An Interface Equilibria-Triggered Time-Dependent Diffusion Model of

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An Interface Equilibria-Triggered Time-Dependent Diffusion Model of the Boundary Potential and Its Application for the Numerical Simulation of the Ion-Selective Electrode Response in Real Systems Vladimir V. Egorov,*,† Andrei D. Novakovskii,†,‡ and Elena A. Zdrachek‡,§ †

Department of Analytical Chemistry, Belarusian State University, Leningradskaya Str., 14, 220030 Minsk, Belarus Research Institute for Physical Chemical Problems of the Belarusian State University, Leningradskaya Str., 14, 220030 Minsk, Belarus § Department of Inorganic and Analytical Chemistry, University of Geneva, Quai Ernest-Ansermet 30, CH-1211 Geneva, Switzerland ‡

S Supporting Information *

ABSTRACT: A simple dynamic model of the phase boundary potential of ion-selective electrodes is presented. The model is based on the calculations of the concentration profiles of the components in membrane and sample solution phases by means of the finite difference method. The fundamental idea behind the discussed model is that the concentration gradients in both membrane and sample solution phases determine only the diffusion of the components inside the corresponding phases but not the transfer across the interface. The transfer of the components across the interface at any time is determined by the corresponding local interphase equilibria. According to the presented model, each new calculation cycle begins with the correction of the components’ concentrations in the near-boundary (first) layers of the membrane and solution, based on the constants of the interphase equilibria and the concentrations established at a given time as a result of diffusion. The corrected concentrations of the components in the boundary layers indicate the start of a new cycle every time with respect to the calculations of diffusion processes inside each phase from the first layer to the second one, and so on. In contrast to the well-known Morf’s model, the above-mentioned layers do not comprise an imaginary part and are entirely localized in the corresponding phases, and this allows performing the calculations of the equilibrium concentrations by taking into account material balance for each component. The model remains operational for any realistic scenarios of the electrode functioning. The efficiency and predictive ability of the proposed model are confirmed by comparing the results of calculations with the experimental data on the dynamics of the potential change of a picrate-selective electrode in nitrate solutions when determining the selectivity coefficients using the methods recommended by IUPAC.

I

membrane−sample solution interface can lead to strong potential drift for highly selective electrodes in the solutions of foreign ions.28−30 On the other hand, progress in computer science and in methods of numerical simulations have promoted the development of electrode potential models,12,31−38 which are characterized by a different level of complexity and demonstrate a certain predictive ability. In some cases, these models can be used to quantitatively estimate the influence of the change in design of ISEs, measurements conditions, and protocols for the main analytical parameters and response characteristics of ISEs in a way which is faster and cheaper than by performing of the experiments. To the best of our knowledge, from the theoretical point of view, the most thorough model among the ones so far described in the literature is the model based on the joint solution of the Nernst−Planck and Poisson equations,32,36,38

t is well-known that the diffusion processes, controlling the rate of ion delivery from the bulk of the membrane and solution to the interface and vice versa, have a significant impact on the response of conventional ion-selective electrodes (ISEs), especially highly selective ones.1−5 The consideration of the diffusion processes’ influence on the ISEs response, at a qualitative level, suggests various techniques for controlling the intensity of ion fluxes and direction by means of optimizing the design of ISEs,6−8 membrane composition,8−10 and inner filling solution10−15 as well as using special algorithms of membrane conditioning (including impulse galvanostatic polarization16−19) and measurements protocols.10,15 As a result, the analytical characteristics of ISEs, including lower detection limit and selectivity, were drastically improved, and the application field of ISEs was significantly expanded.20−24 Furthermore, it was shown that some electrodes provide an analytically significant response only under the condition of the diffusion control of the potential; in particular, the functioning of polyion sensors for determination of heparin and protamine is based on this discovery.25−27 It was found that diffusion processes at the © XXXX American Chemical Society

Received: October 9, 2017 Accepted: December 7, 2017 Published: December 7, 2017 A

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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EXPERIMENTAL SECTION Reagents. Potassium tetrakis(4-chlorophenyl)borate (KTpClPB), t ridodecy lmet hylamm onium chlo ride (TDDMACl), 2-nitrophenyl octyl ether (o-NPOE), and highmolecular-weight poly(vinyl)chloride (PVC) were of Selectophore grade (Fluka, Switzerland). Potassium chloride (KCl), sodium nitrate (NaNO3), sodium chloride (NaCl), and picric acid were purchased from Reakhim in puriss grade. Tetrahydrofuran (THF) was from Vekton in purum grade. 0.1 M sodium picrate (NaPic) solution has been prepared by dissolving picric acid in the solution of 0.1 M sodium hydroxide (Sigma-Aldrich) with simultaneous potentiometric pH control. Membrane Preparation. The membrane cocktail was prepared by dissolving appropriate amounts of PVC, plasticizer and ion-exchanger in the freshly distilled THF. After the cocktails were stirred for 2 h, they were poured into the glass ring fixed on the glass plate. THF evaporated overnight yielding parent membranes of about 400 μm thickness. Disks of 10 mm diameter were cut from the parent membranes and glued to the PVC tubes with a THF/PVC composition. Concentrations of the components in the membrane were as follows: PVC (32.8 wt %), o-NPOE (66.6 wt %), and TDDMACl (0.6 wt % or approximately 0.01 M). Potentiometric Measurements. The functioning of picrate-selective electrode (Pic-SE) in the solutions containing nitrate ions was investigated in different scenarios that corresponded to the algorithms of selectivity coefficient determination according to IUPAC recommendations. On the one hand, the choice of the object for investigation was justified by sufficiently high selectivity of Pic-SE over nitrate ions which is an important condition for the manifestation of the diffusion influence on ISEs response. On the other hand, the choice was also caused by guaranteed absence of the primary ion impurities in salts of interfering ions and this is a necessary condition for the purity of an experiment. Potentiometric measurements were carried out using an eight-channel pH-meter-ionometer Ecotest-120 (Econix, Russia) connected to PC with automatic recording of EMF values with 19.7 s interval. The saturated Ag/AgCl electrode EVL1M3 (Gomel Plant of Measuring Equipment, Gomel, Belarus) was used as a reference electrode. The Ag/AgCl inner reference electrode was dipped into NaCl solution or into a mixture of NaPic and NaCl solution, depending on the type of measurements. A constant rate of stirring (500 rpm) was maintained with an IKA RST basic stirrer. All studies were performed with three electrodes connected in parallel. The unbiased selectivity coefficient was determined using the modified separate solution method (MSSM)41 (see Supporting Information I for more details). The determined Pot MSSM − (K Pic value was (2.8 ± 0.1) × 10−6. ,NO−3 ) Afterward the determination of the selectivity coefficients was carried out according to IUPAC methods: separate solution method (SSM) in different variants (equal potentials and equal activities methods), fixed interference method (FIM), fixed primary ion method (FPM), two solutions method (TSM), and matched potential method (MPM)42 (see Supporting Information II).

which takes into account not only the diffusion processes but also the migration processes in membrane and sample solution phases. However, in practice, the predictive ability of this model is limited by the necessity of using numerous parameters that cannot be easily determined by performing an experiment, in particular, the rate constants of interphase transfer and the mobilities of all species in both phases. Moreover, because in most applications of the model the equilibrium at the interface is postulated and the mobilities are chosen to be the same for most species, so within made simplifications the predictive advantages of this complicated model compared to much more simple ones are questionable. In this respect, the model proposed by Morf seems to be promising.34 This model is based on dividing the membrane and the diffusion layer of the sample solution into a number of conventional layers which are thin enough to assume that the concentration profiles of the components between the midpoints of the adjacent layers are linear and can be calculated by means of a finite difference method. The efficiency of this model has been demonstrated before,34,37 but the direct comparison of the simulation results with experimental data has not been shown yet. However, we have recently discovered that in some realistic scenarios of ISEs operation, when the electrode is conditioned in the way that the primary ion flux is directed from the membrane surface to its bulk, Morf’s model gives failure.39 It was shown that the reason for the observed failure is the fact that during the calculations of mass transfer from zero-layer of the membrane to zero-layer of the solution, the only thing that is being taking into account is the concentration gradient between first and zero layers of the membrane regardless of the feasibility of mass transfer across the interface. In some cases it may result in the oscillations of the calculated concentration values in the surface layers of membrane and sample solution or even obtaining the negative concentration values. Bakker has recently proposed an elegant way to overcome the pitfalls of Morf’s model;40 however, the described approach essentially differs from the one discussed in the present paper. The model described in the present paper is based on the idea that the concentration gradients in membrane and sample solution phases determine only the diffusion inside the corresponding phases. However, the transfer of the components across the interface at any instant of time takes place owing to the establishment of local ion-exchange equilibrium between two sufficiently thin near-boundary layers of membrane and sample solution. A very similar idea was proposed earlier35 to describe the response of Ag+-selective electrode in a presence of ammonium ions under conditions of galvanostatic polarization but in that work the material balance condition for an interfering ion was not fulfilled and it could be crucial for some cases; besides, the algorithm of calculations was not described in detail. In the present paper, a detailed description of the model under consideration is given; by the examples of the numerical simulation of selectivity coefficient determination according to those recommended by IUPAC methods, it was shown that this model remains operational in various realistic scenarios of electrode functioning, including the procedures of conditioning, keeping the electrode out of the solution, and an abrupt change in the composition of the test solution. The simulation results are in good agreement with experimental data.

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THEORY Description of the Model. The discussed model relies on the followings assumptions: 1. The electroneutrality condition for each phase is kept.

B

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry

established local ion-exchange equilibrium and a macron sign is used to distinguish the parameters related to the membrane phase, and ΔCA,1 is a primary ion concentration change in the first sample solution layer which can be calculated as follows:

2. The material balance for each component in the system is observed. 3. An ion-exchanger is localized in the membrane phase, and it is completely dissociated. 4. An ion-exchange process is characterized by fast kinetics, so at any instant of time, the ratio of primary and interfering ions concentrations at the interface satisfies the condition of local equilibrium which applies to sufficiently thin near-boundary layers of the membrane and sample solution. 5. The migration processes in the membrane and sample solution phases as well as the coextraction processes from the sample and inner reference solutions can be neglected. 6. The direction and the intensity of ion fluxes within the membrane and the diffusion layer of the aqueous phase are determined by the concentration gradients of the corresponding ions in each phase according to Fick’s laws. 7. An ion transfer across the interface is being achieved only as a result of an ion exchange between primary and interfering ions. By analogy with Morf’s model,34 to perform the calculations the membrane phase and the diffusion layer of the sample solution are divided into a number of elementary layers, which are sufficiently thin to assume that the concentration profiles established between the midpoints of these layers are linear. However, the significant difference of the discussed model from Morf’s one is that the surface layers are entirely localized in the corresponding phases (see Figure 1), and it does not operate with a physically unclear concept of “zero layer”, which comprises an imaginary part.

KAB

(C = (C

B ,1(t )

A ,1(t )

δ

) )(C

+ ΔCA ,1 δ1 (CA ,1(t ) + ΔCA ,1) 1

−

δ ΔCA ,1 δ1 1

B ,1(t )

− ΔCA ,1)

(3)

KBA

where is a concentration exchange constant of primary ion for interfering ion. The values of the concentration ion-exchange constant used in the calculations were found on the basis of the thermodynamic ion-exchange constant (which was assumed to be equal to the experimentally determined unbiased selectivity coefficient value) and of the individual activity coefficients for picrate and nitrate ions for the corresponding ionic strength values (see Supporting Information III). The expression describing ΔCA,1 in an explicit form was obtained as well (see Supporting Information IV). The corrected concentration values corresponding to the first layers of the phases being in contact are used for the diffusion processes modeling at the following time instant. For the diffusion layer of sample solution the following equations are valid: CA ,1(t + Δt ) = CA*,1(t ) + 2[CA ,2(t ) − CA*,1(t )]

DΔt (δ1 + δ2)δ1 (4)

CA , n(t + Δt ) = CA , n(t ) + [CA , n − 1(t ) − CA , n(t )] + [CA , n + 1(t ) − CA , n(t )]

2DΔt (δn − 1 + δn)δn

2DΔt (δn + 1 + δn)δn (5)

where CA,n(t), CA,n−1(t), CA,n+1(t) are the concentrations of primary ion A in the elementary layers n, n−1 and n+1 at the time instant t; Δt is time interval that was chosen for calculations; D is the diffusion coefficient in the aqueous phase which is assumed to be equal for all ions; δn , δn−1 , δn+1 are the thicknesses of the corresponding layers which are not necessarily should be equal to each other. The calculation of the concentrations in the elementary layers of the membrane is performed in the similar way. The change of the primary ion concentration in the bulk of the sample solution can be calculated according to the eq 6:

Figure 1. Arrangement of the elementary layers at the interface in the disscussed model (a) and Morf’s model (b).

According to the proposed model, each new cycle of calculations starts with the correction of the primary ion concentration in the first layers of membrane and sample solution at the time instant t taking into account a local ionexchange equilibrium on the basis of the initial concentrations of primary and interfering ions in these layers or the concentrations that were established as a result of diffusion by this time: CA*,1(t ) = CA ,1(t ) + ΔCA ,1 CA*,1(t ) = CA ,1(t ) − ΔCA ,1

CA , aq(t + Δt ) = CA , aq(t ) + [CA , aq − 1(t ) − CA , aq(t )] ×

(6)

where VS is a sample solution volume, Vaq−1 is the volume of the last elementary layer of the aqueous diffusion layer that is being in contact with the bulk of the solution. Details of the Calculations. The theoretical simulations were carried out using the following values of main parameters: the thickness of the aqueous diffusion layer, defined as the distance between the extreme points of the calculation (the midpoint of the first elementary layer and the boundary of the last elementary layer facing the solution volume, where the concentration is equal to the concentration in the solution bulk), was 25 μm, the membrane thickness was 400 μm, and

(1)

δ1 δ1̅

2DΔt Vaq − 1 (δaq − 1)2 VS

(2)

where δ̅1 and δ1 are the thicknesses of the first layers of membrane and sample solution, the concentration values denoted as C* correspond to the values corrected in terms of C

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry the diffusion coefficients in membrane and aqueous phases were 5 × 10−12 m2/s and 10−9 m2/s, respectively. The thickness of the first and the last elementary layers in the membrane phase were assumed to be 1 and 4 μm, respectively; the thickness of the rest of them was 5 μm. In the aqueous phase, 10 μm was chosen as the thickness of the first elementary layer, and 20 μm was chosen as the thickness of the second one. The calculations were carried out with the time interval of 100 ms. The volume of the sample solution according to the modeling experiment was 30 mL. At first sight, an evident drawback of the discussed model is that the points for calculation of concentrations, which correspond to local interface equilibrium, do not refer to the interface but to the midpoints of the first layers of the membrane and sample solution, so very thin elementary layers should be used for modeling. However, according to the stability condition, DΔt/δ2 < 0.5,43 the decrease of the elementary layer thickness leads to the necessity of the corresponding decrease of a time increment (see Supporting Information V), which results in the drastic, proportional to (1/ δ3), increase of the duration of calculations. It should be noted here that because the diffusion coefficients in the aqueous phase are much higher than in the membrane one, it is the thickness of the elementary layer in the water phase that determines the maximum acceptable time increment Δt. However, the results of the calculations performed using very thin elementary layers (δ = 0.5 μm) and very small time increments (Δt = 100 μs) show that a linear concentration profile in the aqueous diffusion layer calculated according to eqs 1−6 is established very quickly, within 200 ms (see Figure 2). This suggests that the splitting scheme should not significantly affect the shape of the concentration profile if the measurement time is not too short.

Figure 3. Influence of the thickness of the first elementary layer of the aqueous phase and the aqueous diffusion layer splitting scheme on the concentration profiles of picrate in the aqueous diffusion layer after 0.5 s (a) and 5 s (b) immersion of picrate-SE in 1 M NaNO3. (1) 1 and 24.5 μm (2 layers), (2) 5 and 22.5 μm (2 layers), (3) 10 and 20 μm (2 layers), (4) 50 μm (1 layer), (5) 50 layers, each 0.5 μm except for the last one, which was 0.75 μm. The time increment was 100 μs for the case of 50 elementary layers and 10 ms for all other cases. All other parameters of the simulation corresponded to those presented in the previous section.

One can see that even for the time instant equal to 0.5 s, all the concentration profiles are linear, independently of the splitting scheme, as to be expected. Moreover, the calculated concentration profiles are practically independent of the thicknesses of elementary layers, except for the case when there is only one elementary layer. However, for the time instant equal to 5 s, the concentration profiles for different splitting schemes do not practically differ from each other. This observation makes it possible to use a small number of relatively thick elementary layers if the measurement time is not very short, which allows a drastic reduction in the computation time. In the cases when according to the conditions of the modeling experiment the change of the sample solution concentrations was accompanied by the loss of the contact between an electrode and the aqueous phase (this situation was observed while the calibration curves were obtained by changing concentration from the lower to the higher one), the calculations of the concentration profiles did not stop. During the corresponding period of time required to change the solution, the calculations of the change of the concentration profiles due to the diffusion were carried out for the membrane phase only. At the same time, when the method of 2-fold dilution was used, the contact of the electrode with the aqueous phase was not interrupted and so it was considered that an instantaneous change of bulk concentration in the sample took place but it did not affect the diffusion layer. According to the phase boundary potential model, the potential of Pic-SE at any instant of time was calculated as follows:

Figure 2. Concentration profiles of picrate in the aqueous diffusion layer at different instants of time after immersion of picrate-SE in 1 M NaNO3; output data was obtained with the proposed model.

The data presented in Figure 3 illustrate the influence of the thickness and the quantity of the elementary layers in the aqueous diffusion layer on the shape of the concentration profile. The thickness of the first elementary layer was changed from 0.5 to 50 μm. The thicknesses of other elementary layers were also changed so that the thickness of the aqueous diffusion layer defined as the distance between the extreme points of the calculation (the midpoint of the first elementary layer and the boundary of the last elementary layer facing the bulk solution) remained constant and equal to 25 μm. If the thickness of the first elementary layer was equal to 50 μm, the second elementary layer was absent.

E(t ) = const −

2.303RT ⎛⎜ C Pic,1(t )γPic,1 ⎞⎟ log⎜ ⎟ F ⎝ C Pic,1(t ) ⎠

(7)

where γPic,1 is an individual activity coefficient for picrate ion in the aqueous phase. The const value was determined as the EMF value in the solution with picrate ion activity 0.01 M. Because the experiments for selectivity coefficient determination were being carried out for a long period of time, and with use of several sets of ISEs, the const value varied in the range of 75 ± 15 mV. Because of this, in order to achieve the best agreement between the parameters of simulations and experimental D

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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Analytical Chemistry conditions for each particular case, the mean of const values for three ISEs was used, and it was determined every time before the start of the corresponding series of measurements.

solution without taking into account the feasibility of mass transfer across the interface:

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CA ,0(t + Δt ) = CA ,0(t ) + [CA ,1(t ) − CA ,0(t )]

RESULTS AND DISCUSSION Limitations of Morf’s Model. In Figure 4, the results of Pic-SE potential dynamics for the case of electrode calibration

+ [CA ,1(t ) − CA ,0(t )]

D̅ Δt δ ̅δ

DΔt δ2 (8)

It follows from this equation that in the situation when CA,1(t ) < CA,0(t ), i.e. ion A flux is directed from the interface inside the membrane and difference between CA,1(t) and CA,0(t) is small enough one can easily get negative values of CA,0(t + Δt) and it immediately lead to the failure in the calculations. This situation can arise when, after exposing an electrode to a concentrated interfering ion solution it is afterward moved into the solution with lower concentration, because in this case the CA,0(t ) value calculated according to eq 9 increases abruptly:34 CA ,0(t ) =

CRtotCA ,0(t ) CA ,0(t ) + KABCB

(9)

This problem is discussed in detail in39 (see Supporting Information VI as well). Proposed Model. The model described in the present paper provides nearly the same results as Morf’s model for the cases when the latter keeps its operability (see Supporting Information VII), but it is free of the limitations of Morf’s model. Hereinafter we will show the examples illustrating the predictive ability of the proposed model for numeric simulations of the dynamics of ISEs potential change during selectivity coefficient determination according to the methods recommended by IUPAC. Separate Solution Method (SSM), Equal Potentials Method. This method comprises a wide variability of the algorithms and experimental conditions. First, the dependencies E − loga (a is an activity) can be obtained experimentally in different ways: changing the concentration from lower to higher one using the set of standard solutions or by sequential, e.g., 2fold dilution of the initial solution (the way to obtain E − loga dependencies is not regulated by IUPAC recommendations). Second, the duration of the potential measurements for each step of concentration may vary. And, third, different potential values may be chosen to calculate selectivity coefficients. The data presented in Figure 5 and Table 1 shows that all these factors are significant until a certain extent.

Figure 4. Time dependencies of the potential of the Pic-SE simulated according to Morf’s model in NaNO3 solutions with different n-fold dilution of 1 M solution without (a) and with (b) the background of picrate (10−6 M). The thicknesses of the elementary layers in both phases were 10 μm; the thickness of the aqueous diffusion layer was 30 μm; the time increment was 10 ms. The remaining parameters corresponded to those presented in the previous section.

in nitrate ion solutions by means of graduated dilution of the initial concentrated solution are presented. One can see that the operability of Morf’s model depends on the algorithm and specific conditions of the modeling experiment, namely, on the multiplicity of the dilution and on the background concentration of picrate ion. In the absence of picrate ion in the sample solution (this algorithm can be used for selectivity coefficient determination according to SSM (equal potentials method)), Morf’s model gives failure (oscillations of the potential or even the impossibility of its calculating) for the case of a graduated dilution protocol. The higher the dilution factor is, the earlier the failure in calculations appears. If the dilution is taking place in the presence of the constant picrate ion background (a simulation of FIM), the operability of Morf’s model improves. The observed potential oscillations and failures in calculations are related to the obvious incorrectness of the equation describing the change of primary ion concentration in zero layer of the solution which is based only on the concentration gradients between zero and first layers of the membrane and

Figure 5. Dynamics of the potential change of the Pic-SE in NaNO3 solutions for the increasing concentration protocol from 2.52 × 10−3 M to 1 M (a) and for the case of 2-fold dilution protocol starting from 1 M solution (b). Negative logarithms of the concentrations are indicated. E

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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Table 1. Influence on the Algorithm of the Calibration Procedure, the Level of NO3− Concentration, and the Duration of the Pot − Potential Measurement at Each Step on the Values of the Selectivity Coefficients (log K Pic ,NO−3 ) Determined by SSM (Equal Potentials Method) 2-fold dilution

C NO−3 (M)

t (min)

0.25

2.30 10.20 2.30 10.20 2.30 10.20

0.5 1 a

a

experimental −4.76 −4.90 −4.79 −4.95 −4.80 −5.00

÷ ÷ ÷ ÷ ÷ ÷

− − − − − −

increasing concentration simulation

experimental

−4.92 −4.98 −4.95 −5.03 −4.96 −5.06

4.92 4.97 4.93 4.99 4.88 5.05

−4.76 −4.89 −4.92 −4.98 −5.05 −5.08

÷ ÷ ÷ ÷ ÷ ÷

− − − − − −

simulation −4.79 −4.92 −4.92 −5.05 −5.05 −5.17

4.81 4.95 4.94 5.07 5.07 5.18

Here and in Table 3, all the data on time influence on the selectivity coefficients were obtained from different sets of experiments.

an interfering ion (to provide the reasonable close to Nernstian response) while any concentration of the primary ion corresponding to the linear response range of the electrode can be chosen. After that, the selectivity coefficient was calculated taking into account the ratio of activities of primary and interfering ions according to Nernst equation. Thus, because it was impossible to prepare a picrate ion solution with a concentration of 1 M, we used this variation of the method. The data presented in Figure 6 and Table 2 show the high predictive ability of the discussed model that allows us to

As expected, it turned out that the electrode response in the solutions of picrate ions does not depend on the way in which it was obtained. However, in the case of the nitrate ion, the measured potential values, the potential time-dependency character and the response slope appreciably depend on the algorithm of measurements. It should be noticed that the discussed model not only describes well the resulting dependencies E − log a NO−3 for both above-mentioned algorithms of measurements but also reflects some specific features of the dynamics of the potential change. In particular, one can see that the potential value regularly increases with time for each concentration step in the case if E − log a NO−3 dependency was obtained using the increasing concentrations protocol (see Figure 5a). At the same time, if one follows the 2fold dilution protocol, a significant positive potential drift is observed for the first step of dilution, but all following steps are characterized by a minor drift in the opposite direction (see Figure 5b). The response slope toward nitrate ions is significantly higher for the 2-fold dilution protocol (36 ± 2 mV/decade) than for the increasing concentration protocol (22 ± 2 mV/decade) (see Supporting Information VIII, Figure S-4). Accordingly, the influence of the chosen potential value (or the level of nitrate ion concentration) on the experimentally determined selectivity coefficient values is more profound in the case of using of the increasing concentration protocol (see Table 1). It should be also mentioned that for both measuring protocols, the experimentally determined selectivity coefficients regularly decrease with an increase of the duration of measurements, but despite this, their values remain higher than unbiased ones. The discussed model predicts all abovePot − mentioned nuances, and calculated K Pic ,NO−3 values obtained using this model for all investigated cases are in a good agreement with the experimentally determined ones (the deviation from the average value does not exceed the variation range between two utmost values obtained for three electrodes). SSM, Equal Activities Method. Owing to its apparent simplicity, because the plotting of the electrode functions formally is not necessary, it was the most popular technique still in the early 2000s44 (nowadays it is clearly understood that SSM selectivity coefficients should not be reported without having checked for Nernstian responses to both the primary and interfering ion). Nevertheless, it seems to be more accurate to apply this method in a more rationalized way. This rationale was proposed by Bakker45 when the potential values are measured in the solution with a relatively high concentration of

Figure 6. Dynamics of the potential change (a) for Pic-SE immersed into NaNO3 solutions and the resulting selectivity coefficient change (b).

Table 2. Influence of NaPic and NaNO3 Concentration Levels and of the Duration of the Potential Measurements Pot − on the Values of the Selectivity Coefficients (log K Pic ,NO−3 ) Calculated According to SSM (Equal Activities Method) C Pic− (M)

C NO−3 (M)

0.01

0.01

0.001

0.01

0.001

0.1

0.001

1

t (min) 2.30 10.20 2.30 10.20 2.30 10.20 2.30 10.20

experimental −3.93 −4.09 −3.92 −4.08 −4.47 −4.62 −4.91 −5.01

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷

− − − − − − − −

4.03 4.20 3.99 4.14 4.50 4.65 4.95 5.08

simulation −4.01 −4.15 −4.01 −4.15 −4.49 −4.63 −4.92 −5.05

describe quantitatively the dynamics of the potential change in nitrate ion solutions as well as to predict the values of experimentally determined selectivity coefficients and their dependence on the nitrate ion concentration level and on the duration of measurements. FPM (Fixed Primary Ion Activity Method). Figure 7 presents the experimental and simulated E − log a NO−3 dependencies, which were obtained with a constant background of picrate ions F

DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

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(from the lowest to the highest concentration or vice versa) on the dynamics of the potential change are presented (see also Figure S-6). One can see that the results of simulations for all cases are in a good agreement with the experimental data. The above statement remains true for the obtained selectivity coefficients values (see Table 3), which depend on the choice of protocol for calibration, the concentration level of nitrate ions, and the duration of measurements. The obtained data allows us to conclude that while determining the low selectivity coefficients, the SSM and FIM methods yield practically the same results. In both cases, an increase of the concentration of the interfering ion results in the selectivity coefficient values which are closer to unbiased ones. At the same time, it should be kept in mind that if one uses very high concentrations (several moles per liter), problems may arise due to unpredictable changes in activity coefficients, which can strongly depend on the nature of the ion. An increase in the duration of measurements always leads to a certain decrease in the selectivity coefficients (naturally, the duration should not be excessive in order to prevent diffusion of interfering ions through the membrane). As for the FPM method, it seems to be the least suitable for determination of low selectivity coefficients due to the difficulty of obtaining an electrode response close to Nernstian. It was shown as well that the results of the simulation are also in good agreement with the experimental data obtained by TSM and MPM techniques (see Supporting Information IX).

Figure 7. Dynamics of the potential change for Pic-SE in NaNO3 solutions prepared with 5 × 10−6 M background of NaPic for the case of the 2-fold dilution protocol starting from 1 M solution (a) and for the increasing concentration protocol from 2.5 × 10−2 M to 1 M (b). Negative logarithms of the concentrations are indicated.

using the 2-fold dilution and the increasing concentration protocols. One can see that the electrode response function E − log a NO−3 for 5 × 10−6 M picrate background is very poor, and for this reason it is not possible to obtain relevant selectivity coefficient values on the basis of these dependencies. Strictly saying, under the mentioned conditions, the picrate ions activity is not constant but decreases with the increase of nitrate ions’ activity. Therefore, the presence of maxima on the E − log a NO−3 dependencies (see Figure S-5) can be explained by the superposition of these oppositely directed effects. Nevertheless, the discussed model successfully describes both the general shape of E − log a NO−3 dependencies and the dynamics of the potential change for each separate measurements step, and thus, it predicts an absolute inapplicability of this method for the determination of relevant selectivity coefficients values. FIM (Fixed Interference Method). In Figure 8, the data illustrating the influence of the protocol used for calibration

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CONCLUSIONS By the example of ISEs with highly selective ion-exchange membrane, the applicability and the high predictive ability of the discussed model was demonstrated in relation to the different realistic scenarios of an electrode operation in the presence of an interfering ion. Because the time of simulation is at least 10 times (if one takes into account the preparatory operations time even a hundreds times) shorter than the time of the real experiment and it does not require any material resources, the present model may be of high interest for a priori quantitative estimation of possible influence of interfering ions on ISE potential, the optimization of the conditions and the algorithm of the measurements, and the optimization of membrane composition in order to decrease interfering ions influence. The relative simplicity of the model, which is based on the idea of separate, step-by-step calculation of local equilibria in thin near-boundary layers of the membrane and the sample solution and diffusion processes inside of each phase, allows considering the possibility of its adaptation to more complex systems in which the complex formation, coextraction, transmembrane transport, protolytic processes accompanying

Figure 8. Dynamics of the potential change of the Pic-SE in NaPic solutions prepared with 1 M background of NaNO3 for the case of the 2-fold dilution protocol starting from 10−3 M solution (a) and for the increasing concentration protocol from 5 × 10−7 M to 10−3 M (b). Negative logarithms of the concentrations are indicated.

Table 3. Influence of the Algorithm of the Calibration Procedure, the Level of NO3− Concentration and the Duration of the Pot − Potential Measurement at each Step on the Values of the Selectivity Coefficients (log K Pic ,NO−3 ) determined by fixed interference method 2-fold dilution

C NO−3 (M)

t (min)

0.1

2.30 10.20 2.30 10.20

1

experimental −4.56 −4.69 −5.08 −5.07

÷ ÷ ÷ ÷

− − − −

increasing concentration simulation −4.65 −4.76 −5.11 −5.18

4.67 4.75 5.12 5.12 G

experimental −4.52 −4.66 −4.94 −5.02

÷ ÷ ÷ ÷

− − − −

4.54 4.68 4.96 5.05

simulation −4.52 −4.67 −4.96 −5.08 DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX

Article

Analytical Chemistry

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with the distribution of the molecular forms of the components between phases, and so on, may be encountered. Understandably, in these cases, the equations for the calculation of the local concentrations of the components cannot be obtained in an explicit form and the target concentration values have to be found by means of a numerical solution of the corresponding systems of equations, which take into account a local equilibrium condition for each component, as well as electroneutrality condition for each phase and mass balance for each component for the whole system. One of the relevant tasks is also the adaptation of the discussed model to the cases when the samples contain several interfering ions because it could be important for predicting the influence of the matrix of the complex samples on the results of potentiometric measurements. We believe that the proposed model will be useful for both the prediction of the behavior of ISEs during the real sample analysis and for optimizing the design of ISEs, the conditions, and the algorithm of the measurements.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.7b04134. Additional information as noted in the text (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Vladimir V. Egorov: 0000-0001-9414-0423 Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.analchem.7b04134 Anal. Chem. XXXX, XXX, XXX−XXX