Article Cite This: Anal. Chem. 2018, 90, 9644−9649
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Sensitivity and Selectivity of Ion-Selective Electrodes Interpreted Using the Nernst-Planck-Poisson Model Jerzy J. Jasielec,*,† Zekra Mousavi,‡ Kim Granholm,‡ Tomasz Sokalski,‡ and Andrzej Lewenstam†,‡ †
Anal. Chem. 2018.90:9644-9649. Downloaded from pubs.acs.org by EASTERN KENTUCKY UNIV on 08/14/18. For personal use only.
AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Physical Chemistry and Modeling Department, Al. Mickiewicza 30, 30-059 Kraków, Poland ‡ Johan Gadolin Process Chemistry Centre, c/o Centre for Process Analytical Chemistry and Sensor Technology (ProSens), Åbo Akademi University, Biskopsgatan 8, 20500 Åbo-Turku, Finland S Supporting Information *
ABSTRACT: The Nernst-Planck-Poisson model is used for modeling the sensitivity and selectivity of ion-selective electrodes (ISEs) with plastic membranes. Two pivotal parameters characterizing ISE response are in focus: sensitivity and selectivity. An interpretation of sensitivity, which considers the concurrent influence of anions and cations on the ISE slope, is presented. The interpretation of selectivity shows the validity and limits of approaches hitherto taken to measure the true (unbiased) selectivity coefficient. The validity of more idealized interpretations by the diffusion-layer model is conceived.
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cases, and even further on, that it is another theory than simpler models. This report shows the applicability of the NPP, and is focused on two important cases interpreted very recently by simpler models, namely: (1) The sensitivity of ISEs toward anions and cations, depending on the membrane composition, analyzed by phase boundary potential model (PBM)17 (2) The method for measuring the true (unbiased) selectivity coefficient of ISEs based on diffusion-layer model (DLM).20
on-Selective Electrodes (ISEs) have found widespread use in the direct determination of ionic species in complex samples.1−6 ISEs can be sensitive (permselective) to cations or anions, depending on their ion-selective membrane composition, including the charge of the ion-exchanger dominating in the membrane. The most important quality of ISEs is their selective potential response to a particular ion, known as the primary or preferred ion, in the presence of interfering (discriminated) ion. Selectivity is one of the constitutive parameters of all ISEs, and it is quantitatively characterized by the selectivity coefficients, KIJ of discriminated ions present in the sample. KIJ is interpreted by different ISE response models (with different idealization level) that allow predictions of its value, which can be then compared with experimentally measured values. Correspondingly, two different terms are used in the literature: the theoretical (or thermodynamic) selectivity coefficient (Ktheor IJ ), and the empirical (or apparent) selectivity coefficient (Kpot IJ ). Orders of magnitude differences between the values of these two coefficients have been reported by Umezawa et al.7,8 There are many methods for measuring Kpot IJ , e.g., the separate solution method, the fixed interference method, the matched potential method, etc.9−15 For all of these methods, the results can be influenced by various factors such as detection limit, time, and concentration ratio of the preferred and discriminated ions.16 According to IUPAC, selectivity coefficients calculated using the Nikolskii-Eisenman equation are meaningful only when Nernstian slopes are observed for both ions. However, this recommendation does not provide a tool for estimating the concentration ranges and time frame in which this condition is fulfilled. So far, interpretations have been restricted due to limiting equilibrium approaches.17−19 Application of the Nernst-Planck-Poisson (NPP) model strengthens the theoretical standpoint and allows the validation of the idealized results published so far. The power of NPP may, mistakenly, promote the opinion that this model does not apply for simple analytical © 2018 American Chemical Society
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NERNST-PLANCK-POISSON MODEL The time-dependent potential is the result of concentration/ field changes inside n layers/phases. Each layer has its own thickness dj and dielectric permittivity εj, is flat and isotropic. Therefore, it can be considered a continuous environment, inside which the changes in space and time of the concentrations of r components cji and of the electric field Ej, take place. The Nernst-Planck-Poisson model is an initial-boundary value problem which is in one dimension defined by the set of equations given below. The mass conservation law describes the evolution of concentrations: ∂J j (x , t ) ∂cij(x , t ) =− i ∂t ∂x
(1)
The ionic fluxes are expressed by the Nernst-Planck equation: ÄÅ j ÉÑ Å ∂c (x , t ) ÑÑ F i j jÅ j j Å Ji (x , t ) = −Di ÅÅÅ − zici (x , t )E (x , t )ÑÑÑÑ ÅÅÇ ∂x ÑÑÖ RT (2) Received: June 12, 2018 Accepted: June 22, 2018 Published: June 22, 2018 9644
DOI: 10.1021/acs.analchem.8b02659 Anal. Chem. 2018, 90, 9644−9649
Analytical Chemistry
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where Jji(x, t) is the flux of the ith ion in the jth layer, Dji is the constant self-diffusion coefficient of the ith ion in the jth layer, cji(x,t) is the concentration of the ith ion in the jth layer, zi is the valence of the ith ion, and Ej(x,t) is the electric field in the jth layer. F is the Faraday constant. R and T denote the gas constant and absolute temperature, respectively. The Poisson equation is replaced by its equivalent form, the displacement current equation, to describe the evolution of the electric field: n
I(t ) = F ∑ ziJi j (x , t ) + ε j i
∂E j(x , t ) ∂t
SENSITIVITY (PERMSELECTIVITY) In 2010, Silvester et al.17 described a potentiometric method of monitoring ion-exchange properties (or “purity”) of an organic solvent containing a lipophilic electrolyte. Such a solvent was treated as a liquid membrane between two aqueous solutions with different electrolyte activities. The system analyzed in their work corresponds to an ISE with a membrane containing two ion-exchangers with opposite charges: the cation-exchanger R− (representing tetrakis(4-chlorophenyl)borate ion TClPB−)) and anion-exchanger R+ (representing tetradodecyloamonium ion (TDDA+)). The authors used the phase boundary potential model to interpret the potential response in such a system. In the PBM, the membrane potential is the sum of two interfacial potential drops and is given by the form of the Nernst equation:17
(3)
The overall potential of the system, φ(t), is given by n
φ (t ) = − ∑ j=1
∫d
dj j−1
Article
E j (x , t ) d x (4)
Δφ =
The initial concentrations in all layers fulfill the electroneutrality condition and, consequently, there is no initial space charge in the membrane: ÄÅ É ÅÅ n ÑÑÑ ÅÅ Ñ j j j ci (0, x) = cinit, i(x) and E (0, x) = 0, for x ∈ ÅÅÅ0, ∑ dj ÑÑÑÑ ÅÅ ÑÑ ÇÅÅ j = 1 ÖÑÑ
a RT ln L ziF aR
(7)
where Δφ is the membrane potential, aL and aR are the activities of the analyzed ion in the left and right solutions, respectively. In the presence of cation-exchanger R− in the membrane, the generated potential is positive (zi = +1, i.e., cation-sensitive ISE). When R+ is added at a concentration in excess of R−, the membrane potential switches from cationic to anionic response (zi = −1). If the concentrations of two ion-exchangers are equal, the potential is expected to be zero. The theoretical interpretation was verified experimentally by the titration of a 10−2 M KTpClPB in nitrobenzene solution with 10−4 M TDDACl.17 The experimental results, and their interpretation by PBM and NPP are presented in Figure 1. It is
(5)
The values of fluxes at each boundary λj are calculated using Chang-Jaffe boundary conditions21 in the form: ÷÷÷◊ ’÷÷÷ Ji (λj , t ) = kijcij(λj , t ) − kijcij + 1(λj , t ) (6) where kji ⃗ and kji ⃖ are the first order heterogeneous rate constants used to describe the interfacial kinetics. The kji ⃗ corresponds to the ion i, which moves from layer αj to αj+1, and the kji ⃖ to ⃗ ji ⃖ is the ion i, which moves from αj+1 to αj. The ratio kji/k j equivalent to the ion distribution coefficient ki used in PBM and DLM. Using the Method of Lines (MoL),22 a set of partial differential equations (PDEs) described above, is transformed into a system of nonlinear ordinary differential equations (ODEs) in time variable, forming a Cauchy problem. Due to the stiff nature of these ODEs, a special integrator is needed. The software used in this work uses an external implicit integrator implemented by Ashby.23 The source of this integrator is a C++ interpretation of the procedure RADAU5, written in Fortran by Hairer and Wanner.24 The method used to numerically solve the above problem requires only space discretization. The nonuniform grid, denser near the boundaries, which is presented in detail in ref 25, is used. Because of grid nonuniformity, finite differences with properly selected weights must be applied.26,27 This unconditionally consistent space discretization is one order of approximation higher than the one used by Brumleve and Buck.28 The software used in this work serves as a flexible and powerful tool which enables inspection of numerous cases, i.e., liquid-junction potential in reference electrodes, potentiometric response of ion-selective electrodes with ion-exchanger and neutral-carrier membranes, electrochemical impedance spectroscopy, and other applications.25,29−32 The results, presented in the mentioned papers, show the great potential of possible applications of the NPP model for the description of ion sensors and other electrochemical systems.
Figure 1. Potential response dependence on the amount of anionexchanger. The concentration of cation-exchanger in the membrane is constant. Results obtained from experiment (•), Phase Boundary model () and NPP model (green and red ).
shown that the change from cationic to anionic response observed experimentally is interpreted similarly by both approaches. Additionally, a more general NPP model provides a possibility of envisioning the finite kinetic influence. This influence is manifested by gradual and closer to experiment potential change, in comparison to a rapid jump predicted by the PBM assuming infinite kinetic rates. Both in the experiment and NPP simulation, the liquid membrane is placed between two aqueous solutions: 1 M KCl and 10−2 M KCl (that corresponds to activities 0.601 M and 0.901 × 10−2 M, respectively). 9645
DOI: 10.1021/acs.analchem.8b02659 Anal. Chem. 2018, 90, 9644−9649
Article
Analytical Chemistry
In 2014, Egorov et al.,20 using Diffusion Layer Model principles, established a simple, fast, and theoretically substantiated experimental method for the determination of true (unbiased) selectivity coefficients. Their method is based on the fact that selectivity coefficients determined by the Separate Solution Method (SSM) are timedependent, and on their finding that this dependence is a linear function of time raised to the power of minus one-fourth. If the −1/4 function KPot ) is extrapolated to the intersection with the IJ (t ordinate axis, the selectivity bias can be eliminated and the true (unbiased) selectivity coefficients can be obtained. As the linearity of this function is very good, it is not necessary to obtain a large set of KPot IJ . Two measurements, for the time instants t1 and t2 (where 150 s ≤ t1 10−4), IJ Pot −1/4 the KIJ (t ) function, obtained using the NPP model, is visually linear vs t−1/4 in the range R1 = (0.2; 0.3), which corresponds to the time range: 123 s < t < 625 s (see Figure 2b). When the selectivity decreases, the time required to reach a linear response increases, and therefore the linearity range shortens (see Figure 2c). For medium selectivity (10−7 < Ktheor < 10−4), IJ −1/4 Pot −1/4 the KIJ (t ) function remains visually linear vs t in the range R2 = (0.1; 0.2), i.e., 625 s < t < 10000 s. For high selec−1/4 tivity (Ktheor < 10−7), the extrapolation of KPot ) function IJ IJ (t gives negative results (see Figure 2d). The more thorough analysis of NPP simulated data (using linear regression in the ranges R1 and R2) gives the following results (see Table S4 in the Supporting Information): (1) Model with outward flux: qualitative (within 1 order of magnitude) agreement between Ktheor and Egorov method for IJ Ktheor > 10−3 and Ktheor > 10−5 for R1 and R2, respectively. IJ IJ Lower selectivity cannot be determined (negative results). (2) Model with minimized flux: only intuitive agreement between Ktheor and Egorov method for Ktheor > 10−6 and Ktheor IJ IJ IJ theor −10 KIJ > 10 for R1 and R2, respectively. Lower selectivity cannot be determined (negative results). (3) Model with inward flux: only intuitive greement between Ktheor and Egorov method for Ktheor > 10−11 for R2. Lower IJ IJ selectivity cannot be determined (negative results). For the sake of brevity, these cases (2 and 3) are not shown in the article but are given in the Supporting Information
(Figures S1 and S2, respectively). The NPP predictions were confronted with the experimental results. For more details, see the Supporting Information. For the case of low selectivity (Ktheor > 10−4), the selectivity IJ − of Cl-ISE for F was investigated. Results from this measurement are shown in Table 1 (and Figure S3 in the Supporting Table 1. Selectivity Coefficients Given in the Literature and Obtained Using Egorov Methoda ISE
interferring ion
log KPot IJ (literature)
log KPot IJ (Egorov)
Cl H
F− Na+ K+ Cu2+ Cd2+ Ca2+
−2.1
