10942
J. Phys. Chem. 1995, 99, 10942-10947
Hydrodynamic Voltammetry with Channel Electrodes: Microdisc Electrodes Jonathan Booth, Richard G. Compton,* Jonathan A. Cooper, and Robert A. W. Dryfe Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford University, Oxford OX1 3QZ, U.K. Adrian C. Fisher, Ceri Lyn Davies, and Malcolm K. Walters Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, U.K. Received: February 2, 1995@
Both the strongly implicit procedure and the alternating direction implicit finite difference (ADI)technique are used to simulate mass transport to a microdisc electrode located in a channel flow, taking account of diffusion in three dimensions and the parabolic laminar convective flow over the electrode surface. The effect of the latter on the shape of the disc diffusion layer is qualitatively identified. The flow rate dependence of the transport-limited disc current is predicted and shown to be in quantitative agreement with experiments conducted on the oxidation of ferrocene in acetonitrile solution.
Introduction The many merits of microelectrodes for voltammetric investigations are now widely and for example, the use of fast scan cyclic voltammetry at microdisc electrodes permits nanosecond timescales to be accessed electrochemi... +(+-7 . ... ... call^.^,^ Whilst for the majority of studies microelectrodes are most conveniently employed under diffusion only conditions, \ channel unit in other applications it has proved helpful for the electrode to 2re be located in a flowing stream of electrolyte. This requirement arises naturally in analytical work particularly if electrochemical detection in conjunction with, say, high-performance liquid chromatography {HPLC) or flow injection analysis (FIA) is required5-’ and also in mechanistic photoelectrochemical studies where a flow regime may be required to dissipate heat from Figure 1. Diagrams showing a microdisc electrode positioned in a channel flow cell: (a) practical arrangement used; (b) schematic the interfacial region.*s9 It is therefore of importance to be able representation showing the flow cell and electrode dimensions. Typical to quantitatively predict the effects of a convective flow on the approximate values of the latter relating to the experimental work magnitude of the transport-controlled currents flowing at reported in this paper are d = 0.6 cm, 2h = 0.05 cm, and re = 2.7 x microelectrodes, and the case of a microband electrode located io-’ cm. in a channel flow has been discussed The present paper seeks to establish computational approaches by which the Theory current at a microdisc electrode positioned in a channel flow-as In this section we present a theoretical model for the steadydepicted in Figure 1-may be calculated and to compare the state transport-limited cment at a microdisc electrode of radius resulting numerical predictions with experiment. This is a re positioned in a channel flow. We suppose that the electrode particularly challenging simulation problem since, whereas the is held at a potential such that the electrode reaction corresponding microband problem simply requires the solution in two-dimensions,the case of a microdisc is necessarily a threeAfe--B dimensionalproblem and this dictates the use of computationally efficient approaches. In particular we employ Fist the strongly is driven to completion at its surface and that sufficient implicit procedure (SIP), which we have shown elsewhere to supporting electrolyte is present for migration effects to be be ideally suited to the ready solution of electrochemical mass neglected. transport problems at a level which is quantitative in comparison Under these conditions the steady-state convective-diffusion with experiment,I2 and second, as an independent check, the equation describing the spatial distribution of A is altemating direction implicit finite difference (ADI)techr~ique.~~-~~ We note that an earlier attempt by Tait et al. to solve the problem of interest employed a monstrously CPU-expensive explicit finite difference method which gave answers in rather disappointing agreement with e~periment.~ In contrast the SIP where x, y . and z are defined in Figure 2, a = [A], D is the and AD1 approaches are shown to be both quantitative and diffusion coefficient of A, and v, is the solution velocity profile efficient and should be readily extended to complex electrode in the x direction. Under laminar flow conditions the latter is reactions involving coupled homogeneous kinetics. parabolic, provided that an adequately long lead in section ispresent for the flow to become fully developed.10si6Quanti‘Abstract published in Advance ACS Abstracts, June 15, 1995. tatively, 0022-3654/95/2099-10942$09.00/0
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 27, 1995 10943
Hydrodynamic Voltammetry with Channel Electrodes
respectively, so that
+ +
j = 0, 1 , 2 , ..., N2
y =j A y
Y 4
where Ay = 2h/Nj
i = O , 1 , 2 ,..., Ni ,..., ((1 2y)Ni) = N1
z=ihz
+
2
L
+
k = 0, 1,2, ..., 2aNk, ..., 2(a 1)Nk, ..., where Ax = rJNk {2(a 1 B)Nk = N 3 )
x = kAx
where Az = rJNi
where Nj is the number of boxes that would be required to cover the entire cell depth (2h), if the simulation were to extend beyond a distance NZAy above the disc. With the appropriate choice of N1, N2, and N3, the boundary conditions on the extremities of the simulation grid mirror those in the table above. In both SIP and AD1 simulations, we use the symbol aj,k.i to denote the concentration of A at the coordinate (y, x, z). In the former case we consider eq 1 under steady-state conditions as follows:
X
O=D
I
(b)
aj,k+ I ,i
- 2aj,k.i + aj,k(W2
I ,i
+ 'x aj,k+ 2
2Ni(Az)
1.i
- aj,kAx
1 .i
(3)
This may be rearranged to
= Ak(aj.k+l.i - 2aj,k,i + aj,k-l,i) +
*
2Nk(Ax)
Aj(aj+l,k,i
Figure 2. (a) Three-dimensional finite-difference grid. (b) Grid in the xz plane of the disc electrode. Note that there are 2Ni "boxes" across the electrode and 2Nk boxes along its length. The part of the channel considered, as shown, was subdivided into N1 boxes breadthwise and N2 boxes in height, with 2aNk boxes upstream of the electrode and 2pNk downstream, such that 2(a + 1 p ) N k = N3.
- &j,k,i + 'j-1,k.i) + Ai(aj,k,i+l - 2aj,k,i + a j , k , i - l ) $aj,k+l,i 1 - aj.k-l,i) (4)
or
+
where vo is the velocity at the center of the channel and 2h is the channel depth (height). The boundary conditions pertinent to the problem are as follows:
y=o
disc surface flow cell wall
aa - 0 --
z = f0.5d
flow cell wall
aa = 0
y = 2h
all x, z
all y , z
x-C-00
all y . z
X-+=
y=o
where
Aj = {2h3dAxD/6Vk2h- j A ~ ) j ( A y ) ~ }
(6)
a=O
ar
az
aa - 0 --
ax
In order to solve eq 1 , we represent the flow cell shown in Figure 1 by means of the three-dimensional simulation grid in Figure 2. Note that since the cell is symmetrical about the xy plane at z = 0 (cell center), the computations only need be carried out in half of the flow cell and this accounts for the extent of the grid shown in Figure 2. The grid covers the x-y-z space and has increments hx, Ay, and Az in the x, y , and z directions,
and Vf = 4hdvd3 is the volume flow rate. Equation 5 is in a form directly amenable to solution using the SIP implemented as the NAG FORTRAN Library subroutine D02EBF.I2 The resulting values of 0j.k.i are used to predict the electrode current from the following expression
for all i and k such that
+
[((2a l)Nk - k)AxI2
+ [iAzI25 [rJ2
(7)
and for different flow rates and ceWelectrode geometries.
10944 J. Phys. Chem., Vol. 99, No. 27, 1995
Booth et al.
For AD1 simulations the same grid as employed in the SIP method is adopted but eq 1 is solved in time dependent form: t+ At aj,k,i-
taj,k.i
= ($){t+Ata j.k+l,i
At
-
t+At aj,k,i
AZ
+
= DAt Az2
Vxllt 1; = -
2Ax
leads to
+
- 2t+Ataj,k,i “Atajki-l}- (k) 2Ax x
(q{t+Ataj,k,i+l
Az2
9 .
Application of the AD1 a p p r o a ~ h l ~enables -~~ the above expression to be solved in a fully implicit manner. Rearrangement of eq 8 and substitution of the following ;2” = DAt -
(9)
Ax2
t+At
+
x t+At
aj,k+l,i - 2t+Ataj,k,i“Ataj,k-l,i) + izy(*Ataj+l,k,i - 2f+Ataj,k,it+Ataj-l,k,i}
- taj,k,i= A { Az(t+Ataj,k,i+
1
+ + - 2f+Ataj,k,i + t+Ataj,k,i-l}a t+At aj{ aj,k+l,i
-
t+At aj,k-l,i}
(13)
The AD1 simulation then proceeds by application of the boundary conditions as above for the SIP method and with the additional condition t =0
all x, y , z
a = abulk
Equation 13 may then be solved at any particular time (t
+
r 0
ABOVE 0.90000 0.81000 0.72000 0.63000
-
-
0.99000 0.99000 0.90000 0.81000 0.72000 0.63000
0.54000 0.45000 - 0.54000 0.36000 - 0.45000 0.27000 0.18000 0.09000 BELOW
0
ABOVE 0.90000 0.81000 0.72000 0.63000
- 0.36000
-
-
0.27000 0.18000 0.09000
0.99000
- 0.99000 - 0.90000 - 0.81000 -
0.72000
0.54000 - 0.63000 0.45000 - 0.54000 0.36000 - 0.45000
1$11
%
0.27000 0.18000 0.09000 BELOW
-
0.36000 0.27000 0.18000 0.09000
Figure 3. Concentration profiles of A (normalized to the bulk concentration, ahlk) in the xz plane for electrolysis at a microdisc electrode of radius 27 pm located in a channel flow cell of dimensions 2h = 0.0528 cm and d = 0.6 cm for flow rates of (a, top) 0.0075 cm3 s-’ and (b, bottom) 0.15 cm3 s-l.
J. Phys. Chem., Vol. 99, No. 27, 1995 10945
Hydrodynamic Voltammetry with Channel Electrodes
ABOVE 0.90000 0.81000 0.72000
0.99000
- 0.99000 - 0.90000 - 0.81000
0.63000 0.54000 0.45000 0.36000 0.27000 0.18000 0.09000 BELOW
0
E
0.72000
0.63000 0.54000 0.45000 0.36000
- 0.27000 - 0.18000 0.09000
ABOVE 0.90000 0.81000 0.72000
0.99000
- 0.99000 - 0.90000 - 0.81000 0.63000 - 0.72000 0.54000 - 0.63000 0.45000 - 0.54000 0.36000 - 0.45000 0.27000 0.18000 0.09000 BELOW
- 0.36000 - 0.27000 -
0.18000 0.09000
Figure 4. Concentration profiles of A (normalized to the bulk concentration, abulk) in the xy plane for electrolysis at a microdisc electrode of radius 27 pm located in a channel flow cell of dimensions 2h = 0.0528 cm and d = 0.6 cm for flow rates of (a, top) 0.0075 cm3 s-l and (b, bottom) 0.15 cm3 s-l. Note that the y direction has been amplified by a factor of 3.5 in comparison with the x direction for greater clarity.
+
At/3) the diffusion and convective term in the x direction was solved implicitly asfollows using the Thomas algorithm, whilst the remaining terms are known from either the initial boundary conditions or the previous calculation:
nht) in three steps. In the first step (t
"j,k,i
+ 'z{taj,k,i+l
Ay{taj+l,k,i
- 2taj,k,i + ''j,k,i-l}
+ t+Atl3 - 2"j,k,i + "j-l,k,i} = -(ax - A;>{ 'j,k+l,i} + (2ax+ 1){t+At13aj,k,i} - (lX - a;){ t+Atl3 aj,k-l,i}
In the second time interval (t
+ 2At/3)
the y direction
components were solved implicitly using z *At13
t+Atl3 'j,k,i
-
+
2t+Atl3 'j,k,i t+At13aj,k,i-l} 'j,k,i+l c t+Atl3 t+Atl3 'j( 'j,k+l,i 'j.k-1,i) x *At13 2t+Atl3 aj,k,i t+At13aj,k-l,i} = { 'j,k+l,i -
+a
+ a - P { * ~ ~ ~ ' ~ u ~++( ~ 2, ~+,~1~ }
+
){t+2At13aj,k,i}
-
y t+2Atl3
a
'j-l,k,i}
and for the final time step the z direction components were again solved using
10946 J. Phys. Chem., Vol. 99, No. 27, 1995 t+2Atl3 aj,k,i
+
'
+
-
Y { ~ + ~ ~,k,i~ ' 2t+2At13aj,k,i ~ ~ ~ +
t+2Atl3 x t+2Atl3
c t+2Atl3
1 ,k,i}
- 'j (
{ aj,k+l,i -'z{t+Ataj,k,i+l}
-
t+2Atl3
aj,k+ 1 ,i + t+2Att3
+ (2Az +
l){t+Ataj,k,i}
'j.k-1.i)
Booth et al.
'
aj,k- 1,i} = z { t+At 'j,k,i-*}
-
The simulation proceeds in this manner until a steady-state current is established at the working electrode. The final values of Uj,k,i calculated may then be used to evaluate the steady-state current flowing at the electrode for any particular flow rate using expression 6.
0 1
Theoretical Results and Discussion The behavior of microdisc electrodes of varying radii, re, located in a channel cell of approximate dimensions 2h = 0.05 cm and d = 0.6 cm was examined first for a range of flow rates for a species, A, with D = 2.3 x cm2 s-', corresponding to ferrocene in acetonitrile solution.16 Typical computations were made with the following parameters: N1 = 30, N2 = 100, IV3 = 90, Nj = 1O00, a = 1.2, = 0.8, and y = 0.5. For both SIP and AD1 calculations, increasing the grid size beyond these values produced no significant change (< 1%) in the calculated currents. For the AD1 calculations time steps between 1/100th and 1/1000th of a second were employed depending on the volume flow rate and the electrode size. Comparison of the numerical results obtained using the AD1 and SIP methods showed them to be in good agreement. The results are best considered in terms of the concentration profiles in the three separate planes, xz, xy, and yz. Figure 3 shows the concentration depletion in the xz plane at y = 0.53 pm (J = 1 ) resulting from electrolysis at an electrode of radius 27 pm at two different flow rates: 0.0075 and 0.15 cm3 s-l. The increasing depletion of electroactive material toward the edges of the flow cell ( z f0.5d) in a downstream direction over the electrode is evident. Notice that the size of this depletion is greatest at the lower flow rate and that the concentrations relax slowly back toward their bulk values downstream of the disc. Figure 4 shows the xy plane at z = 0 corresponding to the center of the microdisc as computed for the same set of parameters as used for Figure 3. It can be seen that the effect of the convective flow is to distort the normally symmetrical hemispherically shaped diffusion layer which would arise under diffusion only conditions, resulting in compression of the diffusion layer about the upstream edge of the electrode and expansion of the diffusion layer downstream of the electrode (note the relative increase of the scale in the y-direction as compared to the x direction in Figure 4). The average depth of the diffusion layer is seen to be rather sensitive to solution flow rate, being much more relatively compressed at the faster flow. Last, Figure 5 shows the yz plane at x = 9.2 pm, corresponding to a cross section taken centrally across the electrode and again computed using the same parameters as employed to generate Figure 3. The sensitivity of the diffusion layer depth to the flow rate is again evident.
-
Experimental Section Platinum microdisc electrodes of radii 12 and 27 pm obtained from Bioanalytical Systems (West Lafayette, IN) were polished using a methanovethano1mixture soaked onto a lens cleaning tissue before use. They were assembled into a channel flow cell unit (Figure 1 ) by first mounting them in a PTFE cover plate in which a hole of the correct diameter had been drilled. The microdisc was then sealed into the cover plate using low-
118
ABOVE
- 0.99000 - 0.90000 - 0.81000 - 0.72000 - 0.63000 - 0.54000 - 0.45000 - 0.36000 - 0.27000 - 0.18000
BELOW
0.09000
0
?:r
0.99OOO
0.90000 0.81000 0.72000 0.63000 0.54000 0.45000 0.36000 0.27000 0.18000 0.09000
ABOVE 0.99OOC O.~OOOO - 0.9900~ 0.81000 - O.~OOOC 0.72000 0.8100C 0.63000 - 0.7200C 0.54000 - 0.6300C 0.45000 - 0.5400C 0.36000 - 0.4500C 0.27000 - 0.3600C 0.18000 - 0.2700C 0.09000 - 0.1800C BELOW 0.09OOC
-
Figure 5. Concentration profiles of A (normalized to the bulk concentration, aI-,"lk) in the yz plane for electrolysis at a microdisc electrode of radius 27 pm located in a channel flow cell of dimensions 2h = 0.0528 cm and d = 0.6 cm for flow rates of (a, top) 0.0075 cm3 s-' and (b, bottom) 0.15 cm3 s-I. Note that the y direction has been amplified by a factor of 3.5 in comparison with the z direction for greater clarity.
melting wax, and the cover plate was cemented onto a silica channel unit of conventional design.18-19The latter was 30 mm in length and had approximate cross sectional dimensions of 0.4 mm x 6.0 mm. Precise values for the cell dimensions were found using a traveling microscope, and the cell depth (2h) was obtained from the gradient of the Levich plot (of current against cube root of flow rate) measured with a conventionally sized electrode of approximate dimensions 4 mm x 4 mm located downstream of the microelectrode and mounted on the same cover plate. The assembled cell was incorporated into a flow system so as to provide a means for the delivery of fresh, deoxygenated solution at flow rates in the range to 0.5 cm3 s-l. A saturated calomel reference electrode (SCE)was located in the system upstream of the flow cell, and a platinum gauze counter electrode was located on the downstream side. The solvent used throughout was dried20 acetonitrile (Fisons, dried, distilled), and tetrabutylammonium perchlorate (TBAP, Kodak,puns) served as the background electrolyte. Ferrocene
J. Phys. Chem., Vol. 99, No. 27, 1995 10947
Hydrodynamic Voltammetry with Channel Electrodes
Llal
0.10
0 05
0.5
~ f / c m 3 5.l Figure 6. Experimental (a)and theoretical (A,SIP; A, ADI) behavior of the transport-limited current as a function of electrolyte flow rate for the oxidation of ferrocene at a microdisc electrode of radius 12p m in a channel flow cell with 2h = 0.0546 cm and d = 0.6 cm. The substrate concentration was 0.88 mM. The theoretical behavior was computed using the SIP and AD1 algorithms and a diffusion coefficient of 2.3 x cmz s-I.
appreciable and the current increases with flow rate, but as expected, this increase is much more marked for the larger electrode, although complete independence from the solution flow is theoretically only possible with an infinitesimally small ele~trode.~ Examining the variation of limiting current with flow rate in comparison with the diffusion only limit (see Figures 6 and 7), it can be seen that even low flows can contribute significantly to the current, and it is important that this dependence is recognized, for example, when using microdisc electrodes in flowing systems for analytical purposes where current-concentration calibrations will only apply to particular flow rates. Also shown in Figures 6 and 7 are the currents computed using the SIP and AD1 approaches implemented as described above. Essentially quantitative agreement between both computational methods and experiment is observed over the entire range of flows studied for both electrode sizes. This vindicates the theoretical treatment presented above, and it may be concluded that both the strongly implicit procedure and the alternating direction finite difference technique provide convenient, easy-to-use, and computationally efficient methods for the simulation of mass transport in the cell geometry under discussion. In particular the ready extension to the simulation of electrode processes involving coupled homogeneous kinetics and of voltammetric transients is envisaged.'2q2'
& S I P model
Acknowledgment. We thank the EPSRC and Zeneca for a CASE studentship for J.B. and Bath University for a Summer Studentship for M.K.W.
0 experiment OAOI
I
OD9
I
1
I
0.10
0.15
8
0.05
model
Vf /c m3 s-1
(m) and theoretical (A,SIP A, ADI)behavior of the transport-limited current as a function of electrolyte flow rate for the oxidation of ferrocene at a microdisc electrode of radius 27 p m in a channel flow cell with 2h = 0.0528 cm and d = 0.6 cm. The substrate concentration was 1.07 mM. The theoretical behavior was computed using the SIP and AD1 algorithms and a diffusion coefficient cm2 s-'. of 2.3 x Figure 7. Experimental
was used as received from Aldrich (>99%). Solutions were thoroughly purged of oxygen by bubbling through the solution argon that had been dried with calcium chloride and presaturated with acetonitrile. Supporting theory was generated from programs written in FORTRAN 77 using double precision and executed on an Oxford University CONVEX 220 (SIP) or a HP workstation at the University of Bath (ADI). Each SIP run, corresponding to a particular flow rate, typically required between 10 and 20 min of CPU time; the AD1 calculations required between 10 min and 3 h of CPU time depending on cell geometry/flow rate.
Experimental Results and Discussion Experiments were conducted using platinum microdisc electrodes of radii 12 and 27 pm to study the previously wellcharacterized" oxidation of ferrocene in acetonitrile solution containing 0.1 M TBAP as supporting electrolyte. Well-defined voltammograms with a clear transport-limited plateau were observed with a halfwave potential of +0.37 V (vs SCE),in good agreement with previous work." Figures 6 and 7 show the flow rate dependence of the transport-limited current for the two microdiscs. In both cases the effect of convection is
References and Notes (1) Flesichmann. M.; Pons. S. Ultramicroelectrodes; Datatech: Morganton, NC, 1987. (2) Wang, J. Microelectrodes; VCH: New York, 1990. (3) Adriiux, C. P.; Hapiot, P.; Savbant, J.-M. Electroanalysis 1990,2, 183. (4) Amatore, C.; Lefrou, C. Port. Electrochim. Acta 1991, 9, 311. ( 5 ) Tait, R. J.; Bury, P. C.; Finnin, B. C.; Reed, B. L.; Bond, A. M. J. Electroanal. Chem. 1993, 356, 25. ( 6 ) Matsue, T.; A o k , A.; Ando, E.; Uchida, I. Anal. Chem. 1990, 62, 407. (7) Tait, R. J.; Bond, A. M.; Finnin, B. C.; Reed, B. L. Collect. Czech. Chem. Commun. 1991,56, 192. ( 8 ) Compton, R. G.; Eklund, J. C.; Dryfe, R. A. W. Res. Chem. Kinet. 1993, 1 , 239. (9) Compton, R. G.; Eklund, J. C.; Nei, L. J . Electroanal. Chem. 1995, 381, 87. (10) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J . Phys. Chem. 1993, 97, 10410. (11) Compton, R. G.; Dryfe, R. A. W.; Alden, J. A.; Rees, N. V.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. 1994, 98, 1270. (12) Compton, R. G.; Dryfe, R. A. W.; Wellington, R. G.; Hurst, J. Electroanal. Chem. 1995, 383, 13. (13) Peaceman, D. W.; Rachford, H. H. J. SOC. Ind. Appl. Math. 1955, 3,200. (14) Gourlay, A. R. J. Inst. Math. Its Appl. 1970, 6, 370. (15) Webber, S.; Fisher, A. C. J. Chem. SOC.,Faraday Trans., submitted for publication. (16) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962; p 112. (17) Sharp, P. Electrochim. Acta 1983, 28, 301. (18) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Bethell, D.; Lederer, P. J . Phys. Chem. 1991, 95, 4749. (19) Compton, R. G.; Barbhout, R.; Eklund, J. C.; Fisher, A. C.; Bond, A. M.; Colton, R. J . Phys. Chem. 1993,97, 1661. (20) Coetzee, J. F. Recommended Methods for Purification of Solvents; IUPAC, Pergamon Press: Oxford, 1982. (21) Alden, J. A.; Compton, R. G.; Dryfe, R. A. W. J. Appl. Electrochem., in press.
JP9503369
