Anal. Chem. 1993, 65, 2719-2723
2719
Theory of Steady-State Current at Multiple Microcylinder Electrodes Coupled with a Parallel Planar Electrode Wenfeng Peng and Erkang Wang’ Laboratory of Electroanalytical Chemistry, Chanachun Institute of Applied Chemistry, Chinese Academy of Sciences, Jilin 130h2,’People’a Republic of China -
A novel device of multiple cylinder microelectrodes coupled with a parallel planar electrode was proposed. The feedback diffusion current at this device was studied using bilinear transformation of coordinates in the diffusion space,where lines of mass flux and equiconcentration are representedby orthogonal circular functions. The derived expression for the steady-state current shows that as the gap between cylindrical microelectrodesand planar electrode diminishes,greatly enhanced currents can be obtained with high signal-to-noiseratio. Other important geometrical parameters such as distance between adjacent microcylinders, cylinder radius, and number of microcylinders were also discussed in detail. Since the pioneering work of Adams and his co-workers in the early 19705,’ there has continued to be intense activity in the area of voltammetric microelectrodes. Up to now, a variety of microelectrodedevices were reported, not only those composed of only one microelectrode unit but also those made of multiple unitatermed microelectrodeensembles or arrays28 and those which are microelectrodes coupled with conventional ones as well.9Jo These microelectrode devices have greatly enhanced the power and scope of modern electroanalytical chemistry. Among them, the devices based on socalled “feedbackdiffusion”11mass transport appear to be most attractive. Take interdigitated microband array electrodes (IDAS)- as an example, since very closely spaced bands are bipotentiostated and function as anode and cathode alternatively, reversibleredox speciesoxidized at the anode quickly diffuse over a microscopic distance to the adjacent cathode, and in turn the regenerated species come back to be oxidized, forming a redox recycling between microscopic bands. Due to this diffusion feedback mechanism, IDA electrodes were identified as a class of sensitive devices with fast response characteristics. Other devices of this type include parallel twin band or triple band microelectrodes,l~14narrow gap line
* To whom correspondence should be addressed.
(1) Kminger, P. T.; Hart, J. B.; Adams, R. N. Brain Res. 1973,55,209. (2) Caudill, W. L.; Howell, J. 0.;Wightman, R. M. Anal. Chem. 1982, 54, 2532. (3) Penner, R. M.; Martin, C. R. Anal. Chem. 1987,59,2625. (4) Davis, B. K.; Weber, S. G.; Sylvester, A. P. Anal. Chem. 1990,62, 1ooO. (6) Stemitzke, K. D.; McCreery, R. L. Anal. Chem. 1990,62, 1339. (6) Sanderson, D. G.; Anderson, L. B. Anal. Chem. 1986,57, 2388.
(7) Wehmeyer, K. R.; Deakin, M. R.; Wightman, R. M. Anal. Chem. 1988,57, 1913. (8)Harrington, M. S.; Anderson, L. B. Anal. Chem. 1990,62,546. (9) Ji, H.; Wang, E. Talanta 1991, 38, 73. (10) Lu, J.; Liu, X. J. Electroanal. Chem. 1988,244, 339. (11) Seddon,B. J.;Girault,H. H.;Eddowes,M. J. J.Electroanal. Chem. 1989,266, 227. (12) Foseet, B.; Amatore, C. A,; Bartelt, J.; Wightman, R. M. Anal. Chem. 1991,64,1403. (13) Bartelt, J. E.; Deakin, M. R.; Amatore, C. A.; Wightman, R. M. Anal. Chem. 1988,60,2167. 0003-2700/93/0365-2719$04.00/0
electrodes,15 parallel dual cylinder microelectrodes16 and microdisk array ring/glassy carbon electrodes: etc. The electrochemicalproperties of closely spaced microelectrodes were studied both theoretically and experimentallyby several investigators.6~~~-~3~~~-~~ However, an amperometric study of the diffusion current characteristics at microelectrodes coupled with a conventional-size electrode has not been documented until now. Presently, a cylindricalelectrode is one of the most popular single microelectrode geometries. Because of the availability of quality micrometer wires and fibers and ease of fabrication, over recent years there has been renewed interest in microscopic nobel metal wire and carbon fiber as a versatile amperometric electrode in detector/sensor designs, including enzyme electrode,mliquid chromatography,21-=capillary zone electrophoresisdetectors,u$Band probe electrodes for in vivo measurements.26127 The diffusion current at cylindrical electrodes was first investigated by Laitinen and Ko1thoff.B In step with recent developments in microelectrode research, a reexamination of the cylindrical diffusion current problem has been made, except that different electrochemicalsystems were considered;theoretical basis of chronoamperometry,29,mcyclic voltammetry,2931and pulse voltammet+ were attempted. These studies show that with cylindrical electrodes, unlike sphericalor microdisk electrodes, “true” steadystate behavior is impossible to obtain under diffusioncontrolled conditions. Nevertheless, the current from a nonlinear diffusion process at cylindricalmicroelectrodes can (14) Hill, H. A. 0.;Klein, N. A.; Psalti, I. S. M.; Walton, N. J. Anul. Chem. 1989,61,2200. (15) Brina, R.; Pons, S. J. Electroanal. Chem. 1989,264,121. (16) Pena. - W.; Seddon, B. J.: Zhou. X.: Zhao. Z. Fenxi Huarue 1992, 20, 495. (17) Bard, A. J.; Crayston, J. A.; Kittleson, G. P.; Varco Shea, T.; Wrighton, M. S. Anal. Chem. 1986,58,2321. (18)Magee, L. T., Jr.; Osteryoung, J. Anal. Chem. 1990,62,2625. (19) Aoki, K.; Morita, M.; Niwa, 0.; Tabei, H. J. Electroanal. Chem. 1988, 256, 269. (20) Vadgama, P. NATO ASZ Ser., Ser. C 1988,226,369. (211 Jorgenson. J. W.: Guthrie.. E. J.:, Knecht. L. A. J. Am. Chem. SOC. 1984,56,479. . (22) Goto, M.; Shimada, K. Chromatographia 1986,21, 631. (23) Baur, J. E.; Wightman, R. M. J. Chromatogr. 1989,482,65. (24) Wallingford, R. A.; Ewing, A. G . Anal. Chem. 1988,60,268. (25) OShea, T. J.; Telting-Diaz, M. W.; Lunte, S. M.; Lunte, C. E.; Smyth, M. R. Electroanalysis 1992, 4, 463. (26) Bard, A. J.; Denuault, G.; Lee, C.; Mandier, D.; Wipf, D. 0. Acc. Chem. Res. 1990,23,357. (27) Shannon, C.; Frank,D. G.; Hubbard, A. T. Annu. Rev. Phys. Chem. 1990,42, 393. (28) Laitinen, H. A,; Kolthoff, I. M. J. Phys. Chem. 1941,45, 1061. (29) Kovach, P. M.; Caudd, W. L.; Peters, D. G.; Wightman, R. M. J. Electroanal. Chem. 1985, 185, 285. (30) Aoki, K.; Honda, K.; Tokuda, K.; Matauda, H. J. Electroanal. Chem. 1985,186, 79. (31) Aoki, K.; Honda, K.; Tokuda, K.; Matauda, H. J . Electroanul. Chem. 1986, 182,267. (32) Amatore, C. A.; Deakin, M. R.; Wightman, R. M. J . Electroanal. Chem. 1986,206, 23. (33) Kaneko, H.; Aoki, K. J. Electroanal. Chem. 1988,247, 17. (34) Aoki, K.; Ishida, M.; Tokuda, K. J. Electroanal. Chem. 1988,245,
-”. 29
(36) Murphy, M. M.; ODea, J. J.; Osteryoung, J. Anal. Chem. 1991,
63, 2743.
0 1993 American Chemical Society
2720
ANALYTICAL CHEMISTRY, VOL. 65, NO. 20, OCTOBER 15, 1993
Figure 2.
Representation of feedback dlffusion model at MCM/PE
using orthogonal circular functions: 1, lines of mass flux; 2,
equiconcentration lines. Flgure 1.
Cross section of multiple cylinder microelectrodes coupled
thus at the anode
wlth a parallel planar electrode. r Is cylinder radius; Wis the shortest
COB= c, b
distance of microcyllnders to planar electrode.
be of much greater magnitude compared with that at their disk-shaped counterpart, since the amplitude of current is also dependent on the length of the cylinder. Furthermore, we have demonstrated that when another microcylinder electrode is placed in parallel geometry over a microscopic distance and potentiostated in the limiting-current potential region, enhanced currents can be obtained via a short transient response.16 The purpose of this work is to show that in the same way one observes steady state currents at closely spaced microelectrodeswhen feedback diffusiontakes place,the same effect can be seen a t microelectrode(s) coupled with a conventional electrode. The device discussed here consists of multiple cylinder microelectrodes and a closely spaced parallel planar electrode (MCM/PE), as illustrated in Figure 1. The microcylinders are of the same radius r, they are widely apart from each other, and separated from the planar electrode by a distance of W in practical operation, all the cylinders are connected electrically and work as one electrode. The device we propose has been constructed and used as a highly sensitive and robust electrochemical detector for HPLC.36 This paper presents theory of the steady-state current of the MCM/PE device.
THEORY Firstly we consider the case of one microcylinder electrode coupled with an infinite plane electrode. The device is immersed in a static solution containing species 0 with a uniform concentration Cob and an excess of inactive electrolyte, the microelectrode and the plane electrode work as cathode and anode, respectively. The undergoing electrode reaction is 0 + ne == R, where O/Ris a reversible redox couple and n is the number of electrons involved. Suppose the potential applied to the cathode is sufficiently negative and that to the anode is sufficiently positive so that we have the following boundary conditions, at the cathode (cylinder microelectrode) COC = 0 and at the anode (plane electrode)
(1)
CUB= 0 (2) For the special case when species 0 and R have equal value of diffusion coefficients, DO = DR = D, this relation holds everywhere in the solution at any time37
(4)
Assume 1 >> r, where 1 is the length of the cylinder; mass transport at the microelectrode can then be simplified to a two-dimensional diffusion problem in Cartesian coordinates, as expressed by Fick’s law at
The time-dependent behavior of the diffusion transport at the microcylinder electrode follows a short “Cottrell”response when the diffusion layer is small relative to the electrode radius, then the mass transport deviates from linear behavior as radial flux becomes important, during this phase equiconcentration surfaces lie concentric to the cylinder electrode.28190 At a later time, the boundary of the diffusion layer reaches the positively charged plane electrode, where species R produced a t the cylinder electrode is oxidized and an equal amount of species 0 is regenerated. As the diffusion layer further develops, mass transport at the cylinder electrode surface deviates from radial directions to or from the plane electrode because of the large concentration gradient between the cylinder and the plane electrodes. This results in deformation of the diffusion field from purely cylindrical geometry to a more complex pattern unique to the new environment. For a fully developed diffusion field at the cylinder electrode surface, except the one perpendicular to the plane but in the direction from the plane to the cylinder, theoretically all lines of mass flux link the microcylinder with the plane electrodes. That means there is no mass exchange with the bulk solution and 10096 mass recycling is virtually achieved between the anode-cathode pair. Therefore, a steady state has been effectively established. Under this condition, all points in the diffusion field obey
acoiat = o
(6)
The steady-state feedback diffusion discussed above is a complex mass transport problem that will require the application of mathematical modeling for elucidation. To solve the two-dimensional mass transport problem, the method of conformal representation which has proved so important in other fields like electricity38 and heat conduct i ~ can n be ~ used. ~ In this case mass flux lines are represented by circular arcs and equiconcentration surfaces by noncoaxial cylinders; the equiconcentration surfaces and the mass flux lines cut orthogonally, as described in Figure 2. This simplified mathematical model can be treated using bilinear transformation in the diffusion space from Cartesian coordinates ( x , y)39
(3)
U + iV = i In { [ ( x + c ) + i y l / [ ( x- c ) + iyl) (7) where, c is the confocal distance, the lines of mass flux and
(36) Peng, W.; Wang, E. Preparation and characterization of multiple cylinder microelectrodes coupled with a parallel glassy carbon electrode and ita application to detection of dopamine. Anal. Chim. Acta, in press. (37) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and applications; John Wiley & Sons: New York, 1980; p 161.
(38)Atkin, R. H. TheoreticalElectromagnetism;Heinemann: London, England, 1962, Chapter 11. (39) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed.; Clarendon: Oxford, England, 1986; Chapter XVI.
c,
+ c, = Cob ~~~
~
~
ANALYTICAL CHEMISTRY, VOL. 65. NO. 20, OCTOBER 15, I993
2721
cathode is expressed by
/
/--y taking eqs 17 and 18 it becomes Cob i = nFDl -(VI
- U2)
(20)
vc
where U1 and UZare the values of Uat SIand s2, respectively. Substituting (14) into (201, and considering W = d - r, we get
1
--II
i = 2*nFD1C~/cosh-'(W/r + 1)
Figure 3. Illustration of bilinear transformation of coordinates In the diffusion space of MCM/PE. U denotes a mass flux line and V, is a cross section of the microcylinder, qd,O) is the center of V,, and o(-c,O) and Qc,O) are confocal points on the abssisa.
equiconcentration, denoted by U and V, respectively, are taken as a pair of conjugate functions and are defined by
u = e2 - el
(8)
V = ln(Pl/P2) (9) where, in Figure 3, PI and P2 are distances of any point Ton circle V, from points C(-c,O) and D(+c,O), and e2 are angles of TCD and TDF, and V, represents a cross section of the cylindricalmicroelectrode. Since the ratio of PllPZ is constant for a given V CE CF P,IP2= -
ED=^
let d = W + r, then c
+ (d - r ) -- (d + r) + c
c-(~-P)
(~+F)-c
it follows that c
= (d2- r2)'I2
using (12) in (10) gives
P'IP2 =
d + (d2P
from eq 9
V, = cosh-'(dlr) the dimensionless variable V can be expressed as
V=
c, - COC ,(VC+ Val - v, COB- c,
because V, lies on y-axis, by eq 9 V, = 0, then from eqs 1and 4
V = -co v Cob
(16)
differentiating V to radial variable r' gives (17)
Lets represent an element of the surface of Vacross which mass transport, since for orthogonality
dV=-BU
ar'
as
the steady state current per unit length at the cylindrical
(21)
The current given by eq 22 is a cathodic current, though u-n is not marked on, we mention it here. For a MCMIPE device, where the involved number of microcylinders is m, if the cylinders are infinite from each other, the steady-state diffusion field established between one cylinder electrode and the plane electrode does not interfere with the adjacent, the m cylinders behave as isolated systems and the total current should thus be the sum of individuals, namely
i, = 2~mnFDlCob/cosh-'(Wlr + 1)
(22)
DISCUSSION The device discussed is a system where feedback diffusion takes place between microelectrodes and a macroscopic electrode. Analogous to the rotating ring-disk electrodes (RRDE),the cylinder microelectrodesand the plane electrode work as generator electrode and collector electrode, respectively, but unlike RRDE, when the reduced species reaches the collector electrode, it is oxidized to 0 that can diffuse back to the generator. Thus the planar collector electrode acta as a source and increase the flux of 0 to the microelectrodes. The above derivation of the steady-state current at the cylinder microelectrode(s) is based on a total mass recycling model. From eq 22 it can be seen that the current is in direct proportion to Cob, the concentration of analyte 0, and is dependent on electrode dimensions 1, W, r, and m. Our interests focus on achieving high sensitivity at the present device; therefore, it is necessary to make a further study on optimizing the device geometry. Effects of Gap between Neighboring Microcylinders. As mentioned above, eq 22 is obtained assuming that the microcylinders are infinitely apart from each other. While in practice, the distance between cylinders is of a certain value, because all the cylinders are set at the same potential, according to the present theory, diffusion fields between neighboring cylinders overlap, shielding each cylinder electrode unavoidably. This phenomenon becomes more pronounced as the distance reduces, as a consequence,there exists mass transport to or from the bulk solution. This shielding effect reduces the current at the cylinder microelectrodes in the MCMIPE device when compared to the sum of the currents expected at m infinite apart cylinders. However, if the interelectrode distance is big enough in comparison with Wand P,eq 22 is still applicable; therefore, an approximation can be made when the influence of diffusion field interaction is not apparent. Let us consider a practical MCM/PE device in which adjacent microcylinders are separated by t, the center to center distance, we wish to find an appropriate value to so as to have 95 % of the steady state current calculated from the theoretical model of total mass recycling, rechecking eq 20 shows that circle U tangential to the adjacent must be
uo= *I20
(23)
by a mathematical treatment given in the Appendix,we obtain
2722
ANALYTICAL CHEMISTRY, VOL. 65, NO. 20, OCTOBER 15, 1993 75
Table I. Data of Calculation for the Distance between Two Adjacent Microcylinders (96% of Mass Recycling Is Achieved)* rbm)
Wbm)
10 10 10 10 10 10 20 15 5
20 15 10 5 1 0.5 5 5 5
YO(P~)
R(rm)
tobm)
181.0 146.6 110.8 71.6 29.3 20.5 96.0 84.7 55.4
719.5 582.9 440.7 284.5 116.6 81.5 381.7 336.6 220.3
178.8 144.8 109.5 70.7 29.0 20.2 94.8 83.6 54.7
6.0
“ub
5 6 9 14 34 49 10 11 18
Calculated from eqs 41,42,43, and 26, R and yo are defiied in the Appendix. * Maximum number of microcylinderspermitted for a planar electrode of width or radius of 4 mm. a
1.6 -1.2
1
,
-
2
v
.-CO.8 -
0.4
-
0.0I 0.0
-
-
8.0
4.0
l
-
12.0
-
16.0
-
20.0
-
dum>
Dependence of steady state current on radius at W = 5 pm, obtained by an IBM computer according to eq 22, taking m = 5, n = 1, I = 0.5 mm, C = 1 X lo4 mollL, D = 6.5 X 10-lom2 s-l,and T = 298 K. Figure 4.
to =
2(d + r)(a + r - d ) (r’ - ( a - d)’)’/’
(24)
00 1 00
,
4.0
I
80
12.0
I
160
1
200
W(Un-I)
Figure 5. Dependence of 4 on Wat r = 10 pm, according to eq 22, parameters as given in Figure 4.
only by 157?6, meanwhile, the electrode area increases to 10 times. It should be pointed out though larger current can be obtained by increasing cylinder radius, length, or number of cylinders, the signal-to-noise ratio (SNR) is not improved at all, because the noise is directly proportional to the total area of cylinder electrodes (A = 27rmr1),40 and most importantly, when r is very big with respect to t, steady-state diffusion feedback can no longer be achieved. Effect of Distance of Microcylinders to the Planar Electrode. We now have a look at another geometrical parameter, W , since W is an important operational variable ? in device fabrication, it draws much of our attention. Figure 5 presents an i r W plot of theoretical calculation for a given r; it shows that iT decays as W increases. Unlike that of i r r , the slope of an i r W plot varies a great deal; especially when W approaches zero, the curve becomes very steep. To see the change rate of current, let us differentiate eq 22 with respect to W for constant r
(3) = [cosh-’( W/r + 1)1-’[( W/r)’ + 2 W/rl-1/2 (27) aW r
where (d + 0.988r)(d2- r’) a= (25) d2 - 0.976r’ and d = W + r. The mathematical derivation shows that there will be no significant loss in steady-state current when t > to. However, the critical data to, as listed in Table I for different geometry, are usually bigger than those of W or r by a couple of orders; thus the maximum number of cylinders permitted over a conventional planar electrode of width L should be mma. = int(l/to)
for instance, when r = 10 pm, W = 5 pm, to is 0.28 mm, that means, over a planar electrode of 4 mm width or diameter should be put no more than 14 microcylinders. As W or r reduces, t o decreases rapidly, then more cylinders can be used. Effects of Cylinder Length, Radius, and Cylinder Number. Because i~ a m, i~ a 1, the total current will be enhanced simply by increasing cylinder length and number. It can be seen from eq 22 that the ratio of W/r is a considerable factor which influences the current, the smaller the W / rratio, the larger the steady state current. For a MCM/ PE device with 1 = 0.5 mm, W = 5 pm, and m = 5 in 1 X 1V mol/L analyte solution, the relationship of i~ with r is plotted by an IBM computer in Figure 4. It is obvious that iT increases as r; thus larger current can be obtained by using microcylinders of bigger radius. Nevertheless, the rate of increase in i~value is not fast, when r changes from 1to 10pm, i~ increases
as it can be seen, the rate of change of current is also a function of W. When W < r ,the influence of Won iT is quite significant; a small decrease in Wwill result in a great increase in current, that is to say, the variable W plays a role of Ycurrent amplification”. In Figure 5 where r = 10pm, when Wchanges from 20 to 10pm, only 34% current amplification is obtained, but when W reduces from 10 to 1pm, i~ increases from 0.748 to 2.22 pA by ca. 200%, and the slope of plot changes from -0.328 to -10.9 pA/pm correspondingly. The current can be further increased by reducing W to the submicrometer scale, for instance, at W = 0.1 pm, a theoretical current of 6.97 PA is expected. This value is a factor of 12.5 times compared with that at r = 20 pm. However, when W is very big, to approaches millimeter scale (see Table I); for a device with constant r and t, there exists large mass transport between the cylinders and the bulk solution and it will cause not only significant loss in current but also prolongation of time to reach steady-state response.
CONCLUSIONS Based on a total mass recycling model, theory has been developed for multiple cylinder microelectrodes coupled with a parallel planar electrode. The theory highlights the importance of electrode dimensions to the current response. It shows that the steady state current can be enhanced by (40) Weber, S. G.; Purdy, W. C. Anal. Chim. Acta 1978,100, 631.
ANALYTICAL CHEMISTRY, VOL. 65, NO. 20, OCTOBER 15, 1993
2729
since point T(a,b) is also on the circle of Vc
IY
"\.
(u-d)'+ b2=r2
(32)
b = (r2- (a - d)2)'/2
(33)
sin(d20) =: sin 82 cos 81 - cos 82 sin 8,
(34)
it follows that from eqs 8 and 23 by observing Figure 3
+
+
sin 8, = b/((a cI2 + b2)ll2 Flguro 8. Illustration of boundary mass flux line in MCM/PE where 95 % mass recyclingIsachieved: U, = arcsin(d20);Y0,yo)is center of circle U,; T(e,b)is intersection of U, and V,; others as in Figure 3.
increasing m,1, or r or reducing W, and changing W seems to be the most effective approach to get high sensitivity, especially when W is in submicrometerdimensions, Faraday current can be greatly amplified owing to augmentation of feedback diffusion, while the noise remains in a low level. In addition, to diminishes as W,allowing more microcylinders to be used in the device without significant loss in current; therefore, we can get not only considerable current output but also desirable SNR in electrochemical measurements as well. The device discussed in this paper is suitable for use as a highly sensitive electroanalytical detector.
ACKNOWLEDGMENT This project was supported by the NationalNatural Science
COS
sin O2 = b/((a - c ) ~ b2)'I2
81 = (a + c)/((u + c)' + b 2 ) l t 2 cos 6, = (a - c)/((a - c ) + ~ b2)lt2(35-38)
using (35-38) in (34) gives (d2 - 0.976r2)a2- 2d(d2 - r2)a
+ (d2- r2)' = 0
a=
(d
+ 0.988r)(d2- r") d2- 0.9763
(25)
or
-
a' = (d - 0.988r)(d2 r2)
d2 - 0.976r2
(40)
since a expressed by (25) is reasonable, using it in (33) and (31)
Foundation of China.
(41)
APPENDIX
and according to (30), the radius of UOis
The following derivation is made in Cartesian coordinates. The circle UOcan be expressed as
- 0)2 + (y -yo)2 R2
(28) where, M(O,yo) is the center of the circle and R the radius (see Figure 6). Because &passes throughD(c,O)and T(a,b)(refer to Figure 31, we have (X
(39)
thus
a2+(b-yo)2=R2
+
c2 yo2 = R2
R = (d2- r2 + Y:)'/~ (42) consider UOis tangential to that of an adjacent cylinder to = 2(y,
+R)
(43)
we get finally to =
(29)
2(d + r)(a + r - d ) (r2- (a - d)2)'/2
(24)
(30)
RECEIVED for review March 26, 1993. Accepted June 29,
it gives
1993." (31)
~~
~
a Abstract publiihed in Adoance
~
ACS Abstracts, August 15, 1993.
