2904
Anal. Chem. 1984, 56, 2904-2909
factor can be computed from the following: qa
Big
=~
h
(- CJ1
(-413)
When this calculated value is substituted into eq 18 and noting that C,(O) = 0, then eq 24 is obtained. Registry No. D-Glucose, 50-99-7.
LITERATURE CITED Carr, P. W.; Bowers, L. D. "Immobilized Enzymes in Analytical and Cllnical Chemlstry"; Wiley: New York, 1980. DOrazio, P.; Meyerhoff, M. E.; Rechnltz. G. A. Anal. Chem. 1978, 50, 1531-1534. Arnold, M. A.; Rechnitz, G. A. Anal. Chem. 1980, 5 2 , 1170-1174. Guilbault, G. G.; Montalvo, J. G., Jr. J. Am. Chem. SOC. 1970, 9 2 , 2533-2538. Llenado, R. A.; Rechnitz, G. A. Anal. Chem. 1971, 43, 1457-1461. Tran-Mlnh, C.; Broun, G. Anal. Chem. 1975, 47, 1359-1364. Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 4 7 , 299-307. Brady, J. E.; Carr, P. W. Anal. Chem. 1980, 5 2 , 977-980. Carr, P. W. Anal. Chem. 1977, 49, 799-802. Gough, D. A.; Leypoldt, J. K. Appl. Biochem. Bioeng. 1981, 3 , 175-208. Hameka, H. F.; Rechnltz, G. A. J. Phys. Chem. 1983, 8 7 , 1235-1241. Callanan, W. A. M.S. Thesis, University of Pennsylvanla, 1972. Leypoldt, J. K. Ph.D. Dlssertation, University of California, San Dlego, 1981. Updlke, S. J.; Hicks, G. P. Nature (London) 1987, 214, 988-988. Aris, R. A. "The Mathematical Theory of Diffusion and Reaction In Permeable Catalysts"; Clarendon Press: Oxford, 1975. Viliadsen, J.; Michelsen, M.L. "Solution of Differentla1 Equation Models by Polynomial Approximation"; Prentlce-Hall: Englewodd Cliffs: NJ, 1978. Gough, D. A,; Leypoldt, J. K.; Armour, J. C. Diabetes Care 1982, 5 , 190-1 98.
(18) Gibson, Q. H.; Swoboda, B. E. P.; Massey, V. J. Biol. Chem. 1984, 239, 3927-3934. (19) Finlayson, E. A. "Nonlinear Analysis in Chemical Engineering"; McGraw-Hill: New York, 1980. (20) Leypoldt, J. K.; Gough, P. A. Blotechnol. Bioeng. 1982, 2 4 , 2705-27 19. (21) Gough, D. A.; Leypoldt, J. K. Anal. Chem. 1979, 57,439-444. (22) Gough, D. A.; Leypoldt, J. K. AIChE J. 1980, 2 8 , 1013-1019. (23) Gough, D. A.; Leypoldt, J. K. Anal. Chem. 1980, 5 2 , 11!?6-1130. (24) Gough, D. A.; Leypoldt, J. K. J. Electrochem. SOC. 1980, 727,
---
1278-1286 - -
(25) Engasser, J. M.; Horvath, C. Appi. Biochem. Bioeng. 1978, 1 , 127-220. (26) Cleland, W. W. Adv. Enzymol. Re/at. Areas Mol. Blol. 1967, 2 9 , 1-82. (27) Bright, H. J.; Appleby, M. J. Biol. Chem. 1989, 244, 3625-3634. (28) Weibel, M. K.; Bright, H. J. J . Bioi. Chem. 1971, 248, 2734-2744. (29) Cho, Y. K.; Bailey, J. E. Biotechnol. Bloeng. 1977, 79, 185-198. (30) Guilbault, G. G.; Lubrano, G. J. Anal. Chim. Acta W72, 6 0 , 254-256. (31) Guilbault, G. G.; Lubrano, G. J. Anal. Chim. Acta 1973, 6 4 , 439-455. (32) Lobel, E.; Rlshpon, J. Anal. Chem. 1980, 5 3 , 51-53. (33) Updlke, S. J.; Shults, M.; Ekman, E. Diabetes Care 1982, 5 , 207-212. (34) Fischer, U.; Abel, P. trans.-Am. SOC.Artif. Intern. Organs 1982, 2 8 , 245-248. (35) Romette, J.-L.; Froment, B.; Thomas, D. Clin. Chim. Acta 1979, 9 5 , 249-253. (36) Engaser, J. M.; Hisland, P. J. Theor. E M . 1979, 77, 427-440. (37) Murray, J. D. "Lectures in Nonlinear-Dlfferential-Equatton Models In Biology"; Oxford University Press: London, 1977. (38) Forsythe, G. E.; Malcolm, M. A.; Moler, C. E. "Computer Methods for Mathematical Computatlons"; Prentice-Hall: Englewood Cliffs, NJ, 1977.
RECEIVED for review March 21, 1983. Resubmitted and accepted August 13, 1984. This work was supported by grants Diabetes and the Institutes of Health.
from the
Theoretical Evaluation of Transient Responses of an Amperometric Enzyme Electrode Alain Bergel and Maurice Corntat*
Laboratoire de Chimie Physique et Electrochimie, Laboratoire associt? au CNRS 192, Universitt? Paul Sabatier, 31062 Toulouse, Ct?dex France
Translent responses are calculated for an enzyme amperometrlc electrode based on the substrate oxldatlon by the oxldlzed form of a cofactor catalyzed by an enzyme and the detection of the reduced form of the cofactor by constant potentlal amperometry. The enzyme is In solution In a reaction chamber bounded by the electrode surface and a semlpermeable membrane. An lmpliclt flnlte difference method allows computations for an enzymatic reectlon of first order with respect to the substrate and of zero order wlth respect to the oxldlzed form of the cofactor. Geometrical (depth of the reaction chamber, thlckness of the membrane), kinetic (rate constants for the enzymatlc and the electrochemlcal reactlons), and transport (diffuslon coefficients) parameters are analyzed. The calculation Is extended to enzymatic reactions with more complex rate laws but llmlted to the description of a reagentless electrode based on the constant potential reoxldatlon of the enzymereduced form. The results are applied to an amperometric lactate-specific electrode uslng hexacyanoferrate(I I I ) as a cofactor.
animal tissues are now widespread in the field of analytical chemistry (1-3). Compared to the number of papers dealing with experimental studies or applications of these electrodes, theoretical works are few. Furthermore, they are often limited to the evaluation of the responses of these sensors under steady-state conditions. This is the case for the few works devoted to the enzyme electrodes with potentiometric detection (4-7) where transient analysis is very rare (8). It is also the case for the amperometric enzyme electrodes (9-12). It has to be noted that the analysis of transients for a glucose electrode by digital simulation leads, nevertheless, to the optimization of the performances and to the association with a microcomputer (13). The present work concerns the calculation of transients for an amperometric enzyme electrode, in which the soluble enzyme is confined near an electrode by means of a semipermeable membrane. Such a sensor presents a zone where only diffusion of substrate and products of the reaction OCCWS, in contact with a layer where diffusion is coupled to an enzymatic reaction such as (for instance for an oxidation) substrate S
Enzyme electrodes and more generally bioselective membrane electrodes using cells, microorganisms, or vegetable or 0003-2700/84/0356-2904$0 1.50/0
+ y oxidized cofactor co __
product
P
1984 American Chemical Society
enzyme
+ y reduced cofactor CR
(1)
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
2905
( 7)
4
Solution
%=I
D(LL!
s = S,( crs 0 Reaction
I
4
I
Membrane
Charnber
I
Membrane
Chamber
0
'2
11
Reaction
1
l+L
X
Scheme of the electrode: equatlons and boundary Conditions for the membrane and the reaction chamber.
Figure 1.
where y is a stoichiometric coefficient. The electrode, maintained at a constant potential, allows the amperometric detection of the reduced form of the cofactor (eq 2). Slightly different systems were studied by Gough et CR CO ne (2)
-
+
al. (14, 15). Their systems are composed of a membrane in which the enzyme is immobilized, closely applied to the electrode. Nevertheless, the model uses the same space pattern of two different zones: the membrane where the enzymatic reaction coupled to the diffusion occurs, and a diffusion layer between the membrane and the bulk of the solution. The authors worked out the equations only for the steady state, mainly using a criteria useful to the conception and application of such sensors. It has to be noted that when no enzyme is in the membrane this device niay be used for analytical purpose, to measure transport parameters of the membrane. An analytical approach of this last case is suggested, making the hypothesis of a linear concentration profile in the diffusion layer (16,17). Nevertheless, as far as we know, the computation of transient of such an electrode as described in this work has never been done before. An implicit difference method close to that of Crank and Nicholson was used to solve the equatiohs. The model is then extended to electrodes with more complex enzymatic kinetics and to a reagentless electrode working through the oxidation of the substrate by the oxidized form of the enzyme and the regeneration of this form by constant potential electrolysis of the reduced form. Results of the simulation are compared to the experimental results of an electrode specific of L-(+)-lactate by using the oxidation of lactate by hexacyanoferratefII1) catalyzed by a lactate dehydrogenase and the amperometric detection of the resultihg hexacyanoferrate(I1) (12)
CH,CHOHCOO-
+ 2Fe(CN),3CH,COCOO-
Fe(CN),4-
-
+ LDH
2Fe(CN)64-
Fe(CN):-
+ e-
+ 2H+ (3) (4)
This electrode might also be used without a cofactor (18).
THEORY The electrode is schematically shown in Figure 1;l1 is the depth of the reaction chamber, and l2 is the thickness of the membrane parallel to the plane electrode. The equations and boundary conditions given on this figure correspond to the following hypotheses and experimental conditions: The substrate concentration is less than the Michaelis constant in the reaction chamber (the enzymatic reaction is first order
Flgure 2.
Equations and boundary conditions using dimensionless
variables. with respect to the substrate). The oxidized form of the cofactor is initially in the reaction chamber, and its concentration is high enough in order to transform all the substrate flowing through the membrane; moreover, the enzymatic reaction is zero order with respect to this product. The solution in contact with the right edge of the membrane is stirred. The diffusion coefficients in a given zone are the same for both the substrate and the cofactor. (The simulation could be worked out by using different diffusion and partition coefficients for each species, but this was not useful to fit the experimental results.) The potential applied to the electrode was chosen so the regeneration of the oxidized form is first order with respect to the reduced form. When the dimensionless variables X = x / l l , L = 12/11, T = tD1/l?, S = s/sSo1,C, cr/ys,,, D = D2/D1,and & k,ll/D1 are introduced where t is the time, s and cr the concentrations of substrate (S) and reduced form of cofactor (CR), and 1,s, concentration of the substrate in the solution, the equations and boundary conditions are then written on Figure 2. In the equations of the reaction chamber the dimensionless group 42 = klI2/D1appears, which is in this case a very simple expression of the Thiele modulus, representing the ratio of the potential rate of reaction to the potential rate of diffusion (14).
Discretization of the Equations. The following discussion is related to the S(X,T) function but may be easily transposed to the C,(X,T) function. Evolution equations and their boundary conditions are discretized according to a finite difference method. In electrochemistry the explicit form of this method is often applied (19) because of its simplicity compared to the implicit form, but it is severely limited by its stability, especially when the purpose is to simulate fast homogeneous reaction coupled to diffusion. The implicit variants are rather used because of their accuracy and stability and because it is usually possible to save computing time (20,21). According to these methods 9 becomes at each point of the space grid
0
S,+ln+l
- 2Sin+l AX2
+ Sk-ln+l +
where S," an'd S?+lare the concentrations at the point i for times n and n 1. When the degree of implicitness 0 is equal to 0.5, the method is that of Crank and Nicholson (22) and if it is equal to unity the method is said of Laasonen (20). With X = A T / A X 2the equation may be written
+
2906
ANALYTICAL CHEMISTRY, VOL. 56,
+
NO. 14, DECEMBER 1984
+
((I/D) 20x)s,n - O X S , + ~=~ + ~ h ( l - 0) Sl_ln ((l/D) - 2X(1 - 0))s: X ( l - O)S,+ln (12)
-eXs,-ln+l
+
iE 1 2
+
1-3 1-1 I ficl
3 4
In order to discretize eq 8, a term representing the homogeneous reaction is introduced. This is generally done without any particular theoretical background, only writing the kinetic term at time n (22) (or rarely at time n 1 (21))and keeping the value for 0. In this problem this approach leads to unsatisfactory results because of the high value of 02. So it was better to express the rate at time n 0 and to give 0 a value that minimizes the local truncature error (27). This computation leads to 0 = 0.5. Equation 8 then becomes
Solution
+
+
+
-(X/2)S,-ln+l + (1 + X f12AT/2)SEn+l - (X/2)S,+ln+' = (X/2)S,-ln (1 - X - 02AT/2)SLn (X/2)S,+ln(13)
+
+
and eq 12 becomes
-(X/2)S1-ln+l + ((1/D) + A)S,n+l - (X/2)sL+1"+l = (X/2)sL-1" + ((1/D) - X)S,n + (X/2)Sl+in (14) After discretizing the boundary conditions, one obtains a tridigonal system of N equations with N unknowns solved by a classical method of matrix calculus. Evaluation of Fluxes. It is generally admitted that the evaluation of fluxes a t the interfaces is one of the hardest problems of these methods (23,24),as confirmed by the great number of equations recently reviewed by Lasia (25). This analysis deals with the interface membrane-reaction chamber where
(aslax),= o(as/ax),+
(10)
The simpliest method would consist in using a first-order approximation for calculation of fluxes. For instance at time n + l sln+l
- S,-ln+l AX,
Sl+ln+l
=D
- S n+l 1
(15)
AX2
These equations introduced in the computation program give results with a relative error that decreases as X decreases. For instance, the comparison of the steady-state current values obtained with this model and with the analytical solution *stat
1
=
1 1t - 1 I
)
Ldsh0 + D(ch0 - 1) L0sh0 + Dch0
I=
fic2 I
It2
N-2 N
Flgure 3. Space grid used to discretite4he equations of Figure 2. (1) Space drlven by the equatlons of the reaction chamber. (2) Space driven by the equations of the membrane.
The first method is based on a balance between the points 1 / 2 a t time n 0
I - 'Izand I
flux in I
+
+
accumulation in reaction chamber
t
AX hSn+' bT
+(- 2
reaction in reaction chamber hSn+@
)1+(1/4)
+
(z
)I-(1/2)
accumulation flux in in meinhrane I -I/> When values are calculated in I f 'I4 by means of averages such as l/&31k1+ 3/4SI, it is possible to express the interface equation in finite difference terms. The second method, perhaps more rigorous, consists of using the differences centered at I as a second-order approximation of the flux on both sides. If we take two distinct spaces, one driven by equations in the membrane and the second by equations of the reaction chamber and at the interface, each space overlaps the real spaces by a step (Figure 3). Two fictitious points Ificl and Ificz without any physical meaning are created, the concentration of which verify the equations
(16)
D ' 0s
with
*=
and
ill nF7ssolDl
(17)
leads to a relative error of 82% for AX = and 21% for AX = This could be minimized by inserting the interface between two consecutive points of the network and by adding point I between them. This procedure is equivalent to the local diminution of X but complicates the computation because it is necessary to take into account the X variation between I - 1and I 1 (22). It is also possible to use parabolic approximations, or approximations of highest order (25),but the so-obtained matrix loses its tridigonal form. The two following procedures are more effective (the error between steady-state values obtained by the model and by analytical solution is less than 1%for a AX of l/&). On the basis of more complicated equations, they do not change the flexibility of the model because they do not perturb the equations of the points contiguous at the interface and preserve the tridigonal form of the matrix.
+
When the equality of the fluxes is written Sficln+l
- Sl-p+l
2AX1
s,+,n+l
=D
- sfic2n+l
2AX2
(20)
with the expressions of Sficln+l and SficZn+l deduced from the former equations, the interface condition is obtained with the only unknown values: SI-ln+l, SIn+l,and SIfln+l.It is then necessary to calculate Sficln+l and Sficzn+l for the next iteration. RESULTS AND DISCUSSION Figure 4a shows the variation of the concentrations of the substrate and the reduced form of the cofactor in the reaction chamber vs. time and Figure 4b represents the response curve
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
a
Reduced Concentration Of
Substrate
and
2907
t
a
Cof actorb
Length , pm
b r
u
b
t
1 fl
E
Y
4
E
j
I
.-am
I
1
E
8 c
t 3
0
Time
s
-
Flgure 4. (a) Concentrations of reduced cofactor (- -) and substrate (-) in the reaction chamber at various times. I , = 8 x Cm, 1, =5X cm, D , = 5.0 X 10“ om2s-l,0,= 1.2 X lo-’ cm2s-’, k = 28 s-’. (a) Profile at 2.5 s, (b)at 7.5s,(c)at 20 s,(d) steady state. (b) Response curve of the electrode for the group of parameters of (a). of the same electrode. According to the high value of Qi2 the enzymatic reaction rate is faster than the diffusion rate; so, close to the membrane a reaction layer appears, in which almost all the substrate is transformed. Outside this layer the substrate concentration is about zero and the resulting reduced cofactor undergoes only diffusion. Its concentration profile becomes linear. Taking into account these two species (substrate and reduced cofactor) is sufficient to describe the performances of this sensor. As a matter of fact, according to the hypothesis, the concentration profile of the oxidized form of the cofactor is almost flat because it is always in excess compared to the substrate. Concentration profiles as well as the response curve are not sensitive to the number of steps chosen for the X and T variables in the range used (20-60 steps for A X , for AT). It should be pointed to 5 X out that the occurrence of abnormal oscillations for the concentrations calculated near the interface membrane solution. They are due to the fact that So has its maximum value a t time zero; their amplitude increases when the step chosen for A T is higher and the step on AX lesser; they damp very quickly and do not perturb the profile. On each figure the values of the parameters are given for the electrode specific of lactate (12). The response curve is in good agreement with the experimental one. In this case the very sharp decrease of intensity over the first milliseconds due to the double-layer charging is naturally not observed. The concentration profiles
120 Time , L
Figure 5. (a)Concentration profiles of reduced cofactor in the reaction chamber for various rate constants k : D , and D , are the same as cm, I,= 8 X cm. Curve 1, k that in Figure 4, I,= 18 X = 700 s-’,curve 2, k = 50 s-l; curve 3, k = 0.7 s-’. (a) Profile at 10 s, (b) at 20 s, (c) at 60 s, (d) steady state. (b) Response curves of the same electrode for various k constants (1) 700 s-’and 50 s-’; the curve is not modified. (2) 0.7 s-’. (3) 0.14 s-’. (4) 0.07 s-’;the curve is sensitive to the kinetics of the enzymatic reaction. become rapidly linear in the membrane. Steady-state values are obtained after 20 s for lactate (substrate) and 40 s for hexacyanoferrate(I1) (reduced form of the cofactor). Influence of Some Parameters. Geometrical Parameters. Various values of ll and l2 have been used in the range of 0.030-0.180 mm. When l1 and l2 decrease, intensities naturally increase, whereas the time necessary to obtain the steady-state value decreases. Rate Constant k (or parameter Qi2). k is equal to (k3ENZo)/K, where ENZois the total enzyme concentration, k 3 the dissociation rate constant of the complex enzyme substrate, and K , the Michaelis constant. The results presented on Figure 5 show that the concentration profiles of the cofactor undergo no modification except in that part of the reaction chamber close to the membrane for the highest values of +2. The reaction layer enlarges when +2 decreases, but the concentration profiles remain linear close to the electrode so the intensity does not change. A +2 value of about 100 has to be reached for the kinetics of the enzymatic reaction to be modified. It is then possible to use this electrode to do kinetic measurements. Diffusion Coefficients. Concentration profiles show clearly the predominant role of diffusion both in the mem-
2908
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
equation can be written also as
xSi-ln
+ (1 + 2h)Si" - XSi+l" + 4l2AT
Si"
KM'
+ Si" = O
and
v;.= 42x i - p + (1 + X)Si" + -&+In x 2
prn
Length
b N
E \
E
0.15-
-ra 4a
c u 0 I
0.05. L
a
0
L
Time
,
s
Figure 6. (a) Evolution of the concentration of the reduced cofactor in the reaction chamber for various heterogeneous-transfer rate constants k , at t = 40 s;I , , I,, D,, and D, have the same values as cm s-', (b) 0.5 X cm s-', (c) cm those in Figure 4. (a) s-', (d) 10 cm s-'. (b) Response curves for the same values of the heterogeneous rate constants.
brane and in the reaction chamber; in the range studied (4 x cm2 s-l) the response time and the to 1 X steady-state values of the intensity vary proportionaly to (D,D2)/llDz lzD1)as expected from the analytical solution. Heterogeneous-Transfer Rate Constant k,. For high values of k , (or &) the concentration of reduced cofactor is always zero a t the electrode surface, so the sensor is not sensible to the kinetics of heterogeneous transfer. But when K , decreases beyond approximately 1 cm s-l, the reduced cofactor accumulates in the reaction chamber and its superficial concentration increases (Figure 6a). So the steady-state value of intensity is scarcely modified, but the time necessary to reach it is lengthened (Figure 6b). Various Forms of Enzymatic Reaction Kinetics Laws. In order to describe in a more general way the enzymatic kinetics we tried to introduce in the computation a rate expression such as ( k S ) / ( K M+ S). It is necessary to modify the numerical method to introduce nonlinear equations. The adopted Newton-Raphson method consists in solving the system by iterations after linearization of the equations. Considering eq 13 in the form Fl([S"+l]) = 0, where [Sn+']is the vector describing the substrate profile for time n + 1, a first-order Taylor's development gives
+
where the vector
[HI= [hj]stands for Sjn+'- S?
= hj. This
-
(pr2AT
2
K'M
Si"
+ Si"
I t allows the calculation of [HI and then [S"+l]according to [S""] = [Sn]+ [HI. I t is necessary to run several iterations until the computed profile [SnC1] does not vary. The test of convergence is based on the modulus of [HI which may be taken equal to ![HI1 = max i[h,]. Practically only a few iterations are necessary for each step time (three or four at the beginning of the calculation). The results obtained are in very good agreement with the former when the right values of the kinetic parameters are used. So this kind of approach may be extended to other complex kinetic expressions. Reagentless Electrode. In this kind of electrode the lactate is oxidized by the enzyme, and the oxidized form of the lactate dehydrogenase is regenerated by constant potential electrolysis. The system is quite different because the diffusion coefficient of LDH is higher than the diffusion coefficient of hexacyanoferrate; moreover, the enzyme is trapped in the reaction chamber because it is too large a molecule to diffuse through the semipermeable membrane. Although the numerical solving is no more difficult than the previous systems, it has to be noted that the regeneration of the oxidized enzyme at the electrode is not fast enough (26) to assume an adequate concentration of the oxidized enzyme in the reaction chamber that justifies the use of a simple kinetic equation of first order in substrate. The method allows only an approach of the simulation of the experimental phenomena.
CONCLUSION The present method may be applied to predict the response-time curve of the amperometric enzyme electrode and to optimize such an electrochemical sensor provided the enzyme rate equation is only dependent of substrate concentration in a more or less complicated form. The calculation method may be extended to optimization of electroenzymatic reactors where enzymatic reactions are associated with the regeneration of a cofactor by constant potential electrolysis. In the case of the rate of enzymatic reaction depending on two species whose concentrations are able to vary in the reaction chamber, the computation may be revised. ACKNOWLEDGMENT This work was supported in part by the S. N. Elf Aquitaine. A. Bergel acknowledges this society for a grant. Registry No. Lactic acid, 50-21-5;ferricyanide, 13408-62-3; ferrocyanide, 13408-63-4. LITERATURE CITED ( 1 ) Guiibauit, G. G. "Comprehensive Analytical Chemistry"; Svehla. C., Ed.; Elsevler: Amsterdam, 1977; Voi. V I I I . (2) Fricke, G. H. Anal. Chem. 1980, 52,259R-275R. (3) Cosofret, V. V. "Membrane Electrodes in Drug Substances Analysis"; Pergamon Press: New York, 1982.
Anal. Chern. 1984, 5 6 , 2909-2914 (4) Blaedel, W. J.; Klssel, T. R.; Boguslaski, R. C. Anal. Chem. 1972, 4 4 , 2030-2037. Brady, J. E.; Carr, P. W. Anal. Chem. 1980, 52, 980-982. , Hameka, H. F.; Rechnltz, 0. A. Anal. Chem. 1981, 53, 1586-1590. (7) Pedersen, H.; Horvath, C. Appl. Biochem. Bioeng. 1981, 3, 96. (6) Tran-Mlnh, C.; Broun, 0. Anal. Chem. 1975, 4 7 , 1359-1364. (9) Raclne, P.; Mlndt, W. Experientis, Suppl. 1971, 78, 525-529. (10) Meil, L. D.; Maloy, J. T. Anal. Chem. 1975, 4 7 , 299-307. (11) Mell, L. D.; Maioy, J. T. Anal. Chem. 1978, 4 8 , 1597-1601. (12) Durliat, H.; Comtat, M.; Mahenc, J.; Baudras, A. Anal. Chlm. Acta 1978, 85, 31-40. (13) Kernevez, J. P.; Konate, L.; Romette, J. L. Blotechnol. Bioeng. 1983, 25, 845-855. (14) Gough, D. A.; Leypoldt, J. K. Appl. Biochem. Bioeng. 1981, 3, 175-206. (15) Gough, D. A. ; Leypoldt, J. K. Anal. Chem. 1979, 51, 439-444. (16) Gough D. A,; Leypoldt, J. K. J . Electrochem. SOC. 1980, 127, 1278-1 286. (17) Gough, D. A,; Leypoldt, J. K. AIChE J. 1980, 26, 1013-1019.
.
(18) (19) (20) (21) (22) (23) (24) (25) (26) (27)
2909
Durliat, H.; Comtat, M. Anal. Chem. 1980, 52, 2109-2112. Feldberg, S. W. Nectroanal. Chem. 1989, 3 , 199-296. Winograd, N. J . Nectroanal. Chem. 1973, 4 3 , 1-8. Heinze, J.; Storzbach, M.; Mortensen, J. J . Nectroanal. Chem. 1984, 765,61-70. Booman, G. L.; Pence, D. T. Anal. Chem. 1965, 37, 1366-1373. Sandifer, J. R.; Buck, R. P. J. Nectroanal. Chem. 1974, 4 9 , 16 1- 170. Hanafey, M. K.; Scott, R. L.; Ridgway, T. H.; Rellley, C. N. Anal. Chem. 1978, 50, 116-137. Lasla, A. J. Electroanal. Chem. 1983, 746, 397-412. Durliat, H.; Comtat, M. Anal. Chem. 1982, 5 4 , 856-861. Richtmeyer, R. D.; Morton, K. W. “Difference Methods for Initial Value Problems”; Interscience: New York, 1967.
R~~~~~~~for review ~~b~~~~~23, 1984, ~ ~ ~ ~ jUly b ~ i t t ~ d 24, 1984. Accepted July 24, 1984.
Five-Electrode Thin-Layer Cell for Spectroelectrochemistry Applied to Spectrocoulometric Titrations D. A. Condit,’ M. E. Herrera, M. T. Stankovich? and D. J. Curran* Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003
The spectroelectrochemlcalcell described here Is the first reported whlch is capable of performing dlrect spectrocoulometrlc tltratlons wlthln a thin-layer cavity with negllglble edge effects. Thls was accompllshed by mlnlmlzlng the potential gradlenl and solution resistance effects Inherent to the geometric requirements for slrnultaneous spectral and electrical monltorlng of electrochemically lnltlated processes. Twin reference and counter electrodes and a dual-tapered thlncavlty geometry made a spectrocoulometrlc tltratlon posslMe by decreasing the response tlme whlch, In turn, mlnlmlzed the edge effect. The cell performance characterlstlcs are examined In terms of the devlatlon from theoretlcal thln-layer predictions due to solution resistance and mlnlgrld dimension effects. An emplrlcal cell rate constant of 0.053 s-’ was determined from the spectrocoulometrlc titration of ferrocyanide.
When first introduced (1,2),optically transparent electrodes (OTE) had appeal as a way to dynamically follow heterogeneous and coupled homogeneous reactions a t and near the electrode-solution interface. Titrimetry, with mass transport dependence on convection, came later as an analytical application of OTE’s. Several cells used to perform spectrocoulometric titrations have a standard 1-cm optical path length and a solution volume of a few milliliters. The first cell design reported (3),of this nature, has a tin oxide-coated quartz working electrode which also serves as a cell window while a more recently reported cell ( 4 , 5 )has a quartz or Pyrex cuvette-shaped bottom into which are dipped the counter, reference, and working electrodes. For both cells, the optical path goes through the bulk solution and out of the cells Present address: United Technologies Research Center, Silver Lane, MS-94. E Hartford. CT 06108. Present address: Department of Chemistry, Kolthoff and Smith Halls, 207 Pleasant Street, SE,University of Minnesota, Minneapolis, MN 55455.
unobstructed by other electrodes and stirring devices. From titrations performed with these cells, the formal potential, Eo’, the number of electrons transferred, n,and the total amount of material present can be determined. In contrast to these spectrocoulometric cell designs, thinlayer spectroelectrochemical cells utilizing gold minigrid working electrodes have a shorter optical path length, typically less than 0.3 mm, and mass transport is solely by diffusion (6). The large electrode surface area-to-volume ratio presents an opportunity to carry out rapid direct spectrocoulometric titrations within the volume containing the gold minigrid. However, the existence of edge effects a t the minigrid boundaries led to the development of potentiostatic procedures (7,8) to indirectly determine the number of electrons transferred, n, instead of to direct quantitative determinations of the total amount of material electrolyzed. This paper reporb on a five-electrodesymmetrical sandwich cell comprised of twin reference and twin counter electrodes, a gold minigrid (333 lines/in.) working electrode, and a small-volume (11kL) thin-layer cavity of special geometry. This cell has a response time sufficiently fast to minimize the edge effect and permit direct spectrocoulometric titrations. Cell performance was evaluated by thin-layer cyclic voltammetry and potential step experiments. Results are examined in terms of theoretical predictions.
EXPERIMENTAL SECTION Instrumentation. Solution resistance measurements were made by using an Industrial Instruments Model RC 16B2 conductivity bridge. The electrochemical instruments used were EG&G Princeton Applied Research (Princeton, NJ) Models 173, 175, and 179 along with a Hewlett-Packard Model 7040A X-Y recorder. A Beckman Model MVI UV/vis spectrophotometer has used for the spectrocoulometric titration of ferrocyanide. The most sensitive expanded absorbance range is 0.01 A. Computer programs were written in-BASIC and run on an Apple 11+ with 48K memory. Chemicals, Methyl viologen was purchased from British Drug House, Poole, England. All other chemicals used were ACS reagent grade.
0003-2700/84/0356-2909$01.50/00 1984 American Chemical Society
