Anal. Chem. 1989, 61, 2341-2347 (7) Pakalns, P. Anal. Chim. Acta 1070, 57, 497-501. (8) Hasek,Z. Hutn. Lkty 1070, 25(7), 510, 511. (9) Dorokhova, E. N.; Tikhomorova, T. I.; Cherkasova, 0. 0.; Prokhorova, G. V. Zh. Anal. Khim. 1074, 29(10), 2014-2018. (10) Slickers, K.; WaM, E. Aluminium 1974, 50, 588-591. (11) Schmidt, K. H. Berg-Huettenmaenn. Monatsh. 1970, 775(5), 85-88. (12) Farlnas Gutierrez, J. C.; Valie Fuentes, F. J. An. Quim., Ser. B 1087, 83(3), 310-318. (13) Diemlaszonek, R.; Mouton, J. L.; Trassy, C. Analusis 1979, 7(2), 96-103. (14) Russeva. E.; Havezov, I.; Jordanov, N.; Ortner, H. Fresenius’ 2 . Anal. Chem. 1085, 327(7), 677, 678. (15) Russeva, E.; Havezov, I.; Jordanov, N.; Ortner, H. I n . Khim. 1985, 78(3), 443-448. (16) L’vov, B.; Pelieva, L. A. Zh. Anal. Khim. 1078, 33(8), 1572-1575. (17) Tekula-Buxbaum, P. Mlkrochim. Acta 1981, Z(1-2), 183-190. (18) Mason, B. principles of Geochemistry;Wiley: New York, 1952. (19) Grasserbauer, M. Mkrmhim. Acta 1987, I ( l - 8 ) , 291-305. (20) Sobiecki, A. Rep.-Nati. Inst. Metall. ( S . Aff.) 1079, 2074, 7 pp. (21) Beske, H. E.; Welter, J. M.; Ferlchs, 0.; Melchers, F. G. Fresenius’ 2 . Anal. Chem. 1081, 309, 269-273. (22) Beske, H. E.; Frerichs, G.; Meichers, F. G. Fresenius’ 2 . Anal. Chem. 1087, 329(2-3), 242-246. (23) Caletka, R.; VorwaRer, C.; Krlvan, V. Anal. Chim. Acta 1082, 747, 393-397. (24) Caletka, R.; Hausbeck, R.; Krivan, V. J . Radioanal. Nucl. Chem. 1088, 120(2), 305-318. (25) Yakovlev, Y. V.; Koltov, V. P. Zh. Anal. Khim. 1081, 36(6), 1534-1540.
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(26) Loos-Nescovic, C.; Fedoroff, M. J . Radioanal. Chem. 1083, 80(1-2), 97-108. (27) DeCorte, F.; Speecke, A.; Hoste, J. J . Radioanal. Chem. 1971, 8 , 277-285. (28) Beurton, G.; Beyssier, B.; Angela, Y. J . Radioanal. Chem. 1077. 38, 257-265. (29) Poland, J. E.; Kormali, S. M.; James, W. D.;Schweikert, E. A. J . Radimnai. Nucl. Chem. 1085, 97, 173-178. (30) Kormali, S. M.; James, W. D.; Poland, J. E.; Schweikert, E. A. J . Radioanal. Nucl. Chem. 1985, 97, 179-183. (31) Krivan, V.; Munzel, H. J . Radioanal. Chem. 1073, 75, 575-583. (32) Bauerle, W.; Krivan, V.; Munzei, H. Anal. Chem. 1976. 48, 1434-1440. (33) Schmid, W.; Krlvan, V. Anal. Chem. 1988, 58, 1468-1471. (34) SchmM, W.; Egger. K.-P.; Krlvan. V. J . Radioanal. Nml. Chem. 1088. 723(2), 561-571. (35) Handbook of Nuclear Amation Data ; IAEA Technical Reports: Wen, 1987; No. 273. (38) Ortner, H. M.; Biijdorn. W.; Friedbacher, 0.; Grasserbauer, M.; Krivan, V.; Virag, A.; Wilhartitz, P.; Wunsch, 0. Mkrochim. Acta 1987, I , 233-260.
RECEIVED for review January 25, 1989. Revised manuscript received May 1,1989. Accepted July 11,1989. This work was financially supported by the Bundesministerium fur Forschung und Technology, Bonn, FRG.
Implications of Ion-Exchange Kinetics on Ion-Selective Electrode Responses and Selectivities James R. Sandifer Corporate Research Laboratories, Eastman Kodak Company, Rochester, New York 14650
Slmple principles that have been recognized for some tlme to be Important for the descrlptlon of equlllbrlum (or pseuduequilibrium) responses of ion-selective electrodes (ISEs) are further developed into a theory that explains their tlme-dependent responses. Calculatlons based on the theory provide new Insights into how selectivity coefficients may depend upon kinetic as well as thermodynamlc parameters. The theory Is not conflned to conventlonal ISEs, but also applies to ion-selective field effect transistors (ISFETs) and coated wire electrodes (CWEs) as well.
INTRODUCTION Since the pioneering work of Teorell (1) and Meyer and Sievers (2),theories of ion-selective electrode selectivity have been based predominantly upon thermodynamic considerations, while response times have been attributed primarily to kinetic parameters (3). Kinetics are also used in a general sense to include mass transport. Experimentally measured selectivity coefficients are assumed to correlate with energies of solvation, complex formation constants, and interfacial ion-exchange coefficients. Differences in ion mobilities are generally assumed to account for, at most, only about half of the measured selectivity (3). Furthermore, rates of interfacial ion transport are almost never included in the theoretical calculation of selectivity coefficients. In short, these coefficients have come to be viewed as almost purely thermodynamic in nature, particularly if the membrane is of the “fixed site” variety, in which co-ions are excluded by sites of like sign immobilized in the membrane. A case can be made that ALL conventional ion-selective membranes, constructed from poly(viny1 chloride) (PVC), 0003-2700/89/0361-2341$01.50/0
without the intentional inclusion of mobile sites, MUST have considerable fixed site character. Iglehart et al. (4),and before them Thoma et al. (5), showed that potassium selective membranes that contain valinomycin but no added sites (typically tetraphenylborate or one of its derivatives) polarize when subjected to a large applied voltage. The current maintains a nearly constant value for a few minutes, and then abruptly falls by as much as a factor of 5 before assuming a constant value that is independent of voltage over a wide range. Thoma (5) showed that the concentration profile of valinomycin does indeed tilt during the transition. The initial current is linearly dependent upon applied voltage and the resistance calculated from this linear dependence equals the membrane resistance, R,, as measured using impedance techniques. The initially flat response must indicate a very high concentration of fixed sites in the membrane. Otherwise, concentration polarization of the usually mobile sites themselves would occur, causing the resistance to increase and subsequently causing a decrease in the current. Fixed sites will maintain a flat concentration profile for the potassium/ valinomycin complex (but allow the profile of valinomycin to tilt) because electroneutrality must prevail and the complex is the only other charged species in the membrane ( 6 ) . The resultant “closed circuit” model of the steady-state fluxes in fixed site membranes has been further described in ref 7. The response times of ion-selective electrodes have been explained by using models based on space charge relaxation, ionic diffusion within the bulk of the membrane (or through stagnant layers of adjacent aqueous solution), and slow interfacial ion transport kinetics (3). It seems to have been generally conceded, however, that these processes will eventually reach local interfacial equilibrium (or pseudoequilibrium, because mass transport in the bulk is only steady state) 0 1989 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989
for good ion-selective membranes, and the measured potential will then be close to the true thermodynamic (or steady-state pseudothermodynamic) value. It is becoming increasingly apparent, however, that time dependence is also incorporated into the selectivity coefficient itself, as observed by Arnold et al. using sodium and hydrogen ion selective glass electrodes (8). These electrodes are fixed site and may have much in common with ISFETs and CWEs (9). Selectivity coefficients may appear to be functions of the concentrations of the analyte and interfering ions if potentials are measured during a transient. Such occurrences could complicate chemometric analysis (10-12), introduce considerable error into results obtained from flow injection analysis, and perhaps ultimately require that measurements made during transients of any kind be avoided. Drift compensation during measurements would then become inadvisable. This point of view has been recently elaborated by Berube et al. (13) for silver halide electrodes. Calculations presented here, based on well-established principles and using recently reported data generated by several laboratories (4-7, 14-20) indicate that the issue of time-dependent selectivity coefficients may extend beyond this somewhat obvious problem. In fact, the potential might rapidly assume a stable value that is governed not only by thermodynamic parameters but also by kinetic ones, and may even depend upon the thickness of the ion-selectivemembrane and/or the volume of the sample. As pointed out previously (21), there are fundamental differences between the responses of "conventional" ISEs and those of ISFETs (22, 23) and CWEs (24). The term "conventional", as used in this paper, restricts discussion to liquid or neutral carrier ion-exchange membranes separating sample and internal filling solutions. Solid-state membrane electrodes and electrodes of the second kind are precluded from discussion. ISFETs and CWEs do not generate diffusion potentials, according to conventional wisdom, and therefore should have selectivity coefficients equal to thermodynamic ion-exchangeconstants (24). The results shown below indicate that ISFETs and CWEs DO generate diffusion potentials, which manifest themselves as transients, but that at equilibrium the measured selectivity coefficients should indeed be the thermodynamic values. One must be careful to ensure that these sensors have truly reached equilibrium, however. ISEs should have selectivity coefficients that differ from those of ISFETs and CWEs only by a factor equal to the diffusion coefficient ratio of interfering to analyte ions (3, 25)-assuming that interfacial ion exchange kinetics are fast enough to maintain thermodynamic equilibium. As shown below, experimentally measured values of ion-exchange heterogeneous rate constants (14) can lead to selectivity coefficients that not only are profoundly different from the thermodynamic values but are also concentration dependent, as has been observed experimentally (26,27).
of conventional ion-selectiveelectrodes these surfaces are each contacted by aqueous solutions-the sample at x = 0 and an internal filling solution at x = 1. CJx) is the concentration of the ith species at x . The ionophore is designated by i = 1,and its complex with the ith ion is designated by i = il. V(0) and V(1) are the potential drops across the two interfaces. Although not indicated, gi(x),Ci(x), V(O), and V(1) are all understood to be functions of time as well as distance. Bars have been placed over those parameters that pertain purely to the membrane phase. Co is a normalizing concentration, required to make the units work out. f = F / R T , where F is the Faraday constant, R is the gas constant, and T i s the absolute temperature. ai, k,i, and Vi0 are the charge transfer coefficient, heterogeneous rate constant, and charge transfer standard potential, respectively, of the ith ion. Vio is similar to a parameter defied by Koryta (28,29),except that it includes the thermodynamics of complex formation between Ci(x) and Cl(x). Equations 1 and 2 correspond to an assumed mechanism of ion transport in which the metal ion in solution is sequestered by the ionophore in the membrane C ; ( x ) + C1(x) + C J x )
(3) The interfacial currents, I ( x ) , given by eq 4,must be zero at the two intefaces in order to maintain electroneutrality. In eq 3 and 4,x = 0 or 1. I ( x ) = FCz; 3 J X ) = 0 (4) i
Equation 4 can be evaluated from an initial set of concentrations, using various values of V(0) and V(1), until this condition is met. The corresponding values of V(0) and V(1) are the interfacial potentials required to calculate the total potential drop, V , = V(0) V D + V(1) (5)
+
v,
V , is the potential drop between the solutions (in the case of conventional ion-selective electrodes), measured with identical reference electrodes. v D is the "diffusion potential" across the interior of the membrane. In the present case, it results from differences in mobility between ions of the same sign. Since sites are fixed, both of the mobile ions must move with the same velocity in order to maintain electroneutrality. VD is therefore generated such that the fieldlmobility products (velocities) of the ions are the same, but with opposite sign. The diffusion potential can be calculated by using an algorithm described previously (30)and assuming zero net current flow. The same algorithm will simultaneously adjust the concentration profiles within the membrane, Ci(f), in accordance with the Nernst-Planck equation
THEORY Consistent with experimental observations reported in the literature (4-7,14-20), interfacial ion transport was assumed to be governed by eq 1 and 2, in the case of conventional ion-selective electrodes (those constructed with free standing liquid or ion-exchange membranes), or just by eq 1, with 3,(1) = 0 and V(1) = 0, in the cases of ISFETs and CWEs.
3,(0) = k,,(C ,(0) 6, (0)/ C,) e-a,zJ( V(0)-Vto) - 6, (0)e (1-&J(
,
electric field at f . Of course, 0 < f 1and v D is the negative of the integral of E(%)from f = 0 to f = 1. It can be shown (30) that if f(f) = 0 dei(%)
CZiDiA s
E(x) = V(O)-V,")
(1)
Y,(l) = -kn,,((Cl(1)~ l ( l ) ) / C o ) e ~ ~ z f ( v (-l ) + v ~ )
c
Di is the diffusion coefficient of the ith ion, and E ( x ) is the
11 ( l ) e - ( l - a , ) z f ( V ( l ) + V , ' )
(2)
In these equations 3 , ( x ) is the flux of the ith ion at the left ( x = 0) or right ( x = 1)surface of the membrane. In the case
i
U I
CZi2DiCi(f)
(7)
i
assuming low concentrations so that activity coefficients are close to unity. The algorithm developed for the calculations shown below (see Figure 1)uses the Newton-Raphson method (31)to solve eq 4 separately for V(0) and V(1), calculates Si(0)and Si(1) from these voltages, and then calculates gi(a)for 0 < % < 1
ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989
I-h
2343
total salt concentrations and dielectric constants. Bars distinguish organic phase parameters. Notice in eq 8 that an apparent a value is predicted, which depends upon the actual value of a (probably close to 0.5) and the relative dielectric constants and salt concentrations in the two phases. If a = 0.5, Cmmp= 10 pF, zOrK= 10, and = 0.1 mM, then eq 8 predicts aapp values that range from 0.137 at C* = 1mM to 0.054 at C* = 100 mM. A value of 0.1 was therefore chosen for this parameter. The a that appears in eq 1 and 2 is understood to be this apparent value. Further evidence to support this contention appears in ref 32, which describes a theoretical treatment of high voltage membrane polarization experiments which require a small value of a. Notice that eq 1reduces to the Nernst equation in the h i t that 3i(0)/k8,itends to zero
Newton-Raphson Calculation Of V(0) and V(1) Eqs. 1,2, and 4
I
c*
Calculation 01 lntertacial Fluxes Eqs. 1 and2
Calculation Of Bulk Fluxes and
VD- Ref. 30
1/11) Calculation Of Membrane Potential. Vm
In the event that two ions, i and j , are exchanging at x = 0, the potential drop must also be given by
Flguro 1. Flow chart of the algorithm used for the calculations.
using the algorithm described in ref 30 for solving the Nernst-Planck system of equations. This procedure provides so that V, can be calculated by using eq 5. The procedure is repeated as many times as necessary to advance to time t = nAt, where n is the number of iterations. Obviously, the algorithm is quite simple; however, its simplicity is purchased at the expense of computation time. In order to maintain convergence and accuracy, At must be quite small. If t = 40 8, n is usually in excess of 10000. An IBM 3090 mainframe computer was used to perform the calculations. Input parameters required for the calculations include Di, k,j, ai,and the initial concentrations of ions in the membrane. Values for these parameters can be found in the literature only for various potassium/valinomycin membranes (4-7,14-20). From these references, Di can be assigned a value of 2 X lo4 cm2/s, the concentration of fixed sites in the membrane will be 0.1 mM, and the initial concentration of valinomycin in eq 1 and 2) will be 10.0 mM. These values should be representative of real situations. In addition, heterogeneous rate constants, k8,i,have been measured, with some controversy (14,15),and can generally be assigned values that range between about lo-' and cm/s for different ions. From data presented by Xie and Camman (19),see the K+ curve in their Figure 6, CY should equal about 0.15. This rather low value is probably due to charge transfer between phases that have dissimilar dielectric constants (14),a situation that has been treated by Buck and Bronner (32). They derive eq 8
Equating 11 and 1 2 and rearranging shows that -zFVo = R T In ( K ) = AGO
(13)
where z = zi = zj (although zi = zj is not a necessary assumption), V" = Vio- Vj0,AGO is the Gibbs free energy, and K is the equilibrium constant for the ion-exchange reaction cj
+ Cil ci + Cjl
(14)
Obviously, Vioand Vjoare purely thermodynamic parameters that influence kinetics only because eq 14 is second order and they influence the concentrations of the complexes. The value of Vo was allowed to vary between k0.05 V for illustrative purposes. Much larger variations are possible (28,29); however, they would not be easily accommodated by the simple algorithm employed. Notice that eq 1-7 can be used to derive the well-known result
(cl(x)
where and
In eq 9, Cmmpand Cdl are the compact and diffuse double layer capacitances, respectively, while C* and c in eq 10 are
in the limit that interfacial ion-exchange kinetics are fast (3, 25). Clearly, the present theory is an extension of previous
work into the time domain and reduces to well-known results in the limit of fast interfacial kinetics. Notice that Djand Di virtually never differ by more than a factor of 2, explaining why mobility differences have so little effect on selectivities. The foregoing analysis of interfacial potential differences is virtually identical with that suggested by Janata and Blackburn (341,and Cammann (35),and is the ionic analogue to the mixed-potential problem encountered with electronconducting, metallic, electrodes (36)with one important exception. Metallic electrodes exchange electrons, and zero current means zero net exchange of electrons. The ion-selective electrode is an ion exchanger, and zero current means one ion, the interferent, is coming in at the same rate a different ion, the analyte, is going out. Ion-selective electrodes can use only the ions already within their membranes or internal filling solutions. When these are exhausted the mixed-potential can no longer be maintained. This problem becomes particularly severe in the case of ISFETs and CWEs because they do not have internal filling solutions and their membranes are usually quite thin. These issues will be discussed shortly.
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 21, NOVEMBER 1, 1989
20 t rec
x,cm ( x i 0 3,
40
Flgure 3. (A) Voltage transient for conventional ISE when sample analyte concentration = 10 mM, interferent concentration = 1 M, ks,, cm/s. See text for other parameters. = cmls, and k S J= (6)Concentration profiles after 40 s.
Flgure 2. Simulated concentration profiles through a membrane used
in a conventional ISE. Each profile corresponds to a different time after expowre of the electrode to a sample solution with equal, 10 mM, concentrations of analyte and interferent ions. Times correspond to 0.3125, 0.625, 1.5625, 3.125, 6.25, and 15.625 s from outer to inner curves. See text for further details.
RESULTS Ion-Selective Electrodes. Conventional (free standing membrane) ion-selective electrodes operate on the basis of a transport mechanism (3) in which the transference number of the analyte ion is close to unity while the transference numbers of all other ions, especially those of opposite sign, are close to zero. The ion-selective membrane is then viewed as a "liquid junction" that obeys eq 16.
The ion-selective membrane separates two solutions. One is the internal filling solution, the other is the sample, and the integral extends from a place in the sample where the potential is nominally zero to a place in the internal filling solution where it is V,, relative to whatever reference electrodes are used. Distance dependent activities, a&), ensure that eq 16 is absolutely rigorous. Notice that eq 7 is the finite difference form of eq 16 assuming dilute solutions and concentration-independent diffusion coefficients. Transference numbers, t i , are given by
The transference number of an interfering ion is usually s m d because the concentration of this ion (in the membrane) is limited by its ion-exchange coefficient-see eq 13 and 14. Notice that ti has a bar in eq 17, but not in eq 16. Equation 17 refers only to transference through the bulk of the membrane, while eq 16 is more general, and ti there refers to transference across the interfaces as well. If the sample contains 10 mM concentrations of analyte and interfering ions at time t = 0, the membrane is 10 Km thick, the ion-exchange coefficient is unity, and the diffusion coefficient, heterogeneous rate constant, and a value of the ions are 2 X lo4 cm2/s, cm/s, and 0.1, respectively, then the concentration profiles of the ions should appear as shown in Figure 2, according to the algorithm used in this study. After about 15 s a steady-state profile should be established in which interfering ions transit the membrane from the sample on the left to the internal filling solution on the right and displace analyte ions, which have a concentration of 0.1 M in the internal filling solution, back through the membrane to the sample solution.
Notice that the surface concentrations of analyte and interfering ions are equal at all times, indicating that interfacial kinetics are fast enough to replenish interfering ions that have diffused into the membrane even when the concentration gradients are steep. Under these conditions the potential is expected to reach a "pseudo steady state", which means that it does not vary with time even during the 15 or so s required to reach true steady state. The reason is that the integral in eq 16 is "exact". Basically, we are dealing with a thermodynamic function, V,, which is independent of the path of integration. Therefore, ai(x)can vary in eq 16 without changing the value of V,, a principle that has been understood for some time (25). The algorithm behaved properly and indicated a time-invariant potential of -0.0412 V-exactly what one would expect under these conditions (Ki,J'",= 1). Next, the concentration of interferent was increased to 1 M, while its heterogeneous rate constant was decreased to lo3 cm/s. This is not a "large" decrease, given the results (and arguments) presented by Xie and Camman (15),who claim that the standard apparent exchange current densities of potassium and sodium ions between aqueous solutions and dibutyl sebacate/valiiomycin/PVC membranes is in the ratio of 104:l. The heterogeneous rate constant for sodium exchange could then be as small as lo-' if the heterogeneous rate constant for potassium exchange is Figure 3A shows the predicted potential transient under these conditions and Figure 3B shows the concentration profiles of analyte and interferent ions after 40 s. The transient shows a peak that decays to a steady-state value only after the concentration profiles have established themselves. Such responses are commonly observed (37) when strongly interfering ions enter the membrane. The higher concentration of the interfering ion drives the potential past the steady-state value even though its heterogeneous rate constant is small. Eventually, equilibrium is established at the surface, however, and the potential decreases toward a constant value. Notice that this result is obtained with equal diffusion coefficients. Given that K = 1 (eq 13) and the diffusion coefficients are equal, K,Nt should be unity (eq 15) and the equilibrium potential should be 0.0593. However the calculated potential is 0.0536, indicating a small, but significant, improvement in selectivity. Obviously,the smaller value of the heterogeneous rate constant has influenced the gross transference numbers of the ions, thus enhancing selectivity. Equation 17 no longer applies because it accounts only for the diffusional contributions to ti, and there are kinetic contributions as well. Figure 4 shows a series of "interference curves" for different values of Vo (eq 13),thickness, and the ratio of heterogeneous rate constants (ks,i/hsj,where i is the analyte and j is the interferent). These curves were obtained by calculating
ANALYTICAL CHEMISTRY, VOL.
0 001
0 100
OM0
10
I
Cj, M
Flgure 4. Simulated "interference curves'' for various conventional ISEs. Dlffusion coefficients = 2 X lo-' cm2/s in all cases and k8,, = cm/s: (1) V o = 0.05V, I = 10 pm, k,, = (2)V o = 0 v, 1 = 10 pm, k,J (3) v o = -0.05 v, I = 10 pm, kSJ= lo-'; (4) V" =: 0.05 V, I= 10 pm, k , = lo-': (5)V" = 0 V, I= 10 pm, k,, = lo-'; (6)V o = -0.05 V, 100 pm, k,, =
!=
10.0,
,
.
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,
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.,,
.
,
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. , .,
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.
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61,NO. 21,NOVEMBER 1, 1989
2345
lectivity is large (small KilJOt). Curves 2 and 5 indicate that the effect is larger when KiJwt = 1. Apparently, selectivity will be influenced primarily by thermodynamic effects when V" is large (high selectivity, small &POt) unless the heterogeneous rate constant for exchange of analyte ions is exceptionally slow. Such cases have been observed by workers at Kodak (40),who found that ionophores with high binding constants toward metal ions were sometimes inappropriate for use in ion-selective electrodes because of slow interfacial ion-exchange kinetics. Similar results have been reported for cryptands (41, 42). Curves 3 and 6 show that membrane thickness has an enormous, unexpected, and previously unexplained effect on selectivity. A thicker membrane (curve 6, thickness 100 pm) has close to the thermodynamic selectivity, independent of concentration,while its thinner counterpart (curve 3, thickness 10 pm) shows the maximum variation of selectivity with concentration of any result shown. The explanation is simple. The thinner membrane allows interfering ions to diffuse through it and into the inner filling solution a t a rate that competes with their rate of arrival at the interface by ion exchange. The result is a lower interferent concentration on the membrane side of the sample/membrane interface than would occur at thermodynamic equilibrium. A steady-state situation prevails. Again with regard to the thinner membrane, the sample solution can supply ions faster at higher concentrations and thermodynamic equilibrium can be enforced at the interface. The thermodynamically defined, larger selectivity coefficient then results at these higher concentrations. All of the potentials shown in Figures 4 and 5 were obtained after steady state was obtained. While this required only about 2 min in the case of curve 3 (thin membrane), almost an hour was required in the case of curve 6 (thick membrane), indicating a coupling of interfacial ion-exchange kinetics with diffusional effects. Under these conditions, thinner membranes should provide not only higher selectivity but faster response as well. The selectivities of thicker membranes may appear time dependent. Detection limits of thin membranes may be higher, however, because analyte ions will be forced from the inner filling solution, back through the membrane, and into the sample by the invading interferent. In fact, increased detection limits were observed at Kodak (43) by workers developing the t h i n - f i ion-selectiveelectrode technology used in the Kodak Ektachem clinical analyzer (44, 45). This system employs 10-pL drops of sample solution, the concentrations of which were increased by potassium ions forced out of the electrode (valinomycin in PVC) by high concentrations of sodium. The situation is further complicated by the fact that high osmotic pressure can force salts from the internal reference layer, in the Kodak system, back out into the sample solution when sample ionic strength is low. Although these effects were large enough to be noticeable, they were not large enough to compromise the assay. A final word is in order concerning the calculations shown in Figures 4 and 5. These results indicate a deficiency in eq 17, which only accounts for the transference of ions through the bulk of the ion-selective membrane, implying that ionic concentrations and mobilities alone are enough to define transference numbers. The calculations shown in Figures 4 and 5 indicate that transference numbers are also influenced by interfacial ion transport kinetics (activation resistances) and that these parameters ultimately define selectivities when the thermodynamic ion-exchange parameters, which determine membrane ionic composition, are close to unity. Ion-Selective Field Effect Transistors and Coated Wire Electrodes. Ion-selective field effect transistors (IS-
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ANALYTICAL CHEMISTRY, VOL. 61,NO. 21, NOVEMBER 1, 1989
-oo 4
-0 02
-0 03
2
>' -0 05
-006
3 4
1
I
I
-007 0
20 d
f,
40
t, sec
x,cm ( x ~ o - ~ )
F m e 8. Shnulated concentration profiles analogous to those shown in Figure 2 except for ISFETs or CWEs. Times correspond to 0.3125, 0.625, 1.5625,3.125,6.25,15.625,31.25,62.5,and 125 s from outer to inner curves.
0
I
20
40
sec
Figure 7. Simulated transients for various ISFETs or CWEs. Analyte and interferent concentratlons = 10 mM in all cases and k , = k B / . Membrane thickness = 10 pm, a = 0.1,and V o = 0: (1)d, = 1 X 10-8, D, = 2 x 10-8, k , , = 10-3;(2)D/ = 1 x io", D, = 2 x io", k*,/ = 10-4;(3)D/ = 2 x lo", D = 1 x 10-8,ks,/= 104;(4)0,= 2X 0,= 1 X k8,/= (5) same as curve 1, dlffusion potential contributlon only; (6)same as curve 2, diffusion potential contribution only.
FETs) and coated wire electrodes (CWEs)differ from conventional ion-selective electrodes in that, in principle, they do not have internal filling solutions and therefore cannot support the kinds of concentration profiles shown in Figure 2. Conventional pH-sensing glass electrodes may also fall into this category since they cannot transport protons through their interiors (9,46). The concentration profiles shown in Figure 6, calculated by using the same parameters as used in Figure 2 except that the right-hand interface (with the gate material or the wire) is blocked, should be typical of ISFETs and CWEs. Now the membrane coating becomes depleted of analyte ions, which are displaced by interferent ions, in accordance with thermodynamic requirements. The net effect is that potentials measured after all transients have died out will reflect true equilibrium values, and ion-exchange kinetics can only influence the time required for the sensors to reach equilibrium, not their selectivities. Figure 7 shows some transients calculated with various values of the diffusion coefficients and heterogeneous rate constants. Curves 1 and 2 pertain to the situation where the diffusion coefficient of the complexed interferent is greater than that of the complexed analyte or the ligand. In any case, the effect of changing the value of the heterogeneous rate constant from 10-3 cm/s (curve 1) to 10"' cm/s (curve 2) is
Flgure 8. Simulated transients analogous to those shown In Flgure 7 except the membrane is now employed in a conventional ISE. minimal, and the potential approaches the thermodynamically defmed value after a positive transient. Curves 3 and 4 pertain to the situation where the diffusion coefficient of the complexed interferent is less than that of the complexed analyte or the ligand. The potential again approaches the thermodynamically defined value after a transient, except now the transient is negative. Curves 5 and 6 plot the diffusion potential contributions to the total membrane potentials shown in curves 1and 2 ( VD in eq 5). They reveal that the transients are due only to diffusional parameters, coupled with interfacial kinetics. Obviously, a large number of calculations would be required to determine the relationships between the shapes of the transients and values of the various parameters-the diffusion coefficients, hetereogeneous rate constants, etc. For the present, however, it is enough to point out that the thermodynamic potential will always be ultimately achieved, and that the transients will always require more time in the cases of ISFETs and CWEs than for their ISE counterparts. This is illustrated in Figure 8, which shows ISE responses corresponding to the ISFET/CWE cases simulated in Figure 7. Notice that the final potential, and therefore the selectivity, now depends upon the diffusion coefficient ratios, as expected (seeeq 15). Smaller values of the heterogeneous rate constants result in slower approaches to steady-state potentials, a small enhancement of selectivity when the analyte ion diffuses more slowly than the interferent, and a small degradation of selectivity when the analyte ion diffuses more rapidly. Since the mechanism whereby stable potentials are generated at the interface between the membrane and the gate or wire is not necessarily well-defined, one might entertain the possibility that an "internal filling solution" might form there by default, due to influx of water. Alternatively, the electrode may have a thin internal reference layer (44,45), the composition of which may influence, or be influenced by, ion exchange with the sample, driven by interferents, as described above. These situations represent complications that would require individual attention. For the sake of argument, it is therefore assumed that no internal filling solutions are involved and that the potential drop at the membrane/gate interface can be arbitrarily defined as zero. Conclusions will obviously be influenced by the degree to which this assumption is valid.
CONCLUSIONS The selectivities of ISFETs and CWES should depend solely upon thermodynamic parameters while the selectivities of conventional ISEs should depend upon kinetic parameters and diffusion coefficient ratios as well. The influence of kinetic parameters is greater for thin membranes than for thick ones.
Anal. Chem. 1989, 61 2347-2352
As expected, slow ion-exchange kinetics increase the response times for all types of potentiometric sensors. However, the effect is greater for ISFETs and CWEs than for conventional ISEs because these devices must fill with interferent ions, up to the equilibrium concentration, in order to achieve the equilibrium potential-a prediction which contradicts conventional wisdom. ISEs can achieve analytically useful “pseudoequilibrium” potentials as soon as ion-exchange kinetics establish constant interfacial concentrations because of the “exactness” of the integral in eq 16.
LITERATURE CITED Teorell, T. Roc. SOC.Exp. 6/01.Med. 1935. 33, 282. Meysr. K. H.; Sievers, J.-F. Hdv. Chim. Acta 1938, 19, 649. Morf. W. E. The principles of Ion-Selective E l e c W s and of Membrane Tf8nsPorf;Eisevier: New York, 1981. Iglehart, M. L.; Buck, R. P.; Pungor, E. Anal. Chem. 1988, 60, 1018. Thoma. A. P.; VhrlanCNauer, A.; Arvanitis, S.; Morf, W. E.; Simon, W. Anal. Chem. 1977, 49. 1567. Armstrong, R. D.;Nikitas, P. Electrochim. Acta 1985, 30, 1627. Iglehart, M. L.; Buck, R. P.: Pungor. E. Anal. Chem. 1988, 60, 290. Arnokl, M. A.; Zisman, S. A.: Hise, S. M. Anal. Chim. Act8 1988, 187, 17. Sandifer, J. R. Anal. Chem. 1988, 60, 1553. Otto, M.; Thomas, J. D. R. Anal. Chem. 1985, 57, 2647. Beebe, K.; Uerz, D.; Sandlfer, J.; Kowalskl, B. Anal. Chem. 1988, 60. 66. Beebe, K.; Kowalski, B. Anal. Chem. 1988, 60, 2273. Berube, T. R.; Buck, R. P.; Lindner, E.; (katzei. M.; Pungor, E. Anal. Chem. 1989, 81, 453. Armstrong, R. D. J. Electrwnal. Chem. 1988, 245, 113. Xle, Shengluo; Camman, K. J. Elecmnal. Chem. 1988, 245, 117. Armstrong, R. D. Electrochim. Acta 1987, 32, 1549. Armstrong, R. D.;Todd, M. Electrochim. Acta 1988, 31, 1413. Armstrong, R. D.;Lockart. J. C.; Todd, M. Electrochim. Acta 1988, 31, 591. Xie, Shengluo; Camman, K. J. Electrwnal. Chem. 1987. 229, 249.
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Crawley, C. D.;Rechnitz, G. A. J. Akmbr. Sei. 1985, 24, 201. Sandifer, J. R. Roc. Sens. Expo. 1988, 1 , 179. Janata, J.; Huber, R. J. Chemically Sensitive Field Effect Transistors; Ion-Selective Electrcd8s in Analytical Chemistry;Frelser. H., Ed.; Plenum Press: New York. 1980; Vol. 11, p 107. Sibbaid, A. IEEE Roc. 1980, 130, 233. Cattrali, R. W.; Hamilton, I. C. Ion-Sel. E k t r o a , Rev. 1984, 6 , 125. Eisenman. G. In Ion Selective Electrodes; Durst, R. A., Ed.; NBS S p e cia1 Publication; National Bureau of Standards: Washington, DC. 1969 Chapter 1. Moody, 0. J.; Thomas, J. D. R. T8slenta 1971. 18, 1251. Moody, G. J.; Thomas, J. D. R. Taslenta 1972. 19, 623. Koryta, J. Electrochim. Acta 1979, 24, 293. Koryta, J. Electrochlm. Acta 1984, 29, 445. Sandifer, J. R.; Buck, R. P. J. phvs. Chem. 1975, 79, 384. Hart, H. L. Analytical Geomeby and Ca/ccrhrs, 2nd ed.;D.C. Heath and Company: Boston, MA, 1963; p 264. Buck, R. P.; Bronner, W. E. J. Electroanal. Chem. 1988, 197, 179. Sandifer, J. R.; Iglehart, M. L.; Buck, R. P. Anal. Chem. 1989. 61, 1624. Janata, J.; Blackburn, G. F. Ann. N . Y . Acad. Sei. 1984, 428, 286. Cammann, K. Topics in Current Chemistry; Springer-Verlag: New York, 1985; p 219. Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamntab end Applhtbns; Wiiey: New York, 1980; Chapter 3. Fiedier, U. Anal. Chim. Acta 1977, 89, 101. Buck, R. P. Anal. Chlm. Acta 1974, 73, 321. Buck, R. P.; Sandifer, J. R. J. phvs. Chem. 1973, 77, 2122. Daniel, D. S.; Toner, J. L., Eastman Kodak personal communication. 1989. Lehn, J.-M. Science 1985, 227, 849. Parker, D. I n Advances in Inorganic Chemistry and Radiochemistry; Academic Press: New York, 1983; Vol. 27, p 1. Battaglla, C. J.; Chang, J. C., Eastman Kodak personal communication. 1969. (44) C k m . Eng. News 1980, 33. (45) Wang, J. I n Electrwnalytcal Techniques in Cllnhl Chemistry and Labwatoty Medicine; VCH Publishers, Inc.: New York, 1988. (46) Johanasson, G.; Kariberg, B.; Wlkby, A. T 8 h t a 1975, 22, 953.
RECEIVED for review April 10,1989. Accepted July 26,1989.
Observation of Concentration Profiles at Cylindrical Microelectrodes by a Combination of Spatially Resolved Absorption Spectroscopy and the Abel Inversion Huan Ping Wu and Richard L. McCreery* Department of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, Ohio 43210
A prevlously reported spatially resolved absorptlon technlque was enhanced to determlne concentration vs dlstance profiles, C ( r ) , for optlcal absorbers generated at a cyllndrlcal mlcrowlre electrode. A collimated, monochromatlc beam passed the mkrowlre perpendlcular to Its axls and then was magnlfled and Imaged onto a h e a r diode array detector. Each pixel of the detector monitored the llght lntenslty In a 1.2-pm eegment of the dknenslon perpendkular to and lateral of the wlre axls. Thls spatlally resolved lateral absorbance proflle Is quantltatlvely related to C ( r ) for mlcrowlres wlth radll of 6-25 pm and electrolysis times of 50 ms to several seconds. The dlffueion fleld was extremely sensltlve to convectlon, wlth devlatlons from solely dlffuslve mass transport often belng severe In low-vlscoslty solvents. The results Indlcate that posltlve devlatlons from expected current trandents to mlcrowlre electrodes may be common, wtth 0-25 % errors belng observed for a 6-pm-radlus wlre.
* Author to whom correspondence should be addressed. 0003-2700/89/0361-2347$01.50/0
INTRODUCTION A variety of special features has stimulated a broad research effort into the behavior of very small voltammetric electrodes (1-5). These unusual properties include steady-state diffusion-limited current, a nonplanar diffusion field, high mass flux of solution species, rapid dilution of electrogenerated products, low ohmic potential error, low cell time constant, and applications in very small volumes of electrolyte. The first four of these features are derived from the nature of diffusion when the electrode size becomes small relative to (Dt)1/2, where D is the diffusion coefficient of the electroactive species and t is the time scale of the experiment. The consequences of nonplanar diffusion, particularly the steady-state current at the diffusion limit, have been discussed thoroughly elsewhere and need not be reported here. Of particular relevance, however, is the case of a free-standing microcylinder electrode, such as a carbon fiber or fine metal wire electrode (5-7). An initially unrelated effort in our laboratory is the development of spatidy resolved spectroscopic probes of planar 0 1989 American Chemical Society
