band mode (only one electrode connected) or in generator-collector experiments (each electrode connected at a different potential), reduction of FeIII was achieved by applying a potential step to 0 V vs SCE to the single band or to the generator. In the double-band mode, the generated FeII was collected and oxidized by the collector poised at +0.8 V vs SCE. Simulations. All simulations were performed in the conformal spaces of the systems best fitted to single-band and doubleband assemblies (see text and Appendix) through algorithms adapted from previously described ones.8,10 Programs were written in C language and ran on Pentium III PCs (Dell). Theory Preliminary Considerations. In the present work, microbands are coated by a polymer-film in which the electroactive species are encapsulated. The film of thickness h is supposed uniform over the electrode assembly and of infinite surface in comparison to the electrode area (Figure 1). Leakage of each redox species toward the solution is considered negligible during the time duration of an experiment. Under these conditions, the diffusion process occurs in a semiinfinite space with a geometrical limitation imposed by the film thickness h. Diffusion is assumed here in a general sense, i.e., possibly involving coupling between physical diffusion-migration of free redox species within polymeric channels and diffusion-migration via electron hopping between sites bounded to the polymer structure.1,16 Microband electrodes are of width w and length l. In a double-band assembly, they are supposed of identical dimensions, parallel, and separated by an insulating gap of width g. The two adjacent microbands are operated at different potentials so that species generated at one electrode, called the generator,

8696 J. Phys. Chem. B, Vol. 105, No. 37, 2001

Arkoub et al.

are collected and recycled at the nearby electrode termed the collector (Figure 1B). For large film thicknesses or sufficiently short times so that the diffusion layer does not approach the film external boundary, the assembly experiences necessarily the different diffusional regimes described previously for a semiinfinite solution.10,17 The diffusion layer thus expands with time variations only controlled by the microbandwidth w relative to the diffusion length (Dt)1/2, where t is the duration of the experiment and D the diffusion coefficient of the electroactive species at initial concentration c. Thus, when p ) (w/2)(Dt)-1/2 . 1, the diffusion layer is a thin film adjacent to the electrode surface whose thickness is of negligible value versus the microband and insulating gap widths. The diffusion is mostly planar, and the current ib at a single microband or to the current ig at the generator (Figure 1C) are given by18

ib ) ig ) -nFlDc[1 + 2π-1/2p]

(1)

whereas the current ic at the collector is necessarily null. Note that everywhere in the following ib means the current at a single band (Figure 1A), whereas ig and ic represent the generator and collector currents when the assembly is operated in collectorgenerator mode (Figure 1B). Conversely, when p , 1, the diffusion layer is extremely large vis a vis w and a hemicylindrical diffusion layer centered on the generator axis may develop provided that the film thickness is sufficiently large. The diffusion at a single-band electrode reaches then a quasi steady-state regime.18,19 For generator-collector assemblies, different situations are encountered when p , 1 depending on the respective size of the gap and the diffusion length, viz., depending on the value of the parameter p ) (g/2)(Dt)-1/2 ) (g/w)p.10 When p . 1, the diffusion length is too small for the generated species to reach the collector. Thus, the collector current remains zero and the system behaves as a single band depending only on p values. On the contrary, when p , 1, the diffusion length is much larger than the gap width so that a complete diffusional interaction (i.e., total cross-talk) occurs between the two microbands. All species generated at the first electrode are converted back at the collector into their original oxidation state and diffusionally recycled to the generator (Figure 1B). The currents at the generator and collector become equal and regulated by this cross-talk.10 All situations described above may occur only when h . (Dt)1/2 and h . g. However, when the polymer thickness imposes a physical boundary to the diffusion process, i.e., when h < (Dt)1/2, other situations are elicited both for single-band and double-band assemblies. They are similar in principle to those already reported for microband electrodes operating in channels20,21 although they differ in detail because of the local microfluidics20 or the geometry of the electrode21 considered in these previous studies. Indeed, even if we emphasize here the experimental situation where the limitations of the space available to diffusion are imposed by a thin redox polymer film, they strictly apply to any other experimental situation imposing similar geometrical constraints, e.g., a micrometric channel, provided that the overall hydrodynamics may be neglected during the time scale of an electrochemical experiment. A quantitative evaluation of the diffusion processes occurring at film-coated microbands can be achieved by numerical simulation. The simulations were performed in the conformal space of the device (Figure 2) to take advantage of the great simplification of the problem at hand as established previously for double-band8,10 and triple-band electrodes.22 For single band,

Figure 2. Schematic representation of coated-microband electrodes and of diffusional fluxes operating at their surface. (A) Single microband in real space (left) and in its equivalent in the conformal space used for simulations (right). (B) Same as in A for a double microband assembly. The conformal spaces shown here are based on eqs I and II (A) or III and IV (B; see Appendix). Generator, solid black; collector, hatched shade; insulating materials (glass, Mylar, amd bulk solution), solid gray; space available to diffusion, white.

the conformal space is defined such as the electrode occupies one side of a box (Figure 2A). In the conformal space, the flux lines are almost parallel and the current density becomes almost uniformly distributed over the electrode surface although it is infinite on the edges of the electrode in the real space. No tight simulation grids23 are then required, and the computation is very fast and very precise.8,10 For a double-band assembly, electrodes face each other so that flux and isoconcentration lines are less curved than in the real space (compare Figure 2B). The main difficulty is that in both conformal spaces the film-solution interface is no longer a straight line but rather a parabola-shaped curve (Figure 2A,B). The appropriate boundary conditions are reported in the Appendix as well as the mathematical definitions of the conformal spaces used in all simulations (eqs I-IV). Effect of Film Thickness on Single Microband Current. The simulated electrochemical responses evidence three diffusion regimes depending on the diffusion length (Dt)1/2, the microbandwidth w, and the polymer thickness h. In Figure 3A, the dimensionless current function Ψb ) ib/[-nFlDc] was plotted as a function of p for different dimensionless thicknesses δ ) 2h/w. Three situations are thus clearly identified. On one hand, when p is high enough so that the diffusion length is much lower than h (i.e., δp ) h/(Dt)1/2 . 1), Ψb is a linear function of p with a slope of 2π-1/2. This corresponds actually to semiinfinite planar diffusion because the diffusion layer width is much smaller than that of the generator and no limitation is imposed by the film thickness (Figure 1C). Equation 1 thus still applies and rewrites as

Ψb ) ib/[-nFlDc] ) 1 + 2π-1/2p

(2)

Accordingly, this relation fixes the limit of the highest Ψb value that can be obtained at any p value. On the other hand, when p reaches very low values, i.e., when δp , 1, Ψb varies again linearly with p as it is better seen in Figure 3B in the log-log plot (set of parallel lines with unity slopes). Under these conditions, the flux lines stretch away along the polymer layer being parallel to the insulating plane of the assembly (Figure 1D). With exception of the local area above the electrodes,

Diffusion at Double Microband Electrodes


Figure 4. Generator-collector mode. (A) Dimensionless current Ψg ) ig/[-nFlDc] at the generator versus p ) (w/2)(Dt)-1/2 for R ) g/w ) 0.5. (B) Dimensionless current Ψc ) ic/[nFlDc] at the collector versus p for R ) 0.5. (C) Collection efficiency fc ) -ic/ig ) Ψc/Ψg versus p calculated from data in A and B. (D) 1/fa ) ib/ig ) Ψb/Ψg versus fc also deduced from data in A and B and data in A of Figure 3, the dashed lines represent the theoretical limits achieved when fc approaches 0 (as deduced from eq 4) or 1 (eq 5), respectively. For all curves, from right to left, δ ) 0.5, 1, 2, 4, and 7.

Figure 3. Single-band mode. (A) Dimensionless single-band current Ψb ) ib/[-nFlDc] versus p for different dimensionless thicknesses δ ) 2h/w. (B) log(Ψb) versus log(p) ) log[(w/2)(Dt)-1/2]. From right to left, δ ) 0.5, 0.7, 1, 1.5, 2, 3, 4, 5, 7, 10, and 100. The upper and lower dashed lines correspond to the limits of Ψb as calculated from eqs 2 (i.e., δ f ∞) or 3 (only shown for δ ) 0.5), respectively.

diffusion becomes mostly planar and the diffusion layer progresses laterally in the polymer layer as was reported previously.24 The current is then proportional to 2hl, which is twice the polymer cross section surface area over the microband length. Ψb is given by the Cottrell equation:

Ψb ) ib/[-nFlDc] ) 2π-1/2δp

(3)

Finally, when δp ≈ 1, a third behavior is observed which features the transition between the two previous diffusional regimes. This transition occurs at higher p when δ decreases (Figure 3A,B). In other words, the effect of δ on diffusion appears sooner at the thinner film. Effect of Film Thickness on Generator-Collector Microband Currents. The dimensionless current functions Ψg ) ig/[-nFlDc] and Ψc ) ic/[nFlDc] were defined for the generator and collector microbands respectively and calculated as a function of p for different dimensionless thicknesses δ. This is shown in Figure 4 for R ) g/w ) 0.5. The two limiting cases described above for a coated single microband have their immediate counterparts for a generator-collector assembly. On one hand, when p and δ are large so that δ p . 1 (note that p ) Rp), the diffusion length is so short that no diffusional crosstalk may occur between the bands. The current at the generator is equal to that for a single-band electrode (eq 2), and the collector current is necessarily null. On the other hand, the second limit is reached at long times when δp , 1. Figure 4A shows that Ψg varies then linearly versus p: the thicker δ is,

the higher the slope is. At the same time, Ψc tends toward a constant plateau depending on the thickness δ as shown in Figure 4B. This evidences that when Rp , 1, the collector current reaches a steady state, whereas the generator current still decreases. Eventually, when δp f 0, both currents reach the same limit Ψ∞,δ which depends only on δ (Figure 1A,B). The dimensionless current functions Ψg and Ψc then follow eqs 4a and 4b, respectively:

Ψg ) ig/[-nFlDc] ) Ψ∞,δ + π-1/2δp

(4a)

Ψc ) ic/[nFlDc] ) Ψ∞,δ

(4b)

where Ψ∞,δ is the common dimensionless limit of Ψg and Ψc when t f ∞ (i.e., p f 0) at a given δ value. Ψg has two contributions (eq 4a): the first term Ψ∞,δ results from the feedback between the generator and the collector. As soon as the diffusion length (Dt)1/2 is much larger than the gap width g (i.e., Rp < 1) and the film thickness h (i.e., δp < 1), complete cross-talk occurs between the two electrodes within the restricted space. This component of the diffusion process is then time independent, and the corresponding current reaches its steadystate limit Ψ∞,δ either for the generator or the collector (see below and Figure 4C,D). The second term in eq 4a, viz., π-1/2δp, reflects the lateral diffusion within the film which still proceeds from the generator side opposed to the collector. As for the single-band case, this term follows a Cottrellian-like law but now for an apparent surface area equal to hl because lateral diffusion may proceed only on one side of the generator. This peculiar situation occurs because the finite size of the film over the generator prevents most diffusional lines exiting from the generator side opposed to the collector from finding their way to the collector through long diffusional paths as is observed when the double band placed in an infinite medium.10 As a consequence, the steady state at the collector is reached must faster when δ is smaller (compare Figure 4B).


Arkoub et al.

In the case of generator-collector assemblies placed in a semiinfinite medium, the variations of the inverse of the amplification factor, 1/fa ) ib/ig ) Ψb/Ψg, versus the collection efficiency, fc ) -ic/ig ) Ψc/Ψg, characterize the electrochemical behavior of the device at any time.10 Numerical simulations allow the construction of similar plots for the case of restricted diffusion which is investigated here. These are reported in Figure 4D for different dimensionless thicknesses δ. Two limiting cases, which are identical to those already established for a generator-collector assembly in a semiinfinite solution are observed according to p, whatever the value of the film thickness. Indeed, at times where the diffusion layer does not extend across the gap (i.e., Rp . 1), the current at the generator is equal to that of the single microband giving an amplification factor of 1 and a collection efficiency of 0. In the other case (i.e., Rp , 1), steady-state diffusion is reached: 1/fa varies linearly with fc and a limiting slope of -2 is obtained. This limiting behavior follows from the expressions of Ψb (eq 3) and of Ψg and Ψc, (eqs 4a,b) at small p values. Indeed, elimination of p from this set of equations leads to the following relation:

1/fa ) 2[1 - fc]

(5)

The effect of δ on fa and fc is described by the bundle of 1/fa vs fc curves. The two above asymptotes are reached the faster and the smaller is δ. The thinner the film is, the latter the curves leave the 1/fa ) 1 limit and the sooner they reach the limit in eq 5. For small δ, once the diffusion length exceeds the gap (i.e., Rp < 1), the steady-state diffusion is almost reached at the collector and eqs 4 and 5 apply. For thicker films, the curves become increasingly close to those calculated for a semiinfinite solution because the film thickness is less and less restrictive on the diffusional transport. Effect of Film Thickness on Limiting Current Ψ∞,δ at Double-Band Assembly. To predict semianalytically the steadystate diffusion behavior at the double-band electrode, a second transformation of coordinates must be introduced (see the Appendix, eq XII). In this second conformal map,10 generator and collector electrodes face each other as in a thin layer cell, fully occupying two opposed sides of a closed box (Figure 5C). For an infinite value of δ, the system behaves as a true thin layer cell whose limiting current is given by10

Ψ∞,∞ ) w*/g*

(6)

where w* and g* are the respective width of the electrodes and of the gap in the conformal space (Figure 5D). This transformation is also appropriate to estimate Ψ∞,δ semianalytically. However, the presence of the coated film implies an additional boundary condition, which is materialized by the water-droplike insulating zone located inside the box (Figure 5C). On the basis of this conformal map, the flux lines between the generator and the collector can develop only below the point H (Figure 5C). Indeed, the body of the conformal film blocks part of the diffusion to the collector so that most of the species produced at the generator above the point H may only escape to infinity (Figure 5C). Diffusion proceeds then as if a part of the collector was hidden to the generator as schematized in Figure 5D. This conformal representation of the problem at hand shows the extreme importance of the location of the point H. Below this point, almost straight parallel flux lines develop between the bands, but this may then occur only over a limited width w*′ of the electrodes. On the basis of this conformal space analysis, the limiting current Ψ∞,δ expected for a coated film of thickness

Figure 5. (A-C) Geometries of the space available to diffusion for three different film thicknesses (H: δ ) 0.5. H′: δ ) 1.8. H′′: δ ) 5.) for a double-band assembly in different spaces: (A) in real space, (B) in the conformal space (eqs III and IV the in Appendix) used for the simulation of the time dependent values of Ψg and Ψc, and (C) in the “thin layer cell” conformal space fitted for the analytical approximation of Ψ∞,δ (see text, eq 8, and eq XII in the Appendix). (D) Graphical representation of the space shown in C within the approximation used in eqs 7 and 8. In A-D: generator, solid black; collector, hatched shade; insulating materials (glass, Mylar, and bulk solution), solid gray; space available to diffusion, white. (E) Comparison between the normalized currents Ψ∞,δ/Ψ∞,∞ predicted either from digital simulations performed in the conformal space shown in B (solid symbols), or based on eqs 8 (solid line) or 9 (dashed line; see text).

δ is necessarily smaller than Ψ∞,∞ (eq 6) and is proportional to w*′ at least in a first approximation:

(Ψ∞,δ)approx ) w*′/g* ) Ψ∞,∞w*′/w*

(7)

Nevertheless, this relation affords only a crude estimation of Ψ∞,δ. Indeed, it does not take into account the shape of the film in the conformal space and especially its curvature above point H (compare Figure 5 parts C and D). Actually, because of this smooth curvature, some species generated slightly above point H can still bypass the boundary imposed by the film and reach the so-hidden collector as shown in Figure 5C. In



agreement, the limiting current Ψ∞,δ obtained by numerical simulation is found slightly higher than predicted by the approximation in eq 7 which is rewritten as in eq 8 (see the Appendix, eq XIII):

( ) Ψ∞,δ Ψ∞,∞

w*′ ) ) w* approx

∫0δ[(z2 + R2)(z2 + (2 + R)2)]-1/2 dz ∫R2+R[(z2 - R2)((2 + R)2 - z2)]-1/2 dz (8)

This is confirmed in Figure 5E which reports the current ratio Ψ∞,δ/Ψ∞,∞ (where both currents were determined through simulations in the conformal space shown in Figure 5B) as a function of δ and compares it to the analytical result of the above approximation (eq 8). However, Figure 5E shows that in a logarithmic representation the shift between the simulated values and their analytical approximation in eq 8 remains almost constant for any δ values. This evidences that an excellent adequacy is obtained between the simulated results and those predicted by the simple expression in eqs 7 and 8, provided that these later are evaluated by considering a value of δ 1.41 times larger than its actual value:

Ψ∞,δ ≈ Ψ∞,∞

∫01.41δ[(z2 + R2)(z2 + (2 + R)2)]-1/2 dz ∫R2+R[(z2 - R2)((2 + R)2 - z2)]-1/2 dz

(9)

The excellent agreement between eq 9 (dashed curve) and the simulated results (solid symbols) is evidenced in Figure 5E. Evaluation of h, D, and c. The above theory establishes that the current-time behavior of the single band, or of the generator-collector assembly, reflects several limits and trends that are characteristic of the fact that diffusion occurs within a limited space of a specific geometry. This observation is particularly important because it shows that the geometry and composition of a given confined space may be investigated through transient electrochemistry provided that the time-scale of the electrochemical method allows diffusion to “explore” the limits of the domain under investigation. In this respect, it is worth to recall that this important property makes transient electrochemistry a new kind of topological tool or microtome.25,26 Owing to the further outcomes of the research presented here, it is interesting to show that h, D, and c can be virtually deduced from the chronoamperometric currents at a microband placed under a polymer coating, even when these parameters are unknown because the loaded polymer film has experienced alterations because of its placement in a given environment. Indeed, this affords the product D1/2c (eq 2, with ib proportional to D1/2ct-1/2) from the Cottrellian regime observed at very short times (Figure 1C). The behavior at a long time where lateral diffusion leads again to a Cottrellian regime (Figure 1D)21 gives hD1/2c (eq 3, with ib proportional to hD1/2ct-1/2), from which h may be determined once D1/2c is known. Similarly, the transition time between the two regimes affords the value of Dc, from which D and c can be determined whenever D1/2c is known. It is thus seen that h, D, and c can be virtually determined. We use “virtually” because this supposes that each specific limit can be identified with the required precision in a given experimental situation. However, it is obvious that the observation with the required precision of the Cottrellian regime at short times (Figure 1C) may be extremely difficult because of the Faradaic current corruption by ohmic drop and capacitive currents.

TABLE 1: Values of h, D, and c Determined from Quantitative Analysis of Chronoamperometric Experiments Shown in Figure 8A-C experiment

A

B

C

hest (µm)a h (µm)b D (cm2 s-1)b,c c (mol dm-3)b,c

14 ( 2 12.5 1.9 10-7 0.49

6(1 5.0 1.8 10-7 0.36

2(1 2.5 1.7 10-7 0.37

a

hest: film thicknesses estimated based on the deposited mass of Nafion. b As determined from the fitting procedure of experimental double-band currents (see text), based on n ) 1, F ) 96500 C eq-1, w ) 5 × 10-4 cm, R ) g/w ) 0.5, and l ) 0.35 cm. Data monitored at t < 0.5 s were excluded from the fitting procedure (see text). c Note that because of slow leakage and internal reorganizations, the values of D and c depend slightly on the film thickness and on the time duration during which the film has been exposed to the FeIII-free bulk solution prior to the electrochemical experiments.

In this respect, the use of generator-collector assemblies is more convenient as demonstrated previously by so-called timeof-flight experiments.27 For a given assembly, the collection efficiency (fc) depends mostly on D (Figure 4C), whereas the exact curvature of the relationship between the reciprocal amplification (1/fa) and the collection efficiency depends mostly on h (Figure 4D). Similarly the limits of the generator and collector currents at long times depend on h, D, and c, i.e., afford c when D and h are known based on the above. Therefore, generator-collector assemblies do not require any precise recording of transient Cottrellian regimes at very short times to afford the values of the same set of parameters. Experimental Illustration In the following, we wish to show how these strategies may be adequately used in real experimental situations. For the purpose of this demonstration, we used films of rather defined thicknesses h spanning the range of interest (see hest in Table 1) and controlled loadings of the redox species (c between 0.3 and 0.5 M) in the film. However, we do not input these estimated values in the fitting process, so that the strategy developed hereafter can be generalized to other experimental situations where these parameters are unknown. Furthermore, the excellent agreement between the final results on c and h and their expected values based on the film preparation (Table 1) gives an estimate of the validity of the approach used and in particular demonstrates that the fitting procedures do not converge to an incorrect set of parameters. Effect of Nafion Film Thickness at Coated Single and Double-Band Assemblies. In Figures 6 (voltammetry) and 7 (chronoamperometry) are reported the experimental electrochemical responses measured at single, generator, and collector microbands coated by Nafion films for three different selected thicknesses h (see curves labeled 1-3 in panels A-C of each part). The films were preliminary loaded with FeIII (c between 0.3 and 0.5 M; see the Experimental Section), and the devices were then rinsed and immersed in FeIII-free 0.1M H2SO4 aqueous solution for performing the electrochemical measurements. The voltammetric (Figure 6) and chronoamperometric (Figure 7) curves clearly show the effect of the film thickness onto the measured currents. The thinner the coatings are, the more significant the distortions onto the electrochemical responses are vis a vis a single- or double-band classical behavior, i.e., that observed for a semiinfinite bulk solution of electroactive material. The shape of the cyclic voltammograms obtained at a single band (Figure 6A) is characteristics of planar diffusion in opposition to the steady-state voltammograms usually monitored


Figure 6. Cyclic voltammograms monitored at FeIII-loaded Nafioncoated band assemblies (w ∼ 5 µm, l ∼ 3.5 mm, and g ∼ 2.5 µm) placed in an FeIII-free 0.1 M H2SO4 aqueous solution. (A) Single-band mode. (B and C) Generator-collector mode with (B) generator current ig and (C) collector current ic. Film thicknesses estimated from the mass of Nafion deposited: (1) hest ) 2 ( 1, (2) 6 ( 1, and (3) 14 ( 2 µm. Scan rate 5 mVs-1. In B and C, the generator potential was scanned as indicated, whereas the collector potential was poised at 0.8V vs SCE.

at microelectrodes in solution. Reversible waves are observed from which the apparent standard potential of the FeII/III redox couple is estimated at +0.40 V vs SCE. This value is very close to the one determined without Nafion in an aqueous solution containing FeIII. This result suggests that FeIII and FeII species probably retain in the film their solution chemical structures, i.e., Fe(HSO4)2+ and Fe(H2O)62+, respectively. The voltammograms monitored at a coated generator-collector assembly are given in Figure 6B,C when the collector is held at 0.8V vs SCE. One observes that a steady-state diffusion is achieved at the collector while a wave-shaped voltammogram is still displayed

Arkoub et al.

Figure 7. Chronoamperometric responses for FeIII-loaded Nafioncoated band assemblies under otherwise identical conditions as those mentioned in Figure 6 caption. (A) Single-band mode, (B) generator, and (C) collector currents in collector-generator mode. The singleband (A) or generator (B and C) potentials were stepped from 0.8 to 0 V vs SCE, whereas in (B and C) the collector potential was poised at 0.8V vs SCE.

at the generator. These electrochemical responses thus agree perfectly with the theoretical expectations relative to the possible diffusional regimes which may occur at such devices (compare eq 4). This is even more evident when examining the generator and collector currents in chronoamperometry (Figures 7B,C, curves 1,2). In agreement with the theory, chronoamperometric currents reach their steady-state (collector) or quasi-steady-state (single band and generator) limits at shorter and shorter times the thinner the film is. Fitting of the Electrochemical Responses. The experimental results obtained by chronoamperometry (Figure 7) were compared to those predicted by numerical simulations developed


J. Phys. Chem. B, Vol. 105, No. 37, 2001 8701 for predicting the single-band one observed for the same film (see details in the Appendix). An excellent agreement is observed in Figure 8 between experimental and theoretical currents for the double-band assemblies for the three film thicknesses investigated. Furthermore, the ensuing h values compare satisfactorily well with those estimated based upon the mass of Nafion deposited during the assembly preparation (see Table 1). Similarly, the diffusion coefficient values determined for the ferric ion in Nafion (Table 1) are in good agreement with those reported in the literature, viz., D ) (2 ( 1) × 10-7 cm2 s-1.28 Nevertheless, a slight but systematic discrepancy is always noted for single-band currents at short times. As mentioned above, this may be due to residual ohmic drop and capacitive components which are expected to play a major role when the assembly is operated in the single-band mode compared to the generator-collector case. This may well account for distortions at shorter time scales. Also, even if the same set of parameters (h, c, and D) was used in single- and double-band modes, (i.e., considering identical characteristics of the polymer) it is not necessarily the case experimentally. Indeed, the film properties may be slightly influenced locally by the diffusion-migration of charged species1,16 occurring during one experiment, so that different operations may well lead to different film history. However, this was not a significant problem here because performing the experiments in reverse order did not change appreciably the situation. Finally, a third possible source of bias may result from the differences in the extent of the regions of the film affected by diffusion. Indeed, in the generator-collector mode, the diffusion lines are mainly located between the two electrodes because of their strong diffusional cross-talk (Figure 8). Conversely, in the case of a single-band electrode the diffusion lines extend exclusively laterally toward the solution (compare Figure 1D) although the current densities are comparable (see Figure 8). Thus, the local concentrations of ionic species at the electrodes incorporated within (or expelled from) the film following the redox processes are expected to differ between the two modes. This may probably affect slightly the polymer characteristics.

Figure 8. Comparison between experimental (symbols) and simulated (lines) chronoamperograms for single-band (open circles) or doubleband (generator, solid circles; collector, solid squares) modes. Experimental data are those represented in Figure 7, corrected by residual currents. All currents are shown in absolute values to help the comparison between generator and collector currents. The values of h, D, and c used for simulated currents shown in A-C are reported in Table 1.

along the above theory (Figure 8), both at single-band and double-band assemblies. The experimental currents were corrected by the residual currents, and data at time lower than 0.5 s were not taken into account in the fits because of their corruption by capacitive currents and residual ohmic drop. To fit the curves, one set of parameters was considered as being fixed owing to the known geometry of the assembly (w, l, and g), whereas the other set (h, c, and D) was adjusted without any a priori to reproduce at best the double-band experimental currents. The same set of adjustable parameters (h, c, and D) determined by fitting only the double-band currents was used

Conclusion We have examined here the different diffusional regimes which may control the behavior of thin redox films deposited on a single-band electrode or on a generator-collector assembly. The theoretical limits featuring the different definite diffusional behaviors have been confirmed via simulations, as well as through a semianalytical model for the steady-state limit of the currents in the generator-collector mode. These limits respond differently to the experimentally variable parameters, viz., h, the thickness of the film, c, the concentration of the electroactive material loaded within the film prior to the electrochemical experiments, and D, its diffusion coefficient. This opens therefore an experimental access to the determination of each of these three parameters which characterize the film status, as it has been shown through the experimental illustration presented here. In particular, this shows that whenever swelling/contraction, internal reorganization, and loading of the film are altered because of its exposure to a given chemical environment h, c, and D may be determined by the method examined here. This offers an entry for the design of a new class of artificial noses and artificial tongues, in which any chemical affecting a polymer properties may then be characterized by a set of three parameters instead of a single-point measurement as is usual for operation of such devices. Taking into account that generator-collector devices may also be arranged in such a way that they perform


Arkoub et al.

Boolean algebra (AND or OR gates).7,8 This offers new opportunities and strategies which may help to mimic more closely the actual properties of natural olfactory systems. Work is presently under progress in our laboratory in this direction and will be reported soon elsewhere. Acknowledgment. This work is dedicated to Royce Murray on the occasion of his 65th birthday, in honor of his creativity and humanism in Science. This work has been supported in parts by the CNRS (UMR 8640), the French Ministry of Research (Action Spećifique DGRT N˚ 97.1502), and the Ecole Normale Supe´rieure. Appendix Simulations in the Conformal Space. The conformal space used for the digital simulations of the transport in the polymer is identical to the one developed for semiinfinite solutions.10 Therefore we report here just a brief summary of the principle and the method. We also describe how this method is modified to fit the geometry of the problem. The space variables in the true space are x (parallel to the electrode plane) and y (perpendicular to the electrode plane), whereas they become Γ and θ in the conformal space (see Figure 2). For the singleband and generator-collector assembly, the conformal transformations and transport equations are identical, the only differences being the boundary conditions and scaling factors (vide infra)

x ) (w/2) cos θ cosh Γ

(I)

y ) (w/2) sin θ sinh Γ

(II)

for the single-band electrode and

x ) (g/2) cos θ cosh Γ

(III)

y ) (g/2) sin θ sinh Γ

(IV)

for the double-band electrode. A reversible system A + e- h B is considered, with only A initially present in the film at concentration c. Dimensionless variables are defined as follows: a ) cA/c is the dimensionless concentration, τ ) t/T is the dimensionless time, and T is the maximum duration of the electrochemical experience. The dimensionless parameters p and p are introduced:

p ) (w/2)(DT)-1/2

(VI)

for the double-band electrode. The transport equation derived from the conformal transformation is10,29

)

(VII)

where D* ) p-2[sinh2 Γ + sin2 θ] - 1 for a single-band or D* ) p-2[sinh2 Γ + sin2 θ]-1 for a double-band assembly. In this conformal space, the single-band electrode is defined by Γ ) 0 and 0 < θ < π/2. For the double-band electrode, the gap is defined by Γ ) 0 and 0 < θ < π, whereas the generator is defined by θ ) 0 and Γ < cosh-1(1 + 2/R) and the collector by Γ ) 0 and Γ < cosh-1(1 + 2/R) where R ) g/w. The above partial derivative equation was solved by Hopscotch numerical finite differences.10 However, the simulation

(VIII)

The boundary conditions at the electrodes depend on the electrode as well as on the mode used for the assembly. For chronoamperometry in the single-band mode, a ) 0, on the band surface. In the collector generator mode different potentials were applied to the generator and to the collector. For chronoamperometry, the generator potential was set on the reduction plateau of A species, then a ) 0 at its surface. At the collector, the potential was set on the oxidation plateau of B species, so that a ) 1 at its surface. The current density computation requires the evaluation of gradients versus θ for the double-band electrode or versus Γ for the single-band electrode (see Figure 2A,B). This is performed through a three points parabolic approximation as previously reported.10 The ensuing current is then given by one of the three following integrals, for the single-band electrode, for the generator, and for the collector of the double-band electrode, respectively:

ib ) -2nFlDc ig ) -nFlDc

p ) (g/2)(DT)-1/2

(

∂a ∂a tan θ + tanh Γ ) 0 ∂Γ ∂θ

(V)

for the single-band electrode and

∂2a ∂2a ∂a + ) D* ∂τ ∂θ2 ∂Γ2

grids used here were not regular in the θ direction. A regular grid would difficultly fit the shape of the Nafion/solution interface and therefore would complicate the calculation of the zero-flux conditions valid at this interface. The steps ∆Γ between two mesh points in the Γ direction were then regular, but the steps ∆θ in the θ direction were adjusted so that the Nafion/solution interface passed rigorously through mesh points of the simulation grid. This caused ∆θ to vary slightly around its mean value as a function of the Γ value. This grid combined with the Hopscotch method was used to determine the concentrations at the time τ + ∆τ, once concentrations at time τ were known. To determine the value of a on the boundaries, one must use the proper boundary conditions at each electrode and on each insulated wall. At each lateral insulating wall, a zero flux condition applies; thus, ∂a/∂θ ) 0. For the double band, the same condition applies also at the gap; thus, ∂a/∂Γ ) 0 (see Figure 2). At infinity, the concentration is not affected by diffusion so that a f 1 when Γ f +∞. On the Nafion/solution interface, a zero flux condition applies; thus, ∂a/∂y ) 0 in the real space. In conformal space, this condition becomes

ic ) nFlDc

∂a ∫0π/2 (∂Γ )Γ)0 dΓ

∫0cosh

∫0cosh

-1(1+2/R)

-1(1+2/R)

(∂θ∂a) (∂θ∂a)

θ)0

θ)π

(IX) dθ

dθ

(X) (XI)

Conformal Space Used for the Semianalytical Approximation of the Steady-State Behavior at Double Bands. To evaluate the steady-state current, a different conformal transformation is used. In this case, the generator faces the collector and the space is curved in a way that the system resemble a close box (Figure 5C). To every point (x,y) of the real space is assigned a complex affix z ) x + jy, where j2 ) -1. The transformed affix of this point in the conformal space, Z ) X + jY, is then given by the Schwarz-Christoffel transform:10

Z)K

∫0z[(z2 - u2)(z2 - V2)]-1/2 dz

(XII)

where u ) g/2, V ) w + g/2, and K is an imaginary number serving as a constant scaling factor and allowing a whole space rotation.



For the approximation proposed in Figure 5D, the expression of the dimensionless current for the steady state requires the determination of the value of w*′. This is deduced using the expression above. We evaluate the previous integral between the center of the gap and the point H of the interface. The affix of H is jh, so that

w*′ ) K

∫0jh[(z2 - u2)(z2 - V2)]-1/2 dz ) h K∫0 [(z2 + u2)(z2 + V2)]-1/2 dz

(XIII)

Every affix in the integral is imaginary, which allows us to rewrite it as a real integral. Thus

w*′ ) g*

∫0h[(z2 + u2)(z2 + V2)]-1/2 dz ∫-uu[(z2 - u2)(z2 - V2)]-1/2 dz

(XIV)

which leads to eq 8. Procedure Used To fit Experimental Data. The fitting of chronoamperograms was based on the comparison of the experimental curves i(t) with a series of theoretical λΨ(µp-2) curves simulated for a set of selected values of δ ) 2h/w, where λ and µ are constant scaling factors for a given experiment, being given by

λ ) nFlDc

and µ ) w2/(4D)

(XV)

Therefore, the fitting procedure amounted to determine the set of λ, µ, and δ values that afforded the best agreement between experimental currents and theoretical predictions. Because (w, l, and g) were known for a given geometry of the electrodes and n was known for a given redox couple, it ensued that the fitting procedure afforded the experimental characteristics (viz., h, c, and D) of the film describing a best the experiment at hand:

h ) δw/2, D ) w2/4µ, and c ) 4µλ/(nFlw2)

(XVI)

The correctness of the fits (Figure 8) validates the quality of the model and of the approximations used. In particular, we neglected as explained in the text section, any significant contributions from ohmic drop and capacitive currents. For this reason, the priority in adjusting δ, µ, and λ was given to the currents obtained upon using the device exclusively in the generator-collector mode (see text and Figure 8). The validity of the ensuing set of δ, µ, and λ values was then internally checked by examining the resulting agreement between the predicted single-band current based on these values and the experimental current monitored when the same device was operated in the single-band mode (Figure 8; see below). Starting from this point, we could have proceeded to a global fit based on the three currents. However, for the reasons given in the text, we preferred not to implement further the fitting procedure. Therefore, the results presented in Figure 8, should be considered within this framework, i.e., in which the single-band current serves only as an internal test and not as featuring from the best fitting set considering all three currents. For a given double-band assembly, simulated currents were often larger than the experimental ones (by ca. 5% at most). This occurred because of a bias resulting of geometrical mismatch between the two bands as documented previously.8

Indeed, upon aligning the two metal foils whose cross-sections provide the set of parallel-band electrodes, a slight mismatch may occur, so that a small length at the end of each band may not be facing the other band. The paired band was then separated theoretically into three individual components.8 A first component was the section of the generator not flanked by the collector. It was then treated as a single-band electrode (no feedback: collector current null). The second and main (>95%) component was composed of most of the device in which the generator was flanked by the collector. This component was then treated as a double-band electrode. The third component was the section where the collector was not flanked by the generator. The generator and collector currents for this component were then set to zero because no generation, and therefore no collection, may occur. The whole generator and collector currents were then obtained by summating the currents due to each of these three components, the size of each component being estimated as reported previously.8 Once this correction was taken into account, the agreement was excellent (Figure 8). References and Notes (1) Murray, R. W. Molecular design of electrode surfaces; Wiley & Sons, Inc: New York, 1992; Vol. XXII. (2) Inzelt, G. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1994; Vol. 18, p 89. (3) Geng, G.; Reed, R. A.; Kim, M.-H.; Wooster, T. T.; Oliver, B. N.; Egekeze, J.; Kennedy, R. T.; Jorgenson, J. W.; Parcher, J. F.; Murray, R. W. J. Am. Chem. Soc. 1989, 111, 1614. (4) Albert, K. J.; Lewis, N. S.; Schauer, C. L.; Sotzing, G. A.; Stitzel, S. E.; Vaid, T. P.; Walt, D. R. Chem. ReV. 2000, 100, 2595. (5) White, J.; Kauer, J. S.; Dickinson, T. A.; Walt, D. R. Anal. Chem. 1996, 68, 2191. (6) Kittlesen, G. P.; White, H. S.; Wrighton, M. S. J. Am. Chem. Soc. 1984, 106, 7389. (7) Amatore, C.; Brown, A. R.; Thouin, L.; Warkocz, J. S. C. R. Acad. Sci. Paris 1998, t. 1, 509-515. (8) Amatore, C.; Thouin, L.; Warkocz, J. S. Chem. Eur. J. 1999, 5, 456. (9) Heitner-Wirguin, C. J. Membrane Sci. 1996, 120, 1. (10) Fosset, B.; Amatore, C.; Bartelt, J. E.; Michael, A. C.; Wightman, R. W. Anal. Chem. 1991, 63, 306. (11) Mauritz, K. A.; Hopfinger, A. J. Structural properties of membranes ionomers. In Modern Aspects of Electrochemistry; Bockris, J. O. M., Conway, B. E., White, R. E., Eds.; Plenum Press: New York, 1982; Vol. 14, p 425. (12) Verbrugge, M. W.; Hill, R. F. J. Phys. Chem. 1988, 92, 6778. (13) Shi, M.; Anson, F. C. J. Electroanal. Chem. 1997, 425, 117. (14) Hsu, W. Y.; Gierke, T. D. J. Membrane Sci. 1983, 13, 307. (15) Tatsuma, T.; Ozaki, M.; Oyama, N. J. Electroanal. Chem. 1999, 469, 34. (16) Andrieux, C. P.; Saveánt, J.-M. Catalysis at redox polymer coated electrodes. In Molecular design of electrode surfaces; Murray, R. W., Ed.; Wiley & Sons, Inc: New York, 1992; Vol. XXII, p 207. (17) Amatore, C.; Combellas, C.; Kanoufi, F.; Sella, C.; Thie´bault, A.; Thouin, L. Chem. Eur. J. 2000, 6, 820. (18) Amatore, C. Electrochemistry at ultramicroelectrodes. In Physical Electrochemistry; Rubinstein, I., Ed.; M. Dekker: New York, 1995; pp 131208. (19) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. 1987, 217, 417. (20) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. 1993, 97, 10410. (21) Ferrigno, R.; Brevet, P. F.; Girault, H. H. J. Electroanal. Chem. 1997, 430, 235. (22) Fosset, B.; Amatore, C.; Bartelt, J.; Wightman, R. M. Anal. Chem. 1991, 63, 1403. (23) Shea, T. V.; Bard, A. J. Anal. Chem. 1987, 59, 2101. (24) Rossier, J. S.; Roberts, M. A.; Ferrigno, R.; Girault, H. H. Anal. Chem. 1999, 71, 4294. (25) Amatore, C.; Bouret, Y.; Maisonhaute, E.; Goldsmith, J. I.; Abrunã, H. D. Chem. Eur. J. 2001, 7, 2206. (26) Amatore, C.; Bouret, Y.; Maisonhaute, E.; Goldsmith, J. I.; Abrunã, H. D. Chem. Phys. Chem. Eur. J. 2001, 2, 130. (27) Feldman, B. J.; Feldberg, S. W.; Murray, R. W. J. Phys. Chem. 1987, 91, 6558. (28) Ye, J.; Doblhofer, K. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1271. (29) Amatore, C.; Fosset, B. Anal. Chem. 1996, 68, 4377.

Diffusion at Double Microband Electrodes Operated within a Thin Film

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