J. Phys. Chem. 1993, 97, 12363-12372
12363
Transport Processes in Membranes Containing Neutral Ion Carriers, Positive Ion Complexes, Negative Mobile Sites, and Ion Pairs Tal M. Nahir and Richard P. Buck' Department of Chemistry, The University of North Carolina, Chapel Hill, North Carolina 27599-3290 Received: March 17, 1993; In Final Form: September 14, 1993"
The response of membranes, which are prepared with neutral ion carriers and mobile sites, to an externally applied potential step was investigated theoretically and experimentally. Two cases were identified: the first, in which the neutral carriers were in large excess, showed a gradual decrease in the magnitude of transient currents due to the increase in bulk membrane resistance. Using digital simulations, this observation was related to the polarization of the concentration profiles of sites and counter ions. The second case showed a behavior limited by the small amount of carrier. A characteristic break in a current us time plot in this constant-resistance system corresponded to depletion of the carrier at the interface where the selected ion entered the membrane. Most of the experimental evidence for both processes was obtained from plasticized PVC membranes containing valinomycin and tetraphenylborate-type trapped mobile sites. The results support the presence of a carrier mechanism in these systems and illustrate the transport of ions and ion pairs in the membranes as well.
Introduction Cation-carrier molecules are typically large macrocyclic ionophores, e.g., crown ethers, which selectively bind ions such as K+ or Na+.1-3 The strong interaction between the aqueous ions and the lipophiliccarriers at an aqueous/nonaqueousinterface to produce a positively charged hydrophobic ion complex has been exploited extensively in the preparation of ion-selective electrodes (ISES).~Because of the importance of these analytical tools and possible applications to transport across biological membra ne^,^ there has been interest in the investigation of the transport mechanisms once the ion complex is incorporated into a nonaqueous phase, usually a membrane.6 The analysis of transport processes in cation-selective membranes, which exhibit the carrier mechanism, has usually depended on results from negative fixed-site membranes made from PVC, plasticizer, and an ionophore (see, for example, refs 7 and 8). Yet, common ISEs are often prepared with additional mobile sites such as tetraphenylborate (TPB) anion^.^ Since the latter membranes represent a more complicated system, the corresponding transport studies may show a substantially different behavior from that observed in fixed-site membranes.1° The purpose of this work is the analysis of transient and steady-state responses of mobile-site cation-selectivemembranes when subject to an externally applied potential step. This study focuses on a mobile-site membrane made with the neutral ionophorevalinomycin]I and extendsthe earlier published work on fixed-site membranes. Several active species are present upon equilibrationof a membrane contacting a solutioncontaining KCl(aq): valinomycin (Val), Kval+ (cation complex), TPB(negative mobile site), KvalTPB (uncharged ion pair), and trace amounts of the uncomplexed selected ion K+. The amounts of each of the species, which are assumed to be confined to the membrane phase, depend on the quantities used in the preparation of the membranes and on thermodynamicequilibriumconstants.'* In many cases, and especially when the dielectric constant of the PVC/plasticizer matrix is low, the first four species are present in significant quantities. Therefore, the complete analysis of transport processes must account for the possibility of substantial coupled concentration polarization of each of these active components in the membrane. A similar problem was analyzed previously by Sandblom et 0
Abstract published in Advance ACS Abstracts, November 1, 1993.
a1.13J4Their theoretical systemI3 consisted of two aqueous solutions separated by a water-immiscible liquid. Only a single (positive or negative) ion was allowed to cross the interfaces, and the (negative or positive) sites and ion pairs were restricted to the nonaqueous phase. The results included steady-state expressions for concentration profiles and current us voltage plots. The analysis presented here extends Sandblom's work to a system whichcontains anadditionalneutralcarrier (valinomycin). Furthermore, digital simulations allow calculation of transient properties such as current and bulk resistance as a function of time in response to the application of an external potential step. This approach follows earlier studies of the response of similar systems to ionic steps, constant fluxes, or current steps.lsJ6 Although the focus is on the valinomycin membrane, the processes described apply to other closely related electrochemical systems which exhibit a significant bulk resistance and a substantial concentration polarization of either ionic, neutral, or both types of species. Under such circumstances, the usual Cottrell behavior is not observed, since the interfacial reactant concentration does not drop to zero instantly (at t = O+). Other examples showing this type of response were illustrated ear1ier.l'
Model Equations
Equilibrium. In a cation-carrier system, the only species that is allowed to cross the solution/membrane interface is a selected positive ion. This property, which excludes the transfer of any other species into or out of the membrane, is due to the highly favored and reversible interfacial reaction of a large membraneconfined "trapped" ionophore (a neutral carrier, such as valinomycin) with the permselective ion (K+in this example). Inside the membrane, a thermodynamicequilibrium always exists among three species: a selected positive ion (K+), a carrier (Val), and a cation complex (Kval+). The dissociation constant is given by
K+ = C+Ccarrier/Cwtion
(1) where c+ is the membrane concentration of the selected ion, cC.,,i,, is the concentrationof the ionophore,and cationis the concentration of the membrane-confined cation complex. The value of K+ is typically very small,l* and the concentration of the selected ion in a common membrane is negligible when compared to the concentrations of the other electroactive species. Most carrier-type ion-selectivemembranes are prepared using plasticized PVC as a matrix. Previous analysis showed the
0022-3654/93/2097- 12363%04.00/0 0 1993 American Chemical Society
Nahir and Buck
12364 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
existence of negative immobile (fixed) sites, and their concentration was estimated to near 0.1 mM.19 The mobile-site membranes considered here have at least 10 times more negativesite species due to the addition of tetraphenylborate-based ions. Therefore, the influence of the negative fixed sites is neglected in this analysis. In addition, because the concentration of uncomplexed selected ion, c+, is very small, a relationshipbetween the concentrations of the remaining ions is obtained using electroneutrality:
+ c+ - C f i x d site + Cmobile site
%
%ite
are identical, then the first two log terms on the right side of eq 8 represent the interfacial potential drop, whose magnitude may be rewritten, using the equilibrium constants from eqs 1 and 3 and approximating activities by concentrations, as
(9) or
(2)
where cried lite and CmobiIe ,ite are the concentrations of the two kinds of negative sites inside the membrane. In the present work, c6itealways designates the concentration of free mobile sites only. The low dielectric constant of the membrane matrix, approximately 5-10,20.2* allows for the presence of neutral ion pairs, which are the product of the association of the ion complexes (cations) and sites (anions). The concentrations of these three species in the membrane are related through another dissociation constant:
K = CsiteCcation/Cion pair
(3) Also, since the sites are confined to the membrane at all times (4) where cBite,tot is the total concentration of all sites species (i.e., sites and ion pairs) and d is the membrane thickness. The interfacial potential drop in this system is similar to that obtained for corresponding fixed-site membranes.' Considering the interfacial-charge-transfer equilibrium reaction, carrier(memb) + ion(,,)+ ion complex(memb)+,the equations for the electrochemical potentials at the left and right aqueous/membrane interfaces are, respectively
ion pair
Fluxes. Thefluxofspeciesiatlocationxattimet inareversible one-dimensional nonconvectiveelectroneutral system is given by the Nernst-Planck equation22
where Di is the diffusion coefficient of species i (assuming a constant value everywhere in the system), zi is the charge of i (with sign), and Vis the electrostatic potential. The first term on the right side of the equation describes diffusion, and the second term describes migration of charged species. In the system of interest, two species are neutral (the free ionophore/carrier and the ion pair) and the two species are charged (the positive ion complex and the negative site). Under these circumstances, unlike in many electrochemicalsystems,it is impossibleto assume that the migrational contribution to the flux of the charged species is negligible (this is usually done by providing an excess of an inert supporting electrolyte). Therefore, eq 11 cannot be approximated by the simpler Fick's first law. This simplification is possible, however, in the case of limiting uncharged species (case I1 below). At any point in thesystemandat any timeduring theapplication of an external voltage step, the current density is given by
where A is the cross-sectional area. Combining eqs 11 and 12 yields mcmb x-0 where barriC; is the chemical potential of the carrier inside the membrane at the left interface and similar terms are for the chemical potentialsof theselected ion (+) or ion complex+(cation) inside the membrane or aqueous solution at one of the interfaces; z+ and zationare the charges on the selected ion and ion complex, respectively; Fis the Faraday constant;pembJsois the electrostatic potential inside the membrane at the left interface; and similar terms are for the membrane or aqueous solution at one of the interfaces. x = 0 and x = dare just outside the electrical doublelayer regions. The magnitude of the total system potential drop is
The electric field can be obtained by rewriting eq 13:
a m
I
-E(x) = -I- -
RT
-
(7) Subtracting eq S from eq 6 and applying the definition of the chemical potential yield amcmbx=d cation
amembs=O vIOtaI
= $[In
cation
+ In
F carrier
d
+,,e] +
+
aaq&=d
A@"mb (8) where R is the universal gas constant, T i s the temperature, I+ is the charge of the selected ion, a: is the activity of the selected ion at location x, and similar terms are for the activities of the carrier and ioncomplexspecies. The internal membrane potential drop is A4-b = 4-bJ-dp m b s m o .If the two aqueous solutions
Two distinct contributions to the electric field inside the membrane can be identified. The first term on the right side of eq 14 is the ohmic contribution, which can be integrated over the total thickness of the membrane to yield the ohmic potential drop:
The Journal of Physical Chemistry, ‘01. 97. No. 47, 1993 12365
Transport Processes in Membranes The system bulk ohmic resistance is now defined as ‘ohm
Res = --
@
dx
RT x=d = -sx=o
Ap
(a)
Cz;D,ci(x)
~ 0 0 0 0 0 0 0 0 0 0 ~ 0 carrier
i
If, for instance, the concentrations of the charged species were constant and identical at every point in the membrane (Le., no ionic concentrationgradients exist), then the total inner membrane potential drop would be
V = -IRes =-I
cation complex
(16)
I
I ‘8888888888: 8
lonpalr
aqueous
membrane
aqueous
R Td V
The second contribution to the internal membrane field in eq 14 is due to the presence of concentration gradients of charged species. The resulting potential difference between points in the system is known as the diffusion potential. If no current flows through the system, this is the only contribution to the bulk membrane potential drop. This case was discussed in detail in ref 23 and will not be considered further in this work. The total potential drop across the membrane systems investigated in this work is thus made of three contributions, whose sum is equal to the magnitude of the externally applied potential step, or the measured potential in a current-step experiment: The first two terms are evaluated by integrating the right side of eq 14, and the interfacial contribution is given in eq 9 or eq 10. Digital Simulation The present analysis focuses on transport behavior in the bulk phase of the solution (the membrane) and applies the NernstPlanck equation to describe diffusion-migration processes. It also assumes that the interfacial reactions are not limited by the kinetics of charge transfer and that the system is electroneutral everywhere (see ref 7 for more detail). Therefore, there is no consideration of the Butler-Volmer and the Poisson equations here, A recent example used finite-differencedigital simulations to study the diffuse electrical double layer at a neutral electrode/ electrolyte solution interface.24 When compared to traditional electrochemical systems with one or two active species, the presence of four species (carrier, cation, site, and ion pair) seems to considerably complicate the transport analysis. Figure 1 schematicallyillustrates the transport processes which are expected in the membrane during the flow of current through the systems, as a result of the application of an external potential step. Initially (Figure la), the species are homogeneously distributed inside the membrane. Thus, the concentrations of the cations and anions (sites) must be identical to each other everywhere: the carrier concentration is equal to the amount used in the preparation of the membrane less the amount complexed by the selected ion, and the ion-pair concentration is related to the ionic concentration through a dissociation constant (eq 3) and is not necessarily the same as that of the ions. There are no concentration gradients throughout the membrane. Once current flows through the system, several fluxes of species are identified. To simplify the discussion, Figure 1b depicts the transfer of six elementary charges through a membrane containing valinomycin (a potassium ion carrier). At the membrane/aqueous interfaces, six K+ ions must cross (no other charged species can penetrate this boundary, since the interface selectivityis exclusively determined by valinomycin). Simultaneously,the neutral carrier, valinomycin, is depleted/produced at the interface where K+ enters/exists, and a concentration gradient of this uncharged
I lQ88888888888: x=0
x=d
Figure 1. Schematic diagram of the transport processes in a potassiumselective membrane containing neutral carriers, mobile sites, and ion pairs: (a) shows the initially equilibrated membrane, and (b) illustrates the response of the various species when a potential is applied across the
membrane.
species (from right to left) develops. Everywhere inside the membrane, the net charge transport must also be of six positive charges from left to right. If the mobility of the cation (the ion complex Kval+) is 5 times larger than that of the anion (the site TPB-), the motion of five positive charges to the right and one negative charge to the left adds to a total of six positive charges moving to the right. Consequently, there is an accumulation of ions and ion pairs at the interface where K+ enters, and a concentration gradient of these species (from left to right) develops. A digital simulation of the reaction of this system to an externally applied potential step reveals the transient current and resistance responses, as well as the changes in the concentrations of all species as a function of time. The procedure follows conventional methods for a one-dimensional system, which is divided into a number of volume elements or “boxes” with an equal thickness, Ax.zs More detail and a list of dimensionless parameters are found in the Appendix. Before this simulation can be performed,thesizeof theapplied potential step, thereduced (dimensionless) magnitudes of the dissociation constant in eq 3, and the diffusion coefficientsof the participating species must be assigned. Table I lists values suggested by Armstrong and T0dd2~J~ for a plasticized PVC membrane containingTPB mobile sites and valinomycin in contact with KCl(,, solutions. Figure 2a-c shows the concentration profiles of all membrane species during the period in which an external potential step of 0.5 V is applied. The perturbation of the carrier concentration profiles is similar to that shown in ref 7, and the magnitude of the potential step is not high enough to completely deplete the carrier at the left interface. This implies that the contribution of the carrier species to the interfacial potential drop, eq 10, is small. In addition, the linear steady-stateconcentration gradient of the carrier is directly proportional to the steady-state current (see steady-state analysis of case I1 below). The concentration profiles of the ions and ion pairs show the anticipated depletion at the interface where the selected ion (K+) exits and the accumulation on the oppositeside of the membrane. Unlike the carrier behavior and in contrast with systemscontaining
Nahir and Buck
12366 The Journal of Physical Chemistry, Vol. 97,No. 47, 1993
7.7 mM valinomycin 7.6 mM KtclP8 2v
0'
u 0.2 0.4 0.6 0.8
w O"0 0.2 0.4 0.6 0.8
1
Xr
0.2 0.4 0.6 0.8
I
0'
Xr
t,=0.005
1
Xr
tr=0.075t,=0.995
7
6.5
t-
Carrier mntrolled,
consiant rssisianm
0
-
Ion controlled. variable rmlslanm
I
I
1,000 time
500
I
[SI
1,500
-
I 2,500
I
2,000
Figure 3. Experimentalcurrent us time responseof a membrane to a 2-V potential step, showing regions of two dominating transport processes. TABLE I: Values of Reduced (Dimensionless) Parameters Used in the Digital Simulations parameter value parameter value 5O
0.2
0.4
0.6
0.8
1 -6.2 0
0.2
tr
0.4
0.6
0.6
1
tr
Figure 2. Results fromdigital simulationsof the responseof a membrane = 3 and Csitc,tot,r = 1 to a 0.5-Vpotential step: ( a x ) containing Cvp~,lot,r concentrationprofiles of the carrier, ions, and ion pairs, respectively;(d) current us time; (e) resistance us time (simulation parameters are in Table I). completely dissociated species (no ion pairs),28the perturbation of all of these species is not symmetrical with respect to the center of the membrane in this zCation = -zSite= 1 case, and their steadystateconcentration profiles arenot linear (see steady-stateanalysis of case I below). It is interesting to note that the concentrations of all species at the center of the membrane do not change throughout the experiment. The transient current and resistance responses are shown in Figure 2d,e. Clearly, the increase in membrane resistance corresponds to the decrease in the current. The influence of the interfacial potential drop arising from the presence of the carrier is noticed mainly initially (at tr < 0.05), where the slope of the current-time plot is largest and the resistance has not changed substantially yet.
1 0.2 0.67
DCati0n.r
Dsitc,r
Dim p i r . r
Darricr.r
1
Kr
0.14
Case I: Limiting Behavior due to Ionic Species Initially, a system in which there is a large excess of carrier is considered. Under these circumstances, any limitations on the transport through the membrane will be imposed by the ionic and ion-pair species. Steady-State Analysis. The steady-state current us voltage relationship and corresponding concentration profiles for membranes containing negative mobile sites and ion pairs, which allow the transport of a single cation when immersed between two aqueous solutions, were described in detail by Sandblom et al.I3 Of special interest in the present work are the steady-state resistance usvoltage and current relationships. The general shape of the resistance us reduced current plot as a function of the dissociation constant K (eq 3) is illustrated in Figure 4a using membrane parameters shown in Table I. The dimensionless resistance (y-axis) is given by A'ucstionCsite.tot
Theoretical Analysis of Two Limiting Cases The measured response to an externally applied potential step for a membrane containing all four species mentioned above can be quite involved. For instance, current us time curves may show features which are due to concurrent perturbations in both carrier and ionic species concentration profiles. Figure 3 shows how the two different processes can be separated in time: (1) The break in thecurrent a t -300 s is associated with thecompletedepletion of the carrier a t the interface where the selected ion enters and is followed by a short period in which the current obeys the Cottrell equation.' (2) The more gradual decrease in current later is due to the slow increase in membrane resistance, which is related mostly to the depletion of ionic (ion complex and site) species a t the interface where the selected ion exists. Although it is clear which mechanism is responsible for the observations a t different periods in this example, other membrane compositions may show responses to an applied potential step, which are due to simultaneous influencesof both processes, making it impossible to differentiate between the two mechanisms. Therefore, two limiting cases will be analyzed in this case: case I focuses on the ionic species by looking at membranes which contain an excess of carrier, and case I1 centers on the response of a membrane limited by the low concentration of carrier species.
Res, = Res
d
1 J
where cion(x) is the concentration of either positive or negative ions; u is the mobility given by DIRT; ii is the coupled, interdiffusion mobility of the dissociated species given by Zucationuanion/(ucation + Uanion); 4 is the ratio of the steady-state current to the limiting current, ZI;and 1 and Q are given in ref 13. Reference 13 also provides the steady-state relationship between the voltage and the interfacial ionic concentrations. If the contacting solutions are identical, this expression becomes
Substituting In cion(d)/ciOn(O) from this expression into eq 19 yields a linear relationship between the resistance and the voltage, when the magnitude of the current is near Zi (at 4 = 1). This agrees with the results shown in Figure 4b, which show the resistance us potential curves approaching a region of constant
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12367
Transport Processes in Membranes
11 I
1
high voltage
3.6
-1
t
2
low wnage
2-
l!
M=0 I
0:2
0:4
0:6
0:8
-
3 L ! 2.8
2’60
0.2 0.4
0.6
0.8
1
tr
.....Kd=l ........
1-
MrlQ !
0
0.2
04
0.6
0
08
1
0
02
I
I
0.4 0.6 voltage [VI
I
1
0.8
1
Figure 4. Theoretical steady-state (a) resistance us normalized current and (b) resistance us voltage, as a function of different (dimensionless) dissociation constants.
slope at higher voltages. It also implies that the interfacial potential drop is negligible in this region. Transient Analysis. The main difficulty in finding an algebraic solution for the transport processes in these membranes is due to the presence of ion pairs. While the transient behavior can be described using two systems of diffusion equations-one for the ion pairs and another for the ‘coupled” ions-the resulting boundary conditions for the problem arecomplicated. Traditional theoretical approaches to similar cases showed that besides the finite/semi-infinite boundary conditions, two more boundary equations relating the two diffusing species were needed. For instance,in the example of a diffusion-controlledsystemcontaining a redox couple, these equations were the Nernst equation and a flux-balance expression at the electrode/solution interface where the electron-transfer reaction took place.** Similarly, in this system one can use the equilibrium equation, eq 3, and a new boundary-flux equation. Unlike the case where only two completely dissociated ionic species are present, the total fluxes of the reactive charged species (the ion complex) at the interfaces are not only a function of the current flowing through the system but also depend on the flux of ion pairs. The resulting new boundary condition is
where int denotes the reaction interface, which is an expression containing a migration term (in the cation flux). Under these circumstances, it is impossible to reduce the differential equation system to a diffusion-only problem (see ref 28 for the equations in a completely dissociated system). Thus, the transient analysis of a system containing a cation (counter ion), an anion (site), and an ion pair is carried out using digital simulations. The accuracy of the results from the simulation procedure can be examined by their convergenceafter a long time to the algebraic solutions of the steady state obtained in ref 13. Figure 5 shows the dimensionless current (a) and resistance (b) transients for a series of increasing potential steps, which were obtained using a procedure described earlier in Digital Simulations. The reduced parameters are thoseshown in Table I, and no carrier was included this time. The corresponding steady-state plots us voltage are presented in c and d of Figure 5 , where the solid circles represent dimensionless current and resistance values calculated at tr = 1. The solid lines in the lower two plots were calculated from the expressions given in ref 13 and are in a good agreement with the simulation values. An interesting result, which was confirmed by the simulations, is that membranes containing mobile sites with larger diffusion coefficients are associated with faster depletion of ionic species at the interface where the selected ion exits. Therefore, these membranes will show an earlier drop in the current. This is in
8 3.4
’ . : ”-
3.2 3
2.6 2.80
0.2
0.4
0.6
0.8
voltage M Figure 5. Convergence of transient responses from digital simulations
to-steady-state values. (a) and (b) show the transknt current and resistance, respectively. The values at tr = 1 are shown as solid circles in the respective steady-state plots, (c) and (d), where the lines are from the algebraic solution^.'^
accord with observations of the so-called fixed-site membranes, which can be considered as consisting of very slow moving, or stationary, negative sites that exhibit virtually constant resistance and current for very long periods of time. Another important outcome from the digital simulations is the ability to follow temporal and spatial changes in the magnitudes of the electric fields, which are calculated from the sum of contributions from two types of processes, given on the right side of eq 14. The first term, the ohmic contribution, is larger in the regions where there is a substantial depletion of ionic species, i.e., on the right side of the membrane in this example. As the experiment progresses, this feature becomes more pronounced and the membrane resistance increases. The contribution from the left side of the membrane, where the ionic speciesaccumulate, does not change as much. In the second term on the right side of eq 14, the direction, or sign, of the diffusional contribution depends on the direction of the concentration gradients and on the relative magnitudes of the diffusion coefficients of the ionic species. In this example, there is always an accumulation of these species on the left side of the membrane, and the concentration gradient is, therefore, negative. Also, since the mobility of thecation is larger than that of the anion, the numerator is negative, and the diffusion contribution to the electric field is negative here. Using reduced parameters from Table I, a digital simulation of a mobile-site system without a carrier produces transient responses similar to those shown in Figure 2. The variation in the diffusion portion and the total electric field as the experiment progresses is shown in parts a and b of Figure 6, respectively. Case ik Limiting Behavior Due to Carrier The interpretation of results from this case follows principles applied in the analysis of electrochemical systems, which contain a large amount of supporting electrolyte. Under such circumstances, the electroactive species flux is controlled by diffusion only. Since the perturbation of the concentration profiles of the ionic species is negligible, the resistance is constant (eq 16), and the change in the current us time relationship is determined by the change in the interfacial potential drop, which was given in eq 10. Therefore, this system is identical to that described earlier in the case of a fixed-site membrane.’
Nahir and Buck
12368 The Journal of Physical Chemistry, Vol. 97,No. 47, 1993
Figure6. Electric-fieldprofiles in a 0.5-Vpotential step experiment: (a) diffusion contribution; (b) total electric field.
Steady-State Analysis. When the flux of the reactant (Le., neutral carrier) inside the membrane is controlled by diffusion only, the steady-state concentration profiles of this species are linearly proportional to the distance from the reaction interface. From Fick’s first law, the steady-state current is given by Isteadv state
= -AFJcarrier.steadr
state
--
memb,x=d Ccarrier
-
memb,x=O Ccarrier
d
AFDcarrier
(22)
The expression for the total potential drop over this system is always
since thediffusion potential is zero. Using the steady-statecurrent expression, eq 22, and since the sum of the reactant concentrations at the interfaces is always equal to twice the initial (at t = 0) the concentration at x = 0 is concentration, membs=O Ccarrier
=
Isteady stated
(24)
Carrier,tot - ~ A F D carrier
The steady-state membrane potential drop is then
Vstcady state
- -Isteady
stateReS
-
- -Isteady
stateReS
z+F
memb,x=O 2Ccarrier,tot Carrier memb.x=O Ccarrier
RT
11 4-
RT - -In
-
Isteady state
- TF In I1 - Isteady state
(25)
whereZ1is thesteady-statecurrent when the reactant iscompletely depleted at x = 0, given by
Transient Analysis. Case I1 is similar to the fixed-site membrane system, which was discussed in detail earlier.’ Experimental Part Reagents and Solvents. PVC and potassium tetrakis(4chloropheny1)borate (KTpClPB) were purchased from Fluka. Sodium tetraphenylborate (NaTPB) was obtained from Aldrich. Citric acid, concentrated NaOH, and tetrahydrofuran (THF) were from EM Sciences. Di(2-ethylhexy1)sebacate (DOS) and valinomycin were bought from Sigma. Sodium tetrakis[ 3.5-bis(trifluoromethyl)phenyl]borate (NaTmFMPB) was a gift from Dr. Maurice S. Brookhart (UNC).
Membrane Preparation. THF solutions of PVC and DOS (at 1:2 weight ratio, respectively) were prepared several days in advance and mixed with desired amounts of fresh THF stock solutions of valinomycin and fresh T H F stock solutions of mobilesite salt. The membranes werecast using the procedure described by Craggs et and were approximately 100 X 10-6 m thick. Apparatus. The membranes were placed between two aqueous 0.01 M KCI solutions in an electrochemical cell, designed for the purpose of ionic transport analysis, similar to that described Desired voltage steps were applied by a Solartron 1250 frequency analyzer (Solartron Instrumentation Group, Farnborough, England) through an EG&G 363 potentiostat (Princeton Applied Research, Princeton, NJ) controlled by a Zenith XT computer equipped with a CEC IEEE 488 interface (Capital Equipment Corp., Burlington, MA). This equipment was also used to perform impedance spectroscopy analysis of the membranes. Current os time data were collectedby the same computer using a Data Translation DT2801 A/D board (Data Translation, Marlborough, MA). Procedure. Constant-potential steps wereapplied to thesystem until the current and resistance stabilized. This normally took several hours, depending on the type of membrane under investigation. In the transient analysis, relaxation (at Vappl= 0) periods between consecutive voltage steps were at least as long as the duration of the previous applied potential. Steady-state results were sometimes recorded without any relaxation periods, to significantly reduce the length of the experiments (down to only several days). In these experiments, the magnitude of the potential steps was increased by an increment (usually 0.05 or 0.1 V) after the system reached a constant current reading. During the period in which a potential step was being applied, the impedance of the membrane was measured. The amplitude of the sinusoidal wave was 0.02 V RMS and was added to the constant-bias voltage. In the transient analysis, impedance measurements were done throughout theduration of the constant potential step. The resistance measurements were considered to be at steady state, since the bulk resistance did not change significantlyduring each of these high-frequencyimpedancescans. Experimental Results and Discussion The proposed theoretical model was tested using plasticized PVC membranes containing valinomycin and negative mobile sites. Although the contact with aqueous solutions induced membrane hydration, there was no evidence for any significant changes in conductivities and transport properties throughout the reported results. In addition, the Debye length32is approximately lo-* m, and the space charge regions are small compared to the l e - m membranes. Case I: Limiting Behavior Due to Ionic Species. Steady-State Analysis. A series of potential steps, increasing in size, were applied to a plasticized PVC membrane made with 10.4 mM valinomycin (total concentration) and 1.O mM NaTmFMPB, which was in contact with two 0.01 M KCl(,,) solutions. When the magnitudes of both current and resistance stopped changing (after approximately 1 h at each potential step), these readings were considered the “steady-state” values at the corresponding voltages. Unfortunately, the amount of noise and the breakdown of membrane selectivity did not always allow an easy determination of the location of the limiting current plateau on a current us voltage plot, Figure 7a. Therefore, steady-state resistance us current results were used instead. Figure 7b demonstrates the anticipated gradual growth in the magnitude of the bulk membrane resistance, as the current increases at voltages up to 0.3 V. However, instead of the sharp rise in the magnitude of the resistance, which is expected when the current approaches 11 (Figure 4a), a more gradual slope is observed as the current approaches 1 X 10-6 A. This drop in the rate of change of resistance os voltage was repeatedly noted in measurements up
Transport Processes in Membranes
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12369 (a1
(b)
2,67 2.41 2.2
1 4 0 -
-
0
0.2
0.4 0.8 0.8 current &AI
1
1 0
- 051
2 . W 4,000 6.000 8.000 10,000
0
2.000 4.000 6,000 8,000 10,000
F i p e 7. Experimental steady-state (a) current
us voltage and (b) resistance us current for a membrane containing valinomycin and TmFMPB sites.
[SI lime Is] Figure 8. Experimental transient (a) current and (b) resistance results from valinomycin membranes containing three different types of mobile sites.
to 1.0 V. Hence, while the steady-state results seem to agree with theoretical behavior at lower voltages, the readings at higher potentials suggest a failure to maintain the very low levels of ions (which correspond to larger resistances than those observed) at the membrane/solution interface, where K+ exits the membrane. A possible explanation is the failure to exclude co-ions (e.g., C1-) from entering the membrane together with K+ under these circumstance^^^ (Donnan exclusion failure in plasticized PVC membranes containing neutral carriers was discussed earlier in refs 18 and 3 1). This analysis also suggeststhat the failure occurs when the current is almost at its limiting value, approximately 0.9 X 10-6 A. The applied potential required to achieve the limiting current in this membrane is lower than that predicted by the steady-state results of the model membrane in Figure 5c. This fact can be attributed to a greater extent of dissociation of neutral ion-pair species in the membrane or to a larger dissociation constant-an assumption which can be confirmed by fitting the experimental steady-state results to various combinations of diffusion coefficients and dissociation constants. For example, these transport parameters can be calculated if the magnitudes of the initial resistance (at VapPl= 0), limiting current, and another steadystate resistance are given. However, the possible measurement errors, which are usually correlated with the long duration of experiments, allow only for determination of a range of values. A more accurate estimate uses this range and the results from the transient responses to isolate the best sets which describe the system response. This is shown below in the Transient Analysis subsection. Transient Analysis. A practical consideration in the design of the experiments in this work was that a steady state could be reached in a reasonably short time. Walker et al.I4 showed potential transients in a current-step experiment during approximately 100 h (using a 1.5 X le2m long tube containing 2-propanol and HCI). A corresponding relaxation period of approximately 40 h was also observed. In another experiment, Sandblom e2 ~ 1 presented . ~ ~a similar outcome after 80 h of constant-current steps in a 5 X le2 m long cell containing aqueous 0.01 N HCl. Such long measurements imply that a single set of steady-state results in the present investigation may take months toobtain. It alsobecameclear that long experimentsonplasticized PVC membranes may not be reliable. This was expecially evident in membranes with low resistances (less than lo5 Q for a membrane 1 0 0 X 10-6 m thick with a 1.77 X 10-4 m2 cross section) where even the fluctuations of resistance with time, when no current was flowing, were often significantafter severaldays of immersion in the test solutions and could not be related to any identifiable transport processes. A possible explanation for this behavior is a change in the plasticized PVC matrix due to water permeation. Someof the tested membranes showed cloudiness which increased
with longer soaking times in aqueous solutions. This was most noticeable in membranes containingtridodecylamine, a hydrogenion carrier, but was seldom observed in the valinomycin membranes. A similar hydration process was noticed before in PVC membranes.34 More recently, it was suggested that a 20-40 X 10-6 m water-rich surface layer formed on each side of the plasticized membrane, after it had been immersed in an aqueous solution for several days.35 It was also suggested that two distinct hydration stages were involved: an initial rapid step during which water is dissolved into the plasticized PVC matrix and a later slower process characterized by the formation of isolated water droplets.35 The present observation of variation of membrane resistance with time at zero current, which could be correlated with the slower matrix hydration stage, emphasizes the need for obtaining fast results before such changes become significant. The model presented here does not consider the existence of hydration layers. If, as a result of such conditions, the transport processes deviate from the current model, the experimental data should be viewed as qualitative. A convenient way to control the length of time until a mobilesite membrane shows a significantionic concentrationpolarization was found when different mobile-site species were used. The early carrier-type ion-selective membranes contained
