Impedance and electrical potential across supported liquid membranes: role of interfacial potentials on the active transport of a metal cation

Journal of Membrane Science 163 (1999) 109–121

Impedance and electrical potential across supported liquid membranes: role of interfacial potentials on the active transport of a metal cation L. Canet, P. Vanel, N. Aouad, E. Tronel-Peyroz, J. Palmeri, P. Seta ∗ Laboratoire des Matériaux et Procédés Membranaires, CNRS UMR 5635, 1919 Route de Mende, F-34293 Montpellier, Cedex 5, France Received 30 November 1998; accepted 21 April 1999

Abstract The facilitated transport mechanism of a metal cation across a supported liquid membrane was examined in the case of a Cd2+ flux mediated by the natural ionophore lasalocid; the polypropylene membrane was impregnated with o-nitrophenyl octyl ether. Impedance and membrane potential measurements were carried out along with flux measurements of the Cd2+ /H+ counter transport. A proton flux was inferred from the membrane potential measurements performed under open circuit, zero electrical current, conditions. This proton flux was confirmed by impedance measurements performed under closed circuit, non-zero electrical current, conditions. The main consequence of this proton transfer under open circuit conditions (which are the normal membrane separation conditions) was the building of interfacial potentials. These potentials modify the ionic concentrations at the membrane interfaces, and thus render invalid the mathematical treatment of the cation flux based on the equilibrium ion exchange concept which is generally used. When the potential dependence of the effective interfacial reactions is incorporated into the transport model, the theory can account quantitatively for the experimental fluxes measured under open circuit conditions. ©1999 Elsevier Science B.V. All rights reserved. Keywords: Supported liquid membrane; Impedance; Membrane potential; Transfer modeling; Facilitated transport

1. Introduction Facilitated transport membranes are inspired from natural membrane systems which show high selectivities for metal cation transport. The specificity of the transport is obtained in vivo by the intervention of molecules or molecular assemblies. These molecules or assemblies either form transmembrane pores or channels, or are mobile. In the second case, the molecules, called carriers or ionophores, transport metals by a cyclic transfer mechanism in which diffusion processes intervene. This latter mechanism ∗ Corresponding author. Fax: +33-4-67-04-2820 E-mail address: [email protected] (P. Seta)

has been the main source of inspiration for physical chemists who design increasingly sophisticated carriers in order to get increasingly selective artificial membranes [1,2]. The membranes which are best suited for the incorporation of carriers are supported liquid membranes, either as flat sheets or as hollow fibers. In supported liquid membranes the support must be highly porous, since the pore volume contains the liquid phase, which is the active phase in the extraction process. The liquid phase is retained inside the support by wetting forces; it solvates the complexants that carry the neutral or charged metallic species across the membrane. In a general manner the membrane is all the more stable and efficient as the solvent is non-miscible in water, inert toward the support, and

0376-7388/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 9 9 ) 0 0 1 6 0 - X

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a good solvent for the carrier and for the target metal cation-carrier complexes. If the dielectric constant of the solvent is low, the ions tend to form pairs, whereas they are free if the dielectric constant is high [1,3–5]. The characterization of the transport mechanism is difficult because the accessible parameters are the quantities of species transported across the membrane, measured in the extra-membrane phases and from which are derived the fluxes and the extraction and selectivity coefficients. These global measurements are not rich in information concerning the individual steps which take place at the membrane/solution interfaces and inside the membrane core. Impedance spectroscopy is an electrical relaxation technique which has potentialities in the investigation of the dynamic processes implied in the ionic permeability driven by extractants in supported liquid membranes. This method has already been applied to ion-exchange membranes; it gives the conductivity value and a view of the phenomena which simultaneously intervene inside the membrane and at both interfaces [6–8]. The phenomena which take place inside the membrane or inside the membrane electrode are generally observed in the frequency domain higher than 10 Hz, whereas the interfacial phenomena are observed in the frequency domain lower than 1 Hz [9–13]. When the ratio of ligand concentration in the membrane to metal ion concentration in the source phase is high, the exchange of the metal ion between the aqueous phase and the membrane is fast, the interfacial impedance is low and negligible relative to the impedance of the membrane [10]. Although supported liquid membranes have seldom been studied by the impedance method, this method has occasionally been used to characterize the membrane support [14] or study the membrane stability [15]. In this paper we apply this method, in parallel with measurements of membrane potentials and fluxes, to the case of the facilitated transport of cadmium ions across a polypropylene membrane modified by the incorporation of lasalocid; the lasalocid LH forms with the cadmium ion the complex L2 M and a L2 M/LH shuttle across the membrane is established [16]. We show that a transfer of protons is superimposed upon the counter transport of Cd2+ and H+ . A repercussion of this observation in open circuit conditions (which are generally used in flux measurements) is the creation of interfacial electrical potentials which

influence the interfacial steps of complexation and decomplexation of the transported metal ions by the carrier lasalocid. Within the context of a simple transport model, the presence of interfacial potentials can lead to a natural explanation of the transport mechanism. Experimental data on metal fluxes cannot be explained unless quite different potential dependent effective interfacial reaction rates are taken at each membrane/solution interface.

2. Experimental The cell consisted of two cubic compartments; the membrane was sandwiched between the two compartment walls in which a circular hole was made. For impedance and potential measurements the volume of the solutions was 100 ml and the radius of the hole was 1 cm; for flux measurements the volume was either 1.7 or 50 ml (comparison of open- and closed-circuit values) and the radius 0.29 or 1 cm, respectively. The membranes were Accurel® PP ones kindly given by Akzo Nobel Faser AG (Germany). The organic solvent inside the membrane was o-nitrophenyl n-octyl ether (NPOE from Sigma) which gives particularly stable membranes [1,3,4]. Its high dielectric constant ( = 24) favors the existence of free species. The carrier used, lasalocid, is an antibiotic that belongs to the family of polycyclic carboxylic polyethers. It is a good complexing agent for heavy and transition metals [17,18]. Its concentration in NPOE was fixed at 5 × 10−3 M, except where otherwise stated. The cadmium chloride concentration in the source phase was 10−3 M, except where otherwise stated. The aqueous solutions were prepared with water deionized through a MilliQ Plus column (Millipore); the water resistivity was 18 M cm. For the flux measurements the change in metal concentration in the receiving phase was measured by sampling 1 ml at time intervals of the order of 12 h. These samples were diluted so as to coincide with the measuring range of atomic absorption measurements (Varian Spectra A.A 20). The total volume of the sampling phase was kept constant by addition of an HCl solution. The source solution was buffered. The buffer was a mixture of 10−2 M triethanolamine and 10−2 M N-tris(hydroxymethyl) methyl glycine. The pH of the

L. Canet et al. / Journal of Membrane Science 163 (1999) 109–121

receiving phase was adjusted by HCl addition to water. The cadmium chloride flux J across the SLM is the mole number of metal cation transported from the source solution toward the receiving solution per units of time in seconds and area in square meters. The impedance set-up consisted of a computer driven frequency response analyzer (Solartron FRA 1174) linked to an electrochemical interface (Solartron EC 1286). This interface allowed us to work with the classical four electrode set-up, and thus to obtain spectra perfectly representative of the impedance of the studied membrane. The interfacial potential difference is controlled by a couple of electrodes made of a silver thread carefully covered with silver chloride and wound as a spiral whose surface area was approximately that of the membrane. These electrodes were placed in planes parallel to the membrane, approximately 2–3 mm away from the latter. The values of their potential were frequently checked against reference electrodes. The current was supplied by two counter electrodes which were made of 3 cm diameter platinum disks and set in planes parallel to the membrane at a distance of 3 cm. The cell used for potentiometric measurements was Hg | Hg2 Cl2 | KCl 0.1 N| pH 8.2 buffer | membrane | HCl solution | KCl 0.1 N | Hg2 Cl2 | Hg. Because the pH of the HCl solutions is at most equal to 2, the junction potential between the HCl solution and the KCl solution can be considered as constant. The salts were purchased from Merck.

3. Results 3.1. Potentiometric and related flux measurements The potentiometric measurements were performed under open circuit, zero electrical current conditions. Fig. 1 shows the variation of the membrane potential as a function of the pH of the receiving phase (HCl solution) in the presence and absence of cadmium salt in the source phase (buffer solution) and of lasalocid inside the membrane. The curves have the characteristic shape of curves obtained with a supported liquid electrode selective to H+ ions. They show a pH domain where the variation of the electromotive force with the pH is linear. The limits of this domain are a

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Fig. 1. Membrane potential as a function of the pH of the receiving phase. 䊐 [Lasalocid] = 2.5 × 10−3 M, [CdCl2 ] = 0; 䊊 [Lasalocid] = 2.5×10−3 M, [CdCl2 ]S = 10−3 M; N [Lasalocid] = 0, [CdCl2 ] = 0; H [Lasalocid] = 0, [CdCl2 ]S = 10−3 M. US is the potential in the source phase, UR the potential in the receiving phase.

function of the nature of the solvent, of the ionophore, and of the transported species. In a general manner, the lower limit is determined by the cations in solution and the upper limit by the anions [19–21]. Several results can be inferred from Fig. 1. In all cases the difference between the source potential US and the receiving phase potential UR increases as the receiving phase is acidified. The sign of the potential difference corresponds to the establishment of a proton flux from the receiving phase to the source phase; the potential cannot be due to a cadmium ion flux, because this ion is absent in two of the experiments, and moreover a cadmium flux would give the opposite sign for the potential difference. The slopes of the curves obtained in the absence (filled 5 and 4) and in the presence (䊐 and 䊊) of lasalocid are on the order of 40 and 60 mV, respectively, per log unit and lighten the role of the carrier. For a membrane ideally selective to protons the slope is expected to be 58 mV per pH unit. This is indeed the case in the presence of the complexant. The lower slope observed without complexant is a sign of the concomitant transport of another entity, presumably the chloride ions. In contrast, the lasalocid LH favors a selective proton transport through the LH/L− shuttle; the behavior is analogous to the one of a proton indicator electrode. Curves 䊐 and 䊊 clearly reveal the facilitated transport of cadmium ions from the source phase to the receiving phase by a cation/H+ exchange, because

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the membrane potential diminishes in the presence of cadmium ions; the enhancement of the potential in the source phase due to the proton transport is partly counter-balanced by the decrease of this potential due to the cadmium transport. On the other hand, in the absence of lasalocid the cadmium ions are not transported; the slight enhancement of the potential in the presence of cadmium ions might be due to an adsorption of these ions at the interface (filled 5 and 4 curves). The Cd2+ fluxes were measured in open- and closed-circuit conditions. The source phase was 10−3 M in cadmium chloride buffered to pH 8.2; the receiving phase was 10−2 M HCl. In open circuit conditions the flux was 1.95 × 10−7 mol m−2 s−1 . After 1 h the circuit was closed and the flux was reduced to 1.5 × 10−7 mol m−2 s−1 . In open-circuit conditions the flux results from the counter transport of Cd2+ and H+ complexed by the lasalocid. When the circuit is closed, some of the lasalocid molecules are diverted; they can transport H+ in one direction and return as naked L− in the reverse direction, because a non-electroneutral transport becomes possible. The Cd2+ flux is thus diminished.

3.2. Impedance measurements The impedance measurements were performed under closed-circuit, non-zero electrical current, conditions. 3.2.1. Equivalent circuit Equivalent circuit Nyquist diagrams (imaginary component Z 00 of the impedance as a function of the real component Z 0 of the impedance) were drawn in a large frequency domain (10−2 –10+6 Hz) in which three domains can be distinguished (Fig. 2). 1. At frequencies lower than 1 Hz, the experimental spectrum can be described as a semi-straight line forming a +45◦ angle with the x-axis. 2. In the frequency domain comprised between 10 and 50 kHz the diagram is a semi-circle which is centered slightly under the x-axis. 3. In the domain of frequencies higher than 80 kHz the experimental points stay on a line parallel to the axis of the imaginary values.

Fig. 2. Nyquist diagram; pHS = 8; pHR = 2; no lasalocid and no CdCl2 added.

The diagram observed in the intermediate frequency domain has frequently been attributed to the presence of a constant phase element (CPE); i.e., a capacity variable with the frequency ω, in parallel with a resistance Rp . This element, which corresponds to a dispersion of the electrical capacity with the frequency, would increase with the roughness of the surface and would be due to adsorption effects [22]. We checked if a CPE could account for our results. The impedance of such a circuit is described by relation (1). 1 1 + K(jω)α = A + jB = Z Rp

(1)

with A = 1/Rp + K[ cos(απ/2)]ωα and B = K[ sin(απ/2)]ωα ; K and α are constants, K being the equivalent of a capacitance and α characterizing its dispersion with the frequency. These equations allow us to derive the following relations:    απ  1 = logK + α log ω + log cos log A − Rp 2 (2)  απ  (3) logB = logK + α log ω + log sin 2 This model does not fit our experimental results since we did not obtain linear and parallel variations of log(A − 1/Rp ) and logB versus log ω. We thus eliminate the hypothesis that the equivalent circuit is constituted of a CPE and a resistance in parallel. The electrical scheme equivalent to the spectrum observed at low frequencies is a Warburg impedance that

L. Canet et al. / Journal of Membrane Science 163 (1999) 109–121

Fig. 3. Equivalent circuit.

we denote as W1 (Fig. 3). In the high frequency domain the intersection of the x-axis with the experimental spectrum extrapolated to infinite frequency gives the value of a resistance Rsol ; moreover, we determine directly the value of a capacity C in parallel with all the other elements. After subtraction of the components Rsol and C we observe that the impedance diagram becomes a quarter of a circle centered on a semi-straight line that makes an angle of −45◦ with the x-axis. This spectrum is characteristic of semi-blocking electrodes with charge transfer and diffusion, and the equivalent scheme comprises a resistance R1 and a Warburg impedance W2 in parallel with a resistance R2 (Fig. 3); R1 and R2 can be determined from the spectrum [23]. After subtraction of the R1 contribution, we verify that the resulting admittance is a semi-straight line sloping at 45◦ on the x-axis, whose intersection with the x-axis gives the value 1/R2 . The value of W2 is easily deduced from the imaginary component of the admittance spectrum. Table 1 gives the values of the different elements of the equivalent scheme, determined as explained above for specific experimental conditions. From these values the impedance can be analytically calculated in the whole frequency domain and compared to the experimental values. The good agreement (within 2%) between the values confirms the validity of the proposed electrical scheme. 3.2.2. Attribution of the elements of the circuit The resistance Rsol that changes only with the composition of the aqueous phases external to the membrane can be attributed to the resistance of the aqueous solutions. The capacity C is the membrane capacity because its value was divided by a factor of 2 when two membranes were joined side by side;

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moreover, the experimental value is nearly equal to the capacity that can be calculated from the geometry of the membrane and the solvent dielectric constant. The R1 value is a function of the pH of the phases; the higher the proton concentration, the lower the R1 value. It is independent of the presence or absence of the carrier inside the membrane. By analogy with the electrical scheme proposed in the case of a redox process occurring at an electrode modified by a polyaniline film [24], this resistance can be attributed to the charge transfer resistance at the solution/membrane interface. The variation of R1 might be related to an adsorption prior to the transfer; this adsorption would change the environment of the species relative to their environment in solution. W1 is a Warburg impedance associated with the resistance R1 . It depends in the same way as R1 on the pH of the phases. It tends toward zero when the proton concentration is high (pH 2 on both sides) and is due to diffusion in the vicinity of the interfaces. The resistance R2 and the Warburg impedance W2 represent the contribution of transport through the membrane. Note that their values are divided by a factor between 3 and 4 when the membrane contains the carrier. This result is in accordance with the facilitation of the transport by the lasalocid and corroborates the conclusion drawn from the increase of the slopes of the potentiometric curves in the presence of the lasalocid. The diffusion impedance revealed by the low frequency spectra points to interfacial phenomena [9–13] and might be related to the proton adsorption and desorption steps, coupled to diffusion in the solution layers adjacent to the membrane. We have to stress that the very existence of an equivalent circuit with more elements than a resistance and a capacity points to a transport of charges across the membrane which are not compensated by a transport of charges in the reverse direction. As the addition of metal ions in the source phase did not produce any modification of the spectrum, the impedance measurements confirm the presence of a proton leakage. The absence of influence of the metal ion can be explained by the electroneutrality of the counter transport M2+ /2H+ and by the extremely high value of the selectivity coefficient of the solvent NPOE for protons [25,26].

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Table 1 Values of the impedance components pHS

pHR

Carrier

C (nF)

Rsol (k)

R1 (k)

R2 (k)

W2 (M)

W1 (k)

8 2 8 8 2

2 2 8 2 2

− − − + +

0.31 ± 0.01 0.36 ± 0.01 0.31 ± 0.01 0.35 ± 0.01 0.36 ± 0.01

3.4 ± 0.1 1.1 ± 0.1 6.5 ± 0.1 3.4 ± 0.1 1.0 ± 0.1

50 ± 5 20 ± 2 160 ± 10 50 ± 5 20 ± 2

150 ± 10 36 ± 4 150 ± 10 35 ± 4 11 ± 2

14 ± 2 7±1 8±1 4 ± 0.5 2 ± 0.3

10 ± 0.5 0.6 ± 0.1 17 ± 1 0.5 ± 0.1 0 ± 0.1

4. Cd2+ flux modeling The highly porous supported liquid membrane under study is made up of a loose polymer matrix impregnated with an organic solvent containing the carrier LH (complexing agent). We develop here a simplified theory of divalent metallic cation/proton (M2+ /2H+ ) counter-transport based on equilibrium ionic partitioning at the membrane/external solution interfaces, non-equilibrium (carrier mediated) cation/proton exchange reactions just within the membrane, and finally counter-diffusion of the carrier (LH) and the complex L2 M across the membrane. The feature that distinguishes our theory from previous transport modeling ([27–30] and references therein) is that we take into account both interfacial cation flux resistance and interfacial electrical potentials. The theory can, in a reasonable way, account quantitatively for the following experimental results at fixed carrier concentration in the membrane (5 × 10−3 M) and zero initial cation concentration in the receiving phase (see Fig. 4):

1. At fixed receiving side pH (here pHR ≈ 2), JM is close to zero for low values of source pH (denoted by pHS ) and at a pHS of about 7 it rapidly rises to a high flux plateau. This sharp cross-over behavior defines a critical source side pH, pH∗S ≈ 7.7, at which the cation flux drops from its maximum plateau value. 2. At fixed source side pH (here pHS ≈ 8), JM is close to zero for high values of receiving side pH and for pHR values below 4 rapidly rises to a high flux plateau. This sharp cross-over behavior defines a critical receiving side pH, pH∗R ≈ 3.5, at which the cation flux drops from its maximum plateau value. Our main goal in this section is to show, within the context of our transport theory, how the presence of interfacial electrical potentials can strongly influence the cation flux (which in the present situation is directed from the source phase toward the receiving phase). The principal result that we present here is the following: when the initial cation concentration in the receiving phase is zero, the difference, 1pH∗ , between the two critical pH values defining the locations of the transitions between the high and low flux regions, given by 1pH∗ ≡ pH∗S − pH∗R ,

Fig. 4. Variation of the Cd(II) flux with the pH of the receiving phase at fixed pHS = 8 and pHR < 5, and of the source phase at the fixed pHR = 2 and pHS > 6, [Lasalocid] = 5 × 10−3 M in NPOE; [CdCl2 ]S = 10−3 M. The solid lines are the theoretical ones.

(4)

is non-zero and finite only when the resistance to transport due to the interfacial exchange reactions is non-negligible (with respect to the diffusional resistance within the membrane) and the total potential difference across the membrane (assumed to be located at the interfaces) is non-zero. In this case 1pH∗ depends only on the total potential difference across the membrane (provided, as we argue below, that almost all the potential difference is located at the interfaces) and is simply directly proportional to this potential. On the one hand, if the total potential difference across the membrane were zero and at the same time

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the resistance to transport due to the interfacial exchange reactions were non-negligible, then 1pH∗ would be equal to zero. On the other hand, if the resistance to transport due to the interfacial exchange reactions were indeed negligible (very fast exchange reactions), then at zero initial cation concentration in the receiving phase pH∗R → ∞, and the cation flux remains pinned at its high plateau value; therefore in this case 1pH∗ would tend toward −∞, since pH∗S remains finite. As we show below, the large 1pH∗ of about 4 observed experimentally (Fig. 4) is consistent with the experimentally determined values of the membrane potential (Fig. 1). Our interpretation of the experimental results in terms of the present transport theory thus provides strong evidence that interfacial transport resistance is indeed non-negligible for the system under study and that there are consequential electrical potentials acting at the membrane/external solution interfaces. A more in depth mathematical analysis of our transport model, as well as a more detailed comparison of the theory with experiment, can be found in a subsequent article [31]. Our theory is based on the assumption that only the neutral species LH and L2 M are transported across the membrane (via simple Fickian diffusion) and that resistance to mass transfer in the external aqueous solutions is not important. Neither the transport of neutral ion pairs, nor the transport of free ions, is taken into account in our theory. The low, although not completely negligible, M2+ and H+ fluxes measured in the absence of the carrier support this last assumption. Our theory is based on the following further assumptions: 1. Pseudo-equilibrium ionic partitioning holds at the external solution/membrane interfaces [i = S (source) and R (receiving)]: κiM

[M2+ ]i [M2+ ]i

and

κiH

[H+ ]i [H+ ]i .

(5)

Here κiM = κ0M exp{−zM F1Φi /(RT )} and κiH = κ0H exp{−zH F 1Φi /(RT )} are the electrochemical ionic partition coefficients, which include the effects of the interface potentials 1Φi = Φ¯ i − Φi (zM = +2 and zH = +1 are the ionic electrochemical valences and the overbar denotes membrane quantities). For each ionic species, the ‘zero interface potential’ ionic partition coefficient, κ0M or κ0H , is equal to the ratio of the activity coefficient in the external solutions

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(γ ) to that in the membrane (γ¯ ); i.e., κ0 = γ /γ¯ . In the electrochemical ionic partition coefficients, the coupling between different ionic species arises from the interface potentials. However, we will not attempt to calculate the interface potentials theoretically, but rather estimate them from experimental measurements of the membrane potential (see below). Enforcing pseudo-equilibrium ionic partitioning at each interface leads to [M2+ ]i ≈ κiM [M2+ ]i

and

[H+ ]i ≈ κiH [H+ ]i . (6)

These last results can be obtained by equating ionic electrochemical potentials just inside and just outside the membrane. 2. The kinetics of the ion exchange just inside the membrane at each interface is controlled by the reactions kf

2LH + M2+ L2 M + 2H+ , kb

(7)

where kf and kb are the forward (complexation) and backward (decomplexation) reaction rates, respectively. These reaction kinetics lead to the following metallic cation exchange fluxes in the membrane at the S and R interfaces: JSex = kf [LH]2S [M2+ ]S − kb CS [H+ ]2S

(8)

and JRex = kb CR [H+ ]2R − kf [LH]2R [M2+ ]R ,

(9)

where Ci ≡ [L2 M]i for i = R, S are the complex concentrations at the receiving and source sides of the membrane, respectively. 3. After an induction time ∼ `2 /DC of about 30 min, which is short on the time scale of the experiments, it should be a good approximation to assume that the system is in a pseudo-steady state (` ∼ 102 ␮m is the membrane thickness and DC ∼ 10−7 cm2 /s is the L2 M complex diffusion coefficient). This implies that the carrier (LH) and the complex (L2 M) concentration profiles in the membrane are approximately linear. Therefore the carrier and complex diffusive fluxes are given to a good approximation by the steady-state forms: JLH ≈ −

DLH ([LH]R − [LH]S ) `

(10)

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and JC ≈ −

DC (CR − CS ), `

(11)

where DLH is the carrier diffusion coefficient. Under open-circuit conditions, macroscopic electroneutrality imposes a counter-transport constraint of zero net electric current on the two diffusive fluxes in the membrane: JLH + 2JC = 0.

(12)

4. If we assume that the ionic form of the carrier, L− , is confined to the interfacial exchange regions (with effective volumes much smaller than the bulk of the membrane), then the assumed linearity of the carrier and complex concentrations allows the conservation of carrier to be written approximately as 1 2 ([LH]S

+ [LH]R ) + CS + CR ≈ cL0 ,

(13)

where cL0 is the initial concentration of LH in the membrane. To calculate the initial cation flux, JM , we must solve the pseudo steady-state flux equations, JM ≡ JC = JSex = JRex , along with the electroneutrality and carrier conservation constraints. Once the ionic partition coefficients, the carrier and complex diffusion coefficients, the initial carrier concentration, the membrane thickness, and the exchange reaction rates are fixed, these four equations allow the four unknowns, [LH]i and Ci , to be determined as a function of the initial pH values and initial cation concentration in the source and receiving phases. We find that it is impossible to account even qualitatively for the experimental results if the exchange reaction kinetic equations (Eqs. (8) and (9)) are replaced by an exchange reaction equilibrium: [LH]2i [M2+ ]i Ci [H+ ]2i

kb = K¯ ex ≡ kf

(14)

or, in terms of the external phase ionic concentration, [LH]2i [M2+ ]i Ci [H+ ]2i

= Kex = K¯ ex

(κ0H )2 κ0M

.

(15)

This negative result means that the transfer resistances arising from the ion exchange reactions are not, in general, negligible compared with the resistances

arising from diffusion within the membrane itself. (It is interesting to note that Eqs. (14) and (15) show that the interfacial potentials, 1Φi , cannot modify the exchange reaction equilibriums.) If we assume that almost all of the measured membrane potential difference, 1Φm , is divided, not necessarily evenly, between the two interfaces, then 1Φm ≈ 1ΦS − 1ΦR . This assumption can be justified by observing that within the scope of our model only neutral species diffuse across the membrane, and therefore we do not expect a diffusion potential to arise. With this assumption, our transport theory reveals that, for the case studied, 1pH∗ depends only on 1Φm . Although measurements show that 1Φm is a function of pHR and pHS , we will assume that, in a first attempt at modeling transport, 1Φm can be approximated by the value of −250 mV measured at pHS ≈ 8 and pHR ≈ 2. This assumption can be justified in part by observing that at a fixed pHS of 8, the measured 1Φm is roughly constant over the pHR range (1 < pHR < 4) for which JM is appreciable (see Figs. 1 and 4). In order to get some insight into how the cation flux varies as a function of the experimental parameters, it is convenient to rearrange the system of equations for our model and reduce the number of explicit unknowns. The electroneutrality and carrier conservation constraints together form a system of linear equations that can easily be solved to find the carrier concentration at the R and S interfaces in terms of the cation-carrier complex ones:     1 1 CR − 1 − CS (16) [LH]R = cL0 − 1 + d d     1 1 CS − 1 − CR , (17) [LH]S = cL0 − 1 + d d where d ≡ DLH /DC . These relations show that, due to carrier conservation and electroneutrality, the carrier and complex concentrations are typically at most of order cL0 and are limited by saturation thresholds. These thresholds are reached when pHR is very low and pHS is very high and at the same time transport resistances due to the interfacial exchange reactions are negligible (i.e., the ion exchange reactions are very fast and therefore approximate exchange equilibrium holds at the interfaces). In this situation the following inequalities are obeyed [LH]S , CR  CS and CR , [LH]S  [LH]R and the system of equations given by Eqs. (16)

L. Canet et al. / Journal of Membrane Science 163 (1999) 109–121

and (17) can be solved to find the maximum ‘saturation’ source side complex concentration, CSmax , CSmax ≡

cL0 (1 + 1/d)

(18)

and the maximum ‘saturation’ cation flux [28], max ≡ JM

cL0 DC max DC CS = , ` ` (1 + 1/d)

(19)

both of which depend only on membrane quantities and not at all properties of the external solutions (it should of course be kept in mind that the range of validity of the above ‘saturation approximation’ depends in crucial way on the properties of the external solutions). Since typically d ∼ 1, Eqs. (18) and (19) show max ∼ D c /`. This limiting that CSmax ∼ cL0 and JM C L0 case is called the saturation approximation because the complexing agent is saturated in the form of the complex at the source side and saturated in the form of the carrier at the receiving side [31]. This saturation occurs because the exchange reaction (assumed to be at equilibrium in the present approximation, see Eq. (15)) at the receiving side is heavily skewed in the backward direction due to the high proton concentration there; conversely, the exchange reaction at the source side is heavily skewed in the forward direction due to the low proton concentration there. This situation gives rise to the largest complex and carrier concentration gradients across the membrane, and therefore to the highest fluxes (see Eq. (11)). Since the interface transport resistance is in reality far from being negligible in the system under study, the highest cation fluxes obtained in our transport experiments are always much smaller max . than the maximum saturation flux, JM In order to complete the transformation of the model equations into a form suitable for analytical solution, we introduce the external phase ionic concentration into the steady-state flux equations, JSex = JC and JRex = JC . This substitution leads to the following system of equations: k˜fS [LH]2S [M2+ ]S − k˜bS CS [H+ ]2S = CS − CR

(Source) (20)

for JSex = JC and −k˜fR [LH]2R [M2+ ]R + k˜bR CR [H+ ]2R = CS − CR (Receiving)

(21)

117

for JRex = JC . The effective, potential dependent, forward and backward rate constants, k˜fi and k˜bi , are defined by   ` ˜k i ≡ kf κiM = kf κ0M exp − zM F1Φi (22) f RT DC and

  ` ˜k i ≡ kb (κiH )2 = kb (κ0H )2 exp − zM F 1Φi b RT DC = Kex k˜fi (23) for i = R, S. These effective rate constants have dimensions of inverse concentration squared and depend on the true rate constants, the interface potentials, the ionic partition coefficients, and the transport resistance, `/DC , due to diffusion in the membrane. Due to the exponential dependence, the rate constants are very sensitive functions of the interface potentials. When Eqs. (16) and (17) are used to re-express the [LH]i in Eqs. (20) and (21) as functions of Ci , we obtain two equations in terms of the two unknowns, CS and CR . These two equations can then be solved analytically (or numerically) to find Ci , and therefore the cation flux, JM = JC (Eq. (11)), as a function of the relevant parameters. Since the purpose of the present article is limited to shedding light on the important effects of the interfacial transport resistance and electrical potentials on cation transport, a detailed mathematical analysis of the transport equations will not be presented here (see [31]). Given the experimental input cL0 = 5 × 10−3 M, [M2+ ]S = 10−3 M, [M2+ ]R = 0, and ` ≈ 150 ␮m, we then find, using the solutions to the transport equations, reasonably good agreement between theory and experiment (see Fig. 4) if we take DC ≈ 3 × 10−7 cm2 /s, Kex ≈ 2.2 × 1010 ,

DLH ≈ 1.25, DC

k˜fS = 0.1(m3 /mol)2 .

(24)

The theoretical results presented in Fig. 4 are based on the exact analytical solutions to the transport equations (Eqs. (20) and (21)). Once the effective rate constants at the S interface are known, the values of the effective rate constants at the R interface, are then fixed by the value of the membrane potential, 1Φm :

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  k˜bR k˜fR zM F1Φm = S = exp ≈ 3.6 × 10−9 , RT k˜fS k˜b

(25)

where in obtaining the numerical value we have used the measured value of −250 mV for 1Φm . We thus see that the values of the individual interfacial potentials are buried in the values of the k˜fi and the k˜bi , (multiplied by other, a priori unknown, parameters such as κ0M and kf , etc.), whereas the all important ratio, k˜bR /k˜bS = k˜fR /k˜fS , is determined only by 1Φm itself. If we compare the values of the cation fluxes in the high flux plateau regions with the theoretical maximum flux Eq. (19), we find that the observed plateau values ∼ 2 × 10−7 mol/(s m2 ) are about three times smaller than the expected maximum saturation flux of max ≈ 6×10−7 mol/(s m 2 ). As we now demonstrate, JM this difference reflects the importance of the transport resistance due to the interfacial exchange reactions. The values of the cation fluxes in the high flux plateau regions and the strong crossover behaviors observed in Fig. 4 for the cation fluxes as a function of pH can be explained theoretically by studying the transport equations (Eqs. (20) and (21)) within a high pHS and low pHR approximation. At very high pHS the proton concentration at the source, [H+ ]S , is low enough for the decomplexation (backward) part of the interfacial flux (second term in Eq. (20)) to be negligible compared with the complexation (forward) part (first term in Eq. (20)). As for the R side, at very low pHR the proton concentration, [H+ ]R , is high enough for the decomplexation (backward) part of the interfacial flux (second term in Eq. (21)) to carry the cation flux with a very low value of CR  CS [the complexation (forward) part of the interfacial flux at the receiving side (first term in Eq. (21)) is inoperative here because we have taken [M2+ ]R = 0]. Inserting these two approximations into Eqs. (20) and (21) leads to the following set of equations valid at high pHS and low pHR : k˜fS [LH]2S [M2+ ]S ≈ CS

(26)

for JSex = JC and CR ≈

CS k˜bR [H+ ]2R

 CS

the decomplexation reaction at the R side is negligible compared with both the diffusional resistance in the membrane and the source side interfacial exchange resistance; therefore CR can be set equal to its exchange equilibrium value of zero (Eq. (15) with [M2+ ]R → 0). It is clear that the decomplexation flux at the source side (second term in Eq. (20)) can be dropped only if this term is small with respect to CS , which leads to the second inequality delimiting the range of validity of the present approximation: k˜bS [H+ ]2S  1.

Furthermore, the inequality in Eq. (27) allows [LH]S (Eq. (17)) to be approximated by   1 CS . (29) [LH]S ≈ cL0 − 1 + d This last result can then be substituted into Eq. (26) to find a simple quadratic equation for CS alone, the solution of which is the approximate value of the pl source side complex concentration, CS , in the high flux plateau regions: s 2 pl CS Γ 1 4Γ Γ = 1 + − + max , (30) CSmax 2CSmax 2 CSmax CS where the parameter Γ ≡

1 (1 + 1/d)2 k˜fS [M2+ ]S

pl

for JRex = JC . This last result can be interpreted by noting that at low pHR the transport resistance due to

,

(31)

which has dimensions of concentration, controls the importance of the interfacial transport resistance due to the forward reaction at the S interface. Γ decreases with increasing source side cation concentration and effective forward rate constant. If the forward reaction rate at the S side is very fast, then interfacial transport resistance will be negligible compared with diffusional resistance in the membrane, and Γ /CSmax will pl be very small; in this case CS ≈ CSmax , and therefore the plateau flux, defined as JM ≡

(27)

(28)

DC pl C , ` S

(32)

(Eq. (11) with the inequality Eq. (27)), will be close max . On the other to the maximum saturation value, JM hand, if the forward reaction rate at the S side is slow enough, then Γ can be of order CSmax . In this case Eq.

L. Canet et al. / Journal of Membrane Science 163 (1999) 109–121 pl

(30) shows that CS /CSmax will be much smaller than pl one, and therefore the plateau flux, JM , will be much max . For the smaller than the maximum cation flux, JM system under study, at high pHS and low pHR (i.e., the high flux plateau regions in Fig. 4), the interfacial resistance at the source side due to the complexation (forward) ion exchange reaction is high enough to reduce the limiting plateau flux by a factor of about 3 max . The small difference of about 10% between from JM the theoretical plateau flux obtained at low pHR < 2.5 and pHS = 8 (left-hand curve in Fig. 4) and the theoretical plateau flux obtained at high pHS > 8.4 and pHR = 2 (right-hand curve in Fig. 4) arises because the decomplexation (backward) part of the interfacial flux at the source side (second term ∝ k˜bS in Eq. (20)), while small, is not yet completely negligible in the former case (since pHS = 8 < 8.4). Consequently, at low pHR < 2.5 and pHS = 8 the small backward decomplexation reaction at the source side reduces the actual plateau flux to a value which is about 10% smaller pl than the theoretical limiting value of JM (Eq. (32)). By using the high pHS and low pHR approximation introduced above, we are now in a position to understand the origin of the sharp crossover behavior between the high flux plateaus and the essentially zero flux regions observed in Fig. 4 as the pH is changed. For fixed pHR ∼ 2 and variable pHS the critical value of pHS (denoted by pH∗S ) at which the transition between the high flux plateau and low flux region takes place, comes about when the decomplexation flux at the source side (second term, proportional to k˜bS [H+ ]2S , in Eq. (20)), which is negligible at high pHS (high flux plateau), becomes as important as the complexation flux (∝ k˜fS ). Indeed, it is this decomplexation (backward reaction) flux that eventually reduces the total cation flux to zero as pHS is lowered: the cross-over occurs when the second term in Eq. (8) becomes of the same order of magnitude as the terms retained in the high pHS and low pHR approximation, namely the first term and third terms in Eq. (20) (cf. Eq. (26)). This requirement for the onset of the cross-over behavior translates into k˜bS CS [H+ ]2S ∼ CS or pH∗S =

1 2

log(k˜bS ) ≈ 7.7,

(33)

where we have used Eqs. (23) and (24). For fixed pHS ∼ 8 and variable pHR there also exists a critical value of pHR (denoted by pH∗R ) at which

119

the transition between the high flux plateau and low flux region takes place. This critical pH comes about because as pHR is increased, the transport resistance due to decomplexation at the receiving side (term ∝ k˜bR [H+ ]2R in Eq. (21)) is no longer negligible compared with the diffusional resistance in the membrane. In this case the R-side decomplexation flux can no longer maintain the inequality CR  CS , while at the same time carrying the cation flux (∝ CS within the given approximation), and CR begins to deviate substantially from its reaction equilibrium value of zero. Thus as pHR increases, CR must increase at the same time, until eventually at pH∗R , CR will no longer be negligible compared with CS . This will occur when the inequality in Eq. (27), delimiting the range of validity of the high pHS and low pHR approximation, breaks down; i.e., CR becomes of order CS when k˜bR [H+ ]2R ∼ 1, which leads to pH∗R = 21 log(k˜bR ) ≈ 3.5,

(34)

where we have used Eq. (25). Thus when pHR increases past pH∗R , the complex concentration gradient (∝ |CS − CR |) across the membrane sharply decreases as CR approaches CS in order of magnitude, leading to the rapid decline in flux observed in Fig. 4. It can also be shown that if [M2+ ]R = 0, then pH∗R → ∞ and pH∗S remains finite as the exchange reaction rates kb and kf increase in magnitude (and interfacial transport resistance becomes negligible) provided that the ratio kb /kf = K¯ ex remains finite [31]. The finite value of pH∗R observed experimentally is therefore a direct indication of the importance of interfacial transport resistance. Both of the critical values of pH are clearly observed in Fig. 4. The above arguments also show that when [M2+ ]R = 0 the difference between the two critical pH values is related in a simple way to the total membrane potential: ! k˜bS F1Φm 1 ∗ ∗ ∗ 1pH = pHS − pHR = log R = − 2 RT k˜b ×log(e) ∼ 4.3.

(35)

We can therefore conclude that, within the scope of the present transport model, the observed variation of the divalent cation flux as a function of pHS and pHR provides strong evidence for both the importance

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of interfacial transport resistance and the presence of consequential interfacial electrical potentials (on the order of 100 mV). We have shown that the theory predicts that the difference between the two critical pH values, 1pH∗ , is directly proportional to the total membrane potential (Eq. (35)). The theoretical value of this potential needed to account for the divalent cation flux behavior (Fig. 4) is in good quantitative agreement with the measured values (Fig. 1) over the pH range where this flux is non-zero. Since this agreement does not depend on any adjustable parameters, it can be considered as a severe test of our transport theory.

5. Conclusion The carrier mediated counter flux of protons and metal ions, which is electrically neutral, is effective both in closed and open circuit conditions. Membrane potential and impedance measurements reveal the transport of a charged species across the membrane, which is attributed to a proton leakage. In open-circuit conditions this transport, which is not electrically neutral, can only be transient and leads to the building of a membrane potential. This membrane potential modifies in turn the ion concentrations at the membrane/solution interfaces, thus acting on the metal flux. The closed circuit favors the proton leakage which consequently decreases the metal cation flux. In contrast, the latter is maximum in open circuit conditions which are luckily the conditions usually adopted in the liquid membrane processes. The compensating effect of a superimposed electric potential (as in the case of electrodialysis) could possibly ameliorate the transport phenomenon. Work along these lines is in progress.

Acknowledgements This paper is dedicated to the memory of Daniel Schuhmann a former member of our laboratory, who participated in our early efforts to model cation transport in SLMs and whose results were an inspiration for the theory presented here.

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