J. Phys. Chem. B 2000, 104, 8201-8209
8201
Calculated Effects of Membrane Transport on the Long-Term Response Behavior of Polymeric Membrane Ion-Selective Electrodes Werner E. Morf,*,† Martin Badertscher,‡ Titus Zwickl,‡ Patrick Reichmuth,† Nicolaas F. de Rooij,† and Erno1 Pretsch*,‡ Institute of Microtechnology (IMT), UniVersity of Neuchaˆ tel, Rue Jaquet-Droz 1, CH-2007 Neuchaˆ tel, Switzerland, and Department of Organic Chemistry, Swiss Federal Institute of Technology (ETH), UniVersita¨ tstrasse 16, CH-8092 Zu¨ rich, Switzerland ReceiVed: February 18, 2000; In Final Form: June 13, 2000
Zero-current ion transport through polymeric membranes of ion-selective electrodes (ISEs) alters the composition of the inner solution of these sensors. This causes long-term drifts of the electromotive force that are relevant for miniaturized ISEs as well as for most solid-contacted ISEs in which a thin aqueous layer is formed between the membrane and the metal electrode during conditioning. By a similar mechanism, the composition of the sample can be significantly altered during the measurement if the sample volume is small relative to the volumes of the membrane and the inner solution. This paper provides a formal model that describes the response behavior to be expected for such cases.
Introduction A prerequisite for unbiased measurements with ion-selective electrodes (ISEs) is that the sample solution does not influence the response characteristics of the sensor. Conventional polymermembrane ISEs largely fulfill this condition.1-3 Although water4,5 and various ionic6 or nonionic7,8 components may readily diffuse through the sensor membrane to and from the inner reference solution, the composition of the latter is not significantly altered by these transport processes due to the relatively large volume. For miniaturized ISEs, however, the volume of the inner solution is often very small,9-12 and, therefore, changes in the composition of the reference electrolyte can be of concern. In solid-contacted ISEs,13 no aqueous layer is incorporated between the membrane and the inner electrode. However, due to the relatively fast diffusion of water through the membrane, a thin aqueous film may nevertheless be formed.14-16 Because of its extremely small volume, its composition will be most susceptible to sample influences. Similar phenomena can be encountered when very small sample volumes are measured with ISEs. In this case, fluxes across the membrane may alter the composition of the sample. Zero-current transport of ions in ISE membranes has been observed earlier.17,18 In the past, we have shown by theory and experiment that transmembrane ion fluxes may significantly alter the composition in the boundary layer of the sample solution near the electrode.19-22 By appropriate control of such ion fluxes, the lower detection limits have been improved by up to 6 orders of magnitude.23 In these cases, however, it was assumed that the composition of the bulk of the sample is not affected by the ion fluxes. In the present paper, formal models are developed for describing the changes in the response of ISEs due to alteration of the composition of the inner solution. The model covers various types of ISEs (Figure 1) differing in the thickness of the internal aqueous layer and in the configuration of the inner reference element. The latter can be an electrode of the second † ‡
University of Neuchaˆtel. Swiss Federal Institute of Technology (ETH).
kind (e.g. Ag/AgCl in contact with a chloride containing aqueous solution) or an ion-selective element such as an IS-FET9,24-26 or another sensor membrane. The volume of the aqueous layer is assumed to be very small if the membrane is directly placed on the inner reference element. The present model will also be applied to the opposite case where the sample volume is small relative to that of the inner solution.27,28 In both cases, transmembrane ion fluxes are shown to lead to middle- and longterm effects, such as potential drifts and changes in the apparent ion selectivities. Effects of this kind were indeed found in very recent experimental studies.29,30 Theory For convenience, a series of model assumptions is used below in order to simplify the theoretical description. (i) The ISE membrane is treated as an idealized electroneutral phase which contains an excess of neutral cation-selective ionophores as well as a given amount of anionic sites as stationary species, and which is interposed between the sample solution and an inner solution layer (see Figure 2). (ii) The only exchangeable species considered in the membrane are cations of the same charge (usually as ionophore complexes), whereas anions originating from the aqueous phases are largely excluded. (iii) The whole system is in a zero-current state. (iv) The mobility of all cations in the membrane is characterized by the same average diffusion coefficient, which is a very good approximation for ionophore complexes of cations of the same charge. (v) For all cations, partitioning equilibria are assumed to hold between the membrane surfaces and the contacting aqueous solutions, although transmembrane diffusion fluxes may arise between the two interfaces. (vi) The sample solution contains primary or interfering cations, their total activity being well above the detection limit of the ISE, and hydrophilic, poorly extractable anions. (vii) The inner solution layer is assumed to have a fixed volume and a well-defined, sufficiently high, total activity of cations. The application of these idealizing assumptions greatly simplifies the theoretical treatment, while the model still leads
10.1021/jp000655m CCC: $19.00 © 2000 American Chemical Society Published on Web 08/09/2000
8202 J. Phys. Chem. B, Vol. 104, No. 34, 2000
Morf et al.
Figure 1. Schematic representation of the various types of ISEs discussed in this work: A, conventional ISE with a large volume of inner solution; B, “coated-wire” ISE with a very thin aqueous film; C, ISE with an IS-FET as internal reference element; D, miniaturized ISE with incorporation of a relatively thick aqueous layer (e.g. a hydrogel) as inner solution. For B and C, it is assumed that the inner aqueous film is formed during conditioning of the sensor.
established at the two interfaces according to assumption v, the total concentrations of exchangeable cations Mz+ at the membrane boundaries, cm,T(x)0) and cm,T(x)d), are related to the ion activities in the sample solution and the inner solution layer, am′ and am′′, respectively:2,22,33-35
Figure 2. Schematic representation of the model system.
cm,T(x)0) ) Kmam′(aq)Ψ′-z
(1)
cm,T(x)d) ) Kmam′′(aq)Ψ′′-z
(2)
where
results.2,3
to very useful and realistic When assumptions i-iv are used with the Nernst-Planck equations for the ion fluxes, it strictly follows that the membrane-internal diffusion potential is set equal to zero.2,31,32 Hence, the fluxes of all cations in the membrane simply obey Fick’s laws, and the stationary anionic sites are distributed at a uniform concentration. According to assumptions v-vii, the interfacial equilibria are implicitly extended to the whole solution volumes since it was shown that diffusion-induced deviations between bulk and boundary solution activities are negligibly small at higher concentrations.22 From assumptions iii, v, and vii, it becomes evident that cations in the inner solution layer can be exchanged by cations from the sample owing to ion countertransport processes at zero current. The following treatment focuses on long-term influences of membrane transport on the composition of the aqueous solutions and analyzes the resulting effects on the potential response of ISEs. The electromotive force (emf) response of ion-selective membrane electrodes is mainly determined by the phase boundary potentials between the membrane surfaces at x ) 0 and x ) d and the contacting aqueous phases (′) and (′′), respectively (see Figure 2). When partition equilibria are
Km ) km(1 +
∑ν ∑n βMLν,ncL (x)n) γ ν
1
(3)
m(x)
Ψ′ ) eF(φ(0)-φ′)/RT
(4)
Ψ′′ ) eF(φ(d)-φ′′)/RT
(5)
Km is the so-called overall partition coefficient of the indicated ion, which accounts for all effects of ion-ionophore complexation in the membrane,2,22 βMLν,n is the stability constant of a given 1:n complex M(Lν)nz+, cLν(x) is the concentration of the free ionophore Lν in the membrane (see assumption i), km is the partition coefficient of the uncomplexed ion Mz+, γm(x) is the activity coefficient of this species in the membrane, Ψ′ and Ψ′′ are functions of the boundary electric potential differences φ(0) - φ′ and φ(d) - φ′′, respectively, and R, T, and F have their usual meaning. It should be pointed out that eqs 1-5 are not restricted to ionophore based electrodes, but they also apply to pure liquid ion-exchanger membranes if cLν(x) ) 0 is inserted throughout. The potential functions Ψ′ and Ψ′′ defined in eqs 4 and 5 enter into the description of the membrane potential
Ion Transport through Polymeric Membranes of ISEs
J. Phys. Chem. B, Vol. 104, No. 34, 2000 8203
EM and of the emf response E of the ISE cell, respectively:2
E ) ER + E M ) E R +
RT RT ln Ψ′ ln Ψ′′ F F
(6)
where ER is a reference potential summarizing all emf contributions in addition to EM. According to assumptions i-iv, the membrane-internal diffusion potential was set equal to zero in eq 6. Since the membrane phase is considered to contain a constant total concentration RT of unexchangeable anions R- (see assumptions i-iv), the electroneutrality condition requires that
cm,T(x)0) ) ∑cm,T(x)d) ) RT/z ∑ m m
Ψ′′z )
z
Kmam′ ∑ m
(8)
∑ Kmam′′ R m
(9)
RT z
cm,T(x)0,t
