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Evaluation of Egorov’s Improved Separate Solution Method for Determination of Low Selectivity Coefficients by Numerical Simulation
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Evaluation of Egorov’s Improved Separate Solution Method for Determination of Low Selectivity Coefficients by Numerical Simulation
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Department of Inorganic and Analytical Chemistry, University of Geneva, Quai E.-Ansermet 30, 1211 Geneva, Switzerland
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Analytical Chemistry

Cite this: Anal. Chem. 2014, 86, 16, 8021–8024
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https://doi.org/10.1021/ac502638s
Published August 6, 2014

Copyright © 2014 American Chemical Society. This publication is licensed under these Terms of Use.

Abstract

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The group of Egorov has recently proposed an elegant method to determine unbiased selectivity coefficients of ion-selective electrodes (Egorov et al. Anal. Chem. 2014, 86, 3693). Once the electrode is exposed to a solution containing only interfering ions, the time-dependent experimental selectivity coefficients are plotted as a function of the inverse fourth root of time and extrapolated to zero. The principal assumption of the approach is the progression of the diffusion layer in the membrane phase with square root of time. This letter critically evaluates the usefulness of this methodology by finite element analysis. The results suggest that the improvement of observed selectivity values are highly significant for an initially uniform distribution of primary ions across the membrane, strongly supporting the methodology. When strong inward ion fluxes of primary ions are present instead, a modification of the method by extrapolation of logarithmic selectivity coefficients appears to give the best results.

Copyright © 2014 American Chemical Society

The selectivity coefficients measured for membrane electrodes of high selectivity are often biased when the membranes are conditioned with a solution of the preferred primary ion. This is because this ion must be quantitatively exchanged by the interfering ion within the diffusion layer of the membrane to obtain a potential that depends only on the interfering ion in a nernstian manner. (1) A range of methods has been proposed to overcome this limitation, including conditioning the membrane with a less preferred ion (1) and imposing strong inward fluxes by adding a complexing agent in the inner solution during measurement of the interfering ion. (2)
When a membrane is exposed to a solution containing interfering ion salt, the primary ion activity at the membrane surface (position 0) is approximated by counterdiffusion processes as follows (see Supporting Information for full details): (3)(1)where KI,Jpot is the selectivity coefficient, aJaq is the interfering ion activity, cIm* is the primary ion concentration in the bulk of the membrane, and q is the permeability ratio, defined as(2)where D indicate the diffusion coefficients for I and δ the diffusion layer thickness in the indicated phase. If one assumes that δIm(t) increases with time according to δIm(t) = (πDImt)1/2, this relationship is inserted into eq 2 and then combined with eq 1 to obtain (4)(3)
Extrapolating experimental selectivity coefficients plotted against t–1/4 to a value of t–1/4 = 0 (t → ∞) should approach the situation where aIaq(0) = 0, thereby eliminating the experimental selectivity bias. (4)

Results and Discussion

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We simulate experimental behavior by finite difference analysis, based broadly on the approach described by Morf (see the Supporting Information for details). (5) In a first set of calculations, the concentration of primary ion at the inner side is assumed to be equal to the ion-exchanger concentration. This case is known to exhibit the lowest operational detection limit of the ion-selective electrode since it minimizes zero current counter- and codiffusion fluxes of primary ion across the membrane. (3, 6) Figure 1 shows the apparent selectivity coefficients as a function of time upon exposure to interfering ion solution for the indicated unbiased selectivity values. It is apparent from the diminishing vertical distance of the curves that the observed selectvity coefficients start being biased for logKI,JpotcJ values smaller than −4.5.

Figure 1

Figure 1. At time zero, the sample solution is changed to one containing interfering ions only, and the potential is monitored over time, displayed here to show the apparent selectivity coefficient. The curves correspond to different unbiased selectivities in steps of 0.5 orders of magnitude with a membrane containing a primary ion concentration equal to the ion-exchanger concentration (balanced membrane).

Egorov’s extrapolation approach is demonstrated with the same data set in Figure 2, where the values are replotted as a function of the inverse fourth root of time. The values extrapolated to zero appear to be generally quite close to the unbiased selectivity values shown as open circles. In this graph, the correlation is difficult to discern for values better than about log KI,Jpot = −7.5, but we will see that it is still adequate for very large selectivities.

Figure 2

Figure 2. Egorov’s extrapolation approach dashed lines according to eq 3, shown for the data in Figure 1. Open circles mark the unbiased selectivity coefficients for the series.

Egorov’s methodology assumes that the membrane diffusion layer follows the behavior δIm(t) = (πDImt)1/2. The simulated time-dependent membrane concentration profiles for the case of best selectivity is shown in Figure 3 (10 min time interval shown). From this data set, the diffusion layer thickness was calculated by linear extrapolation of the interfacial membrane concentration gradient (element 0 and 1 in the simulation) to the bulk membrane concentration. This distance was plotted as a function of the square root of time, confirming linear behavior (see Figure 4). All other selectivities gave linear behavior as well (not shown).

Figure 3

Figure 3. Time-dependent membrane concentration profiles (10 min interval shown) relative to the ion-exchange concentration, cRm, for the indicated selectivity and with the data from Figure 1

Figure 4

Figure 4. Confirmation of the key assumption of the approach, δIm(t) = (πDImt)1/2 using the simulated membrane concentration data for the case of best selectivity.

Nonetheless, it was found that the extrapolation approach fails for extremely large selectivies if the potentials are not sampled for sufficiently long times (see Figure 5). While a 10 min sampling time appears to be sufficient for log KI,JpotcJ = −7.0, longer experimental times of up to 60 min are required for systems of excellent selectivity. Otherwise, one may obtain physically implausible negative selectivity coefficients. In these more extreme cases, the extrapolated selectivity is more than 2 orders of magnitude smaller than the value suggested from the last potential reading, see Figure 2.

Figure 5

Figure 5. Experimental selectivity coefficients for the data shown in Figure 1 obtained from Egorov’s approach (Figure 2) and comparison to ideal behavior (no bias). Increasing experimental times reduces the uncertainty of the method.

Figure 6 compares the experimental selectivity coefficients obtained by Egorov’s approach with the values obtained from traditional potential readings after 60 min of exposure time. With the new method, the bias is generally acceptable, at most half an order of magnitude, while the traditional method fares significantly worse.

Figure 6

Figure 6. Experimental selectivity coefficients for the data shown in Figure 1 using simple potential readings and Egorov’s approach (Figure 2), and comparison to ideal behavior (no bias).

Two more cases were considered in this study. If one assumes a strong outward flux of primary ions from the membrane by cotransport from the inner solution (see Supporting Information for details), the results are initially comparable to those discussed above, but at high selectivities of log KI,Jpot < −5.0, the membrane potential becomes dictated by the outflux of primary ion salt and the methodology starts to fail (Figure 7).

Figure 7

Figure 7. Experimental selectivity coefficients for the data shown in Figure 1 using simple potential readings and Egorov’s approach (Figure 2) and comparison to ideal behavior (no bias).

We now consider a case with a strong inward ion flux owing to quantitative displacement of primary ion by interfering ones at the inner solution side of the membrane. Such compositions have earlier been suggested to be very useful for the determination of unbiased selectivity coefficients, (2) but a rigorous treatment has not yet been put forward.
As shown in Figure S6 in the Supporting Information, Egorov’s extrapolation procedure fails to give chemically meaningful values for membranes of good selectivity, as the simulated data for log KI,JpotcJ = −7.0 are clearly nonlinear. This is likely due to the strong inward gradient of the primary ion within the membrane, which makes it difficult to fulfill the assumptions required in the approach.
Surprisingly, an adaptation of the methodology gives excellent results for situations exhibiting a strong inward flux. If the time-dependent logarithmic experimental selectivity coefficient is plotted on the y-axis, extrapolation to the unbiased values is successful for all selectivities studied in the simulation (see Figure S7 in the Supporting Information). This modified approach appears to give much better selectivity values than traditional potential readings. The reason for this is not immediately clear. Figure S8 in the Supporting Information shows the simulated concentration profiles for the case of highest selectivity studied here to show how the membrane concentrations evolve over time. Figure 8 compares this modified method with traditional readings, suggesting the significant reduction of any bias even with membranes of strong inward ion fluxes.

Figure 8

Figure 8. Simulated logarithmic selectivity coefficients for membranes exhiting a strong inward ion flux, using the modified method of extrapolating logarithmic selectivity coefficients (open circles) and the potentials after 10 min experimental time (black circles).

In conclusion, Egorov’s methodology is elegant and useful and should be clearly adopted by researchers in the field. The results appear to be quite spectacular for so-called balanced membranes where transmembrane ion fluxes are kept to a minimum. For membranes exhibiting strong inward fluxes it appears more appropriate to extrapolate logarithmic selectivity coefficients, but the reasons for this are not yet understood. In all cases, the reliability of the method should be confirmed with different interfering ion concentrations giving the same selectivity value, as originally proposed by Egorov and co-workers. (4)

Supporting Information

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Derivation of Egorov’s approach, details of simulations, and additional data. This material is available free of charge via the Internet at http://pubs.acs.org.

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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    • Author
      • Eric Bakker - Department of Inorganic and Analytical Chemistry, University of Geneva, Quai E.-Ansermet 30, 1211 Geneva, Switzerland Email: [email protected]
    • Notes
      The authors declare no competing financial interest.

    Acknowledgment

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    This work was supported by the Swiss National Science Foundation.

    References

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    This article references 6 other publications.

    1. 1
      Bakker, E. Anal. Chem. 1997, 69, 1061 1069
    2. 2
      Sokalski, T.; Ceresa, A.; Zwickl, T.; Pretsch, E. J. Am. Chem. Soc. 1997, 119, 11347 11348
    3. 3
      Ceresa, A.; Bakker, E.; Hattendorf, B.; Gunther, D.; Pretsch, E. Anal. Chem. 2001, 73, 343 351
    4. 4
      Egorov, V. V.; Zdrachek, E. A.; Nazarov, V. A. Anal. Chem. 2014, 86, 3693 3696
    5. 5
      Morf, W. E.; Pretsch, E.; De Rooij, N. F. J. Electroanal. Chem. 2007, 602, 43 54
    6. 6
      Peper, S.; Ceresa, A.; Bakker, E.; Pretsch, E. Anal. Chem. 2001, 73, 3768 3775

    Cited By

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    Analytical Chemistry

    Cite this: Anal. Chem. 2014, 86, 16, 8021–8024
    Click to copy citationCitation copied!
    https://doi.org/10.1021/ac502638s
    Published August 6, 2014

    Copyright © 2014 American Chemical Society. This publication is licensed under these Terms of Use.

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    • Abstract

      Figure 1

      Figure 1. At time zero, the sample solution is changed to one containing interfering ions only, and the potential is monitored over time, displayed here to show the apparent selectivity coefficient. The curves correspond to different unbiased selectivities in steps of 0.5 orders of magnitude with a membrane containing a primary ion concentration equal to the ion-exchanger concentration (balanced membrane).

      Figure 2

      Figure 2. Egorov’s extrapolation approach dashed lines according to eq 3, shown for the data in Figure 1. Open circles mark the unbiased selectivity coefficients for the series.

      Figure 3

      Figure 3. Time-dependent membrane concentration profiles (10 min interval shown) relative to the ion-exchange concentration, cRm, for the indicated selectivity and with the data from Figure 1

      Figure 4

      Figure 4. Confirmation of the key assumption of the approach, δIm(t) = (πDImt)1/2 using the simulated membrane concentration data for the case of best selectivity.

      Figure 5

      Figure 5. Experimental selectivity coefficients for the data shown in Figure 1 obtained from Egorov’s approach (Figure 2) and comparison to ideal behavior (no bias). Increasing experimental times reduces the uncertainty of the method.

      Figure 6

      Figure 6. Experimental selectivity coefficients for the data shown in Figure 1 using simple potential readings and Egorov’s approach (Figure 2), and comparison to ideal behavior (no bias).

      Figure 7

      Figure 7. Experimental selectivity coefficients for the data shown in Figure 1 using simple potential readings and Egorov’s approach (Figure 2) and comparison to ideal behavior (no bias).

      Figure 8

      Figure 8. Simulated logarithmic selectivity coefficients for membranes exhiting a strong inward ion flux, using the modified method of extrapolating logarithmic selectivity coefficients (open circles) and the potentials after 10 min experimental time (black circles).

    • References


      This article references 6 other publications.

      1. 1
        Bakker, E. Anal. Chem. 1997, 69, 1061 1069
      2. 2
        Sokalski, T.; Ceresa, A.; Zwickl, T.; Pretsch, E. J. Am. Chem. Soc. 1997, 119, 11347 11348
      3. 3
        Ceresa, A.; Bakker, E.; Hattendorf, B.; Gunther, D.; Pretsch, E. Anal. Chem. 2001, 73, 343 351
      4. 4
        Egorov, V. V.; Zdrachek, E. A.; Nazarov, V. A. Anal. Chem. 2014, 86, 3693 3696
      5. 5
        Morf, W. E.; Pretsch, E.; De Rooij, N. F. J. Electroanal. Chem. 2007, 602, 43 54
      6. 6
        Peper, S.; Ceresa, A.; Bakker, E.; Pretsch, E. Anal. Chem. 2001, 73, 3768 3775
    • Supporting Information

      Supporting Information


      Derivation of Egorov’s approach, details of simulations, and additional data. This material is available free of charge via the Internet at http://pubs.acs.org.


      Terms & Conditions

      Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.