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J. Phys. Chem. B 1999, 103, 852-859
Modeling of the Electrified Interface of Liquid Membrane Ion-Selective Electrodes A. Vincze,*,† G. Horvai,‡ and F. A. M. Leermakers§ Department of NBC and EnVironmental Protection, Miklo´ s Zrı´nyi UniVersity of National Defense, H-1101 Budapest, Hunga´ ria krt. 9-11, Hungary, DiVision of Chemical Information Technology, Technical UniVersity of Budapest, H-1111 Budapest, Gelle´ rt te´ r 4, Hungary, and Laboratory of Physical Chemistry and Colloid Science, Wageningen Agricultural UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ReceiVed: August 27, 1998; In Final Form: NoVember 16, 1998
Computer modeling of the interface between two immiscible electrolyte solutions confirms that non-Nernstian behavior of cation-sensitive ion-selective electrodes (ISEs) can occur by the following molecular mechanisms: (i) extraction of weakly hydrophilic anions into the membrane, known as the anion effect or Donnan exclusion failure; (ii) leaching of the membrane’s hydrophobic anion at low aqueous electrolyte concentrations; (iii) interference effects of competing cations. All three effects are known to occur under experimental conditions. Our theory employs a lattice-based self-consistent field model extended to include ion-carrier complexation by means of a multistate mechanism. The interface is created in the computer model by a self-assembling mechanism controlled by an unfavorable interaction parameter between the (PVC-lookalike) polymer of the membrane phase and the aqueous solution. Electrostatic potential profiles and charge-density profiles are available for the whole range of parameters considered in the computation of the calibration lines. The width of the polar/apolar interface is only a few molecular diameters thick, but at low ionic strength conditions the electric double layer extends far into both water and the apolar membrane phase.
Introduction Ion-selective electrodes (ISE) based on polymer membrane matrixes are widely used for the selective determination of ions in analytical practice. Among the investigated membrane materials, plasticized PVC has had the most favorable properties. Adding selective complexing agents (called ligands or carriers) to the membrane controls ISE selectivity. The most common composition is roughly 33 wt % PVC, 66% plasticizer [such as dioctylsebacate (DOS) or o-nitrophenyloctyl ether (o-NPOE)], and 1 wt % carrier, such as valinomycin or crown ethers. Occasionally, lipophilic salts are also included. A great effort has been directed toward understanding the selective extraction and transport of ions in these systems and toward explaining the origin of the membrane potential.1-16 It has been established that the interface between the membrane and the aqueous phase (sample) is one of the most relevant factors in the behavior of these electrodes.17,18 Thermodynamic and quasi-thermodynamic theories of liquid ISE membrane behavior have been developed, and they proved to be very useful in improving ISE technologies.19,20 These theories are necessarily based on a simple physical model of the membrane. Certain relevant phenomena such as the adsorption of surfactants at the interface and the molecular structure of the interface can usually not be treated at depth by these models. Yet there is growing evidence that adsorption at or near the interface can be important. Surfactant contaminants of the membrane material may for instance be adsorbed at the interface and cause a large interfacial resistance.21 Different spectroscopic studies22-26 also indicate adsorption processes at the interface. * Author to whom correspondence should be addressed. † Miklo ´ s Zrı´nyi University of National Defense. ‡ Technical University of Budapest. § Wageningen Agricultural University.
The plasticized PVC membrane may be considered as a gelled liquid. For this reason the membrane/water interface can be considered as a liquid/liquid interface. Different models have recently described the molecular structure of such an interface. These include a molecularly sharp interface between the two phases,27 an interfacial range with a gradual concentration change of the two solvents,28 an interface as an “elastic” inner layer,29 and a dynamic interface with capillary waves superimposed on a molecularly sharp interfacial region.30 Some of the newly developed models of the liquid-liquid interface are more sophisticated than those used in earlier ISE theories. Computer models of the interface have recently become capable of providing a detailed and pictorial view of the interfacial distribution of different species and of the potential drop at the liquidliquid interface.31-33 In a previous paper34 we described a new theory (called the multistate self-consistent-field theory), which proved to be useful in modeling the interface between an electrolyte solution and a liquid-phase membrane incorporating a neutral carrier. The model is an extension of the self-consistent-field theory developed by Scheutjens and Fleer (SF-theory).35 The SF-theory is basically the extension of Flory’s polymer solution theory,36 which allows for compositional inhomogeneities in (usually) one direction, for instance perpendicular to the interface. It enables therefore the modeling of spontaneous formation of the interface between two immiscible solutions by self-assembling of the components in the system without the presumptions considering the exact place and width of the interface. Although the SF-theory proved to be very useful in describing colloidal systems,37-39 our previous attempts to use the theory for the modeling of liquid membrane neutral carrier ISEs pointed out the necessity of the extension of the theory to account for complexation equilibrium known to occur in the membrane.40 In the new theory, local complexation equilibrium is considered
10.1021/jp9835420 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/20/1999
Interface of Liquid Membrane Ion-Selective Electrodes by introducing the multistate character of the molecule segments. The detailed description of the new theory was given in our previous publication together with simple calculations on a model system consisting of a liquid-membrane phase doped with a carrier molecule in contact with a simple electrolyte solution.34 The first results showed that the model could be used to interpret or predict important analytical features of ISEs. The goal of this paper is to present a detailed analysis of a system with a composition typical for neutral carrier ISE membranes generally used in practice. The behavior of the model system is compared to that of real systems, first of all with respect to the dependence of the interfacial potential drop on the amount and nature of the components present in the system. In this way some important and experimentally observable nonideal features of these electrodes can be explained on a semi-molecular level. Multistate Self-Consistent-Field Theory for the Membrane-Solution Interface Since the theory has been described in our previous paper,34 here we present only its basic features in order to give a background for the results. The membrane-sample interfacial region is considered as a collection of molecules each composed of one or more spherical units (“segments”) of equal size, which are confined to lattice sites. The lattice consists of parallel layers, which contain the same number of sites each. The width of these layers is equal to the diameter of a segment of a molecule, which was chosen here to have the dimension of a water molecule, i.e., 0.3 nm in diameter. The lattice sites are organized into a crystal-like structure. In this study a hexagonal lattice is used. The layers are numbered by z ) 1, 2, ..., M, and the lattice is sized and positioned in such a way that any concentration gradients should disappear at each end, i.e., at z ) 1 and z ) M, respectively. In the case of a two-phase distribution, this condition makes sure that layer z ) 1 is in the bulk of one of the phases and layer z ) M is in the bulk of the other phase. The lattice is completely filled with molecule segments, i.e., there are no holes in the lattice and the system is incompressible. The components are modeled either as single segments (e.g., water molecule, simple ions, monomers, etc.) or as chains of segments (PVC chains, carrier molecules, etc.). No branching of the chains is allowed. Complexation reactions between carrier molecules and cations are considered as described in ref 34. The necessary parameters for the calculations are (1) the volume of the lattice, (2) the segmental structure (chain architecture) of the components, (3) either the total amount of a component in the system or its concentration in one of the bulk phases, (4) the charge and the dielectric permittivity of the segments, (5) the stability constants (taken in the bulk of one of the phases) for the complexes of each complexable ion, and (6) the Flory-Huggins interaction parameters for each segment pair. The interaction parameter between molecule segments A and B, (designated as χAB) is the change in the free energy in kBT units, due to the process in which segment A in the bulk of pure A phase is exchanged with segment B in the bulk of pure B phase provided that the segments are neutral. (Note that χAB is defined as a dimensionless parameter). After performing calculations with different compositions of the system, one can obtain the respective potential drops between the bulk phases and the corresponding density profiles across the whole lattice. Using these data it is possible to set up what would be a calibration curve in a real experiment. In this way one can study the effects of various components (interfering
J. Phys. Chem. B, Vol. 103, No. 5, 1999 853 TABLE 1: Segment Representation of the Components Present in the Model component
segment representation
PVC-chain carrier molecule hydrophobic anion water hydrophilic cations hydrophilic anions
(A)15 (A)8V(A)8 (B)3C-(B)3 W P + , K+ S - , N-
TABLE 2: Segment Parameters and Segment-Segment Interaction Parameters (χ) Used in the Model Calculationsa
χ
W A B
W A B C V K N P S
0
2 0
C
V
K
N
P
S
2 10 -0.1 -5 -5 -5 -5 0 0 0 2 2 5 2 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
relative dielectric valence constant 0 0 0 -1 0 +1 -1 1 -1
80 2 2 20 40 80 80 80 80
a The values in italics were used in the initial calculations and have been varied later as described in the text.
ions with the same or opposite charge as the analyte ion, etc.) on these calibration lines. From the density profiles one can investigate the adsorption of the different species at the interface, giving rise to the interpretation of the different anomalies on a molecular level. When doing these calculations we obviously neglect all sorts of sample-dependent potential drops in parts of the galvanic cell other than the membrane-sample interface. In the following sections we present the details of such an analysis. Calculations of Ion-Selective Membranes Based on Neutral Carriers Choice of Parameters. Below we consider a system consisting of two contacting phases: (a) an organic liquid phase composed of PVC-(lookalike) chains blended with neutral carrier molecules and with a salt having a hydrophobic anion, and (b) an aqueous solution containing a mixture of hydrophilic salts. This type of system is of practical importance, since it is the basis of liquid membrane electrodes doped with neutral carriers. The segment representation of the components in this model system is given in Table 1. The segment-segment interaction parameters in units of kBT, the valences and the relative dielectric constant of the segments are given in Table 2. We modeled the membrane material with a simple linear polymer chain consisting of 15 segments of type A, which can be considered as CH2 units. Our choice to model plasticized PVC by a single type of a short polymer chain without plasticizer was dictated by the wish to keep the system as simple as possible. The carrier is also modeled as a chain, comprising 16 segments of type A and one active segment V in the middle of the chain. The hydrophobic anion is modeled as an oligomer of B and C segments. According to Table 2, the chemical nature of segment B is identical to that of segment A. The introduction of segment B allows a convenient way to change the hydrophobicity of the hydrophobic anions (see below). The interaction parameter between water (segment W) and CH2 groups (segment
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Figure 1. Typical concentration profiles across the membrane-sample interface. Left side: bulk membrane; right side: bulk sample. Note the logarithmic scale on the concentration axis.
A) was estimated to have the value of about 2.39,43 This value ensures phase separation between water and the membrane material. The Flory-Huggins interaction parameters for ionwater and ion-membrane material are approximated from the available data for the Gibbs energy of transfer of different ions from water to an organic solvent.6,18 Negative χ values between water and ions are selected to account for the hydrophilicity of the ions. For the active (i.e., cation-binding) segment V of the carrier molecule, χ values are chosen to make it slightly hydrophilic. By the nature of the mean field approximation the geometry of the lattice dictates the geometry of the membrane-water interface. In the present model the interface is thus flat. The system size is set to 50 lattice layers with a characteristic layer thickness of d ) 0.3 nm. At both sides of the lattice, reflecting boundary conditions are imposed. The complexation constant for the hydrophilic cation (K+) and the carrier molecule is 105 in the bulk of the membrane phase, while it is only 10-5 for the other hydrophilic cation (P+). This means that the carrier molecule and hence the membrane is specific for cation K+ over cation P+. K+ is therefore considered here as the primary or analyte cation. The system is open for all components except the membrane material, the amount of which is chosen to fill approximately half of the lattice (i.e., its amount can fill 25 lattice layers). The membrane bulk concentration of the carrier molecules (A)8V(A)8 and that of the hydrophobic anions (B)3C-(B)3 were set to 0.05 M and 0.025 M, respectively. These values were estimated from the composition of liquid membranes frequently used in practice. The aqueous bulk concentrations of the hydrophilic ions were also set to the desired values (see below). The amount of water was chosen to completely fill the rest of the system. In the process of analyzing the behavior of the model system the following method was used. For the parameter set presented
above, the bulk aqueous concentration of the primary electrolyte (KN) was varied, while that of the background electrolyte (PS) was fixed. In each step the equilibrium distribution of all species in the system was calculated. The distribution of the bulk components (polymer and water) has always shown the spontaneous formation of two phases. This was due to the choice of the parameter χWA ) 2. The charge density and the electrostatic potential were also calculated as a function of the z coordinate. The potential drop between the two bulk phases was calculated and plotted against the logarithm of the primary ion concentration in the bulk of the aqueous phase. In this way “calibration curves” were calculated for a given parameter set. Some of the other parameters were then systematically varied to analyze the behavior of the model. Results Concentration, Potential, and Charge-Density Profiles. Typical concentration profiles of the components in the model system in the vicinity of the interface (given in mole percent/100) are shown in Figure 1. The aqueous bulk concentration of KN is 1 M and that of the PS is 0.01 M in the system plotted. It can be seen that the water and the membrane material separate, forming an interface spontaneously, across which all the other components are distributed according to their chemical nature and charge. Note that the concentration of water in the membrane phase is close to 10 mole percent, which corresponds to approximately 1 wt %. In the bulk of the organic phase, the number of K-complexes are equal to that of the (B)3C-(B)3 anions; however, there is positive adsorption of the K-complexes at the organic side of the interface, from where the (B)3C-(B)3 anions are expelled. This causes the organic side of the interface to be positive, which is compensated with negative charges at the other side of the
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Figure 2. Charge-density and electrostatic potential profiles in the system where χAW ) 2 and aqueous bulk concentration of PS is 0.01 M.
Figure 3. Concentration gradient (dc(z)/dz) profiles across the membrane-sample interface, where χAW ) 2 and the aqueous bulk concentration of PS salt is 0.01 M. Note: for better viewing the gradient values for water and A15 are divided by 100.
interface. The corresponding charge-density profile and the electrostatic potential profile of the same system are shown in Figure 2. The interface between the two bulk solutions can generally be defined as the region where the concentrations of the components are significantly different from their bulk values. The interface is more easily visualized, however, by showing the gradients of the concentration profiles as in Figure 3, which corresponds to Figure 1. It can be seen that concentration gradients only differ from zero in a narrow region. Typically this occurs for about 8 layers (between layers 25 and 33 in this
figure), which represents a thickness of about 2.5 nm in this model. By comparing this value to that determined for the water/ nitrobenzene system by scanning electrochemical microscopy (SECM) of 2-4 nm,44 the agreement is fairly good. Potentiometric Calibration Lines and the Anion Effect. The results presented so far are for a given set of parameters. The total potential drop in this case between the bulk phases (i.e., between the left and the right edge of Figure 2) provides one point in a calibration curve for this system. From the calculations with different aqueous bulk concentrations of the primary electrolyte (KN) the full calibration curve can be
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Figure 4. Simulated calibration curves for the primary ion K with phase transfer free energy (∆χN-) of anion N being varied from 9 to 0 kBT.
obtained. A series of such calibration curves is plotted in Figure 4. These calibration curves have been generated by varying the concentration of KN and the hydrophilicity of the counterion N-. The hydrophilicity is expressed here as ∆χN (from aq to org.) ) χN-A - χN-W. It can be shown that the difference in ∆χN- of two ions N-1 and N-2 is twice the difference in the respective free energies of transfer from the aqueous to organic phase and therefore can be taken as a measure of hydrophilicity. Besides the concentration of KN and the hydrophilicity of Nall other parameters are kept the same as in the previous section. (In subsequent examples only those parameters will be mentioned for which the values are changed.) The results in Figure 4 illustrate the experimentally well-known “anion effect”. It can be seen that for KN concentrations less than 0.01 M the curves are identical, following nearly Nernstian response (slope ) 59 mV/decade). This response is, however, distorted at higher salt concentrations. This effect is more pronounced for more hydrophobic (lower value of ∆χN) anions in agreement with experimental observations. The anion effect has been generally attributed to the anomalous extraction of N- ions into the membrane. To understand this anomaly it should be recalled that cation selective ISEs are in general selective cation exchangers. In the system considered in this paper the (B)3C-(B)3 anions serve as cation exchange sites in the membrane. Cation exchange membranes do in principle exclude aqueous counteranions such as N-. However, if N- is very hydrophobic and its concentration is high, it may be partly extracted into the membrane. This is called the breakdown of Donnan exclusion. Our calculations confirm this explanation. The membrane bulk concentration of N- as a function of the aqueous bulk concentration of the KN salt in case of different ∆χN- values is plotted in Figure 5. It can be seen that the more hydrophobic the anion N- is, and the higher the concentration of KN in the aqueous phase, the more N- is extracted into the membrane phase. It is also known from experiments, that (B)3C-(B)3 can be effective as suppressor of the anion effect only if its concentra-
tion in the membrane is sufficiently high and if it is sufficiently hydrophobic, that is also important to keep this ion in the membrane phase since it is not covalently bound to the membrane. These observations are again confirmed by our calculations as illustrated in Figure 6. In the systems plotted in Figure 4, the volume of the total amount of (B)3C-(B)3 is 0.28% of the whole system. By reducing the amount of (B)3C-(B)3 by 1 order of magnitude (0.028%) a calibration line is obtained, where the anion effect is seen to be much more pronounced. The (B)3C-(B)3 anion is withheld in the membrane as a salt of the K(A)8V(A)8 complex cation. Thus it is expected to leach out of the membrane phase if it is not sufficiently hydrophobic and the concentration of K in the aqueous phase is low. This assumption is supported by our calculations in Figure 6. To change the hydrophobicity of the hydrophobic anion, the interaction parameter of segment B with the aqueous phase (i.e., with segment W) was varied. The hydrophobicity of segment B can also be expressed via ∆χB, which is defined analogously to ∆χN-. By reducing the hydrophobicity of segment B (increasing ∆χB), calibration lines were calculated as plotted in Figure 6. It is clear that when the anion (B)3C-(B)3 is not hydrophobic enough (∆χB ) 0), the calibration line deviates from ideality, which causes a higher limit of detection. The reason for this can be concluded from Figure 7, which presents the density profiles of (B)3C-(B)3 in the different systems. It can be clearly seen from Figure 7 that a significant extraction of (B)3C-(B)3 from the membrane to the aqueous phase occurs at low primary salt concentration when ∆χB ) 0. This is evident from the significant concentration of this ion on the aqueous side of the interface. Interference by Cations. A cation-selective ISE is expected to measure one kind of cation, the so-called primary ion, as selectively as possible against other cations. There are two reasons why another cation may interfere with the measurement of the primary ion. One is that the interfering cation forms a relatively stable complex with the neutral carrier in the membrane. The other possible reason for interference by a cation
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Figure 5. Dependence of the membrane bulk concentration of anion N- on the water bulk concentration of KN salt for different values of ∆χN-.
Figure 6. Effect of the amount and the ∆χB value of the hydrophobic anion B3CB3 on the calibration line with ∆χN ) 7 and ∆χN- ) 9, respectively.
is that it is so lipophilic that it may be extracted into the membrane even without forming a stable complex (or any complex at all) with the carrier. We have investigated both types of cation interference by varying the complex stability constant and the ∆χP value of the secondary cation P+, respectively. The effects of these parameters are shown in Figure 8. In this figure, three almost identical calibration lines are shown in the middle of the graph. One of them (solid line) is the same as the calibration line marked with empty diamonds in Figure 4. In this case the complex stability constant of the interfering cation was 10-5, i.e., the complex formation with cation P+ was negligible. Cation P+ was also very hydrophilic
with ∆χP ) 10. The second calibration line (empty triangles) is for the case when the complex stability constant of P+ was increased by 8 orders of magnitude to 103 (which is still 2 orders of magnitude less than the corresponding value for the primary cation K+). For the third calibration line in the middle (empty diamonds), the complex stability constant of P+ was lowered back to 10-5, but its ∆χP value was varied from 10 to 5, i.e., it was made more hydrophobic. For each case the water bulk concentration of P+ was fixed to 0.01 M. The latter calibration line proves that a sufficiently liphophilic cation may also cause substantial interference if its concentration is not sufficiently low compared to the primary ion. In agreement with experi-
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Figure 7. Concentration profiles of the hydrophobic anion B3CB3 for different ∆χB values (-2 and 0) of segment B at different KN concentrations.
Figure 8. Effect of the complex stability constant (log K(P)) and ∆χP value (s ) 5, m ) 10, u )10) of cation P and the complex stability constant of the primary cation K+ (log K(K)) on the calibration curve in mixed solution. The aqueous bulk concentration of cation P was 0.01 M.
mental observations, the calibration line in the second case deviates from the 59 mV/decade line at low primary cation concentrations. Calculating the distance of the two lines with the Nikolsky equation45 (with c(K,w) ) 0.0001 M, log K(K,w) ) 5, c(P,w) ) 0.01 M, log K(P,w) ) 3), we get 17.82 mV difference, compared to 17.98 mV obtained with our model. The values for c(K,w) ) 0.0001 M and c(P,w) ) 0.1 M are
61.65 and 61.56 mV for the Nikolsky equation and our model, respectively. Note that the otherwise small difference between the values calculated by the Nikolsky equation and our model, respectively, is higher for lower ionic strength (0.17 mV for the first case and 0.09 mV for the second). This is caused by the finite size of the system. At lower ionic strength of the aqueous phase the double layer reaches the boundary of the
Interface of Liquid Membrane Ion-Selective Electrodes lattice, which results in the calculated potential being slightly different from the aqueous bulk value. This has been proved by repeating the same calculations using large enough lattice (M greater than 50), which can only be done at the expense of significant computational time. The sensitivity of an ISE to various interferences depends also on the complex stability constant of the primary cation with the carrier. This effect was studied by varying the stability constant of the primary cation K+ from 105 to 107 and103, respectively. The corresponding calibration curves (empty circles, full triangles, and full diamonds, respectively) are also plotted in Figure 8. It can be seen that at a given concentration the interfacial electrostatic potential drop is more positive when the stability constant of the primary cation is higher, i.e., the higher the stability the better the potentiometric selectivity for the primary ion. We should also note however, that the higher the stability constant of cation K+, the greater the anion-effect. Thus the use of very strong complexing agents (e.g., cryptands) is unfavorable from the potentiometric point of view. Numerous extensions of these model calculations are possible. For example one can introduce various types of surfactants in this system. These surfactants will naturally influence the structure of the interface. Surfactant adsorption may induce kinetic barriers for the transport of ions through the interface. Even from equilibrium considerations one can obtain insight in the kinetic barriers since the latter depend on the concentration and potential distributions in the interfacial region. The lateral homogeneity of a surfactant film at the membrane-water interface may also be an important kinetic factor. These studies will be dealt with in our future publications. Summary We have shown that our SCF model captures experimental observations very well. As in the real experiments Nernstian behavior with upper and lower limits has been found for the model systems. Deviation from the Nernstian response at low concentration was found to be generated either by the presence of interfering cations or by the leaching of the hydrophobic anion into the aqueous phase. The upper limit of Nernstian response is governed by the nature of the anions in the aqueous phase and the concentration of the hydrophobic anion additive in the membrane phase. The experimental findings mentioned here had been explained before in terms of classical thermodynamics. The explanations were the result of about two decades of extensive work in several research groups. The SCF model presented here is a fast tool to verify such thermodynamic results. Even more importantly, however, it is suitable to address unsolved problems, e.g., about the structure of the membrane-water interface, foreseeably even in the presence of surfactants. Acknowledgment. The authors acknowledge financial support from the OTKA T 1996 and PECO 1079E6 grants, and the Z. Magyary scholarship to A.V. References and Notes (1) Ammann, D.; Morf, W. E.; Anker, P.; Meier, P. C.; Pretsch, E.; Simon, W. Ion-Sel. Electrode ReV. 1983, 5, 3.
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