Anal. Chem. 1999, 71, 1204-1209
Lowering the Detection Limit of Solvent Polymeric Ion-Selective Electrodes. 1. Modeling the Influence of Steady-State Ion Fluxes Tomasz Sokalski,*,†,§ Titus Zwickl,† Eric Bakker,*,‡ and Erno 1 Pretsch*,†
Department of Organic Chemistry, Swiss Federal Institute of Technology (ETH), Universita¨ tstrasse 16, CH-8092 Zu¨ rich, Switzerland, and Department of Chemistry, Auburn University, Auburn, Alabama 36849
The processes determining the lower detection limit of carrier-based ion-selective electrodes (ISEs) are described by a steady-state ion flux model under zero-current conditions. Ion-exchange and coextraction equilibria on both sides of the membrane induce concentration gradients within the organic phase and, through the resulting ion fluxes, influence the lower detection limit. The latter is shown to improve considerably when very small gradients of decreasing primary ion concentration toward the inner electrolyte solution are created. By merely altering the concentration of the inner electrolyte, detection limits may vary by more than 5 orders of magnitude. Very large gradients, however, are predicted to lead to significant depletion of analyte ions in the outer membrane surface layer and thus to apparent super-Nernstian response. The currently recommended IUPAC definition of the lower detection limit leads to nonrealistic values in such cases. Small changes in the concentration profiles within the membrane may have large effects on the response of the ISE at submicromolar levels and enhance its sensitivity to interferences during trace determinations. The model studies presented here demonstrate that trace level measurements with ISEs are feasible but often require higher membrane selectivities than expected from the Nicolskii equation. In most cases described so far, the lower detection limit of solvent polymeric membrane-based ion-selective electrodes (ISEs) lies in the micromolar range.1-3 Significantly lower values were found only when analyte ion concentrations were kept under control with the help of ion buffers,4,5 whose effect is most likely due to their complexing the analyte ions that leach from the membrane. In their absence, the lower detection limit is governed †
Swiss Federal Institute of Technology. Auburn University. § On leave from Department of Chemistry, Warsaw University, ul. Pasteura 1, PL-02-093 Warsaw, Poland. (1) Umezawa, Y. Handbook of Ion-Selective Electrodes: Selectivity Coefficients; CRC Press: Boca Raton, FL, 1990. (2) Bakker, E.; Bu ¨ hlmann, P.; Pretsch, E. Chem. Rev. 1997, 97, 3083-3132. (3) Bu ¨ hlmann, P.; Pretsch, E.; Bakker, E. Chem. Rev. 1998, 98, 1593-1687. (4) Schefer, U.; Ammann, D.; Pretsch, E.; Oesch, U.; Simon, W. Anal. Chem. 1986, 58, 2282-2285. (5) Sokalski, T.; Maj-Zurawska, M.; Hulanicki, A. Mikrochim. Acta 1991, 285291. ‡
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by this leaching process, which may cause analyte ion concentrations at the phase boundary to be significantly higher than in the sample bulk.6,7 One possible origin of this bias is the salt coextraction from the inner electrolyte solution into the membrane, which generates a primary ion flux toward the sample.8 Based on experiments with various inner solutions of relatively high concentration and/or containing lipophilic salts, such coextraction processes were shown to be the reason the lower detection limit was shifted upward.8 With other internal filling solutions, however, no correlation between calculated and observed detection limits was found, showing that additional effects must be taken into account. Indeed, there is experimental evidence for analyte ions being transported through the membrane owing to ion exchange and countertransport of interfering ions.9,10 Very recently, detection limits in the picomolar range were obtained with inner solutions whose concentration of analyte ions was buffered to a low level while keeping that of the interfering ones high.11 On the other hand, if the latter was kept low as well, no improvement was reached.12 These observations were interpreted in terms of an ion-exchange process causing a flux of analyte ions toward the inner solution, so that the above-mentioned leaching into the sample can no longer occur.11 In this paper, a quantitative model is presented that describes the underlying processes in ISE membranes at steady state. Although such a steady-state model is valid in limited cases only, it can be used to qualitatively understand the basic factors influencing the lower detection limit of ISEs. The companion paper presents experimental results with Pb2+- and Ca2+-selective ISEs that can be interpreted in terms of this model.13 THEORY The present study is based on the phase boundary potential model considering ion-exchange and coextraction equilibria2 on (6) Morf, W. E.; Kahr, G.; Simon, W. Anal. Chem. 1974, 46, 1538-1543. (7) Hulanicki, A.; Lewenstam, A. Talanta 1976, 23, 661-665. (8) Mathison, S.; Bakker, E. Anal. Chem. 1998, 70, 303-309. (9) Erne, D.; Morf, W. E.; Arvanitis, S.; Cimerman, Z.; Ammann, D.; Simon, W. Helv. Chim. Acta 1979, 62, 994-1006. (10) Maj-Zurawska, M.; Erne, D.; Ammann, D.; Simon, W. Helv. Chim. Acta 1982, 65, 55-62. (11) Sokalski, T.; Ceresa, A.; Zwickl, T.; Pretsch, E. J. Am. Chem. Soc. 1997, 119, 11347-11348. (12) Bakker, E.; Willer, M.; Pretsch, E. Anal. Chim. Acta 1993, 282, 265-271. (13) Sokalski, T.; Ceresa, A.; Fibbioli, M.; Zwickl, T.; Bakker, E.; Pretsch, E. Anal. Chem. 1999, 71, 1210-1214 (following paper in this issue). 10.1021/ac980944v CCC: $18.00
© 1999 American Chemical Society Published on Web 02/12/1999
one hand, and on flux equations under stationary-state and zerocurrent conditions on the other.14,15 To investigate basic effects, the number of parameters is kept low by making a series of simplifications, namely that ionic phase-transfer reactions are much faster than corresponding diffusion processes so that the actual sample-membrane phase boundary is at local equilibrium, that ion pairs within the organic membrane are not considered, that the ISE response can be described by the phase boundary potentials only, that lipophilic ionic sites are homogeneously distributed across the membrane, and that all complexes in the membrane are sufficiently strong so that fluxes of uncomplexed ions can be neglected. It is also assumed that the system is always at steady state so that all concentration profiles are linear. These assumptions were already used in the phase boundary potential model2 or in treatments describing detection limits for solid-state6,7 and liquid membrane ISEs.16 The assumption of steady-state conditions is here not problematic as long as the phase boundary equilibrium is fast enough. The analogous treatment of polyion sensors in which ion fluxes are essential for the response also shows the validity of this approach.17 In this work, the following additional simplifying assumptions were used: (1) The system contains only two monovalent cations (I+, J+) that form 1:1 complexes with the ionophore. (2) Only one kind of monovalent anion (A-) is present in the aqueous solutions. Its concentration in the Nernst layer is constant and the same as in the bulk of the aqueous phases. The lipophilic anionic sites in the membrane are also singly charged. (3) Activity coefficients are set to unity for the entire system so that concentrations are used in all calculations. (4) The ionic concentration of each species in the inner filling solution is constant (e.g., by using a buffer or high concentration). (5) Diffusion coefficients are constant in each phase; moreover, they are the same for all diffusing species in the membrane phase. In the following, the equations for the phase boundary potential and the equilibria are given first using the common notations.15 The analyte ion, I+, and interfering ion, J+, form the complexes IL+ and JL+ with the ionophore L. As in earlier treatments,2 the measured potential may be described by taking into account the phase boundary potentials at the two membrane/aqueous solution interfaces:
EMF ) E° +
RT cI′[L]′ RT cI′′[L]′′ ln ln F [IL+]′ F [IL+]′′
(1)
At the aqueous solution/membrane boundaries, the ion exchange of analyte and interfering ion between sample and membrane may be described by the overall ion-exchange constant, Kexch:
Kexch )
[JL+]′cI′ [IL+]′cJ′
)
[JL+]′′cI′′
(2)
[IL+]′′cJ′′
The coextraction of analyte cations and counterions from the aqueous phases into the membrane is quantified using the overall coextraction constant, Kcoex:
Kcoex )
[IL+]′[A-]′ [IL+]′′[A-]′′ ) cI′cA′[L]′ cI′′cA′′[L]′′
(3)
Further, the mass balance of the ionophore and the charge balance is applied by considering the simultaneous occurrence of the complexes of both ions, the coextracted anion A-, and lipophilic anionic sites. The general Nernst-Planck flux equation reduces to Fick’s first law of diffusion by neglecting convection and migration terms. For a one-dimensional steady-state system,
∆ci Ji ) -Di ∆x
(4)
where, for each phase, Ji is the flux of the species i (mol cm-2 s-1), Di its diffusion coefficient (cm2 s-1), and ∆c/∆x its concentration gradient. Since the law of mass conservation requires equal fluxes in each phase, it follows from eq 4 that
DI,aq DIL,m ) ([IL+]′′ - [IL+]′) δaq δm
(5)
DJ,aq DJL,m ) ([JL+]′′ - [JL+]′) δaq δm
(6)
(cI′ - cI(bulk)) (cJ′ - cJ(bulk))
with δaq and δm denoting the thicknesses of the aqueous Nernstian boundary layer and of the membrane, respectively. By applying assumptions 1 and 5 to the general form of the law of mass conservation for a ligand giving 1:ni complexes with any i, the flux equation reduces to
[IL+]′′ - [IL+]′ + [JL+]′′ - [JL+]′ + [L]′′ - [L]′ ) 0
(7)
where EMF is the measured potential, all constant potential contributions being included in E°, and cI, [L], and [IL+] are the sample concentrations of the primary ion and the membrane concentrations of free ionophore and its analyte ion complex, respectively, with R, T, and F having their usual meanings. The sample boundary layer and the inner side of the ISE membrane are symbolized by prime and double prime, respectively.
Finally, the zero-current condition for species i of charge zi requires
(14) Brumleve, T. R.; Buck, R. P. J. Electroanal. Chem. 1978, 90, 1-31. (15) Morf, W. E. The Principles of Ion-Selective Electrodes and of Membrane Transport; Elsevier: New York, 1981. (16) Ishibashi, N.; Imato, T.; Yamauchi, M.; Katahira, M.; Jyo, A. In Ion-Selective Electrodes, 4; Pungor, E., Buza´s, I., Eds.; Akade´miai Kiado´ and Elsevier Science Publishers: Budapest and Amsterdam, 1985; Vol. 22. (17) Fu, B.; Bakker, E.; Yun, J. H.; Yang, V. C.; Meyerhoff, M. E. Anal. Chem. 1994, 66, 2250-2259.
[IL+]′′ - [IL+]′ + [JL+]′′ - [JL+]′ + [A-]′′ - [A-]′ ) 0
∑z J ) 0
(8)
i i
i
which with the model used here simplifies to
(9)
CALCULATIONS All calculations were done with Mathematica 3.0 on a Silicon Graphics Octane, Irix 6.4, workstation. Explicit results for selected Analytical Chemistry, Vol. 71, No. 6, March 15, 1999
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cases are given below. The basic example was computed using the following parameters: RT ) 1 × 10-4 M, LT ) 2 × 10-4 M, -5 log Kpot IJ ) log Kexch ) -11, log Kcoex ) -1, DI,aq ) 1.356 × 10 2 -1 + + -5 2 -1 + + cm s (Na as I ), DJ,aq ) 9.321 × 10 cm s (H as J ), DIL,m ) DJL,m ) DA,m ) DL,m ) 5 × 10-7 cm2 s-1, δm ) 100 µm, δaq ) 60 µm, aJ′′ ) 0.1 M, and aJ(bulk) ) 10-7 M. The Kexch value mimics the high selectivity of ISEs for heavy metals such as silver ions,18 while Kcoex values were previously determined to be in this range for valinomycin-based electrodes.8 The concentration of the primary ion in the inner solution, cI′′, was varied in steps of one decade from 1 to 10-15 M. For each ISE, the EMF response was calculated by varying the primary ion concentration in the sample bulk, cI(bulk), from 1 to 10-15 M in steps of 0.1 logarithmic units. First, the surface equilibrium concentrations of all relevant species in the inner boundary layer of the membrane ([IL+]′′, [JL+]′′, [L]′′, [A-]′′) were computed by solving eqs 2 and 3, with appropriate mass and charge balances. Using these values, the corresponding concentrations on the sample side ([IL+]′, [JL+]′, [L]′, [A-]′) and in the adjacent Nernstian diffusion layer (cI′, cJ′) were obtained with eqs 2, 3, and 5, 6, 7, 9. Finally, eq 1 yielded the EMF values as a function of cI(bulk). They were normalized to 200 mV for cI(bulk) ) 10-2 M. RESULTS AND DISCUSSION Under zero-current conditions, the flux of primary ions across the membrane is accompanied either by that of coextracted counterions in the same direction or by a counterflux of interfering ions entering the membrane through an ion-exchange process. This is represented in Figure 1, which schematically shows the influence of coextraction and ion-exchange processes on the lower detection limit of ISEs. Scheme A depicts the situation where cations and anions are coextracted from the relatively concentrated inner solution, transported across the membrane, and finally reach the dilute sample. The detection limit is given by the analyte concentration at the phase boundary, which may be substantially higher than that of the sample bulk. Scheme B shows that ion fluxes in direction of the sample may occur even when coextraction from the inner solution is negligible. In this case, the exchange of primary ions by interfering ones induces a concentration gradient and, hence, fluxes through the membrane. In scheme C, on the other hand, a partial ion exchange at the inner side of the membrane may induce a concentration profile of primary ions that decreases toward the internal solution. The corresponding reverse flux may, in principle, counteract the effects shown in A and B and lead to substantially lower detection limits than usually observed. By considering coextraction and ion-exchange processes on both sides of the membrane, EMF functions have been calculated for a series of ISEs having the same membrane but different primary ion concentrations in the range of 1-10-15 M, in the inner solution (Figure 2, curves are labeled with the corresponding negative logarithm). The concentration of the highly discriminated -7 M in the sample and interfering ion, J+ (log Kpot IJ ) -11), is 10 -1 10 M in the inner filling solution. For easy comparison, the curves are shifted so as to have the same EMF value at cI(bulk) ) 10-2 M. At concentrations >10-3 M, the composition of the inner solution has no effect on the ISE response, so that all curves (18) Bakker, E. Anal. Chem. 1997, 69, 1061-1069.
1206 Analytical Chemistry, Vol. 71, No. 6, March 15, 1999
Figure 1. Schematic representation of the processes influencing the lower detection limit of ISEs based on an ionophore (L) forming 1:1 complexes with the monovalent primary (I+) and interfering ions (J+). Gradients are generated in the aqueous Nernstian phase boundary (thickness δaq) because of coextraction of I+ and A- from the inner solution (A), or partial exchange of primary ions by interfering ones at the sample or reference side (B or C, respectively).
Figure 2. Calculated EMF functions for a series of ISEs having the same membrane but different primary ion concentrations, cI′′, from 1 to 10-15 M in the inner solution. Curves are labeled with the corresponding negative logarithm. See Calculations for the other parameters.
coincide. The upper detection limit, given by the coextraction of I+ and A- from the sample into the membrane, is at cI(bulk) ) 10-1.5 M. At low sample concentrations, on the other hand, the primary ion concentration of the inner solution has a considerable effect: by reducing it from 1 to 10-7 M (ISEs 0-7), the lower detection limit gradually improves from 10-5.5 to 10-11.8 M. A further 10-fold reduction in cI′′ (ISE 8) causes an abrupt change in the response function’s shape. For cI(bulk) between 10-9 and 10-10 M, the EMF suddenly drops by about 200 mV and remains
Table 1. Molar Concentrations (c) for the ISEs 0, 2, 7, 8, and 15 as a Function of the Sample Primary Ion Concentration, cI(bulk) sample inner membrane side ISE 0 (cI′′ ) 1 M) [IL+]′′ ) 1.99819 × 10-4 M [JL+]′′ ) 1.99819 × 10-16 M
ISE 2 (cI′′ ) 10-2 M) [IL+]′′ ) 1.43408 × 10-4 M [JL+]′′ ) 1.43408 × 10-14 M
ISE 7 (cI′′ ) 10-7 M) [IL+]′′ ) 1.00000 × 10-4 M [JL+]′′ ) 1.00000 × 10-9 M
ISE 8 (cI′′ ) 10-8 M) [IL+]′′ ) 9.99901 × 10-5 M [JL+]′′ ) 9.99901 × 10-9 M
ISE 15 (cI′′ ) 10-15 M) [IL+]′′ ) 9.99001 × 10-8 M [JL+]′′ ) 9.99001 × 10-5 M
cI(bulk) 1 10-3 10-6 10-9 10-12 10-15 1 10-3 10-6 10-9 10-12 10-15 1 10-3 10-6 10-9 10-12 10-15 1 10-3 10-6 10-9 10-12 10-15 1 10-3 10-6 10-9 10-12 10-15
membrane boundary layer on sample side
c I′
cJ′
1 10-3 3.21 × 10-6 2.21 × 10-6 2.21 × 10-6 2.21 × 10-6 1 1.00 × 10-3 1.96 × 10-6 9.61 × 10-7 9.60 × 10-7 9.60 × 10-7 1 1.00 × 10-3 1.00 × 10-6 1.00 × 10-9 2.07 × 10-12 1.49 × 10-12 1 1.00 × 10-3 1.00 × 10-6 7.81 × 10-10 1.01 × 10-14 1.01 × 10-14 1 1.00 × 10-3 1.08 × 10-11
