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Langmuir 2007, 23, 5216-5226
Roles of Dynamic Metal Speciation and Membrane Permeability in Metal Flux through Lipophilic Membranes: General Theory and Experimental Validation with Nonlabile Complexes Zeshi Zhang,† Jacques Buffle,*,† and Herman P. van Leeuwen†,‡ CABE, Department of Inorganic, Analytical and Applied Chemistry, UniVersity of GeneVa, Sciences II, 30 quai E. Ansermet, CH-1211 GeneVa 4, Switzerland, and Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, NL-6703 HB Wageningen, The Netherlands ReceiVed December 9, 2006. In Final Form: January 29, 2007 The study of the role of dynamic metal speciation in lipophilic membrane permeability in aqueous solution requires accurate interpretation of experimental data. To meet this goal, a general theory is derived for describing 1:1 metal complex flux, under steady-state and ligand excess conditions, through a permeation liquid membrane (PLM). The theory is applicable to fluxes through any lipophilic membrane. From this theory, fluxes in the three rate-limiting conditions for metal transport are readily derived, corresponding, namely, to (i) diffusion in the source solution, (ii) diffusion in the membrane, and (iii) the chemical kinetics of formation/dissociation of the metal complex in the interfacial reaction layer. The theory enables discussion of the reaction layer concept in a more general frame and also provides unambiguous criteria for the definition of an inert metal complex. The theoretical flux equations for fully labile complexes were validated in a previous paper. The general theory for semi- or nonlabile complexes is validated here by studying the flux of Pb(II) through PLMs in contact with solutions of Pb(II)-NTA and Pb(II)TMDTA at different pHs and flow rates.
1. Introduction In natural waters, trace metals are present in a large number of different forms in addition to the “free”, hydrated metal ion. For instance they may be complexed with small inorganic ligands such as carbonate or OH-, complexed with small or colloidal natural organic ligands such as humic substances, adsorbed on inorganic colloids or particles, or adsorbed at the surface of or incorporated inside microorganisms.1 These various species play different roles in the circulation of a given trace metal in aquatic systems, particularly in their biouptake and ecotoxicological input.2 Biouptake has long been related to the free ion activity model (FIAM) or in its more recent form to the biotic ligand model (BLM), according to which the metal uptake rate is supposed to depend only on the thermodynamic activity of the free metal in solution, irrespective of the other species. A number of papers,2,3 however, have shown that, under natural conditions, a part of the metal complex may contribute to the uptake flux, particularly the dynamic ones,4 that is, those which are sufficiently mobile (sufficiently large diffusion coefficient) and labile (sufficiently large association/dissociation rates compared to diffusion rates). * To whom correspondence should be addressed. Fax: +41.22.379.6069. E-mail:
[email protected]. † University of Geneva. ‡ Wageningen University. (1) Buffle, J. Complexation Reactions in Aquatic Systems; Ellis Horwood: Chichester, 1988. (2) Wilkinson, K. J.; Buffle, J. Critical evaluation of physico-chemical parameters and processes for modelling the biological uptake of trace metals in environmental (aquatic) systems. In Physicochemical kinetics and transport at biointerfaces; van Leeuwen, H. P., Ko¨ster, W., Eds.; IUPAC Series on Analytical and Physical Chemistry of Environmental Systems; Wiley: Chichester, 2004; Vol. 10, Chapter 10. (3) Campbell, P. G. C. Interactions between trace metals and aquatic organisms: A critique of the free ion activity model. In Metal speciation and bioaVailability in aquatic systems; Tessier, A., Turner, D. R., Eds.; IUPAC Series on Analytical and Physical Chemistry of Environmental Systems; Wiley: Chichester, 1995; Vol. 3., Chapter 2. (4) van Leeuwen, H. P.; Town, R. M.; Buffle, J.; Cleven, R. F. M. J.; Davison, W.; Puy, J.; van Riemsdijk, W. H.; Sigg, L. EnV. Sci. Technol. 2005, 39, 85458556.
Metal uptake by microorganisms may involve very complicated mechanisms of transfer through the plasma membrane.5 The permeation liquid membrane (PLM), employed for metal flux measurements and metal speciation analysis, is a much simpler technical device, even though the metal flux through this membrane is influenced by the same environmental factors as that through biomembranes (Figure 1). Thus, PLMs can be used to study the role of these factors, as well as the key elements required for a “bioanalogical” sensor to determine the bioavailable metal in environmental systems.6 In PLMs,7 (Figure 1) the free metal ion, M, can exchange between the test (source) solution and the membrane composed of a lipophilic organic solvent with a dissolved metal complexant, C, which serves as the carrier. The complex MC is transported by diffusion through the membrane, and M is released into the strip solution on the other side by forming a complex MS, which is more stable than MC, with a strong ligand, S. The free carrier then diffuses back toward the source solution interface, and M is thus accumulated into the strip solution. Metal enrichment by PLMs (also called SLMs) was first proposed in the 1970s and then developed by Danesi.8 It has been used by many researchers to selectively enrich certain chemicals7 such as metals,9-11 proteins,12 or organic acids.13,14 (5) Ko¨ster, W. Transport of solutes across biological membranes: prokaryotes. In Physicochemical kinetics and transport at biointerfaces; van Leeuwen, H. P., Ko¨ster, W., Eds.; IUPAC Series on Analytical and Physical Chemistry of Environmental Systems; Wiley: Chichester, 2004; Vol. 10, Chapter 6. (6) Buffle, J.; Tercier-Waeber, M.-L. TrAC, Trends Anal. Chem. 2005, 24, 172-191. (7) Buffle, J.; Parthasarathy, N.; Djane, N. K.; Matthiasson, L. Permeation liquid membrane for field analysis and speciation of trace compounds in waters. In In situ monitoring of aquatic systems; Buffle, J., Horvai, G., Eds.; IUPAC Series on Analytical and Physical Chemistry of Environmental Systems; Wiley: Chichester, 2000; Vol. 6, Chapter 10. (8) Danesi, P. R. Sep. Sci. Technol. 1984-1985, 19 (11-12), 857-894. (9) Juang, R.-S.; Lee, S.-H. J. Membr. Sci. 1996, 110, 13-23. (10) Campderros, M. E.; Acosta, A.; Marchese, J. Talanta 1998, 47, 19-24. (11) Breembroek, G. R. M.; van Straalen, A.; Witkamp, G. J.; van Rosmalen, G. M. J. Membr. Sci. 1998, 146, 185-195. (12) Tsai, S.-W.; Wen, C.-L.; Chen, J.-L.; Wu, C.-S. J. Membr. Sci. 1995, 100, 87-97. (13) Juang, R.-S.; Huang, R.-H.; Wu, R.-T. J. Membr. Sci. 1997, 136, 89-99.
10.1021/la063568f CCC: $37.00 © 2007 American Chemical Society Published on Web 03/29/2007
Flux of Pb(II) through Lipophilic Membranes
Figure 1. Schematic representation of the PLM and the process of metal complex speciation. l ) membrane thickness; δso and δst ) diffusion layer thicknesses in the source and strip solution, respectively; DM, DML, and DMC ) diffusion coefficients of M, ML, and MC, respectively; ka and kd ) association and dissociation rate constants for ML formation; KML ) stability constant of ML; Kp ) partition coefficient of M ) [MC]/[M][C].
However, the most attention has been paid to industrial applications and only a few studies were devoted to trace metal analysis in natural waters.7,15-17 Parthasarathy et al.7,17 were the first to show that PLMs enable the determination of the free metal concentration in the presence of various synthetic and natural hydrophilic ligands. Later, it was shown7,18,19 that, by tuning the diffusion layer thickness in the source solution (through hydrodynamic control), as well as the nature and concentration of the carrier, C, PLMs can be used to determine either the concentration of the free metal ion or that of the total labile complexes. Various technical devices such as flat sheet PLMs,16 emulsion PLMs,20 hollow fiber PLMs,21 flow-through PLMs,18 and PLM-integrated voltammetric microsensors22 have been developed, and environmental applications have been reported.23-25 The important feature depicted in Figure 1 is that M is accumulated into the strip solution at a rate which depends, in particular, on the physicochemical processes occurring in the source solution. The amount of accumulated metal therefore enables the determination of the overall flux through the source solution and the PLM and thus the nature of the rate-limiting factors. For example, if the transport through the membrane is slow compared to that in the source solution, the depletion of the metal concentration at the interface is negligible and the flux through the PLM only depends on the free M concentration in solution (as assumed in the FIAM model for microorganism (14) Schafer, A.; Hossain, M. M. Bioprocess Eng. 1996, 16, 25-33. (15) Papontoni, N. K.; Djane, K.; Ndungu, J.; Jonsson, A.; Mathiasson, L. Analyst 1995, 120, 1471. (16) Parthasarathy, N.; Buffle, J. Anal. Chim. Acta 1994, 284, 649. (17) Parthasarathy, N.; Buffle, J.; Gassama, N.; Guenod, F. Chem. Anal. (Warsaw) 1999, 44, 455. (18) Tomaszewski, L.; Buffle, J.; Galceran, J. Anal. Chem. 2003, 75, 893900. (19) Zhang, Z.; Buffle, J.; van Leeuwen, H. P.; Woiciechowski, K. Anal. Chem. 2006, 78, 5693-5703. (20) Li, L.-Q.; Zhang, Y.-H.; Gan, W.-E.; Li, Y.-D. At. Spectrosc. 2001, 22, 290-294. (21) Parthasarathy, N.; Pelletier, M.; Tercier-Waeber, M.-L.; Buffle, J. Electroanalysis 2001, 13, 1305-1314. (22) Salau¨n, P.; Buffle, J. Anal. Chem. 2004, 76, 31-39. (23) Sigg, L.; Black, F.; Buffle, J.; Cao, J.; Cleven, R.; Davison, W.; Galceran, J.; Gunkel, P.; Kalis, E.; Kistler, D.; Martin, M.; Noel, S.; Nur, Y.; Odzak, N.; Puy, J.; van Riemsdijk, W.; Temminghoff, E.; Tercier-Weaber, M.-L.; Toepperwien, S.; Town, R. M.; Unsworth, E.; Warnken, K. W.; Weng, L.; Xue, H.; Zhang, H. EnV. Sci. Technol. 2006, 40, 1934-1941. (24) Ndungu, K.; Hurst, M. P.; Bruland, K. W. EnViron. Sci. Technol. 2005, 39, 3166-3175. (25) Shkinev, V. M.; Fedorova, O. M.; Spivakov, B. Y.; Mattusch, J.; Weinrich, R.; Lohse, M. Anal. Chim. Acta 1996, 327, 167-174.
Langmuir, Vol. 23, No. 9, 2007 5217
uptake). When, on the other hand, the transport processes in the source solution are slower than those in the membrane, the former become rate limiting. This may be due, in particular, to the slow diffusion of the complex ML (low value of the diffusion coefficient of the complex, DML, as for colloidal complexants) and/or to the slow dissociation reaction of ML (low value of the dissociation rate constant, kd, as, e.g., for some organic metal complexes). A complete study of the roles of these factors is thus essential to understand metal biouptake as well as to develop well-controlled bioanalogical PLM sensors. Even though the roles of the kinetics of interfacial reaction26,27 and mass transport in the water phase28,29on ion transfer at the liquid/liquid interface have been studied, a rigorous theory for the combination of mass transport in both phases and the chemical kinetics at the water side of the interface has not yet been reported and checked experimentally. In the present work, we develop such a general theory and mathematical expressions, providing the steady-state flux of metal ions through a liquid/liquid interface, by considering both the kinetic and thermodynamic properties of the metal complex, in combination with its transport in solution and inside the PLM. A general analytical expression for the flux is obtained, which simplifies into the expected limits for the fully inert and fully labile complexes. The concept of a reaction layer at the interface is extended by this general theory and is discussed in detail. Experimental systematic verification for the labile systems has been reported in a previous paper,19 using copper(II)-diglycolic acid, lead(II)-diglycolic acid, and copper(II)-N-(2-carboxyphenyl)glycine complexes. Here, the validation of the general equation is complemented by studying the metal flux in the presence of semilabile Pb(II)-TMDTA and Pb(II)-NTA complexes. The impact of the pH and diffusion layer thickness has been investigated. The general theory leads to a mathematical definition of inert complexes and, in turn, to two clearly different types of physical conditions which lead to inert behavior.
2. General Theory for the PLM Steady-State Metal Flux in a Complexing Solution 2.1. Derivation of the Flux Equation. Let us consider the metal complexation reaction (eq 1) with the association and dissociation rate constants ka and kd, respectively, and the equilibrium constant K ) ka/kd ) [ML]/[M][L]: ka
M + L {\ } ML k
(1)
d
In the following, the ligand will be assumed to be in large excess compared to M; that is, CL . CM, where CL and CM are the total concentrations of L and M in the source solution, respectively. In addition, the possible protonation of L will be assumed to always be at equilibrium, so that the free concentration of L, [L], can be expressed as
CL
[L] )
n
1+
(2)
βiH[H]i ∑ i)1
(26) Nitsch, W.; Weigl, M. Langmuir 1998, 14, 6709-6715. (27) Salvador, J.; Puy, J.; Cecilia, J.; Galceran, J. J. Electroanal. Chem. 2006, 588, 303-313. (28) Plucinski, P.; Nitsch, W. Langmuir 1995, 11, 4691-4694. (29) Levadny, V.; Aguilella, V.; Belaya, M.; Yamazaki, M. Langmuir 1998, 14, 4630-4637.
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and is proportional to CL at constant pH. (The βiH values are the cumulative acid-base equilibrium constants of L). In the most general case, the association and dissociation rates of ML are not very fast compared to the diffusion rates of M and ML in solution. The continuity equations for the metal species are
this condition is fulfilled for at least several hours after the start of the accumulation step.7,19 On the other hand, the time required to get steady-state concentration profiles in solution and in the membrane are of the orders of δ2so/DM ) 10 s (with δso ) 100 µm) and l 2/DMC ) 30 s, respectively. Under those conditions, the flux through the membrane is given by
∂[M] ∂2[M] ) DM 2 + kd[ML] - ka[M][L] ∂t ∂x
J ) DMC([MC]1 - [MC]2)/l
(3)
2
∂[ML] ∂ [ML] ) DML - kd[ML] + ka[M][L] ∂t ∂x2
(4)
where t is time and x is the distance from the membrane/source solution interface (for the other symbols, see Figure 1). Let us introduce the dimensionless variables
θ)
[M] [M]
b
and ψ )
[ML] [ML]b
0) 0)
J)
ka[L] d2θ ka[L] + ψ θ DM DM dx2
(6)
kd kd d2ψ ψ + θ DML dx2 DML
(7)
ka[L] d2ψ ka[L] ψ+ θ 2 D DM dx M
dx2
)0
(9)
which shows that the sum of the gradients of [M] and [ML], weighted by their diffusion coefficients, is constant over the diffusion layer. This implies that, due to the chemical coupling of M and ML, a single depletion layer thickness applies to both M and ML despite their different diffusion coefficients. The equations for the profiles of θ and ψ are obtained by using boundary conditions, which correspond to a well-stirred solution, in which a fixed reactive diffusion layer with thickness δso is created at the membrane surface:
At x ) 0: θ ) θ0 )
[M]0 dψ )0 , [M]b dx
At x g δso: θ ) 1, ψ ) 1
(13)
This flux is equal to that on the source solution side of the membrane:
J ) DM[M]b
(dθdx)
(14)
x)0
0
Kp[C]DMCθ dθ ) Fθ0 ) dx x)0 DMl
( )
(10) (11)
In the following, we shall compute the flux at times long enough (typically >1 min) so that steady-state concentration profiles are established, but short enough for the free M concentration to be much smaller in the strip than in the source solution, that is, for the PLM system to remain far from equilibrium. With the flow-through cell described in this paper,
(15)
with
F)
(8)
The summation of eqs 6 and 8 cancels out the kinetic terms to give
d2(θ + K[L]ψ)
[M]bKp[C]DMC 0 θ l
Combining eqs 13 and 14 gives
By multiplying the left-hand and right-hand sides of eq 7 by K[L] (with ) DML/DM) and rearranging with ka[L] ) kdK[L], one obtains
0 ) K[L]
where [MC]1 and [MC]2 are the concentrations of the metal carrier complex in the membrane at the source/membrane (subscript 1) and strip/membrane interfaces (subscript 2; Figure 1), respectively. The strip ligand, S, is so strong that, during the steady state, [MC]2 ) 0. In addition, [MC]1/[M]0 ) Kp[C], where Kp is the partition coefficient of M between the source solution and the membrane and [C] is the carrier concentration in the membrane phase, in large excess compared to MC. Thus, eq 12 can also be written:
(5)
where [M]b and [ML]b are the concentrations of M and ML in the bulk of the source solution. At steady state, eqs 3 and 4 can be rewritten as follows:
(12)
KP[C]DMC DMl
(16)
By integrating eq 9 with the boundary conditions 10, 11, and 15, we obtain
θ + K[L]ψ ) (x - δso)Fθ0 + 1 + K[L] for 0 < x < δso (17a) and
θ + K[L]ψ ) 1 + K[L]
for x g δso
(17b)
which reflects that the weighted sum of the normalized concentrations of M and ML is a linear function of x in the whole diffusion layer. The concentrations profiles of M and ML and the flux equation are found by combining eqs 6 and 17a to eliminate ψ:
ka[L] d2θ ka[L](1 + K[L]) θ+ Fθ0x + 2 DMK[L] DMK[L] dx ka[L] (1 + K[L] - Fθ0δso) ) 0 (18) DMK[L] In eq 18, a combination parameter, λ, appears as a physically meaningful parameter:
λ)
x
DMK[L]
ka[L](1 + K[L])
x
)µ
K[L] 1 + K[L]
(19)
Flux of Pb(II) through Lipophilic Membranes
Langmuir, Vol. 23, No. 9, 2007 5219
where µ is given by
µ)
x
DM
(20)
ka[L]
µ is the so-known reaction layer thickness.30 It is the maximum distance from the interface, from which M can still diffuse to and be consumed at the surface, instead of being associated again with L into ML. For that reason, we shall call it the “association” reaction layer. λ is an even more general expression for the reaction layer thickness, as shown by its two limiting cases. For very stable complexes, when K[L] . 1, λ ) µ, and the reaction layer is controlled by the diffusion of free M and its reassociation with L, as discussed above. On the other hand, for large size, stable complexes, for which K[L] , 1, λ ) (DML/kd)1/2; that is, the reaction layer thickness is now controlled by the diffusion and dissociation of the complex ML (see section 2.5 for more details). It will be called the “dissociation” reaction layer. Typically, the reaction layer thickness is much smaller than δso for labile complexes, while it is much larger for inert complexes. The general solution of eq 18 is as follows:
θ ) C1 exp(-x/λ) + C2 exp(x/λ) +
Fθ x + 1 + K[L] Fθ0δso (21) 11 + K[L]
(
C1 )
)
θ0(1 + Fδso + K[L])
1 + K[L] 1 - exp(2δso/λ)
θ0(1 + Fδso + K[L]) 1 + K[L] 1 - exp(2δso/λ)
C2 )
exp(2δso/λ) (22)
[
δso
1 + K[L]
+
-1 (23)
( )]
δso λK[L] 1 tanh + F 1 + K[L] λ
-1
(24)
or
[
J ) [M]b
δso
DM(1 + K[L])
(
K[L] µ DM 1 + K[L]
+
)
3/2
l + Kp[C]DMC
(x
δso tanh µ
(26)
Under this condition, the flux is proportional to the free metal concentration in the bulk of the source solution, [M]b, irrespective of the degree of lability of the complex. 2.3. Limiting Case 2: Labile Complexes. Complexes are fully labile if the association/dissociation rates of ML are much faster than the diffusion rates of M and ML in solution. This means that the third term of eqs 24 and 25 is much smaller than the first one; that is,
λK[L] tanh(δso/λ) , δso For labile complexes, δso . µ30-32 so that, when K[L] . 1 (i.e., λ ∼ µ), tanh(δso/λ) f 1, and one obtains
δso . µK[L] The expression for the flux of fully labile complexes then simplifies into
[
J ) [M]b
δso
DM(1 + K[L])
+
l Kp[C]DMC
]
-1
(27)
and, when diffusion in the membrane is much faster than in solution,
DM(1 + K[L]) D h CM ) δso δso
J ) [M]b
(28)
where D h is the average diffusion coefficient of M and is defined as30
From eqs 21-23, we get (dθ/dx)x)0 and θ0 which, combined with eqs 13 or 14, provides the expression of J:
J ) DM[M]b
Kp[C]DMC J ) [M]b l
0
where C1 and C2 are constants which can be found by using the boundary conditions 10 and 11:
1-
2.2. Limiting Case 1: Membrane-Limited Flux. When the second term in eq 25 is much larger than the other two terms, diffusion inside the membrane is the only flux-limiting process and eq 25 simplifies into eq 26:7,19,22
)]
1 + K[L] K[L]
D h ) (DM[M]b + DML[ML]b)/CM
(29)
Under this last condition, J is proportional to the total concentration rather than to the free concentration of M. Equations 26-28 have been experimentally checked in detail.19 Note that, even for fully labile complexes, the flux can still be fixed by eq 26, when the second term in the denominator of eq 27 is much larger than the first one (see limiting case 1). When δso ≈ λK[L] tanh(δso/λ), the behavior of the complex is between that of a fully labile complex and a kinetically controlled complex, and it is called semilabile. 2.4. Limiting Case 3: Kinetically Controlled Complexes. Under the condition
δso , λK[L] tanh(δso/λ)
-1
(25)
with tanh(x) ) [exp(2x) - 1]/[exp(2x) + 1]. The denominators of eqs 24 and 25 contain three terms, each one representative of a resistance to the flux. From left to right, they correspond to (i) the physical diffusion in the source solution, (ii) the physical diffusion in the membrane, and (iii) the chemical rate of dissociation/formation of the metal complex in solution. From eq 25, a few important limiting cases can be distinguished. (30) Heyrovsky, J.; Kuta, J. Principles of Polarography; Publishing House of the Czechoslovak Academy of Sciences: Prague, 1965.
the association/dissociation rates of ML are slow compared to its diffusion rate in the source solution and become the ratelimiting steps, provided the diffusion in the membrane is fast, that is, 1/F is small compared to the third term of the denominator of eq 25. For the often-encountered condition K[L] . 1, which also implies δso/λ ∼ δso/µ, three cases can be discriminated: (i) When δso/λ < 0.1, the complex can be considered as inert (see discussion (31) Galceran, J.; Puy, J.; Salvador, J.; Cecilia, J.; van Leeuwen, H. P. J. Electroanal. Chem. 2001, 505, 85-94. (32) Van Leeuwen, H. P. Electronanlysis 2001, 13, 826-830.
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Zhang et al.
in section 2.5). (ii) When δso/λ > 10, tanh(δso/λ) ∼ 1 in eq 25. By also taking K[L] . 1 into account, eq 25 then reduces to
[
J ) [M]b
l + Kp[C]DMC
x
1 DMka[L]
]
-1
(30)
that is, the contribution of the complex ML (second term inside the brackets of eq 30) is limited by its chemical kinetics. A pure kinetically controlled flux is obtained (J ) DM[M]b/µ) when the first term in the brackets is negligible, that is, the diffusion in the membrane is very fast with respect to the chemical kinetics. (iii) When 0.1 < δso/λ < 10, the behavior of the complex is between that of an inert complex and that of a kinetically controlled complex (see section 2.5). Note that the values of δso/λ ) 0.1 and δso/λ ) 10 are somewhat arbitrary, with the only important conditions being that tanh(δso/λ) ∼ δso/λ and tanh(δso/λ) ∼ 1, respectively. 2.5. Dissociation Reaction Layer, Lability Index, and Inert Complexes. The lability index, L, expresses the ability of a complex ML to contribute to the supply of M to the interface, via diffusion and dissociation. It is defined4 as the ratio between the purely dissociation rate-controlled flux and the flux determined by diffusion in solution only. It is usually stated4 that, when L . 1, the complex is fully labile, whereas for L , 1 the complex is nonlabile. In intermediate cases, ML is semilabile (but see discussion below). The above ratio of fluxes can be computed as the ratio of the first term over the third term of the denominator of eq 25. By introducing the definition of µ (eq 20), making an algebraic rearrangement, and defining
χ ) K[L]
(31)
σ ) δso(kd/DML)1/2
(32)
and
one obtains
L )
σx1 + χ χ tanh(σx1 + χ)
(33)
Note that σ is the ratio between δso and the “dissociation” reaction layer, ω, given by
ω ) x(DML/kd)
(34)
The physical meaning of ω is understood by analogy to that of µ (eq 20)30. It is the thickness of a solution layer in which ML can travel by diffusion before redissociating, that is, during its lifetime. When ω > δso (σ < 1), the complex has a very low probability to dissociate before reaching the consuming interface. When this probability is zero, the complex is inert by definition. Thus, σ is a good parameter to evaluate whether or not a complex is inert. It is also easily verified that the condition ω > δso is equivalent to kdt < 1, that is, a condition for inertness in the time domain. Equation 33 is a fully general expression for the lability index for inert, semilabile, and labile complexes. Figure 2 shows the general relationship between L, σ, and χ. Two limiting cases are interesting to discuss. For σ(1 + χ)1/2 > 3 (roughly the right side of Figure 2), tanh(σ (1 + χ)1/2) ∼ 1, and eq 33 reduces to
Figure 2. Lability index, L, as function of σ ) (DML/kd)1/2 for various values of χ ) K[L]. The points on the curves corresponds to eq 39. Their envelope is the threshold of inertia, and the hatched zone corresponds to the domain where the complexes are inert.
L ) σx1 + χ/χ )
1/2 k1/2 d (1 + K[L]) δso D1/2 MLK[L]
(35)
which is the usual expression reported in the literature.4 The lability index as defined in eq 35 indeed applies to the usual case of noninert complexes (σ > 1) and allows the distinction between labile and nonlabile complexes within the dynamic regime. For σ(1 + χ)1/2 < 0.1 (roughly the left side of Figure 2), the approximation tanh(x) ∼ x can be used so that
L ) 1/χ
(36)
This case corresponds to inert complexes. Interestingly, for such complexes, not only is L independent of the formation/ dissociation rate constants, but it may even be larger than 1 when χ ) K[L] < 1. Since only cases with K[L] > 1 are practically interesting, χ < 1 implies that ) DML/DM , 1. This condition applies, in particular, for complexes with macromolecular or colloidal complexants.39 Equation 36 shows that L ) 1/χ ) DM[M]b/DML[ML]b is just the ratio of the diffusive flux of M over that of ML (i.e., of total M for K[L] . 1) by assuming that they are independent from each other. This limiting value of L is expected for an inert complex, since no dissociation occurs in the reaction layer. The threshold conditions for inert behavior can be found as follows, by referring to the flux equation specifically developed for fully inert complexes:7,22
J ) [M]b
[
]
δso l + DM Kp[C]DMC
-1
(37)
Equating the denominators of eqs 25 and 37 shows that an inert complex obeys the condition:
tanh(δso/λ) ∼ δso/λ
(38)
This equality is never exactly fulfilled, but tanh(x)/x f 1 when (33) Basolo, F.; Pearson, R. G. Mechanisms of Inorganic reactions; Wiley: New York, 1967. (34) Morel, F. M. M.; Hering, J. G. Principles and applications of aquatic chemistry; Wiley: New York, 1983. (35) Martell, A. E.; Smith, R. M.; Niotekaitis, R. J. NTST Metal Complexes, version 5.0; 1998. (36) Guyon, F.; Parthasarathy, N.; Buffle, J. Anal. Chem. 1999, 71, 819-826. (37) Rabenstein, D. L. J. Am. Chem. Soc. 1971, 93 (12), 2869-2874. (38) Tarelkin, A. V.; Kravtsov, V. I.; Kondrat’ev, V. I. Vestn. Leningr. UniV., Ser. 4: Fiz. Khim. 1989, 3, 112-116. (39) Pinheiro, J. P.; Minor, M.; van Leeuwen, H. P. Langmuir 2005, 21, 86358642.
Flux of Pb(II) through Lipophilic Membranes
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Table 1. Predictions of the Free Ligand Concentration Corresponding to the Threshold of Inertia, [L]Τ, for Various Charges of L and 1:1 Metal Complexesa log ([L]Τ/M)
metal
DM (×10 cm2 s-1)
k-w (s-1)
0
-1
-2
-3
-4
Pb(II) Hg(II) Cu(II) Cd(II) Zn(II) Co(II) Ni(II) Fe(III) Al(III)
9.5 7.6 7.2 7.2 7.2 7.0 6.9 5.9 6.0
7 × 109 2 × 109 1 × 109 3 × 108 7 × 107 2 × 106 3 × 104 2 × 102 1
-12.37 -11.92 -11.64 -11.12 -10.49 -8.96 -7.14 -5.03 -2.72
-13.19 -12.74 -12.46 -11.94 -11.31 -9.77 -7.94 -6.25 -3.94
-14.00 -13.55 -13.27 -12.75 -12.12 -10.59 -8.75 -7.48 -5.17
-14.92 -14.47 -14.19 -13.68 -13.04 -11.41 -9.67 -8.70 -6.39
-15.63 -15.18 -14.90 -14.38 -13.75 -12.22 -10.38 -9.92 -7.61
a
-6
δso ) 100 µm, I ) 0.1 M. Values of k-w are from ref 34.
x f 0. This is expected since the dissociation rate of a reversible complex is never zero. However, eq 38 allows the definition of an arbitrary threshold for inertia. For instance, it is valid within e1% error for δso/λ e 0.1, which can be accepted as a limit of inertia. Introducing the expressions for λ and µ (eqs 19 and 20) and expressing them in terms of σ and χ gives the general condition for the so-called threshold of inertia, ΤI:
ΤI ) σx1 + χ )
δsox1 + K[L]
xDML/kd
e 0.1
(39)
By applying eq 39 to each curve of Figure 2, one obtains a set of points whose envelope allows the definition of inert (left side) and noninert domains. Two cases can again be discriminated: (a) K[L] , 1. As discussed above, this implies , 1. Equation 39 then becomes
kd e 0.01DML/δso2 which shows that the dissociation rate constant is the critical factor, irrespective of the specific nature and concentrations of M and L. (b) K[L] . 1. This condition applies to the small size, stable complexes. Rearranging eq 39 then leads to the threshold ligand concentration, [L]T:
[L]T e 0.01DM/(δso2ka) The corresponding inert behavior can be understood by realizing that K[L]. 1 implies that [ML]b ≈ CM ) cte, irrespective of [L]. Let us explore Figure 2 at a fixed value of σ < 0.01 (i.e., a fixed dissociation rate). If we start in the noninert domain and decrease [L], L will increase (even though it will stay below 1), reflecting an increase of the rate of reestablishment of the equilibrium between ML, M, and L. This is due to an increase of the association rate, because of the increase of [M] (corresponding to the decrease of K[L]), which more than compensates for the decrease of [L]. However, if [L] decreases below its threshold value (envelope in Figure 2), the probability that this reaction occurs inside the reactive diffusion layer is so low that ML can be considered as inert. The combination of the Eigen mechanism33 with the above considerations enables us to predict that [L]T should be independent of the chemical nature of the ligand or the complex. Indeed, as a first approximation, ka can be estimated by
ka ) Kosk-w
(40)
where k-w is the rate constant for the removal of a water molecule from the inner shell of the hydrated metal ion M and Kos is the
Figure 3. Flow cell system and its dimentions. L ) length of the channel (12 cm); W ) width of the channel (1 mm); hso) depth of the source channel (0.6 mm); hst ) depth of the strip channel (0.4 mm); effective strip volume ) 192 µL.
stability constant of the outer-sphere complex. The values of k-w are known for each metal ion, and those of Kos can be computed.33,34 They depend on the number of electric charges of M and L and on the ionic strength. Table 1 gives values of [L]T thus computed for various metals and charges of L, at I ) 0.1 M. Note that, in Table 1, [L]T is the free ligand concentration. In the case of protonation, the total ligand concentration may be much larger and composite effective rate constants should replace the association and dissociation rate constants discussed above (section 4.1). 3. Experimental Section 3.1. Reagents and Membranes. The following reagents used for transport experiments were analytical grade: 2-(N-morpholino)ethanesulfonic acid (MES, Sigma), trans-1,2-diaminocyclohexaneN,N,N′,N′-tetraacetic acid monohydrate (CDTA, Fluka), sodium hydroxide and lithium hydroxide (Merck), and palmitic acid (PA, Fluka). 1,10-Didecyl-1,10-diaza-18-crown-6 ether (Kryptofix 22DD, Merck) was pure for synthesis (>98%). The solvents, toluene and phenylhexane, were Fluka analytical grade products. MilliQ water was used for preparing all the aqueous solutions. The source solution contained the desired concentration of Pb(II) (usually 5 µM) in 0.01 M MES (pH ) 6 adjusted with LiOH). The strip solution was composed of 0.001 M CDTA (pH ) 6.20 adjusted with NaOH). The membrane solution was made of 1:1 22DD/PA in a 1:1 (v/v) mixture of phenylhexane and toluene. Note that, compared to previous work, palmitate was used instead of laurate in the present study, for the reason given in ref 19. Celgard 2500 polypropylene hydrophobic membranes (Celanese Plastic, Charlotte, NC) were used as flat sheet support. The membrane characteristics are as follows: porosity ) 0.45; thickness ) 25 µm; average pore diameter ) 0.04 µm. 3.2. The PLM Device and its Characteristics. A schematic view of the flow-through cell system used in this work and its corresponding characteristics are shown in Figure 3. A peristaltic pump (Gilson, Omnilab) was used to control the source volume flow rate. Standard tubings (Tygon R3607, ID ) 2.29 mm and 3.175 mm, Biosystems, Omnilab) were used to connect the source solution to the flow-through cell, whose characteristics and advantages have been described in detail.18 The source solution flows through four parallel channels. The metal from the source solution passes through
5222 Langmuir, Vol. 23, No. 9, 2007
Zhang et al.
the flat membrane (Figure 1) and accumulates in the stagnant strip channel solution. The strip channel is made of four arms positioned exactly opposite to the source channels and connected to each other to form one single channel. The strip solution was collected by a pump at the end of the accumulation period, and the corresponding metal concentration was measured by flame or flameless atomic absorption spectrometry (AAS) using a Pye Unicam SP9 spectrometer or a Perkin-Elmer 4100 spectrometer with an HGA 700 graphite furnace. This system enables us to define the optimum geometries for the strip and source channels. The strip channel depth, hst, should be sufficiently small to obtain a short response time. The depth, hso, width, W, and length, L, of the source channel (Figure 3) together with the solution volume flow rate, ν, provide a well-defined thickness of the diffusion layer, δso18: δso )
1/3 2/3 1/3 -1/3 0.901D h 1/3 so d hso L ν
(41)
where d ) W/2 ) half-width and the numerical coefficient is valid for centimeter-gram-second (cgs) units. The driving force of the PLM is the free M activity gradient between the two solutions; thus, the complexant S in the strip solution (Figure 1) should be strong enough to maintain a free M activity lower than that in the source solution during the whole measurement period. In this paper, this was ensured, by using CDTA as ligand S, with a very large degree of complexation for Pb(II). Based on the stability constants35 (Table 2), Rst ) CPbst/[Pb++]st was calculated to be 1012.7. Due to this large value, the strip solution behaves as a perfect sink for Pb(II) under the experimental conditions used here. The values of Kp) 4.2 × 105 M-1 and DMC ) (3.6 ( 1.2) × 10-8 cm2/s, for Pb(II) with the system using 0.1 M 22DD/0.1 M palmitate,19 were used. 3.3. Conditions of Metal Transport Experiments. 3.3.1. FlowThrough Cell Preparation and Function. A membrane was impregnated with the membrane solution as outlined before.18,19 After rinsing it with water to remove the excess solution, the membrane was clamped tightly and evenly between the source and strip halfcells by 10 screws. The strip solution was injected into the strip channel with a syringe, and then the source solution with a given metal concentration was circulated with a constant flow rate. After accumulation during a given period of time (typically 10-30 min), the strip solution was taken out and Pb(II) was measured by AAS or flameless AAS. The source channel was then washed by flowing MilliQ water, while the strip channel was washed by injecting a new strip solution. Circulation of MilliQ water was performed for up to 25 min to minimize any carry-over effect. A new transport experiment was then performed with the test source solution and a new strip solution. Usually, the initial metal flux and permeability, under one given set of conditions, were computed from four accumulation experiments performed with different accumulation times. J was obtained from the slope of the linear plot of the accumulation factor, F (see section 4.3 for definition), as a function of time2,18,19 (eq 50). Typically, such plots do not pass through the origin, since, at the beginning of the experiment, there is an ∼1 min time lag which corresponds to the transient conditions required to establish the steadystate concentration gradients in the membrane and in solution.36 3.3.2. Transport Experiments in the Presence of Complexants. The nature of the complexes used as well as the experimental conditions (pH, ligand concentration) were chosen in such a way that only 1:1 complexes were formed, and for which, as much as possible, the value of lability index, L, was
