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Electric Relaxation Processes in Chemodynamics of Aqueous Metal Complexes: From Simple Ligands to Soft Nanoparticulate Complexants Herman P. van Leeuwen,† Jacques Buffle,‡ and Raewyn M. Town*,§ †
Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands CABE, Section de Chimie, University of Geneva, Sciences II, Quai Ernest-Ansermet 30, CH-1211 Geneva 4, Switzerland § Institute of Physics and Chemistry, University of Southern Denmark, Campusvej 55, DK-5230 Odense, Denmark ‡
ABSTRACT: The chemodynamics of metal complexes with nanoparticulate complexants can differ significantly from that for simple ligands. The spatial confinement of charged sites and binding sites to the nanoparticulate body impacts on the time scales of various steps in the overall complex formation process. The greater the charge carried by the nanoparticle, the longer it takes to set up the counterion distribution equilibrium with the medium. A z+ metal ion (z > 1) in a 1:1 background electrolyte will accumulate in the counterionic atmosphere around negatively charged simple ions, as well as within/around the body of a soft nanoparticle with negative structural charge. The rate of accumulation is often governed by diffusion and proceeds until Boltzmann partition equilibrium between the charged entity and the ions in the medium is attained. The electrostatic accumulation proceeds simultaneously with outer-sphere and innersphere complex formation. The rate of the eventual inner-sphere complex formation is generally controlled by the rate constant of dehydration of the metal ion, kw. For common transition metal ions with moderate to fast dehydration rates, e.g., Cu2+, Pb2+, and Cd2+, it is shown that the ionic equilibration with the medium may be the slower step and thus rate-limiting in their overall complexation with nanoparticles.
’ INTRODUCTION In environmental and biological media there is a great diversity of nanoparticulate complexants, both natural and engineered ones. Their physical nature ranges from soft and permeable to hard and impermeable, with mixed coreshell type entities in between. There is concern about the impact of the increased loading of nanoparticles in ecosystems. Many natural and engineered nanoparticles can complex various metal ions13 and sorb a range of organic compounds.47 Thus they may play a significant role in the chemodynamics of essential and toxic chemicals in aquatic systems. In this regard, it is crucial to quantify the kinetic features of formation and dissociation processes of nanoparticulate complexes with small molecules and ions. In the present work we focus on metal ion complexation processes and consider ligands in the range from oligoions [after Morel’s use of oligoelectrolyte to describe fulvic acids (FA)],8 typified by FA, to soft nanoparticles, e.g., large humic acids (HA). In practice there is not a sharp distinction between oligoions and soft nanoparticulate entities. The Eigen mechanism is often used to describe aqueous metal ion complexation by simple ligands.9 In this approach, the overall reaction is considered to take place via two successive steps: formation of an outer-sphere reactant pair (or “ion pair” if both reactants are ionic), followed by inner-sphere coordination r 2011 American Chemical Society
between the metal ion and the ligand (eq 1). kos a
kisa
kd
kd
MðH2 OÞnzþ •S ais MðH2 OÞnd S MðH2 OÞnzþ þ S a os ð1Þ
þ dH2 O
os where S is a specific binding site of the complexant, kos a and kd are the respective outer-sphere association and dissociation rate constants, kisa and kisd are the inner-sphere association and dissociation rate constants, d is the denticity of S, and • denotes outer-sphere associate. Often, for complexation with simple ligands, dehydration of the metal ion (rate constant kw) is rate-limiting for innersphere complex formation. Still, especially for metal ions with fast dehydration rates, other processes such as ligand reconformation may also be rate-limiting in the eventual inner-sphere complex formation.9 The Eigen approach has recently been applied to the case of soft nanoparticulate complexants.10 Expressions were derived for the rate constants for the intraparticulate individual outersphere and inner-sphere association and dissociation steps for the
Received: September 14, 2011 Revised: November 27, 2011 Published: November 29, 2011 227
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Langmuir limiting cases of low and high charge densities. A defining feature of nanoparticulate complexants is that their binding sites are spatially confined to the particle body. As a consequence, the time scales of various steps in the overall complexation process differ from those for simple ligands which are homogeneously dispersed in solution. The framework developed10,11 demonstrates that particle shape/size, reactive site density, and charge density are key determinants of the pertaining chemodynamic properties. The particular kinetic features of nanoparticulate binding sites can explain observations such as an apparent decrease in voltammetric lability of metal complexes when they are sorbed to particles.12 An important feature is that for rapidly dehydrating metal ions, such as Pb2+ and Cu2+, there are steps in the nanoparticulate complex formation process that proceed at rates comparable to or even lower than those for water-release and thus may be rate-limiting.10 Here we consider the nature and impact of accompanying electric relaxation processes in the overall complex formation of metal ions with charged soft nanoparticles and oligoion complexants, in comparison with simple ligands. Any ion, molecule, or particle carrying charge forms around/in it an electric field which results in the building-up of a counterionic atmosphere by the electrolyte. Several ligand categories can be discriminated based on their electrostatic properties. 1 Simple anions, such as Cl-, PO3 4 , or CH3COO for which the size of complexant is comparable to the size of the hydrated Mz+ aq . 2 Larger complexing oligoions, with sizes a few times greater than Mz+ aq and carrying several charges and one or two complexing sites, e.g., FA-type ligands. 3 Charged, gel-like, permeable particles of radius rp, containing many charges and complexing sites distributed throughout the particle volume. The size of the particle is much greater than that of Mz+ aq , whereas the dimension of the ionic atmosphere or double layer is similar to or smaller than that of the particle. At equilibrium, the concentration of free Mz+ aq inside the charged particle body will be different from that in the bulk solution. 4 Charged, compact, impermeable particles with many surface charge sites and many surface complexing sites, e.g., a solid metal oxide particle or a liquid micellar particle. Usually the ionic atmosphere is smaller than the particle, but because the particle is not permeable, generally only the relaxation processes in the extraparticulate double layer are relevant for the present purposes. 5 Charged, porous aggregates comprised of nonpermeable nanoparticles. Here the radius of the porous aggregate is generally much larger than the Debye screening length k1 of the medium. In this case the time scale of relaxation in the ionic atmosphere as compared to that associated with diffusion inside the aggregate depends on the relative magnitude of k1 and the interaggregate pore size. The higher the charge density of the nanoparticle, the greater the time needed to set up the equilibrium electric potential profile with respect to the bulk medium. Accordingly, fast reactions of small ions with specific sites within a nanoparticle generally are interrelated with the kinetics of their mere electrostatic partitioning into the charged nanoparticulate body and/or surrounding ionic atmosphere. The pertaining distinction between chemical and electrostatic processes needs to be resolved for the specific nanodomain where dimensions of double layers and diffusion
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layers are of similar magnitude. Our present focus will be on the 3D case of soft permeable nanoparticles and oligoions; with appropriate reformulation of the geometrical conditions, the concepts will also apply to hard (impermeable) nanoparticles and coreshell particles.
’ THEORETICAL FRAMEWORK The theory for metal complexation with soft nanoparticles has been detailed previously,10 and the basic concepts are briefly recalled below. Here we consider class 2 (oligoions) and class 3 (soft nanoparticulate) ligands, for which several steps are involved in the overall complexation process as illustrated in Figure 1. Any one of these steps may be ratelimiting, and for a given reaction the pertaining rate-limiting step will depend on the nature of both the complexant (size, charge density, and binding site density) and the metal ion (rate of dehydration, kw). The rate constant ka,p in Figure 1 represents the electric association between Mz+ aq and the charged complexing entity: this is essentially a diffusive rate constant that includes the conductive modification by the electrostatic field encountered over the diffusion path. The rate constant z+ kos a includes the additional step of Maq distributing between the outersphere volume and the remaining particle volume. The SmoluchowskiDebye treatment of ka,p,1315 leads to the equation ka, p ¼ 4πNAv rp ðDM þ Dp ÞU=½expðUÞ 1
ð2Þ
where rp and Dp are the radius and diffusion coefficient of the particle, respectively, and DM is the diffusion coefficient of the metal ion. The factor U/[exp(U) 1] represents the conductive accelerating (U < 0) or retarding (U > 0) effect of the electrostatic field on the diffusive transport of Mz+ aq toward the particle. It should be emphasized that eq 2 is based on Coulomb type interaction between the charged particle and the incoming ion. For multicharge particulate ligands, the electric energy U is given by U = zMFψ/RT, with ψ being the position-dependent potential in/around the particle or oligoion. For oligoions, the Mz+ aq is considered to penetrate mostly the extraparticulate diffuse double layer region and hence the effective potential for Mz+ aq is probably approximated fairly well by the average surface potential in the diffuse double ̅ 0, and U is denoted by layer, ψ̅ 0, i.e., the potential applicable to eq 2 is ψ U0. For soft nanoparticles, the Mz+ aq can generally penetrate the particulate entity and accordingly the potential applicable to eq 2 then is the average one in the particle body, ψ ̅ p, and U is denoted by Up. As outlined in Figure 1, the outer-sphere complexation process involves several steps. Under the conditions considered herein, the diffusion of Mz+ aq within the particle can be neglected since it is much faster than that outside the particle.16 For charged particulate complexants, the kos a for an individual site should also account for the distribution of Mz+ aq between the outer-sphere volume and the remaining particle body, and differences in potential, if applicable.10 Basically, kos a is a diffusive rate constant, modified by the electrostatic field encountered over the diffusion path; it does not include the electrostatic Boltzmann partitioning equilibrium coefficient, fB. As elaborated below, fB is a concentration ratio that expresses the extent to which the Mz+ aq ions accumulate in the vicinity of a negatively charged entity. For small ions and molecules the electrostatics can be described by a differentiated discrete approach for a multicharged ligand such as EDTA, in which the electrostatic reactant pair interaction energy Uos accounts for all localized charges present.17 For larger oligoions and nanoparticles, the use of a smeared-out potential to describe the electrostatics is appropriate because the entities are large enough to define a ψ ̅ p and a surface potential ψ̅ 0.18,19 The relatively small size of oligoions means that the presence of only a few charged groups already brings them into the high charge density regime where the chargecharge separation is smaller than the Debye length. For these complexants, the metal ion binding site is more likely to be close to the surface of the molecule. 228
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Figure 1. Stepwise complexation of a hydrated metal ion, Mz+ aq , with (A) a class 2 oligoion (rp ≈ 1 nm) typically containing one charged or uncharged binding site (S) and ca. 5 charged sites () and (B) a class 3 nanoparticulate complexant containing charged or uncharged binding sites (S) and charged sites (). The +’s denote the extraparticulate counterion atmosphere. The steps are (1) diffusion of Mz+ aq from the bulk solution to the surface of the complexant, (2) (A) penetration into the diffuse double layer of the oligoion and possible partial incorporation within the oligoion as a free hydrated ion (the dashed circle around Mz+ aq denotes its location somewhere within the extraparticulate spherical surface) or (B) passage over the energy barrier and incorporation within the particle as a z+ free hydrated ion (3) outer-sphere association of Mz+ aq with S, and (4) inner-sphere complex formation, including the loss of water of hydration by Maq and formation of a chemical bond with site S. The transport across the complexant/medium interface is considered infinitely fast. For a soft nanoparticle, the Mz+ aq generally can fully enter the particle body. Thus if the extraparticulate counter charge (in the diffuse double layer in the medium) is negligible with respect to the counter charge inside the particle body, the relevant partitioning to consider is between Vos and the remaining particle volume, Vp NSVos, where NS is the number of specific sites. The ka,p for the entire particle (eq 2) is converted into kos a for an individual site by accounting for the distribution of the incoming Mz+ aq between the outer-sphere volume and the remaining particle body. The resulting expressions for the high and low charge density cases are10
Furthermore, the size of the complexant is so small that the partitioning of Mz+ aq into the extraparticulate counterion atmosphere will be larger than that into the particle body. Thus the particle-associated Mz+ aq will be distributed between the outer-sphere volume around the specific site S, i.e. Vos, and the remaining volume of the counterion atmosphere, Vatm. As a qualitative approximation, one can represent the true potential distribution in the counterion atmosphere by a two-state model (analogous to the Manning counterion condensation model for polyelectrolytes),20 with Vatm ≈ (4/3)π[(rp + O(k1))3 (rp)3], where O(k1) represents a length parameter of order k1. Accordingly, kos a can be written as kos a ¼ ka, p
V os expð UÞ Vatm
kos a ¼ 4πNAv rp ðDM þ Dp Þ
ð3Þ
Up V os Vp expðU p Þ 1
½high charge density regime
where exp(U) reflects the preference of Mz+ aq for the specific site, i.e., ̅ pψ ̅ 0)/RT], where ψ ̅ 0 approximates the exp(U) = exp[zMF(ψ average, smeared-out potential in the counterionic atmosphere and ψ ̅ p is the potential at the specific site (high charge density regime). The expression for kos a can be readily adapted to account for, e.g., partial penetration of Mz+ aq into the complexant body.
kos a ¼ 4πNAv rp ðDM þ Dp Þ
expð U os ÞV os ½expð U os Þ 1NS V os þ Vp
½low charge density regime 229
ð4aÞ
ð4bÞ
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In the high charge density case, the magnitude of Up is determined by the average electrostatic potential difference between the particle body and the solution, ψ̅ p (Up = zMFψ̅ p/RT). The equilibrium stability constant Kos for the outer-sphere Mz+ aq •S reactant pair compares Mz+ aq , with its fully intact inner hydration sphere in contact with S, to Mz+ aq at the potential of the bulk medium (cf. Figure 1). It is given by21 4 K os ¼ πa3 NAv expð UÞ 3
We continue for the more or less usual situation of a minor target reactant ion in a large excess of background electrolyte. If the charge number of the minor reactive species, Mz+ aq , is higher than that of the cations of the background electrolyte, the electric relaxation of the former is more involved. In the initial stage of the relaxation process, all ions in the medium participate in proportion to their (mobility and) concentration. In the final electrostatic equilibrium, however, counterions with higher charges are preferred over those with lower charge. Consequently, for mixed electrolytes the composition in an electric double layer differs from that in the medium. In our case of Mz+ aq (with z > 1) in a 1:1 background electrolyte, Mz+ aq will to some extent accumulate in the vicinity of the negatively charged entities. It will do so in an extraparticulate double layer, in the case of smaller negative ions and oligoions, as well as within the body of a large Donnan particle with negative structural volume charge. The effect is completely similar to what happens in electrochemical systems with minor electroactive species in an excess background electrolyte.23 The maximum extent of accumulation is simply given by the pertaining Boltzmann equilibrium factor
ð5Þ
where a is the center-to-center distance between reactants in the outersphere reactant pair. For the low charge density regime for nanoparticles (and for small ions/molecules), U = Uos. For oligoions, U = U0, and for nanoparticles in the high charge density regime, U = Up. The corresponding outer-sphere dissociation rate constant for soft nanoparticles is os obtained from kos a /K using eqs 4 and 5 rp Up V os kos d ¼ 3 3 ðDM þ Dp Þ a Vp 1 expð U p Þ ½high charge density regime
ð6aÞ
rp os os os þ Vp Þ kos d ¼ 3 3 ðDM þ Dp ÞV =ð½expð U Þ 1NS V a ½low charge density regime ð6bÞ
fB ¼ expð zM Fψ=RTÞ ¼ expð UÞ
where the applicable ψ depends on the nature of the entity involved, e.g., for a large Donnan particle it is ψ̅ p, and for an oligoion it is the representative average value in the counterion atmosphere, ψ̅ 0. The Boltzmann factor is a concentration ratio, which for a nanoparticle corresponds to cM,p/c/M, where cM,p is the free metal ion concentration in the particle body and c/M is the free metal ion concentration in bulk solution; for an oligoion the Mz+ aq accumulation is predominantly extraparticulate and thus fB corresponds to the average excess concentration in the counterionic atmosphere. The relaxation time constant for Mz+ aq to reach equilibrium with an ionic atmosphere is typically of order r2p/DM (cf. Equation 7 for kos d which reflects the reciprocal residence time at the potential Up). Thus, for rather small particles such of Mz+ aq as oligoions, with rp of 0.3 1 nm, this leads to relaxation times on the order of 1010 to 109 s. This time scale is inside the 1/kw regime of fast dehydrating metal ions such as Cu(H2O)2+ 6 and Pb(H2O)2+ 6 . The applicable potential for the electric relaxation process in the counterionic atmosphere of an oligoion is that pertaining to the diffuse double layer which largely extends outside the particle body. The electric relaxation processes up to formation of Mz+ aq •S thus may take place concurrently with the sequence of complexation reactions outlined above.
For oligoions, kos d is obtained by combining eqs 3 and 5. The relevant expression is analogous to that for the high charge density case (eq 6a), in which Vatm replaces Vp, and for the two-state approximation with k1 as the characteristic thickness (see above) Uo corresponds to the smeared-out potential in the counterion atmosphere which holds over the distance from rp to rp + O(k1). For nanoparticles for which rp . a, with Vp . NSVos and Dp , DM, and considering Vos = (4/3)πa3, the following limits are obtained: high charge density :
kos d ¼
Up 3DM rp2 1 expð U p Þ
ð7aÞ
low charge density :
kos d ¼
3DM rp2
ð7bÞ
Equation 7b shows that in the low charge density limit, kos d approaches the conventional rate constant for diffusion from an uncharged sphere of radius rp, i.e., 3DM/r2p. This conforms with the r2p dependence of a diffusion controlled time-constant for a spherical particle with sites homogeneously distributed within the particle body volume.11
’ ELECTRIC RELAXATION TIMES Complex formation/dissociation processes inherently require establishment/adjustment of the electric potential of the ligand particle body with respect to the medium. In the low charge density case the potential inside the particle body is virtually zero, except in the vicinity of the binding sites, if charged.11 Accordingly, the concentration of Mz+ aq in the bulk of the particle is the same as that in the medium, except for the local counterionic atmospheres around the few charged sites. The time scale for electric relaxation of these atmospheres is as fast as in the bulk medium, with a time constant τ = 1/Dk2, also known to apply to classical double-layer relaxation at electrode/solution interfaces22,23 or in dispersions of hard colloids.24 For a particle at high negative charge densities, with a radius rp that is well above the Debye length k1, the setting up of the full counterion distribution equilibrium with the medium takes more time than for the low charge density regime.25 For the present work it is essential to quantify the pertaining rate and time characteristics.
ð8Þ
’ RATE OF ACCUMULATION OF Mz+ aq IN THE PARTICLE VOLUME OR COUNTERION ATMOSPHERE For accumulation of ions in the ionic atmosphere of an oligoion, a characteristic potential is that at the distance k1 from the surface, and for accumulation inside a nanoparticle it is the volume average particle potential, ψ ̅ p. For nanoparticles in the low charge density limit, the ionic atmosphere pertains essentially only to the individual charged sites, i.e., the electrostatic considerations are similar to those for simple ligands (case 1 in the Introduction), albeit with a lower rate of diffusive transport from the medium to the particle-bound sites, and thus a lower kos a . The rate at which Mz+ aq from the bulk solution attains partition equilibrium with the charged entity is primarily governed by diffusion. The steady-state diffusive flux, Ja,p, of Mz+ aq from the 230
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bulk solution to the surface of a charged spherical particle (oligoion or nanoparticle) with ionic atmosphere surface k1 is given by
interface can be expressed as c0M ðtÞ ¼ cM ½1 expð t=τrel Þ
where τrel for the soft particle case is the characteristic time constant
Ja, p ¼ ðDM þ Dp ÞðcM c0M Þ=ðrp þ k1 Þ ½mol m2 s1
ð9aÞ
which for a nanoparticle with large rp (rp . k1) reduces to Ja, p ¼
ðDM þ Dp ÞðcM c0M Þ=rp 2 1
½mol m
s
τrel ¼
where rp + k1 is the effective radius for transport in the double 0 layer, c/M is the concentration of Mz+ aq in the bulk solution and cM is z+ the concentration of Maq at the medium side of the particle/ medium interface. The corresponding diffusive rates, Ra,p, are Ra, p ¼ 4πðDM þ Dp Þðrp þ k1 ÞðcM c0M Þ ð10aÞ
Ra, p ¼ 4πðDM þ Dp Þrp ðcM c0M Þ ½mol s1 , per particle
ð10bÞ
Over time, cM in the particle body (or double layer) and c0M gradually grow so that both Ja,p and Ra,p slow down. The concentration of the free reactant species Mz+ aq within the counterionic atmosphere or particle body increases until it reaches its final equilibrium value, which is up to a factor fB higher than it would be for an uncharged particle (cf. eq 8). Hence, the concentration of the outer-sphere intermediate Mz+ aq •S also grows with time, in instantaneous equilibrium with free Mz+ aq,p. The relationship between the two follows either the high charge density Vos/Vp distribution or the low charge density exp(Uos) one, as relevant (cf. Equation 4, with the low charge z+ density regime tacitly taken as NSVos , Vp). Since Mz+ aq a Maq •S inside the particle, then in the high charge density regime cM•S/cM,free = 1, while in the low charge density regime cM•S/cM,free = exp(Uos). For the case of a nanoparticle with rp . k1, the total concentration of Mz+ aq inside the particle at equilibrium is given by the time integral of the rate (eq 10), divided by the particle volume Z ∞ 0
ðRa, p ðtÞ=Vp Þ dt ¼
0
ðRa, p ðtÞ=Vp Þ dt ¼
ð11aÞ
3ðDM þ Dp Þðrp þ k1 Þ Z ðrp þ
¼ fB cM
k1 Þ3
rp3
∞ 0
ð14Þ
holds over the time scale of formation of the inner-sphere complex MS. That is, the concentrations of both the free metal ion in the particle and the outer-sphere complex grow with time to their equilibrium values, prior to innersphere complex formation. In this case, for a diffusive steady-state in the medium, the steady-state concentration of Mz+ aq •S is attained when the rate of inner-sphere complex formation is equal to the rate of diffusive supply, Ra,p = [DM(c/M c0M)/rp]A, where c0M is cM at the medium side of the medium/particle interface cM(r = +rp), and A is the surface area of the particle. However, for the limiting case of inner-sphere complexation being rate-limiting, diffusion is fast compared to inner-sphere complex formation, thus(c/M c0M) , c/M. This implies that c0M ≈ c/M, and consequently, cM•S is approximately constant as prescribed by the outer-sphere complex equilibrium with bulk c/M. This situation arises for slowly dehydrating metal ions, e.g. Ni(H2O)2+ 6 , and consequently the rate of inner-sphere complex formation is the rate-limiting step. Under these conditions, any electric relaxation processes occur on a much faster time scale than inner-sphere complex formation. The time-independent concentration of the outersphere complex allows application of the ensuing steadystate relationships for ka and kd, i.e.19
where fB is the Boltzmann factor, eq 8. The corresponding expression for the case of an oligoion, with negligible accumulation within the body of the molecule, is Z ∞
ð13Þ
3ðDM þ Dp Þ
zþ zþ zþ Maq ðbulkÞ a Maq ðparticleÞ a Maq •S
3ðDM þ Dp Þ Z ∞ ðcM c0M ðtÞÞ dt 0 rp2
¼ fB cM
fB rp2
z+ Thus the concentrations of Mz+ aq •S and free Maq inside the charged entity exponentially grow with time according to a [1exp(3Dt/r2pfB)] dependence. Compared to the case without accumulation, the relevant rate constant, krel = 1/τrel = 3(DM + Dp)/r2p, is modified by the factor 1/fB. For oligoions, expressions analogous to eqs 12 and 13 are obtained from eq 11b. Let us now return to the final step in the process of complex formation between Mz+ aq and S (Figure 1) and consider which of the steps in the overall complexation reaction may be ratelimiting. Whether or not the kinetics of electric relaxation impact on the overall reaction rate depends on the relative magnitude of the outer-sphere and inner-sphere complex formation rates: (i) the rate of inner-sphere complex formation is rate-limiting. In this situation there is equilibrium up to formation of the outer-sphere associate Mz+ aq •S, i.e.
ð9bÞ
½mol s1 , per oligoion
ð12Þ
ka ¼
ðcM c0M ðtÞÞ dt
ð11bÞ
kos a kw þ kw
ð15Þ
is kos d kd kos d þ kw
ð16Þ
kos d
and
1
which holds for k much greater than the radius of the metal ion. For nanoparticles, for conditions of zero cM in the particle at t = 0, and insignificant depletion of the bulk solution (i.e., sufficiently dilute dispersion), the time evolution of the metal concentration at the medium side of the particle/medium
kd ¼
(ii) the rate of outer-sphere complex formation is rate-limiting. In this case the diffusion step is very slow compared to the inner-sphere complex formation. This implies that all 231
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species within the particulate entity are continuously in equilibrium, i.e. zþ Mzþ aq ðparticleÞ a Maq •S a MS
ð17Þ
Then cM(particle)/cM•S and cM•S/cMS(=1/Kis) remain constant while there is an ongoing flux of free metal. That is, the diffusive kos a (i.e., ka,p) is in control and the outer-sphere complex formed is continually partly consumed in order to maintain equilibrium with the inner-sphere complex. This diffusive flux is of a steady-state nature, JM = DM(c/M c0M)/rp, for times beyond O(r2p/DM). In this situation none of the relevant concentrations are approximately constant. Inside the particle, the equilibrium z+ z+ Mz+ aq •S a MS holds, in addition to Maq a Maq •S. All forms of M z+ , M •S, and MS) then follow the 1expinside the particle (Mz+ aq aq (t/τrel) dependence until the eventual equilibrium with c/M in the medium has been achieved. This situation arises when the formation of the inner-sphere complex is fast relative to that of the outer-sphere, e.g. for rapidly dehydrating metal ions such as Cu(H2O)2+ 6 . In this case the overall complex formation rate constant, ka, corresponds to krel, which equals kos a decreased by the Boltzmann equilibrium distribution factor, fB (eq 13). It is important to realize that the rate of outer-sphere formation is formulated in terms of c/M, while the thermodynamic stability constant KMS = KMS (zero charge)fB. That is, KMS has the c/MfB included. Thus the rate constant kos a remains the same in the presence of accumulation, but the attainment of the eventual equilibrium is fB times more demanding.
’ COMPARISON OF THE LIMITING RATES OF OUTERSPHERE AND INNER-SPHERE COMPLEX FORMATION The rate of the overall complex formation process comprises (i) outer-sphere complex formation which is generally governed by the rate of diffusive supply of Mz+ aq to the particle or oligoion, Ros a , followed by (ii) the rate of inner-sphere complex formation, Risa . Thus we seek to establish whether outer-sphere or innersphere complex formation is rate-limiting under given conditions (nature of the metal ion, particle size, applicable potential). Therefore we compare the limiting values of the individual rate is Ros a for electric relaxation and the individual rate Ra for inneris sphere complex formation. Thus Ra is computed for full ion distribution equilibrium with the medium, and the concentration of free metal in the nanoparticle body or oligion counterion atmosphere, cM,p, equal to its equilibrium value fBc/M. Similarly, Ros a is computed on the basis of full equilibrium between all species inside the particle or oligoion, i.e., cM,p a cM•S a cMS. In order to make the rate of outer-sphere complexation, Ros a , directly comparable to the apparently monomolecular rate of innersphere complexation, Risa , it is necessary to formulate both rates in terms of the total number of reactive sites per unit volume of dispersion. The ensuing rate expressions for soft nanoparticles in the high charge density regime are obtained as follows: os For outer-sphere complex formation, Ros a equals ka (eq 4) / times cMcS, where cS is the smeared-out site concentration equal to NScp, where cp is the concentration of particles in the bulk dispersion, i.e. Raos
¼ 4πNAv rp ðDM
Figure 2. Comparison of computed rate constants for outer-sphere kos a and inner-sphere kisa complex formation as a function of potential (high charge density regime) for (A) a nanoparticle of radius, rp = 5 nm, NS = 20, ̅ and (B) an oligoion of radius rp = 1 nm, NS = 1. For the nanoparticle case ψ corresponds to the average smeared-out potential in the particle body. For the oligoion case, ψ ̅ corresponds to the difference between that in the extraparticulate counterion atmosphere (up to rp+k1) and that at the is 1 in place local site, k1 = 1 nm, and kos a and ka are computed with rp + k 1 3 of rp, and Vatm in place of Vp (i.e., (4/3 )π[(rp + k ) (rp)3]). In all cases DM is taken as 1 109 m2 s1 and assumed to be much greater than Dp. The simple ligand case, with Uos corresponding to a 2+,1- ion pair, is shown by the blue horizontal bars on the y-axis: the solid line corresponds 9.6 1 is to kos s . a , and the dashed line is ka for kw = 10
For inner-sphere complex formation Rais ¼ kw cM•S NAv cp Vp
where cM•S = (NSVos)/(Vp)cM,p, with cM,p = fBc/M. Accordingly, this leads to Rais ¼
kw NS V os fB cM NAv cp Vp ¼ kw NS V os fB NAv cM cp Vp
¼ kisa cM cp ½mol m3 s1 where
kisa
is now formulated in the same dimensions as
kisa ¼ kw NS V os fB NAv
Up NS V os c cp þ Dp Þ Vp expðU p Þ 1 M
½mol m3 s1
ð19Þ
½m3 mol1 s1
ð20Þ kos a ,
i.e.: ð21Þ
1 is in For the oligoion case, kos a and ka are computed with rp + k 1 3 place of rp, and Vatm in place of Vp (i.e., (4/3) π [(rp + k ) (rp)3]). If a process other than dehydration of the metal ion is
ð18Þ 232
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Langmuir rate-limiting for inner-sphere complex formation then its rate constant appears in eq 21 instead of kw. Figure 2 compares computed limiting values of kos a (eq 4a,4b) and k isa (eq 21) as a function of ψ̅ , for oligoions and nanoparticulate complexants. For oligoions and particulate ligands it is seen that with a fast dehydrating metal ion such 9.6 1 s ), outer-sphere complex formaas Cu(H2O)2+ 6 (kw = 10 tion is always rate-limiting. Even for the case of a simple ligand with charge 1 forming a complex with Cu2+, the kos a is only slightly greater than kisa (see horizontal bars on y-axis in Figure 2). At the other extreme, for a slowly dehydrating metal 4.5 1 s ), inner-sphere complex ion such as Ni(H2O)2+ 6 (kw = 10 formation is generally rate-limiting for simple ligands and smaller particles and moderate ψ̅ values. But with increasing particle size, Risa and Ros a approach comparable magnitude for strongly negative particle charges (Figure 2). Thus for larger, highly charged soft particles, outer-sphere complex formation may be rate-limiting even for relatively slowly dehydrating metal ions.
’ CONCLUSIONS It is shown that for charged oligoionic and nanoparticulate complexants the magnitude of the electric field has consequences for the chemodynamics of their aqueous metal complexes. The more negatively charged the complexant, the greater the extent to which Mz+ aq will accumulate in the counterion atmosphere and in the particle volume, as determined by the Boltzmann equilibrium partition coefficient, fB. The accumulation process is generally diffusion controlled, thus the greater the electric field, the longer time it takes to set up partition equilibrium between Mz+ aq in the particle and in the medium. As a consequence, two different situations arise depending on the relative magnitudes of the rate constants for outer-sphere and inner-sphere complex is formation. If Ros a is much greater than Ra , then a constant steadystate concentration of the outer-sphere complex is achieved before significant inner-sphere complex formation takes place, and cM•S remains constant over the time scale of MS formation. is At the other limit, with Ros a much less than R a , the concentrations of all species within the particulate entity are increasing with time, and are in continuous equilibrium with each other during the ongoing inner-sphere complex formation. In this case the overall complex formation rate constant is governed by the process of attaining eventual ionic distribution equilibrium. For metal ions with moderate to fast dehydration rates, outer-sphere complex formation may thus become overall rate-limiting. The electric relaxation process is an issue of primary importance in the complexation kinetics of a range 2+ of common transition metal ions, e.g., Cu(H2O)2+ 6 , Cd(H2O)6 , . and Pb(H2O)2+ 6 ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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’ SYMBOLS AND ABBREVIATIONS a center-to-center distance of closest approach between a hydrated metal ion and a complexing site S (m) bulk concentration of free metal ion M (mol m3) c/M concentration of the outer-sphere reactant pair Mz+ cM•S aq •S (mol m3) diffusion coefficient of the metal ion (m2 s1) DM difffusion coefficient of the particle (m2 s1) Dp Boltzmann equilibrium partitioning factor fB steady-state diffusive flux of Mz+ Ja,p aq from the bulk solution to the surface of a charged spherical particle (mol m2 s1) overall rate constant for complex formation ka (m3 mol1 s1) rate constant for electric association between Mz+ ka,p aq and the charged complexing entity (m3 mol1 s1) rate constant for outer-sphere reactant pair association kos a (m3 mol1 s1) rate constant for inner-sphere complex formation kisa (m3 mol1 s1) os rate constant for outer-sphere reactant pair dissociation kd (s1) rate constant for inner-sphere complex dissociation (s1) kisd overall rate constant for complex dissociation (s1) kd rate constant for electric relaxation (s1) krel rate constant for water substitution (s1) kw overall stability constant of the complex MS (m3 mol1) KMS 1 Debye length (m) k hydrated metal ion Mz+ aq reactant pair between a hydrated metal ion and a Mz+ aq •S complexing site S number of complexing sites in a particulate comNS plexant rate of outer-sphere reactant pair formation (mol m3 s1) Ros a is rate of inner-sphere ion complex formation (mol m3 s1) Ra S site that covalently binds a metal ion characteristic time constant for electric relaxation (s) τrel dimensionless interaction energy between Mz+ Uo aq in the bulk medium and within the oligion counterion atmosphere dimensionless interaction energy between Mz+ Up aq in the bulk medium and inside the particle body volume of the counterion atmosphere around an Vatm oliogion (m3) os outer-sphere volume for an ion pair between Mz+ V aq and individual site S (m3) volume of the particle body (m3) Vp particle radius (m) rp average electrostatic potential difference between the ψ̅ 0 solution and the diffuse double layer (V) average electrostatic potential difference between the ψ̅ p solution and the particle body (V) ’ REFERENCES (1) Wonders, J. H. A. M.; van Leeuwen, H. P. Electrochim. Acta 1998, 43, 3401. (2) Buffle, J. Complexation Reactions in Aquatic Systems: an Analytical Approach; Ellis Horwood: Chichester, U.K., 1988. (3) Barton, L. E.; Grant, K. E.; Kosel, T.; Quicksall, A. N.; Maurice, P. A. Environ. Sci. Technol. 2011, 45, 3231. (4) Benhabib, K.; Town, R. M.; van Leeuwen, H. P. Langmuir 2009, 25, 3381.
’ ACKNOWLEDGMENT This work was performed within the framework of the BIOMONAR project funded by the European Commission’s seventh framework program (Theme 2: Food, Agriculture and Biotechnology), under grant agreement 244405. 233
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