Kinetic Aspects of Donnan Membrane Technique for Measuring Free

Mar 30, 2005 - Addition of ion complexation ligands in the acceptor solution in the Donnan membrane technique (DMT) can lower its detection limit for ...
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Anal. Chem. 2005, 77, 2852-2861

Kinetic Aspects of Donnan Membrane Technique for Measuring Free Trace Cation Concentration Liping Weng,* Willem H. Van Riemsdijk, and Erwin J. M. Temminghoff

Department of Soil Quality, Wageningen University and Research Centre, P.O. Box 8005, 6700 EC, Wageningen, The Netherlands

Addition of ion complexation ligands in the acceptor solution in the Donnan membrane technique (DMT) can lower its detection limit for free metal ion concentration in natural samples. In this paper, the influence of added ligands on the transport behavior of trace ions in DMT was studied using numerical and analytical models and experimental tests. The results show that addition of ligands in the acceptor can significantly influence the time to reach the Donnan membrane equilibrium. Depending on several factors, the flux can be controlled by the diffusion in the stagnant solution film at the solutionmembrane interface, by the diffusion in the membrane, or by both. The conditions under which the diffusion in the solution film or in the membrane becomes the ratelimiting step are discussed and approximate analytical solutions for some special cases are presented. Very low concentrations of free metal ion can be measured using the ligand complexation DMT. Depending on the degree of complexation in the sample, the measurement can be based on either the Donnan membrane equilibrium (when the complexation degree is low) or the kinetic interpretation of the ion transport (when the complexation degree is high). In environmental systems such as soil and water, elements such as heavy metals can exist in various chemical forms (species). The chemical speciation of the elements is a critical factor influencing their mobility, bioavailability, and toxicity. Various analytical techniques such as stripping chronopotentiometry,1,2 competitive ligand exchange/anodic stripping voltammetry, and competitive ligand exchange/adsorption stripping voltammetry,3 diffusion gradients through thin-film gels,4 permeation liquid membranes,5,6 and Donnan membrane technique (DMT)7,8 have * To whom correspondence should be addressed. E-mail: liping.weng@ wur.nl. Fax: 0031-317-483766. (1) Town, R. M.; van Leeuwen, H. P. Electroanalysis 2004, 16, 458-471. (2) Van Leeuwen, H. P.; Town, R. M. Environ. Sci. Technol. 2003, 37, 39453952. (3) Xue, H.; Sigg, L. In Environmental Electrochemistry: Analysis of Trace Element Biogeochemistry; Rozan, T. F., Taillefert, M., Eds.; ACS Symposium Series 811; American Chemical Society: Washington, DC, 2002; p 336. (4) Davison, W.; Zhang, H. Nature 1994, 367, 546-548. (5) Buffle, J.; Parthasarathy, N.; Djane, N.-K.; Matthiasson, L. In In Situ Monitoring of Aquatic Systems. Chemical Analysis and Speciation; Buffle, J., Horvai, G., Eds.; John Wiley & Sons Ltd.: Chichester, 2000; Vol. 6, pp 407493. (6) Tomaszewski, L.; Buffle, J.; Galceran, J. Anal. Chem. 2003, 75, 893-900.

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been developed to measure a certain chemical fraction of one element. A review of the recent progress in metal speciation in water can be found in Batley et al.9 It has been established that, in many circumstances, the free metal ion (Mz+) concentration (or activity) is the key factor in determining metal bioavailability and toxicity.10,11 Moreover, it is a well-defined thermodynamic entity, and its measured value can easily be compared to calculations done with modern geochemical speciation codes such as MINTEQA2,12 PHREEQC,13 ECOSAT,14 and ORCHESTRA.15 The DMT uses a cation exchange membrane to separate the cationic free and monomeric species from the others. The concentration of the free ion can then be derived from the total concentration of these cationic species. Application of the DMT to measure the free ion activity/concentration of trace metal ions in environmental samples can be found since the early 1980s,7,8,16-21 and promising results have been obtained. An example can be found in Weng et al.,22 in which the DMT was used to measure simultaneously the free ion activity of Cu2+, Cd2+, Ni2+, Pb2+, and Zn2+ in 2 mM Ca(NO3)2 solution in equilibrium with various soil samples. However, as also revealed in these data, the free metal ion concentrations in some soil samples studied were too low to (7) Weng, L. P.; Temminghoff, E. J. M.; Van Riemsdijk, W. H. Eur. J. Soil Sci. 2001, 52, 629-637. (8) Temminghoff, E. J. M.; Plette, A. C. C.; Van Eck, R.; Van Riemsdijk, W. H. Anal. Chim. Acta 2000, 417, 149-157. (9) Batley, G. E.; Apte, S. C.; Stauber, J. L. Aust. J. Chem. 2004, 57, 903-919. (10) Parker, D. R.; Pedler, J. F. Plant Soil 1997, 196, 223-228. (11) Campbell, P. G. C. In Metal Speciation and Bioavailability in Aquatic Systems; Tessier, A., Turner, D. R., Eds.; John Wiley & Sons: Chichester, U.K., 1995; pp 45-102. (12) Allison, J. D.; Brown, D. S.; Novo-Gradac, K. J.; MINTEQA2, Geochemical Assessment Model for Environmental Systems. Version 3. Environmental Protection Agency: Athens, 1991. (13) Parkhurst, D. L.; Thorstenson, D. C.; Plummer, L. N.; User’s Guide to PHREEQC (Version 2), a Computer Program for Speciation, Batch Reaction, One-Dimensional Transport and Inverse Geochemical Calculations. U.S. Geology Survey, 1990. (14) Keizer, M. G.; Van Riemsdijk, W. H. ECOSAT, a Computer Program for the Calculation of Chemical Speciation and Transport in Soil-Water Systems. Agricultural University of Wageningen: Wageningen, 1994. (15) Meeussen, J. C. L. Environ. Sci. Technol. 2003, 37, 1175-1182. (16) Cox, J. A.; Slonawska, K.; Gatchell, D. K.; Hiebert, A. G. Anal. Chem. 1984, 56, 650-653. (17) Lampert, J. K. Measurement of Trace Cation Activities by Donnan Membrane Equilibrium and Atomic Absorption Analysis. Ph.D. Thesis, University of Wisconsin, Madison, 1982. (18) Fitch, A.; Helmke, P. A. Anal. Chem. 1989, 61, 1295-1298. (19) Minnich, M. M.; McBride, M. B. Soil Sci. Soc. Am. J. 1987, 51, 568-572. (20) Oste, L. A.; Temminghoff, E. J. M.; Lexmond, T. M.; Van Riemsdijk, W. H. Anal. Chem. 2002, 74, 856-862. (21) Salam, A. K.; Helmke, P. A. Geoderma 1998, 83, 281-291. 10.1021/ac0485435 CCC: $30.25

© 2005 American Chemical Society Published on Web 03/30/2005

be measured using the DMT in combination with inductively coupled plasma mass spectrometry (ICPMS) for concentration analysis. The detection limit of DMT for the trace metal ions is in the range of 10-7-10-11 M confined by ICPMS. It is expected that the problem of the detection limit encountered in the soil sample analysis will constrain also the application of DMT to the surface waters, where the trace metal ion concentrations are often much lower than in the soil solution. One way that can lower the detection limit of the DMT is to add ion complexing ligands in the receiving solution (acceptor) to accumulate the ions of concern to a measurable concentration. As far as the amount of the chosen ligand and the complexation constants are known, the free ion concentration in the acceptor can be calculated using the total ion concentration in the acceptor. When the Donnan membrane equilibrium is reached, the free ion concentration in the sample can be derived from the free ion concentration in the acceptor based on the Donnan membrane equilibrium principle,23,24 which says that, at the Donnan membrane equilibrium, the activity ratios (corrected for charge) of the ions in the solution on the two sides of the membrane are equal:

(

) (

ai,donor ai,acceptor

1 zi

)

aj,donor ) aj,acceptor

1 zj

(1)

However, to be able to apply such an approach, the condition of the (pseudo) Donnan membrane equilibrium has to be obeyed and the time needed to reach the equilibrium is in practice an important issue. It is expected that addition of ligands in the acceptor will influence the time to reach equilibrium. In natural samples, inorganic and organic ligands are also present that form complexes with the ions in the sample solution. It is not well understood how the presence of these ligands will influence the speed of ion transport in the DMT measurement. For the application of the DMT to natural samples, knowledge of the basic transport phenomena of trace ions in the membrane analysis under conditions relevant to environmental systems is essential. In this study, a numerical model is first developed to simulate the transport behavior of ions at trace concentration in the DMT. The aim of the development of such a model is not to get a perfect description of the data but to gain better understanding of some critical factors that influence the transport of trace metal ions over the cation exchange membrane. This should lead to new approaches to measure the free ion at very low concentration. The numerical model developed is calibrated using experimental data. With the help of the model, the effects of ligands on the transport kinetics of trace metal ions are simulated and illustrated. Conditions under which the ion diffusion in the solution film or in the membrane becomes the rate-limiting step are discussed, and approximate analytical solutions for some simple situations are derived. In the last part of the paper, strategies for measuring the free ion concentration in natural samples using DMT are discussed. (22) Weng, L. P.; Temminghoff, E. J. M.; van Riemsdijk, W. H. Environ. Sci. Technol. 2001, 35, 4436-4443. (23) Donnan, F. G. Chem. Rev. 1925, 73, 3-79. (24) Helfferich, F. G. Ion Exchange; McGraw Hill: New York, 1962.

DESCRIPTION OF THE NUMERICAL MODEL The theory of ion exchange kinetics has been developed over a long period of time.25 To describe ion transport through ion exchange membranes, different models have been proposed and tested. Depending on the purpose of the modeling, the ion transport process is represented in these models with various degrees of detail. The numerical model developed in this work simulates the transport of ions through a cation exchange membrane in the DMT analysis. The concept of the model is general and can in principle also be applied to anion exchange membranes. The major assumptions made in this model are as follows: (1) The concentration of ions under concern is much smaller than the concentration of the major ions in the background electrolyte solution. The transport of the trace ions under this condition will not influence the electrostatic neutrality and the electrostatic potential to any significant extent. (2) The concentration of the major ions is the same in the solution on both sides of the membrane. Therefore, there is no concentration gradient of the major ions and no electrostatic potential gradient in the membrane. (3) The adsorption of ions in the membrane is controlled only by electrostatic force. At the interface, equilibrium is preserved between the membrane and the solution film adhering to it (i.e., there is no interfacial resistance to diffusion). (4) The chemical equilibrium in the system is achieved much faster than the transport, and therefore, the local equilibrium assumption is used. Figure 1 illustrates schematically the DMT system as represented in the model, in which five zones are indicated, i.e., bulk solution in the donor side, diffusion layer at the donor-membrane interface, the cation exchange membrane, diffusion layer at the acceptor-membrane interface, and the bulk solution in the acceptor side. There is no concentration gradient in the bulk solution of both the donor and acceptor parts due to mixing of the solution by pumping. Membrane Characteristics. The cation exchange membrane used in this work (55165 2U, BDH Laboratory Supplies) has a matrix of polystyrene and divinylbenzene with sulfonic acid groups, which are fully deprotonated at pH >2. The thickness of the membrane (δm) is 0.16 mm, and the charge density of the membrane is 0.8 mmol g-1 (dry weight). In the DMT analysis carried out in this work, the surface area of one membrane that is in contact with the solution (Am) is 7.0 cm2 with a dry weight of 1.1 g. The basic parameters of the membrane are listed in Table 1. The electron microscope image of the membrane surface (Figure 2) shows that there are irregular-shaped nanosized pores in the material. The rest of the membrane around the pores is a tight matrix that is probably not accessible for the diffusion of water and ions. The nature of the membrane structure will lead to a reduction of the surface area available to diffusion. The effective surface area is denoted as Ae. In the model, it is assumed that the membrane is a Donnan phase: a phase with immobile charge in the matrix and placed in electrolyte solution. The volume of the Donnan phase (VDonnan) is derived from the product of the effective surface area (Ae) and (25) Helfferich, F. G. In Mass Transfer and Kinetics of Ion Exchange; Liberti, L., Helfferich, F. G., Eds.; Martinus Nijhoff Publishers: The Hague, 1983; pp 157-180.

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Figure 1. Schematic representation of the numerical model. Table 1. Parameters of the Cation Exchange Membrane (55165 2U, BDH) weight density (g cm-2)

charge density (mmol g-1)

thickness (δm) (mm)

surface area of a membrane (Am) (cm2)

volume of a membrane (Vm) (mL)

0.014

-0.8

0.16

7.0

0.112

the membrane thickness (δm). All the charge carried by the membrane material is supposed to be homogeneously distributed in the Donnan phase. The overlapping double layers in the narrow pores will lead to a Donnan potential (ψ) that depends on the charge density of the membrane (q), the Donnan volume, and the salt level. Ion Concentration in the Membrane. The concentration of ions in the Donnan phase of the membrane is calculated using the Boltzmann equation:

Ci,m ) CiBZi -Fψ/RT

B)e

(2) (3)

In the Donnan phase of the membrane, the charge carried by the membrane sites (q) will be neutralized by the charge of the ions in the membrane:

q + ΣZiCiBZi ) 0

(4)

Because the electrolyte concentration on both sides of the membrane is the same (assumption 2), there will be no electrostatic potential gradient in the membrane. Compared to the thickness of the membrane (0.16 mm), the thickness of the electrostatic diffuse layer extending into the solution from the membrane interface is much thinner (a few nanometers); therefore, the electrostatic potential in the membrane is treated as a block function and the thickness of the electrostatic diffuse layer outside the membrane is ignored. 2854

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Figure 2. Electron microscope image of the membrane surface.

Ion Transport in Solution. In the DMT analysis carried out in this study, the solution on both sides of the membrane was mixed by pumping (2.5 mL min-1). Therefore, there is no concentration gradient in the bulk solution. The transport of ions between the solution and membrane will be controlled by the diffusion in the solution film at the membrane interface, i.e., the diffusion layer. The thickness of the solution diffusion layer is generally in the range of 0.03-0.1 mm for a solution stirred from rigorously to mildly. In the model, a value of 0.05 mm was chosen for the thickness of the diffusion layer (δ) at the donor-membrane and acceptor-membrane interface. In the numerical model, ion diffusion in solution is calculated using Fick’s second law:

∂Ci ∂2Ci ) Di 2 ∂t ∂δ

(5)

The diffusion coefficients of ions in solution (Di) are assumed to be the same as in water. The values of the diffusion coefficients used in this work are taken from the literature and are listed in Table 2.

Mt ) JAe

Table 2. Diffusion Coefficients in Water (Di) ion

K+

Mg2+

Cu2+

Al3+

NO3-

NTA-Cu-

Di (× 10-9 m2 s-1)

1.96a

0.71a

0.71a

0.54a

1.90a

0.54b

a

Reference 27. b Reference 28.

Ion Transport in the Membrane. Because there is no electrostatic potential gradient in the membrane (assumption 2), there is no need to use the Nernst-Plank equation25 to calculate the diffusion. In the membrane, the diffusion of ions is driven by their concentration gradients in the Donnan phase. The diffusion in the membrane is also calculated with Fick’s second law:

∂Ci,m ∂2Ci,m ) Di,m ∂t ∂δ 2

(6)

m

The irregularity of the pores in the membrane (Figure 2) will result in some tortuosity effect on ion diffusion in the membrane, which will contribute to a smaller apparent diffusion coefficient in the membrane (Di,m) than that in the solution (Di). The diffusion coefficient in the membrane can also be reduced due to the interaction of the ions with the functional groups on the walls of the pores and due to the heterogeneity of the charge distribution. All the effects on the diffusion coefficient due to the presence of the membrane are lumped here into a tortuosity factor λi that relates Di,m to Di:

Di,m ) Di/λi

(7)

Computation. The calculation was done using the computation algorithm ORCHESTRA.15 Equations 5 and 6 were solved numerically by a mixing cell method.26 The diffusion layers in the donor and acceptor solution at the membrane interface were represented by five cells (layers) each (∆δ ) 0.01 mm), while the cation exchange membrane was divided into 10 layers (∆δm ) 0.016 mm). With this approach, the integrated form of eqs 5 and 6, i.e., Fick’s first law, can be used to calculate the diffusion flux between two adjacent layers. The flux in solution Ji,sol and in the membrane Ji,m can be calculated as

Ji,sol ) -Di Ji,m ) -Di,m

∆Ci ∆δ

∆Ci,m ∆(BZiCi) ∆Ci ) -Di,m ) -Di,mBZi ∆δm ∆δm ∆δm

(8)

(9)

in which ∆δ and ∆δm are equal to the thickness of one small layer in the solution and membrane diffusion parts, respectively. Mass exchange between two neighboring layers (Mt in mol s-1) was calculated by multiplying the flux with the effective surface area of the membrane (Ae): (26) Van Beinum, W.; Meeussen, J. C. L.; Van Riemsdijk, W. H. Environ. Sci. Technol. 2000, 34, 4902-4907. (27) Markus, Y. Ion Properties; Marcel Dekker Inc.: New York, 1997.

(10)

There are two unknown parameters in the model as formulated above, i.e. the effective surface area (Ae) and the tortuosity factor (λi). These two parameters will be derived by fitting the model to experimental data. EXPERIMENTS Donnan Membrane Technique. The design of the ion exchange cell has been described by Temminghoff et al.8 A schematic drawing of the setup is given in Figure 3. All cells, bottles, and test tubes were washed before use with 0.1 M HNO3 and ultrapure water. Before analysis, the membranes were prepared by shaking several times successively with 0.1 M HNO3, 1 M Ca(NO3)2, and the background electrolyte solution at the concentration that was going to be used in the experiment. During the DMT analysis, both the donor and the acceptor solutions were circulated constantly by pumping (2.5 mL min-1) (peristaltic pump, Gilson Miniplus III). At a certain time interval after the start of a DMT experiment, the donor and the acceptor solutions were sampled. The pH was measured in all the samples. Subsamples of the donor and the acceptor solutions were acidified with HNO3 to a final HNO3 concentration of 0.14 M. These acidified solutions were used to analyze the ion concentrations. The concentrations of major cations were measured by inductively coupled plasma atomic emission (ICPAES, Eppendorf Elex 6361), and the concentrations of the trace cations were measured by ICPMS (Perkin-Elmer, Elan 6000). The concentration of nitrate was measured with an automated continuous flow spectrophotometric method after acidification to pH 2 with HCl. Experiment 1. Experiment 1 measures the transport of K+, Mg2+, Al3+, and NO3- with 2 and 20 mM Ca in the solution. The experiment was done in two parts. In the first part, transport of the cations (K+, Mg2+, Al3+) was measured in the background of Ca(NO3)2. About 10 µM K, Mg, and Al were added in the form of a nitrate salt to the donor solution. The pH of the donor and acceptor solutions was adjusted to pH 3. At this pH, the cations will be present mostly in the form of free hydrated ions. The volume of the donor and acceptor solution is respectively 1.0 L and 15 mL. Each time a 2-mL sample was taken and the solution was not refilled after sampling. In the second part of the experiment, the transport of NO3- was measured in CaCl2 background. To the donor side, 0.14 mM NO3- was added in the form of NaNO3. The volume of the donor solution was 1.5 L and that of acceptor was 25 mL. Experiment 2. In this experiment, the transport of Cu in the presence of synthetic ligand nitrilotriacetic acid (N(CH2COOH)3, NTA) was measured. The background solution in both the donor and acceptor was 0.5 mM Ca(NO3)2. The same concentration of NTA was added to both the donor and acceptor solutions in one cell. Three NTA concentrations were chosen, i.e., 1.1, 2.5, and 16.1 µM. To the donor solution ∼1.0 µM Cu was added in the form of nitrate salt. The pH in the donor and acceptor solutions was adjusted to 4.8 ( 0.3. The volume of the donor solution was 1 L and that of acceptor was 15 mL. Each time, a 5-mL sample (28) Norkus, E. J. Appl. Electrochem. 2000, 30, 1163-1168.

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Figure 3. Schematic picture of the DMT setup.

was taken from both the donor and acceptor. The acceptor solution was refilled with the original blank acceptor solution after sampling. RESULTS AND DISCUSSION Model Calibration, Effects of Ion Charge, and Background Solution. Figure 4 shows the results of experiment 1. For the cations, the transport rate decreased with the increase of the ion charge. The transport of K+ was faster than Mg2+, and the trivalent cation Al3+ was the slowest among the cations measured. The transport speed of Al3+ was most significantly affected by changing the Ca concentration. Aluminum transportation was faster at 20 mM than at 2 mM Ca(NO3)2. In the experiment, no measurable transport was found for the monovalent anion NO3- during the experimental time (data not shown). In the following part, the numerical model will be used to simulate the transport of ions with various charges at two Ca concentrations. The unknown model parameters will be calibrated using the data of experiment 1. In the model simulation, it was found that the predicted transport rate of the bivalent and trivalent trace cations is very sensitive to the effective surface area of the membrane (Ae) (thus also the Donnan volume of the membrane VDonnan) and much less sensitive to the apparent diffusion coefficient in the membrane (Di,m, λi). The model prediction of the transport rate of the monovalent trace cations and anions is sensitive to both parameters, i.e. Ae and λi. Therefore, the model was first calibrated with the data of the bivalent and trivalent cations to fix Ae. In Figure 5, model predictions of the change of Mg2+ concentration in the acceptor solution under the conditions of experiment 1 at 2 mM Ca(NO3)2 were shown using various values of Ae and λi. At Ae ) Am and λi ) 1 (thus Di,m ) Di), the predicted time to reach 95% equilibrium concentration (t95%) is 4.5 h, which is much shorter than the measured results. By reducing Ae to 20% of the original membrane area (Am), the predicted t95% is 21.7 h, which is in good agreement with the experimental data (Figure 5). The change of the apparent diffusion coefficient in the 2856 Analytical Chemistry, Vol. 77, No. 9, May 1, 2005

Figure 4. Change of concentration of K, Mg, and Al in the acceptor at 2 and 20 mM Ca background (results of experiment 1).

Figure 5. Change of Mg concentration in the acceptor at 2 mM Ca(NO3)2. (Symbols are data from experiment 1; lines are model predictions.)

membrane (Di,m) does not influence much the predicted transport of Mg2+. At λi ) 10 and 40, the calculated t95% is 22.0 and 22.1 h, respectively (Figure 5). Both the model calculations (results not shown) and the experimental data suggest no significant difference in the transport rate of Mg2+ between 2 and 20 mM Ca(NO3)2. Using these data, the model parameters Ae and VDonnan can be calibrated. The fitted Ae and VDonnan equal to 20% of their original values, i.e., Am and Vm. These parameters will be kept constant in the rest of the simulation.

Figure 6. Change of Al concentration in acceptor measured and predicted, (Symbols are data from experiment 1; lines are model predictions using Ae ) 0.2Am; λi ) 20).

Similar to the transport of Mg2+, the transport rate of Al3+ predicted by the model is strongly dependent on the value of the effective surface area Ae. With the values of Ae and VDonnan fitted using the Mg2+ data, the model gave a reasonable prediction of the transport of Al3+ at 2 and 20 mM Ca(NO3)2 (Figure 6). Changing the apparent diffusion coefficient in the membrane does not influence the predicted transport rate of Al3+ much. In this figure, only the calculation at λi ) 20 is given. The predicted transport of K+ and NO3- is sensitive to both the effective surface area Ae and the apparent diffusion coefficient in the membrane (Di,m). Figure 7 shows the predicted and measured transport of K+ using Ae ) 20%Am and various values of λi. At λi ) 1, the calculated t95% for K+ is 5.1 and 5.5 h, respectively, at 2 and 20 mM Ca(NO3)2 (Figure 7), which is too fast in comparison with the experimental data. Increasing the λi value (decreasing Di,m) leads to slower predicted K+ transport. The influence of changing the apparent diffusion coefficient in the membrane (Di,m) on the predicted transport of K+ is larger at 20 mM than at 2 mM Ca(NO3)2. The best description of K+ transport data for the two Ca concentrations is found at λi ) 20 (Di,m ) 5%Di). The transport of NO3- is very slow, and its concentration in the acceptor is not measurable during the experiment. Therefore, it is not possible to derive a more accurate value of the apparent diffusion coefficient in the membrane for NO3-. Using the parameters fitted above (Ae ) 20%Am, λi ) 20), the predicted concentration of NO3- in the acceptor solution after 24 h is only 1% of that at equilibrium at 2 mM CaCl2 background. It will take ∼60 days to reach 95% equilibrium. This slow transport predicted by the model is in agreement with the experimental data. Using the model, the differences in the rate of transport of ions with various charges and at various Ca concentrations in the background can be understood conceptually. At VDonnan equal to 20% of the initial membrane volume (Vm), the volume charge density of the membrane Donnan phase (q) is -3.5 mol L-1. The Boltzmann factor (B) in the membrane calculated using eq 4 is 29.6 and 9.4, respectively, at 2 and 20 mM Ca(NO3)2 background. The concentration of K+, Mg2+, Al3+, and NO3- in the Donnan phase (Ci,m) at equilibrium will be respectively 29.6, 876.2, 25 934.3, and 0.034 times of that in the bulk solution at 2 mM Ca(NO3)2

Figure 7. Change of K concentration in acceptor measured and predicted. (Symbols are data from experiment 1; lines are model predictions using Ae ) 0.2Am.)

background. The concentration gradient in the Donnan phase for cations is much larger than the concentration gradient for the same cation in the solution diffusion layer. The difference between the concentration gradient in the membrane and in the solution diffusion layer increases with the increase of the charge of the cations. Often, the larger concentration gradient in the membrane results in a larger flux in the membrane than in the solution diffusion layer, despite a lower diffusion coefficient in the membrane. Therefore, for cations, the diffusion in the solution diffusion layer is usually the rate-limiting step in the DMT analysis under the experimental conditions used in this paper when there is no ligand in solution (see below) and when the salt level in the background solution is not very high. With increasing salt level, the concentration gradient of the cations in the membrane will be reduced. When diffusion in solution is the rate-limiting step, the time to reach equilibrium will increase with the increase in the amount of the ion that is accumulated in the membrane when the volume of the acceptor solution is fixed. This explains why the time to reach equilibrium is shorter for K+ and Mg2+ than for Al3+ in experiment 1. This can also explain the observation that the equilibrium time for Al3+ is most significantly influenced by the Ca concentration (Figure 4). The concentration of Al3+ in the Donnan membrane varies much more with the change of the Boltzmann factor than Mg2+ and K+. A decrease of the Ca Analytical Chemistry, Vol. 77, No. 9, May 1, 2005

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Figure 8. t50% predicted using the numerical model by assuming the complexed ion species do not diffuse or diffuse with the same diffusion coefficient of the free ion. (2 mM Ca(NO3)2 background, Vacceptor )15 mL, Ae ) 0.2Am, λi ) 20, Di ) 7 × 10-10 m2 s-1.)

concentration leads to more Al3+ accumulation in the membrane, which results in a slower equilibrium. In a previous paper about DMT,8 it was assumed that ion transport in the membrane is the rate-limiting step. By neglecting the amount of ions accumulated in the membrane, it was predicted that a bivalent cation would reach the Donnan membrane equilibrium faster than a monovalent cation. As discussed above, the assumptions made in that work are not justified; therefore, the order of equilibrium time for monovalent and bivalent cations was not correctly predicted in that paper. For anions such as NO3-, the concentration gradient in the membrane is much smaller than the gradient in the solution. The diffusion in the membrane is the rate-limiting step for anion transport across the cation exchange membrane. A reduced concentration gradient in combination with a reduced apparent diffusion coefficient in the membrane leads to strongly retarded transport of anions in the cation exchange membrane. Effects of Complexing Ligand. To lower the detection limit of the DMT, one possibility is to add ligands that form complexes with the ion of interest to the acceptor solution to increase the total ion concentration in the acceptor. Because the DMT analysis is so far based on the Donnan membrane equilibrium, it is crucial to understand the effects of ligands on the kinetics of ion transport. Figure 8 shows the predicted time to reach 50% equilibrium concentration (t50%) in the acceptor for a bivalent trace cation at 2 mM Ca(NO3)2 background in the presence of a ligand. The calculation was done for the situation in which the same ligand is present at the same concentration in both the donor and acceptor solutions and the ratio of the total to free ion concentration, referred to as complexation factor Pi, ranges from 1 (no complexation) to over 1000. In the calculation, the complexed ion species are assumed either not to diffuse or to diffuse in the solution with the same diffusion coefficient as the free ion (in this calculation Di ) 7 × 10-10 m2 s-1, λi ) 20, Vacceptor ) 15 mL). The calculation shows that if the complexed ion species do not diffuse in both the donor and acceptor solutions, the time to reach equilibrium will increase almost proportionally to the increase of the degree of complexation (Pi). The effect of adding ligand to the acceptor on the equilibrium time, under this assumption, is comparable to the effect of increasing the volume of acceptor solution. When 2858 Analytical Chemistry, Vol. 77, No. 9, May 1, 2005

Figure 9. Transport of Cu in the presence of NTA. (Symbols are data from experiment 2; lines are model simulations using Ae ) 0.2Am, λi ) 40.) Table 3. Constants Used for NTA Speciation Calculation (Database ECOSAT14) species

NTACa

NTACu

NTA2Cu

NTAH

NTAH2

NTAH3

Log K

8.2

14.5

17.1

10.5

13.5

15.9

the complexed species are allowed to diffuse in both the donor and acceptor, there is no significant effect of the ligand on the equilibrium time when Pi is relatively small (50), the equilibrium time increases almost linearly with the increase of Pi but with a smaller slope than when the complexed species do not diffuse. Figure 9 shows the results of experiment 2. Using the complexation constants (Table 3), the ratio of total to free Cu2+ (Pi) in the donor solution can be calculated from the measured total Cu and Ca concentration and pH. At 1.1, 2.5, and 16.1 µM NTA, Pi is respectively 21, 1142, and 14 170. As shown in Figure 9, respectively 100%, 48%, and 3% of the equilibrium concentration was reached after 72 h for the three NTA concentrations. Equilibrium is reached faster when Pi is smaller. The lines in Figure 9 are the model predictions for the changes of the concentration over time. In this calculation, λi ) 40 was used. When using the same λi as found for K+ transport data (λi ) 20), the model overestimated the transport speed of Cu in comparison with the experimental results (calculation results not shown). Using λi ) 40, the model predictions are in reasonable agreement with the data. The relatively small difference (2 times) between the fitted values of λi for K and Cu may indicate that the value of λ i for various cations is relatively similar. The results of the model prediction shown in Figure 8 and experiment 2 (Figure 9) suggest that if the complexed ion species diffuse with a mobility comparable to the free ion and when the complexation factor Pi is relatively small, the Donnan membrane equilibrium for the trace cations can be achieved in a time period similar to when no ligand is present. However, when Pi is relatively large, the diffusion in the membrane becomes the rate-limiting step (see below), and the time to reach equilibrium will increase linearly with the increase of Pi, which implies that application of

the DMT based on the Donnan membrane equilibrium principle will become impractical considering the time needed. An alternative approach is to see if the free ion concentration can be derived from the data when the system is not yet in the Donnan membrane equilibrium (kinetic approach). This will be analyzed below. Transport Controlled by Diffusion in Solution and in the Membrane. As has been suggested above, the trace ion transport in the DMT can be limited by diffusion in the solution or by diffusion in the membrane. In this section, a more general discussion will be given with respect to the factors that influence the transition between the two cases. When all the species of ion i in the solution diffuse, the diffusion flux of ion i at steady state in the solution diffusion layer is a sum of the flux of all the species:

Ji,sol ) -ΣDi,j

∆Ci,j ∆δ

(11)

If it is assumed that the diffusion coefficients of all the species of ion i are the same as the diffusion coefficient of the free ion Di, eq 11 becomes

Ji,sol ) -Di

∆Ci,tot ∆Ci ) -DiPi ∆δ ∆δ

(12)

As has been discussed above, the ion diffusion in the membrane can be calculated with eq 9:

∆Ci Ji,m ) -Di,mBZi ∆δm

(9)

It follows from eq 9 and 12 that when DiPi/δ , Di,mBZi/δm, the solution diffusion is the rate-limiting step (Ji,sol , Ji,m). If DiPi/δ . Di,mBZi/δm, the diffusion in the membrane is the rate-limiting step (Ji,sol . Ji,m). If the values of DiPi/δ and Di,mBZi/δm are comparable in magnitude, the transport rate is influenced by diffusion in both the solution and the membrane. The comparison suggests that (1) an increase in degree of complexation Pi, (2) a decrease in the ion valence Zi, and (3) an increase in background salt concentration, thus decrease in B, will result in a transition to a more membrane diffusion controlled transport. For instance, at 2 mM Ca(NO3)2 background, the Boltzmann factor (B) is 29.6. The sum of the thickness of the solution diffusion layers (δ) on both sides of the membrane is 0.1 mm and the thickness of the membrane δm is 0.16 mm. If λi ) 20, the comparison of the flux of a bivalent cation can be written as

DiPi Di,mPi DiPi DiBzi : : ) ) Pi:27.4 δ δm δ λiδm

(13)

If we chose arbitrarily that one flux has to be more than 10 times slower than the other to be considered as the ratelimiting step, for the situation that eq 13 applies, the ion transport will be solution diffusion-limited when Pi < 2.7 and membrane diffusion-limited when Pi > 270. At Pi between 2.7 and 270, the transport is controlled by diffusion in both the solution and the membrane.

Approximate Analytical Solutions. For some special cases, ion transport in the DMT can be approximated using the classical linear driving force approximation, for which analytical solutions can be found. Under the conditions that (1) the solution diffusion is limiting the transport, (2) all the species of ion i diffuse with the same diffusion coefficient Di, (3) Pi is the same in both the donor and acceptor solutions, and (4) mass of ion i in the membrane phase is neglected, the change of ion concentration in the acceptor solution over time can be calculated by

Ae dCi,tot,acceptor ) (C - Ci,tot,acceptor) dt Vacceptor i,tot,donor

(14)

The analytical solution for eq 14 is

Using eq 15, the equilibrium time can be calculated. For instance

t95% ) -Ln(0.05)

Vacceptorδ AeDi

(16)

Please note that the thickness of the diffusion layer (δ) indicated in eqs 14-16 should be the sum of the diffusion layer on both sides of the membrane. The t95% calculated for Mg2+ using eq 16 for Vacceptor ) 15 mL is 12.7 h, which is less than the t95% calculated numerically (22.0 h). The discrepancy is mainly caused by ignorance of the time needed to accumulate the ion in the membrane in the analytical equation. The error will become smaller when the mass of ion i in the membrane becomes relatively less compared to the amount of the same ion in the acceptor, i.e., when Vacceptor and Pi is relatively large. Equation 16 suggests that when the diffusion in solution is limiting the transport, the time to reach equilibrium will increase with the increase of the ratio between the acceptor volume and membrane surface area (Vacceptor/Ae). When (1) diffusion in the membrane is the rate-limiting step, (2) Pi in acceptor solution remains constant (linear complexation), the change of the free ion concentration in the acceptor can be calculated using the linear driving force approximation:

dCi,acceptor Ae Di,m BZi ) (C - Ci,acceptor) dt Vacceptor δm Pi,acceptor i,donor

(17)

The analytical solution to eq 17 is

From this analytical solution, the equilibrium time can be calculated. For instance

t95% ) -Ln(0.05)

VacceptorδmPi,acceptor AeDi,mBZi

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Equation 19 suggests that, when the membrane diffusion is the rate-limiting step, the time needed to reach equilibrium increases proportionally with the increase of the complexation factor in the acceptor solution Pi,acceptor and decreases with the increase of the Boltzmann factor B. The equilibrium time calculated using eq 19 is in good agreement with the equilibrium time calculated using the numerical model for the situation that the diffusion in the membrane is the rate-limiting step (results not shown). These equations also indicate that waiting for the Donnan membrane equilibrium at large Pi is in practice not feasible, as has been shown in experiment 2. Instead, the measurement can be based on the interpretation of the transport kinetics as discussed below. Strategies for Measuring Free Ion in Natural Samples Using DMT. Based on the understanding of the effects of ion complexing ligands on ion transport kinetics, the following suggestions can be made regarding the application of DMT in measuring free ion concentration in natural samples. If the free ion concentration in the sample is above the detection limit of the ICPMS (situation 1), there is no need to add ligands to the acceptor solution. The Donnan membrane equilibrium can be reached in 1-2 days, and the measurement can be based on the equilibrium principle. When the free ion concentration in the sample is below the detection limit of the ICPMS, and if the ratio of the total to free ion concentration in the sample (Pi) is relatively small (situation 2), for instance, Cd, Ni, and Zn at close to neutral pH, a ligand with not too high affinity and at a relatively low concentration can be added to the acceptor. Donnan membrane equilibrium can be expected within a few days time and the free ion concentration in the sample can be calculated based on the equilibrium. Under this situation, ion transport in the DMT will be controlled by diffusion in both the solution and the membrane. The time needed to reach equilibrium in situations 1 and 2 can be reduced by reducing the ratio of Vacceptor/Ae. The third situation that can happen is for metal ions with a free ion concentration lower than the detection limit of the ICPMS and a relatively high degree of ion complexation (large Pi) in the sample (situation 3), for instance, Cu and Pb in surface water at near-neutral or slightly alkaline pH. For this situation, a ligand with a high affinity can be added at relatively high concentration to the acceptor solution. This treatment will result in a membrane diffusion-limited transport process, and the free ion concentration in the sample can be derived from the ion flux measured; therefore, there is no need to wait for the Donnan membrane equilibrium. Under situation 3, the free ion concentration in the sample can be calculated from the total ion concentration in the acceptor measured at a certain time interval using either the numerical model developed or the approximate analytical solutions. From eq 18, it can be derived that the free ion concentration in the donor Ci,donor can be calculated from the total ion concentration in the acceptor measured at time t:

To be able to use eq 20, Pi in the acceptor has to be known and to be almost constant. 2860

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If the free ion concentration in the acceptor is much smaller than the free ion concentration in the donor, i.e., Ci,acceptor , Ci,donor, which is a condition that can be met when it is far from the Donnan membrane equilibrium, from eq 17 it can be derived that the ion flux is linearly related to the free ion concentration in the donor:

Ci,donor )

Vacceptor δm Ci,tot,acceptor Ae D BZi t

(21)

i,m

Using eq 21, the calculated free Cu2+ concentration in the donor solution from the measured total Cu concentration at time intervals of 15, 24, 48, and 72 h in experiment 2 is 0.61 ( 0.23 and 0.058 ( 0.018 nM at 2.5 and 16.1 µM NTA, respectively. These results are in fair agreement with the free Cu2+ concentration calculated based on the chemical speciation calculation using the constants in Table 3, i.e., 0.89 and 0.074 nM. At these two NTA concentrations, the degree of complexation is high (Pi ) 1142, 14 170, respectively) and the transport of Cu is membrane diffusion limited. At 1.1 µM NTA (Pi ) 21), it is likely that diffusion in both solution and the membrane are affecting the transport rate. This leads to a faster transport and after 3 days the system is in equilibrium, as can be seen in Figure 9. When we still use eq 21, we arrive at an estimated free Cu2+ concentration of 2.2 ( 1.2 nM instead of the real activity, which equals 52 nM. The above analysis shows that low free ion concentration in the sample can be measured using kinetic interpretation as shown in eq 21 when the assumptions implicit in its derivation are met. Please note that, in natural samples, metal-ligand complexes dissociate with various kinetic characteristics. In practice, the dissociation kinetics of the metal complexes have to be considered to distinguish situations 2 and 3, but the principles discussed above are still applicable. ACKNOWLEDGMENT This research was funded by the Research Directorate General of the Commission of the European Commission, Contract EVK1CT-2001-00086 (BIOSPEC project). The authors thank Freerk Dousma and Gerlinde Vink for carrying out experimental work, and Erwin Kalis for offering a photo of the membrane. We also thank Wendy van Beinum for her help with the numerical modeling. GLOSSARY ai,donor

activity of ion i in donor solution (M)

ai,acceptor

activity of ion i in acceptor solution (M)

aj,donor

activity of ion j in donor solution (M)

aj,acceptor

activity of ion j in acceptor solution (M)

Am

surface area of a membrane (m2)

Ae

effective surface area of a membrane (m2)

B

Boltzmann factor

Ci

free ion concentration of ion i in solution (M)

Ci,j

concentration of species j of ion i in solution (M)

Ci,m

concentration of ion i in membrane (M)

Ci,tot

total concentration of ion i in solution (M)

Di

diffusion coefficient of ion i in water (m2 s-1)

diffusion coefficient of species j of ion i in water (m2 s-1)

Vm

volume of a membrane (m3)

VDonnan

volume of the Donnan phase in a membrane (m3)

Di,m

apparent diffusion coefficient of ion i in membrane (m2 s-1)

Vacceptor

volume of the acceptor solution (m3)

F

Faraday’s constant (C mol-1)

Ji,sol

flux of ion i in solution (mol m-2 s-1)

Ji,m

flux of ion i in membrane (mol m-2 s-1)

Mt

mass exchange due to transport (mol s-1)

Pi

complexation factor. Ratio of total to free concentration of ion i

q

volumetric charge density carried by the membrane sites (mol L-1)

R

gas constant (J mol-1 K-1)

t50%

time to reach 50% equilibrium concentration (s)

t95%

time to reach 95% equilibrium concentration (s)

T

absolute temperature (K)

Di,j

Zi

charge of ion i

Zj

charge of ion j

δ

thickness of diffusion layer in solution (m)

δm

thickness of the membrane (m)

ψ

electrostatic potential in the membrane Donnan phase (V)

λi

tortuosity factor of ion i transport in membrane

Received for review October 1, 2004. Accepted February 17, 2005. AC0485435

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