Modeling diffusion-limited, neutral-macrocycle-mediated cation

Rational Design of Liquid Membrane Separation Systems ... Ronald L. Bruening , Bryon J. Tarbet , Krzysztof E. Krakowiak , Merlin L. Bruening , Reed M...
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Anal. Chem. 1989, 67, 1140-1148

Modeling Diffusion-Limited, Neutral-Macrocycle-Mediated Cation Transport in Supported Liquid Membranes Reed M. Izatt,* Ronald L. Bruening, Merlin L. Bruening, Gypzy C. LindH, and James J. Christensen'

Departments o f Chemistry and Chemical Engineering, Brigham Young University, Provo, U t a h 84602

A membranediffusion-limited transport model has been found to accurately describe catlon transport by neutral macrocycles In thin sheet supported llquld membranes. Analysls of this model was performed by examining transport of CdBr,, CdCI,, and Cd(SCN), by a hydrophobic analogue of 18crown-6 (18C6). Macrocycle loss from the membrane is negligible In these systems. The model parameters are membrane geometry, solute diffuslon coefflclent in the membrane, Initial phase constituent concentrations and speciation, and an equlllbrlum constant for the extractlon process. This model has also been revised to include consideration of macrocycle loss from the membrane by Including the partition coeffklent of the macrocycle and aqueous catlon-macrocycle equlllbrlum constants as fundamental parameters. Transport of KNO, and Pb(NO,), by the 18C6 analogue was examined by uslng the revised model. Finally, the model has been extended to multiple catlon systems and studied with respect to competltlve alkali, alkaline-earth, and group I I B metal catlon transport uslng the 18C6 analogue. All of the model parameters are either readily measured or calculated. The model correctly quantifies the effects on transport and selectivity of varying anion concentration and source phase anion type as observed In previous macrocycle-mediated membrane studles. Finally, the fundamental parameters involved In the modellng equatlon allow for the effects on transport rates and selectlvltles of catlon, anion, and macrocycle type and concentration to be dlstlngulshed. This makes possible the correlatlon of transport rates and selectlvltles with the molecular properties of the species Involved and the more rapid screening of new macrocycles as possible membrane carriers.

Macrocyclic ligands such as certain crown ethers show excellent cation selectivity when used as carriers in liquid membrane transport. Furthermore, the observed cation selectivity order is often similar to that found for cation binding by the same ligands in homogeneous solution (1). Most previous membrane transport experiments involving macrocycles involved bulk or emulsion liquid membrane systems. The studies done with these systems have been reviewed (2-4). Liquid membranes containing macrocycles have potential applications in ion-selective electrodes (5-2 I ) and in making specific separations. These applications require a quantitative understanding of the transport process. Recently, macrocycles have been incorporated successfully into thin sheet and hollow fiber supported liquid membranes (12, 13). The selective interaction properties of the macrocycles were used to make cation separations in these new systems as has been done in bulk and emulsion liquid membranes. The thin sheet supported liquid membrane (TSSLM) system has advantages over each of the other three liquid *To whom correspondence should be addressed at the Department of Chemistry. Deceased, September 5 , 1987. 0003-2700/89/036 1-1 140$01.50/0

membrane systems in modeling transport in a t least two of the following areas. First, the TSSLM has the best-defined geometry. Second, both aqueous phases of a TSSLM can be stirred in order to maintain equivalent solute concentrations throughout these phases and minimize aqueous boundary layers. Third, the membrane phase is not stirred allowing the important transport step to be diffusion through the bulk of the phase rather than diffusion through the organic boundary layer. Fourth, small samples of both aqueous phases can be removed several times during the course of a timed experiment with minimal effect on phase volumes and concentrations. A few attempts to model macrocycle-mediated cation transport in liquid membranes have been made. Building on the work of Ward (14),Reusch and Cussler derived equations for the diffusion-limited transport of a cation and cotransporting anion by a neutral macrocycle and showed that several empirical trends in transport with a particular macrocycle are predicted by the equations (15). Lamb et al. showed that the Reusch and Cussler model predicts several other observed trends with the same macrocycle (16). Later, Lamb et ai. attempted to modify this model to include kinetic reaction terms to explain the eventual decrease in transport with increasing stability of the macrocycle-cation complex (17). However, Behr et al. (18)and Fyles (19)have since shown that this phenomenon can be explained by using a diffusion-limited transport model. Stolwijk et al. even took into account empirically the loss of the macrocycle to the aqueous phases in a diffusion-limited model (20). However, no modeling studies have been made in which the macrocycle-mediated cation fluxes and selectivities are numerically predicted under a variety of conditions from an appropriate model and a few fundamental and measurable parameters. In this paper, the easily modeled TSSLM system will be used to show the ease and accuracy of such modeling. In particular, the knowledge of extraction and aqueous equilibrium constants (K, and K ) will be shown to be essential to the modeling process. The various approaches that have been used to model the effect of the cotransporting anion on transport (16,21,22)will be shown to be equivalent. The issue of modeling macrocycle loss from the membrane will also be tackled, although some experimental systems are modeled easily without this consideration because the macrocycle is maintained virtually quantitatively in the membrane phase. This retention is due to the batch-type experiments performed, the hydrophobicity of the macrocycle used, and the cationanion extractability and small aqueous cation-macrocycle K value designed into the source phases of these systems. However, many needed cation separation conditions and certainly continuous separation systems will create situations where some loss of the macrocycle to the aqueous phases in liquid membrane systems will occur. It will be shown that the model can be modified by using fundamental and measurable parameters to account quantitatively for macrocycle loss from the membrane. With this modified model, not only will the need for replacement of the macrocyclic ligand in the membrane be quantified and understood, but numerical cation fluxes will still be predictable and understandable under system conditions where macrocycle loss from the membrane C 1989 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

r

Stirrers

-1

I

membrane

I -

U Figure 1. Thin sheet supported liquid membrane (TSSLM).

occurs. Finally, the model will be extended to competitive transport systems. Both selectivity and cation flux will be accurately predicted by using the same measurable fundamental parameters.

EXPERIMENTAL SECTION Extraction and Transport Determinations. The TSSLM (Figure 1) experiments were carried out as described previously (12). The K,, values for extraction of the various neutral ion pair cation-anion combinations by the macrocycle from the aqueous to the organic phase were determined according to the method of Ouchi et al. (23). However, additional experiments where the concentration of the macrocycle was held constant, but the initial aqueous phase neutral cation-anion ion pair concentration was varied, were also performed and showed that stoichiometries involving more than one cation per macrocycle were not operative. The K,, experiments were conducted with equal volumes of aqueous and organic phases so that loss of the macrocycle to the aqueous phases would be negligible. The ionic strengths of the solutions used in the K,, measurements matched those used in the TSSLM experiments when activity coefficient data were not used. Measurement of cation disappearance from aqueous phases containing 0.001 M Cd(N03),and excess SCN-, Br-, or C1-; 0.001 M Hg(N03), or Zn(NO,), and excess SCN-; or 0.001 M Pb(N03), and excess Mg(N03), were used in the Cd(SCN),, CdBr,, CdCl,, Hg(SCN),, Zn(SCN),, and Pb(N03),K, experiments, respectively. Aqueous phases containing various concentrated solutions of either CdCl,, ZnBr,, NaNO,, Ca(NO,),, Sr(N03),, Ba(N03)2,or 1 M KNO, were used in these respective K,, measurements. The appearance of the cation in back extractions to pure H 2 0 was monitored. Reagent grade chemicals were used in all experiments. Phenylhexane (Eastman Kodak) was used as the organic solvent and bis(l-hydroxyheptylcyclohexano)-18-crown-6(R2DC18C6) (Parish Chemical, Orem, UT) (see structure) was used as the macrocycle in all experiments. Phenylhexane was used as the

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tioning is necessary to maintain a significant fraction of hDCl8C6 in the membrane for the TSSLM systems studied. We desired to model systems where there can be some macrocycle loss from the membrane but where sufficient cation transport occurs for accurate measurement of experimental fluxes for comparison with those predicted from the model. Partition Coefficient Determination. The partition coefficient, K,, for hDCl8CS partitioning between phenylhexane and water was measured. This could not be done by conventional means (24,25)since in aqueous-organic solvent systems with no cations present R2DC18C6 is partitioned nearly quantitatively to the organic phase even when the volume ratio of the aqueous to the organic phase is quite large (12). Hence, 0.33 M Ba(N03), was added to the aqueous phase of an extraction experiment to enhance loss of R2DC18C6 to the aqueous phase. The enhancement occurs as the R2DC18C6 partitioned to the aqueous phase interacts with aqueous Ba2+. This experiment involved equal volumes of organic and aqueous phases. Barium was chosen as the cation for this experiment since the K value for its interaction with 18-crown-6 (18C6) sized macrocycles is large (26), Ba(N03), extraction by R2DC18C6 is small (see Results and Discussion), and the K value for Ba2+-NO; ion pairing a t 1 M ionic strength is small and has been quantified (27)allowing the calculation to be made of the amount of Ba(I1) present as Ba2+, Ba(N03)+,and Ba(NO,),(aq). Experiments to determine K , involved mixing for 1 h 8 mL of aqueous 0.33 M Ba(N03), with 8 mL of 0.05 M R2DC18C6 in phenylhexane in a small, capped weighing bottle using a magnetic stirrer and stirring bar. The extraction experiments have shown that this time period is sufficient for equilibrium to be reached. The phases were then allowed to separate. A 4-mL aliquot of the organic phase was removed and similarly equilibrated with 4 mL of water containing 0.001 M Pb(NO3I2and 0.166 M Mg(N03)2. The final aqueous phase was analyzed for Pb2+ and Ba2+. A similar experiment involving 4 mL of the above lead-containing aqueous phase and 4 mL of fresh 0.05 M R2DC18C6 in phenylhexane was also performed. The disappearance of Pb2+from the source phase (the original aqueous phase is also analyzed for Pb2+) is measured in the Pb2+experiments. Loss of R2DC18C6 to the aqueous phases is negligible in the Pb2+extractions as expected and determined from the similar Pb(N03), K,, experiments with varying organic phase R2DC18C6concentrations. Enhanced ion pairing of Pb2+with the excess NO3- is also a major factor in negating R2DC18C6 loss under these conditions (27). An equation to calculate K , for R2DC18C6 using K , and K expressions as well as the above experiments will now be derived. A mass balance for the R2DC18C6 a t the beginning and end of the Ba(N03), extraction involves the initial amount of hDC18C6 and that present either as free R2DC18C6 or as the Ba(II)-R2DC18C6 complex (Lorgtod,find) in either the aqueous or organic phase a t equilibrium. The amount of Ba(N03),-R2DC18C6 complex in the organic phase is measured experiment,ally from the amount of Ba2+in the back extraction. Expressions for the concentrations of aqueous and organic phase free R2DC18C6 (Laqand Lorg)as well as aqueous phase complex (MLm+*qwhere Mm+= Ba2+)can be derived and substituted from the K , and K expressions given as

K , = [Llorg/[Llaq

(1)

and

K = [MLm+]aq/[Mm+]aq[L]aq

(2)

The original mass balance can now be written as [Linitidlorg - [Ba(N03)2L]org= [L]"'g(l

+ l/K,(l + K[Ba2+]*q)] (3)

R = l -hydroxyheplyl

bis-(l-hydroxyheptylcyclohexano)-1&crown4

solvent in order to maintain membrane integrity (12). The macrocycle R2DC18C6 was selected for use because of its large distribution to phenylhexane over water (12). This large parti-

where only [LIorgand K , are unknown parameters. Only concentrations of the species are involved in the mass balance since the volumes of both phases are equal. The values of [Linitidlorg and [Ba2+Iaqare known from the experimental conditions since [Ba2+Iaqremains constant (little extraction). The K value for Ba2+-18-crown-6 interaction was used rather than that for Ba2+-R2DC18C6interaction (26)since R2DC18C6 is too insoluble in water for K measurement. Hence, the K, measurement must

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ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

be considered to be only an estimate. However, the presence of alkyl or cycloalkyl macrocycle substituent groups has a minimal effect on cation-macrocycle interaction (26),which suggests that this estimate is a reasonable one. Dicyclohexano-18-crown-6 (DC18C6) is more similar in structure to the macrocycle of interest than 18C6. However, some uncertainty is involved in the value of K for Ba2+-DC18C6interaction (26). Hence, the corresponding Ba2+-18C6K value was used. The [L]"'g value can be obtained by subtracting the amount of [Ba(N03),Log]from [Lorgtotal,fmd]. The value of [Lorgmtal,find] is obtained by analyzing the amount of Pb(N03)2back extracted and comparing this with similar Pb(N03)*extraction using fresh RzDC18C6-containingorganic phases in the K,, expression. Once all other parameters in eq 3 are known, the value of K , can be calculated. A detailed presentation of the algebra involved in the above derivations is given elsewhere (28). Materials and Calculations. The TSSLM source phases and solvent extraction aqueous phases included at times the salts Cd(NO& (Baker & Adamson), Mg(NO& (Baker & Adamson), MgBrp (Coleman & Bell), LiSCN (Alfa), MgCI, (Mallinckrodt), Pb(NO& (Spectrum), Ba(N03)2(J. T. Baker), KNOB (Mallinckrodt), Hg(N03)2 (G. Frederick Smith), Zn(NO& (Mallinckrodt), NaN03 (J.T. Baker), LiN03 (Mallinckrodt), (Fisher),and Ca(N03j2(J.T.Baker). The Li+ and Mg2+salts were used as the additional reagents in obtaining excess anion concentrations and/or to adjust ionic strength since these cations do not interact with 18C6-sized crown ethers (26). The single cation transport studies of Cd(SCN)2, CdBrz, and CdC1, were conducted in excess concentrations of the anions so that little Cd(I1) would be present as the free cation and, hence, available to enhance macrocycle leaching to the aqueous source phase via complexing with the macrocycle. Cadmium(I1) was chosen for study because (1)the K value for its interaction with DC18C6 in H20 is small, (2) it can be readily transported in the membrane system in excess concentrations of the anions, and (3) transport of Cd(NO3), is minimal (22). The combination of the characteristics of R2DC18C6 and Cd(I1) allowed experiments to be performed where the ionic strength of the source phase could be controlled with Mg(NOB), or LiN03 and virtually all of the R2DC18C6 will remain in the membrane phase. Some of the Pb(N0J2 single cation TSSLM experiments were performed with identical source phases, but with varying RzDC18C6concentrations, while the KNOBsingle cation experiments were performed with varying source phase conditions, but the same RzDC18C6 concentration. These two TSSLM systems were chosen for study since (1) significant transport occurs; (2) macrocycle loss from the membrane occurs; and (3) the effects of varying concentrations of both macrocyclic ligand and source phase constituents on macrocycle loss from the membrane as well as cation fluxes can be assessed. The competitive cation systems were chosen so that the effect of each of these possible factors on transport could be examined. The TSSLM receiving phase was always distilled, deionized water. Aqueous phase cation analyses were performed with a Perkin-Elmer Model 603 atomic absorption spectrophotometer. Aqueous phase anion analyses were determined by ion chromatography (Dionex Model 2010). The anion analyses were performed in both K,, and transport measurements to show that the transport mechanism involves a single type of anion when more than one anion type is present. This was empirically found to be the case in each applicable instance. All experiments were performed in triplicate and the standard deviations are presented with the results. The experimental fluxes were calculated by obtaining the experimental slope and slope standard deviation for receiving phase cation concentrations vs. time and multiplying these quantities by the receiving phase volume and dividing by the membrane surface area. A least-squares linear regression analysis was used to determine the above slope values. The flux values were calculated by using the entire surface area of the membrane and not just the porous area. This practice has been followed in previous SLM studies with other types of carriers (29-35). The flux values for the porous surface area can be calculated by dividing the fluxes listed by the porosity given by the manufacturer (0.38). Aqueous equilibria calculations involved all appropriate stoichiometric combinations of cation-anion interactions described by Smith and Martell (27).

R E S U L T S A N D DISCUSSION Membrane Diffusion-Limited T r a n s p o r t Modeling. The first modeling equation desired must describe single cation transport without loss of the macrocyclic carrier from the membrane. T o obtain such an equation, we will start with the solution to the differential equations for mass transport assuming transport to be membrane diffusion-limited as derived by Ward (14). The result can be expressed as

where the first term represents non-carrier-mediated transport of any neutral species S and the second term involves transport of S mediated by the ligand L. The superscripts s, r, and org represent the source, receiving, and organic membrane phases, respectively, D is the diffusion coefficient for the indicated species, 1 is the transport path length through the membrane, Ltod is the sum of L present as the free carrier and as the complex SL, and K is the equilibrium constant for the formation of the SL complex in the membrane. It must be remembered that the model of eq 4 applies only when diffusion through the membrane is the transport rate determining step. The model must be modified when either interfacial diffusion or kinetics is important. The utility of the model lies in the fact that macrocycle-cation single phase and solvent extraction interaction kinetics are extremely rapid (3, 15, 18, 19, 26). Empirical observations allow the incorporation of several assumptions used in obtaining the Reusch and Cussler model (15). First, comparison of experiments with and without R2DC18C6 showed that carrier-mediated fluxes are much greater than non-carrier-mediated fluxes with phenylhexane as membrane solvent. This allows the first term to be dropped. Second, (SIs>> (S)' and 1 >> K(S)' during the course of the experiments allowing the smaller terms to be neglected. Hence, we obtain JA

=

~sL~~L,wlo'g[sls 1[1 + K(S)S]

(5)

Transport of cations by neutral macrocycles requires that anion(sj accompany the cation in transport in order for electrical neutrality to be maintained. Hence, the transported species S is really M,A,,, where M"' is the cation and A"- is an anion. We will be dealing with monovalent anions in this study and so MA, transporting species will be presented in the equations to follow, although the equations can be written for multivalent anions as well. At this point, Reusch and Cussler tailored their model to 1:l salts with only a single salt present in the source phase and divided the extraction constant into a partitioning and a reaction coefficient for M+saq + A-,aq = MAmembrane constant for MA + L = MAL interaction in the membrane (15). We wished to have a more general diffusion limited model to deal with more complicated source phases including those with significant ion pairing such as in our recent Zn(II), Cd(II), and Hg(I1) separations (22). Furthermore, the above partitioning coefficient and MA + L interaction terms involving very hydrophobic solvents such as phenylhexane are difficult to measure because they are very small and very large, respectively. Hence, our model considers the transport of an MA,,, species from the aqueous phase and involves the single extraction equilibrium expression MAmaq + Lmembrane = MAm Lmembrane, which can be described by an appropriate K,, value. Values of K,, are readily measurable (23) and show less activity coefficient variation than the above partition coefficients since only neutral species are involved. Furthermore, the aqueous phase neutral ion pair concentrations

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

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Table I. Predicted and Experimental Cd(I1) Fluxes (J)" 108J

A-

1% Ke,b

[AI, M

BrBrBrBrSCNSCNSCN-

0.61 f 0.04 0.61 f 0.04 0.61 f 0.04 0.61 f 0.04 2.01 f 0.07 2.01 f 0.07 2.01 f 0.07 -1.3 f 0.2 -1.3 f 0.2

0.56 0.30

c1c1-

0.20

0.10 0.85 0.40 0.10 0.56 0.20

0.27 0.30 0.24 0.10 0.40 0.46 0.14 0.40 0.20

104(0/1)d

predictede

observed

3.42 3.54 3.39 3.20 3.01 2.69 3.10 d d

83 110 74 35 1100 1100 670 1.6 0.8

89 f 5 10 f 2 79 f 1 35 f 1 1010 f 10 940 f 20 650 f 20 2.5 f 0.30 2.0 f 0.20

"Units of mol.s-1.m-2 in a 1 M ionic strength, 0.05 M Cd(II)/0.05 M R2DC18C6 in phenylhexane on Celgard 2400/H20 thin sheet supported liquid membrane. *CdAZaq+ Lort= CdAzLorginteraction. cFraction of the total amount of Cd(I1) present as the neutral CdAz complex calculated using the following Cd2+-A-aqueous interaction constants given as log p, values: Br-, PI = 1.57, p2 = 2.1, p3 = 2.6, p4 = 2.6; SCN-, PI = 1.32, p2 = 1.99, p3 = 2.0, p4 = 1.9; C1-, p1 = 1.35, pz = 1.7, p3 = 1.5 (27). dunits of L.s-l.m-z. Experimental value of the diffusion coefficient divided by the diffusion path length calculated using the observed flux and membrane diffusion limited transport model. The average D/1 value is 3.19 X lo-' L.s-.m-2. The values for the CdC12systems were not used in the average since there is greater inaccuracy in measuring the much smaller K,, and J values. 'Predicted from the average D/1 value and membrane diffusion limited transport model. can be calculated at particular ionic strengths by using the total cation and anion concentrations along with appropriate K values (27). These adjustments to eq 5 result in D M A , K e x [Ltntal]0rg[MAm18

JMA,

=

l(1 + Kex[MAmI')

(6)

which is the modeling equation used in this study. I t is obvious from the above discussion that eq 6 and the Reusch and Cussler model are identical under the particular case of a single 1:1salt since the concentration of the neutral ion pair is related to the concentrations (actually activities) of the free cation and anion by a constant, the K value for ion pairing. The advantage of using eq 6 over earlier models (14, 15, 17-20) is that all of the terms can be readily measured and/or calculated. Hence, if the model is found to be effective in predicting MA,,, fluxes, only a few measurements are necessary in order to predict membrane behavior under a number of different conditions. Effectiveness of the Model in Predicting CdBrz, Cd(SCN),, and CdClz Fluxes. Equation 6 was tested to see if experimental fluxes can be predicted accurately from a knowledge of the five fundamental parameters K,,, CY,, D/1, [LtOdlorg,and [Cd(II)tob#, where CY, is the fraction of Cd(I1) present as the neutral ion pair. In Table I, values for these parameters are presented together with predicted and experimental flux values for the several Cd(I1)-A- systems. The log K,, values in Table I were measured as part of this study. The [CdAzIs values are obtained by multiplying 0.05 M, [Cd(II)tod]s,by CY,. The [LtntalloTterm is 0.05 M in all of the systems studied since the bDC18C6 is retained quantitatively in the membrane phase. This situation, which is not characteristic of most systems studied, was the reason for choosing these particular experimental conditions as described in the Experimental Section. This leaves only the D/1 term to be obtained. The diffusion coefficients for neutral solutes in the same solvent show minor variations (36,37) and can be considered to be constants. The membrane diffusion path length, 1, is held constant in all of our experiments. The value of I , however, is very difficult to obtain for supported liquid membranes since the porous path of the support is tortuous. Furthermore, the entire surface area of the support is not available to the liquid since a certain percent porosity is involved. It was decided that under these conditions it would be better to include the porosity in the D/1 term of our model and to calculate this constant from a few experimental flux values. There is some disadvantage to this calculation method in that D/1 must be obtained ex-

perimentally for any change in membrane solvent and/or support. Furthermore, in order for our model to be effective, a single D/1 value must be used to predict flux values successfully under conditions where flux, K,,, [MA,,$, and [Ltodlorgvalues vary, but the membrane solvent and support are the same. The D / 1 values for the CdBrz and Cd(SCN)z systems necessary to predict the experimental flux values exactly were calculated by using eq 6 and the other equation parameters (Table I). Corresponding D / l values for the CdCl, systems were not calculated because of the greater inaccuracy in measuring the much smaller K,, and empirical flux values. The small variation in the calculated D / l values is evidence that the membrane diffusion limited transport model of eq 6 describes well the transport process being observed. Membrane diffusion limited transport in these situations is not surprising considering the rapid kinetics of cation-macrocycle interaction (26) and the high interfacial stirring rates (12). The calculated D/1 values were averaged to obtain a value, 3.19 x L-s-1-m-2,for use in predicting numerical fluxes using eq 4 and the other equation parameters. Such predicted fluxes for all nine systems studied are also given in Table I. The fact that the back calculated D / 1 values for the CdBr, and Cd(SCN)zsystems showed little variation ensured that excellent agreement between the predicted and experimental fluxes would be observed for these systems. However, the well within an order of magnitude agreement of the predicted and experimental CdClz fluxes, despite the difficulties of obtaining the values, gives further credence to the model used. The expected transport of two anions of one particular type with each Cd2+cation was tested and confirmed for the CdBrz experiments where the agreement was always within 5%. Use of the Model to Understand the Various Effects on Transport. The model also offers advantages in separating the effects on transport and selectivity of cation, anion, and macrocycle type and concentration. The concentration effects are separated into the [LtndloVand [MAmIsterms. In the K,, term, all concentrations are normalized. The linear effect of RzDC18C6 concentration on transport involving the macrocycle used in this study as predicted by eq 6 has been demonstrated (12). The cation and anion concentration effects can also be separated in the calculation of [MA,,$ from the total concentrations and interaction equilibria. We must remember, however, that the equilibria are also a function of cation and anion type. Fortunately, these type effects are generally well understood and have been quantified in compilations such as that of Smith and Martell (27). The other effect of anion type on transport is the ease of extractability of the anion from the aqueous to the membrane phase. This

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ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

effect can be singled out by looking at K,, values for systems containing the same macrocycle and cation but varying anions as in the present study. For example, the more hydrophilic C1- is much more difficult to extract into the membrane phase and, hence, Cd(I1) transport rates involving C1- are reduced in comparison to those involving Br- and SCN- a t the same [CdAzjSlevels. Both of these anion type effects have been observed in liquid membranes (16,21,22)and are now easily modeled and understood using eq 6. The effects of macrocycle type on cation transport can be seen by comparing K,, values for the same organic solvent and anion type while varying cation or macrocycle type. This effect will be examined more rigorously in the section on competitive transport. Modeling Loss of Macrocycle from the Membrane Phase. In modeling loss of macrocycle from the membrane, we will also assume that the kinetics of macrocycle partitioning are rapid. Phase distribution of the macrocycle will be defied in terms of species concentrations in the two well-stirred bulk aqueous phases and in the two interfacial regions of the membrane as well as the K values for macrocycle partitioning, macrocycle-mediated cation and accompanying anion extraction, and aqueous phase macrocycle-cation interaction. The derivation of such an equation to model macrocycle loss from the membrane to the aqueous phases necessarily begins with a mass balance equation. The amount of macrocycle originally present in the membrane phase is equated with the various macrocycle species present during the transport process. These species are the free macrocycle in any of the phases, cation-macrocycle complex in the aqueous phases, and cation-accompanying anion-macrocycle complex in the membrane phase. In membrane diffusion limited transport, the parameter of interest is the total concentration of ligand in the membrane phase present either as free macrocycle or a complex (eq 4-6). Hence, such an additive term is used in the mass balance expression. Volume times concentration terms are involved in the mass balance since the phase volumes are unequal. The mass balance expression is solved for a steady-state condition as is the case for modeling of cation fluxes. The TSSLM is ideal for such modeling since the source phase concentrations change little over time and the receiving phase cation and anion concentrations are relatively negligible (12). Some assumptions reduce the complexity of the mass balance for the TSSLM, although the modeling equation can be derived and used without these assumptions (28). These assumptions are that the concentrations of the cation and cation-anion ion pair in the source and receiving phases are the initial concentrations and zero, respectively. The receiving phase assumption implies that cation-macrocycle extraction and aqueous interaction are negligible at the receiving phase membrane interface and that all the membrane-contained macrocycle at this interface is present as free macrocycle. These assumptions hold for the 6 h, concentrated source phase TSSLM studies of this paper. After the above assumptions are made, the mass balance is given as

where the superscripts org and aq are the organic membrane and either aqueous phase, respectively, L is the free ligand, V is the volume; and MLm+is a cation-macrocycle complex. The volumes of the two aqueous phases are equal in the TSSLM experiments and so a single aqueous volume term is used. Expressions for [L] in all three phases and [MLm+]' can be derived by using a mass balance for the organic phase only and the K,, K , and K,, expressions given, respectively, as eq 1, 2, and

[MA,LIorg

Kex =

[ MA,Jaq[L J o r g

The K,, expression is given for monovalent anions since these were used in our K,, and TSSLM studies. Substituting these expressions into eq 7 and rearranging terms, we obtain

in which the actual total concentration of the macrocyclic ligand in the organic membrane phase, [LtoM,findlorg,is given in terms of the original amount of macrocyclic ligand, three measurable equilibrium constants, and two known concentrations. When additional cations or cation-anion combinations are present in the source phase, additional terms are added to the two summation terms in the far right side of the denominator. The fraction of the initial total amount of macrocyclic ligand which still remains in the membrane, f , can be calculated by using eq 9. A detailed presentation of the algebra involved in deriving eq 9 is given elsewhere (28). In order to model cation fluxes with loss of macrocycle from from eq 9 the membrane phase, one calculates [Ltod,findlorg and uses this value as [Lto+Jorg in eq 6. One advantage of modeling cation transport using eq 9 and 6 is that all of the terms can be measured and/or calculated. Hence, if the model is found to be effective in predicting MA, flux, only a few measurements are necessary in order to predict membrane behavior under a number of different conditions where macrocycle loss from the membrane may or may not occur. Furthermore, just as the various parameters in eq 6 can be used to identify macrocycle, cation, and anion type and concentration effects on the transport process, the parameters in eq 9 can be used to understand why and when loss of macrocycle from the membrane occurs from numerical, predictive, and molecular viewpoints. Effectiveness of the Model in Predicting Pb(N03)2 Fluxes with Varying Macrocycle Concentrations. The derived model involving eq 9 and 6 was tested to see if experimental fluxes could be predicted from the model equations and knowledge of the few fundamental parameters in the equations. The first system tested involved Pb(NO&, using a constant source phase content, and a systematic variation of the initial concentrations of macrocycle in the membrane. Excess NO3- was present to enhance Pb2+-N03- ion pairing so that transport would be enhanced and leaching of the macrocycle from the membrane due to aqueous Pb2+-macrocycle interaction would not be overwhelming. The TSSLM phase conditions, the parameters used in and calculated from the predictive equations, and the predicted and observed fluxes for these systems are given in Table 11. The predicted fluxes in Table I1 were calculated in the following manner. The amount of macrocycle remaining in the membrane was first calculated by using eq 9. This parameter is presented in Table I1 as f , the fraction of the macrocycle remaining in the membrane. The log K p value for the macrocycle was estimated to be 4.2 according to the method described in the Experimental Section. The K values used for eq 9 are for interaction of cis-syn-cis-DC18C6 with the cations. As discussed earlier, K values for the actual macrocycle used cannot be measured, but macrocycle-cation K values show little variation with addition of alkyl and cycloalkyl substituent groups. The macrocycle closest in structure to R2DC18C6 is DC18C6. The cis-syn-cis isomer was chosen for use since K values for this isomer are slightly higher than those for other isomers and the presence of the OH group on the side chain of R2DC18C6might be expected

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

1145

Table 11. Predicted and Observed Pb(NO& and KNOs Fluxes ( J ) " M(m) Pb(I1) Pb(I1) Pb(I1) Pb(I1) K(I) K(I)

source phase [M(m)l,M [NOCl,M 0.05 0.05 0.05 0.05 0.2 0.1

0.66 0.66 0.66 0.66 0.2 0.1

01, b

0.28 0.28 0.28 0.28 0.663 0.739

P

"m

0.32 0.32 0.32 0.32 0.062 0.039

membrane [Lil,d M

0.12 0.12 0.12 0.12 0.60 0.71

0.005 0.01 0.03 0.05 0.10 0.10

108J

log Ke/ 2.60 2.60 2.60 2.60 -0.25 -0.25

f 0.04 f 0.04 &

0.04

f 0.04 f 0.13 f 0.13

predicted'

observed

16.0 33.0 98.0 165.0 13.0 4.9

12.0 f 2.0 31.0 f 4.0 91.0 f 11.0 191.0 f 20.0 12.0 f 0.5 5.0 0.1

*

OUnits of moldrn-*. Receiving phase is always distilled, deionized water. *Fraction of the total concentration of M(m) present as the (M(N03)ispecies, where i = o or m, in concentration and activity terms for the Pb(I1) and K(1) cases, respectively. The calculated Pb(I1) values are valid at an ionic strength of 1 M (log of the p, interaction constants: = 0.33, pz = 0.4). The calculated K(1) values were obtained by using KNOBactivities and the 0 M ionic strength K value for ion pairing (log K = -0.15) (38,27). The excess NO< is present as Mg(N0J2. cFraction of the macrocyclic ligand (L), R2DC18C6, remaining in the phenylhexane membrane phase. Calculated by using eq 9, the other data in this table, the estimated partition coefficient of the macrocycle (log K , = 4.21, and the K value for aqueous Mm+-cissyn-cis-dicyclohexano-18-crown-6 interaction (log K = 4.95 and 2.02 for Pb2+and K+, respectively) (26). Initial concentration of the macrocyclic ligand, L, in the membrane phase. 'log K,, values valid for M(N03)msq+ Low = M(NO3),Lorgextraction at 1 and 0 M ionic strengths for Pb(N03)zand KNOB,respectively. 'Predicted by using eq 6, the other data in this table, and the empirical D/1 value of 3.19 X L.s-'.m-2 determined for Dhenvlhexane on Celeard 2400 sumorts. to slightly enhance cation-macrocycle interaction in some cases. In a n y case, little variation is observed in K values for 18C6 vs the isomers of DC18C6 (26). We felt that whenever possible, K values for a macrocycle similar to the one actually used should be employed in the modeling calculations for consistency. The log K (H,O) value for Pb2+-cis-syn-cisDC18C6 interaction is 4.95. The log K,, value (Table 11) involving Pb(N03)2and R2DC18C6was measured a t the ionic strength (1 M) present in the source phases of the Pb(I1) TSSLM experiments. The standard deviation of this log K,, value is small. Once f is known, the Pb(N03)2flux values can be predicted easily by using eq 6. The [Ltotallorgterm is now obtained by multiplying [Linitial]OrK by f. The D/1 value used in the flux predictions is that obtained empirically for phenylhexane on Celgard 2400 supports, 3.19 X L.s-1.m-2, using the Cd(SCN)2 and CdBrz experimental data. The other terms necessary to use eq 6 are given in Table 11. The calculated f value for these Pb(I1) systems is 0.12. Almost 90% of the macrocycle is lost from the membrane despite our efforts to enhance Pb2+-N03- ion pairing with excess NO3-. Furthermore, the value o f f is independent of the amount of macrocycle present as seen in eq 9. Thus, a plot of cation flux vs the amount of macrocycle initially in the membrane is linear whether or not leaching occurs (eq 6). This linear relationship has been observed (15,16,18,19). The fact that macrocycle loss from the membrane has seldom been considered in modeling is probably due to the linearity in either case. Macrocycle loss occurs because the extraction process with this particular macrocycle and membrane solvent cannot compete with aqueous macrocycle-cation interaction under the source phase and volume ratio conditions. This inability to compete is easily understood by comparing log K , log K,,, and log K , values. The hydrophobicity of the macrocycle is sufficient to maintain the macrocycle in the membrane phase when mixed with pure water, even when the water phase is present in large excess as seen in the large K , value. The affinity of Pb(I1) for the macrocycle is high as seen in both the K and K,, values. However, the difficult extractive process involving Pb(I1) and NO3- changing phases is not sufficiently enhanced by the hydrophobicity of the macrocycle to compete with aqueous Pbz+-macrocycle interaction as seen by comparing K and K,, values. The large volume ratio of the phases offsets the large K , value, and leaching driven by cation-macrocycle interaction occurs. This example illustrates the effectiveness of using an equation comprised of only fundamental parameters in understanding, on both a quantitative and qualitative molecular basis, loss of macrocycle from the membrane phase.

Once f is known, the Pb(N03)2flux values are predicted easily by using eq 6. The agreement between observed and predicted fluxes shown in Table I1 is quite good. This agreement further confirms that the membrane diffusionlimited transport model involving a single D/1 term for a particular solvent-support combination predicts cation transport effectively. The agreement using an f value of 0.12 demonstrates that modeling of macrocycle loss from the membrane is an essential part of the modeling process. If the macrocycle were assumed to remain quantitatively in the membrane, all the predicted flux values would be multiplied by 8.3 and predicted and observed agreement would be poor. The agreement also shows that the effect of changing membrane macrocycle concentration both initially and during transport is well understood. Effectiveness of the Model in Predicting KNOBFluxes with Varying Source Phase KNOB Concentrations. Equations 9 and 6 were similarly used to model K N 0 3 fluxes. In this modeling, activity coefficients (38) and the resulting activities were used in the modeling process and in obtaining a K,, value valid a t zero ionic strength. This was done since KNO, activity coefficient data are more prevalent than K+-N03- ion pairing data a t various ionic strengths. Hence, activities were used rather than concentrations in eq 9 and 6. The activity coefficients of the neutral macrocycle in either water or phenylhexane and of the transporting complex in phenylhexane were assumed to be one. The experiments were done with identical initial macrocycle concentrations, but varying KNOB concentrations, to test whether or not the general model is valid under these conditions. The parameters used in the KNOBmodeling are also given in Table 11. The log K ( H 2 0 )value for K+-cis-syn-cis-DC18C6 interaction is 2.02 (26). The fraction of the macrocycle that remains in the membrane (Table 11) decreases with increasing K+ activity since the aqueous K+-macrocycle interaction largely responsible for the leaching increases with increasing K+ activity. However, the amount of leaching does not vary linearly with K+ activity a t these KNO, levels because of the principles of equilibrium for nonquantitative reactions. Maintenance of the macrocycle in the membrane due to KNOBextraction and loss of macrocycle from the membrane due to partitioning to pure water are virtually negligible effects in the KNOBsystems. However, all of these effects are taken into account by use of eq 9. Furthermore, the relative influence of these effects is understood easily as with the Pb(N03)2systems by comparing the numerical values for each of the terms in the equation. The KNOBflux values for these systems are also predicted

1146

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

accurately (Table 11). The log Ke, value for KNOBextraction has a large standard deviation due to the small amount of extraction. However, even with use of the two standard deviation limits of the Ke, value in eq 9, the predicted flux values for the lower and higher limits of the 0.2 M KNOBand 0.1 M KNO, source phase systems are 9.8 and 3.7 or 17 and 6.6, respectively. This agreement is still satisfactory for the individual fluxes and the ratio of the two fluxes remains the same. The agreement obtained in the flux value ratios is further evidence for the applicability of the model. An important test in modeling neutral macrocycle-mediated transport has been to determine the linear relationship between cation flux and cation concentration or activity raised to an integer power (15, 16). In the earlier studies, only the salt of interest is present in the source phase, but at varying salt concentrations. The KNO, systems of this study meet this description. However, the relationship between cation flux and salt concentration or activity raised to an integral power does not fit. The experimental ratio of the two fluxes is 2.4. The ratios predicted by using salt concentrations are 2 and 4 for integral powers of one and two, respectively, while the ratios are 1.8 and 3.2 for powers one and two using what should be the more accurate salt activities. In a membrane diffusion limited system one would expect that salt activities raised to the second power should be predictive of transport trends (14-16). The problem with these approaches is that the effect of either salt concentration or activity on macrocycle loss from the membrane has not been considered. The predicted flux ratio using eq 9 and 6 is 2.7. Hence, the equations derived for this study can be used to predict transport trends accurately with the expected second power dependence of flux on salt activity or concentration. Furthermore, the equations are general and allow for usage either with or without loss of macrocycle from the membrane phase. Competitive Transport Modeling. Expressions like eq 9 and 6 can be derived in a similar manner for competitive systems containing two or more different cations in the source phase. More than one anion and polyvalent anions can also be included but are not in this study since single monovalent anion types were examined. The flux equation for competitive transport is

to be the relative concentrations of the extracted species, the degree of ion pairing, and the interaction (Kex)selectivity of the macrocycle. As was the case with single cation transport modeling, these effects are singled out and quantified from fundamental parameters using the model. It should be reemphasized that the above equations are valid only when 1:l macrocycle-cation interaction occurs. This is the case for the great majority of cation-macrocycle combinations (23, 27). However, in this study we came upon a case where 2:l macrocycle-Zn(I1) interaction occurs. A modeling equation for such transport is easily derived in a manner similar to that for 1:l interaction. However, the final equation involves a quadratic expression. The equation used to obtain the amount of free macrocycle at the source phase interface of the membrane phase for a Zn(SCN)2vs Cd(SCN)2 competitive transport system is given as

2~ex,~n(s~~)z[Zn(SCN),1"([L]org~s)2 + (1 + K ~ X , C ~ ( S C N ~ ~ )[Linitiallorg [ L I ~ ~ ~ '=~ o (13) Loss of macrocycle from the membrane phase is virtually negligible in the Zn(SCN), vs. Cd(SCN)2system due to weak aqueous macrocycle-cation interactions, small concentrations of free cations, and the large Kp value for the macrocycle. However, such loss could easily be included in eq 13. The squared or 2:1 macrocycle-cation interaction term of the quadratic expression involves Zn(I1) as seen in the equation. The superscript org,s identifies the source phase interface of the organic membrane phase. The membrane concentration of free ligand at the receiving phase interface has again been assumed to equal the total membrane concentration of L elsewhere in the membrane as done in the previous TSSLM studies (12). Once the value of [LIorg*'has been calculated, the Cd(SCN)2and Zn(SCN)2flux values are predicted, respectively, to be

and JZII(SCN)~

=

D

I

(15)

while predicted Cd(I1) vs Zn(I1) selectivity is given by

-

*Cd(SCN)z/Zn(SCN)z-

Kex,Cd(SCN)z[Cd(SCN) 1' Kex,Zn(SCN)z

In those cases involving macrocycle loss from the membrane

r

I

[ L t o t a ~ , f i n a l l O= ~ ~ [Linitiallorg/

Vaq 1+ -

VorgKp

is used to calculate the [Ltotal,find] value for use in eq 10. The numerical subscripts indicate the different cations. Selectivity is defined as the ratio of the flux values for two or more different cations in a competitive transport experiment involving these cations. Thus, in a two cation system, selectivity for one cation over another is calculated by dividing the form of eq 10 for the first cation by that for the second cation. Most of the terms in eq 10, including [Ltotal,find]org obtained from eq 11, divide out and the cation l/cation 2 selectivity (a,,z) is given as

From eq 10, the parameters that affect selectivity are seen

[Zn(SCN)2IsLL1

(16) erg's

Notice that the concentration of the macrocycle has now become an important selectivity variable unlike in eq 12. A detailed presentation of all of the algebra involved in deriving eq 10-16 is given elsewhere (28). Effectiveness of the Model in Predicting Competitive Alkali and Alkaline-Earth Cation Transport. The derived model for competitive cation transport involving eq 1G-12 was tested to see if experimental fluxes could be predicted by using the equations and knowledge of the few fundamental parameters in the equations. The first system tested involved KNOB vs NaNO, vs LiN03 transport using R2DC18C6as the carrier. Each of the three NO3- salts was present in the source phase at 0.3 M. The TSSLM phase conditions, the parameters used in and calculated from the predictive equations, and the predicted and observed fluxes for this and other systems are given in Table 111. To predict the fluxes in Table 111, the fraction of the macrocycle remaining in the membrane V, was first calculated by using eq 11. The K values for eq 11 in these experiments are for the interaction of either cis-syn-cis-DC18C6 or a mixture of DC18C6 isomers with the cations. The K,, values needed for use in eq 11 are also presented in Table 111. Once f is known, cation fluxes can be predict'ed by using eq 10. The

ANALYTICAL CHEMISTRY, VOL. 61, NO. 10, MAY 15, 1989

1147

Table 111. Predicted and Observed Competitive Cation Fluxes (J)" source phase [M(m)l,M

no.

M(m)

A-

1

K(I) Na(I) Li(1) Ba(I1) Ca(I1) Sr(I1) Hg(I1) Cd(I1) Cd(I1) Zn(I1)

NO3-

2 3 4

NOy NO< NO3NO