6666
J. Phys. Chem. B 2001, 105, 6666-6674
Binding of Metal Ions to Polyelectrolytes and Their Oligomeric Counterparts: An Application of a Generalized Potts Model† Ger Koper Laboratory of Physical Chemistry, Delft UniVersity of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
Michal Borkovec* Department of Chemistry, Clarkson UniVersity, Box 5814, Potsdam, New York, 13699 ReceiVed: January 25, 2001
Competitive binding of metal ions and protons to polyelectrolytes and the corresponding oligomers is studied within a statistical mechanics framework. The problem is formulated with a generalized Potts model, which describes the different states of the macromolecules in terms of appropriate multivalued state variables. The interactions are assumed to be short ranged and can be parametrized in terms of nearest neighbor pair energies. We focus on linear chains, which can be treated with transfer matrix techniques in a straightforward fashion. Binding isotherms for polyelectrolytes and binding constants of different solution complexes are evaluated.
Introduction The present study focuses on interactions of metal ions with macromolecules. The topic is relevant in surprisingly different areas; let us just illustrate their diversity by a few examples. (i) Metal ions are central in living organisms, and their biological function is usually mediated by interactions with proteins. For example, the calcium complexing protein calbindin plays an important role in bone formation and neurotransmission.1-3 (ii) Supramolecular materials offer much potential for novel applications in electronics and material sciences, and self-assembly represents a major route to various two and three dimensional organized structures. This process sensitively depends on the nature of metal ions present, and rather unique structures can be obtained by exploiting this approach.4-6 (iii) Toxic or radioactive metal ions are of major concern when released into the natural environment, and their concentrations can be controlled by macromolecules. Various polyamines have been discussed as promising agents for novel wastewater treatment processes because of their strong metal complexing capacity.7-9 Such metal-ligand interactions have been extensively studied by coordination chemists in the past.10-13 Attention was focused on the interactions of single metal ions with possibly several low-molecular weight ligands. Besides the increasing popularity of spectroscopic tools, classical potentiometric titration still represents a key technique to study metal-ligand interactions. Metal ions usually compete with bound protons, and therefore metal binding is often assessed by measuring proton binding isotherms at various metal loading. Such data are typically analyzed in terms of a chemical equilibrium model. The metal binding characteristics of a ligand are then rationalized in terms of a sequence of metal-ligand complexes.11,12 This approach was successfully extended to study metalprotein interactions, because there is usually a single metal binding center within a protein.1-3 In the mentioned example of calbindin, two calcium ions bind to the active center in a †
Part of the special issue “Bruce Berne Festschrift”. * To whom correspondence should be addressed.
cooperative fashion. The salt dependence of this complexation reaction could be fully explained in terms of an electrostatic model.3 The situations discussed so far share an important simplificationsonly a few metal ions bind to the ligand. On the other hand, high-molecular weight ligands (i.e., macromolecules) may complex a large number of metal ions.8,14 For example, up to one hundred copper ions may be associated with a single dentritic polyamine mentioned above.8 Although this situation is generic for many different macromolecules, our understanding of such metal-macromolecule interactions is rudimentary on the fundamental level. So far, only a simplified continuum charge-condensation model was used to rationalize the maximum metal binding capacity of some linear polyelectrolytes.15 However, this description is hardly satisfactory, as the binding of metal ions is highly localized. Recently, it has been shown that metal ions bind to linear poly(ethylene imine) in various well-defined complexes with coordination numbers up to six (see Figure 1).11,16 Quantitative models of such binding processes are lacking. The aim of this paper is to explore this uncharted territory with statistical mechanics techniques further. Thereby, we shall build on an important analogy to the well-studied case of proton binding. When a small number of protons bind to an acid or a base (i.e., “ligand”) the situation can be described with simple chemical equilibria. In the case of a weak polyelectrolyte, however, many protons do bind to the macromolecule and the proton binding isotherm can be only obtained by considering the cooperative effects of all interactions within the chain. The latter situation can be approached with an Ising model, as initiated by Kirkwood and Shumaker17 and Steiner.18 This framework was put on a broader footing recently by us.19-24 Particularly, we have shown that this model properly describes the proton binding characteristics of polyelectrolytes as well as the corresponding low-molecular weight oligomers.20 This model represents an important starting point for the present work. Metal ions compete for binding sites with protons, and in their absence, the known proton binding isotherms must
10.1021/jp010320k CCC: $20.00 © 2001 American Chemical Society Published on Web 06/13/2001
Binding of Metal Ions to Polyelectrolytes
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6667 average becomes
〈 f 〉 ) Ξ-1
}) -βF({s }) e ∑ f ({si})an({s H i
i
(3)
{si}
where the (semi)grand partition function can be written as
Ξ)
}) -βF({s }) e ∑an({s H i
i
(4)
{si}
or using eq 2 one obtains N
Ξ)
Figure 1. Pictorial representation of the interactions of linear poly(ethylene imine) with protons and divalent metal ions.
be recovered. Although our development is general, we focus on linear polyelectrolytes to illustrate the effects of competition between protonation and metal complexation in various coordination geometries. Extensions of the present model and comparison with experimental data will be discussed elsewhere.
To introduce the basic philosophy, recall the well-studied case of proton binding.17-24 Protons bind to a macromolecule at localized (ionizable) sites. To enumerate these binding states, one introduces a set of state variables si with i ) 1, 2, ..., N, where N is the number of sites in the molecule. When the site i is protonated, si ) 1, whereas for the deprotonated state, si ) 0. The collection of all state variables {s1, s2, ..., sN} (or briefly {si}) specifies the protonation (micro)state of the macromolecule. To each microstate, we can assign a free energy F({si}). A cluster expansion of the free energy up to pair interactions leads to the well-known Ising model or lattice gas model and will be discussed in more detail below.19,23 With this free energy at hand, all quantities of interest can be evaluated by thermal averaging. Two kinds of averages are of interest: canonical and (semi)grand canonical. In the canonical-like average the overall number of bound protons is constrained to a fixed number n′ (0 e n′ e N). The expectation value of a any function f({si}) of the state variables can be written as
∑ f ({si})e-βF({s })δn′,n({s }) i
i
{si}
(1)
N si is a function giving the number of where n({si}) ) ∑i)1 bound protons for a given configuration {si} and δi,j is the Kronecker delta. The partition function is defined as
K h n′ )
∑e-βF({s })δn′,n({s }) i
{si}
(5)
The latter expression corresponds to the fugacity expansion of the partition function known in statistical mechanics or to the binding polynomial as described in the protein literature.25 In solution chemistry, it is customary to introduce pH ) -log10 aH. Knowing the partition function, various thermal averages can be computed. For example, the average degree of protonation is given by
θH )
〈n〉 N
(6)
which is the thermal average of the number of bound protons. This expression can be also written as the derivative
Ising Model and Proton Binding
-1 〈 f 〉n′ ) K h n′
K h nanH ∑ n)0
i
(2)
In the above equations, we have introduced the inverse thermal energy β ) 1/(kT). Within the (semi)grand canonical ensemble, the activity of the protons aH is kept constant, and the corresponding to thermal
θH )
aH d ln Ξ 1 d ln Ξ ) N daH N d ln aH
(7)
Inserting the binding polynomial eq 5 into eq 7, one recovers the isotherm N
θH )
nK h nanH ∑ 1 n)0 N
N
(8)
∑ Kh nanH
n)0
This expression is well-known in solution chemistry as the titration curve of a polyprotic acid, and confirms that K h n indeed corresponds to the to the equilibrium constant of the reaction
A + nH h HnA
(9)
where A is the fully deprotonated molecule. The charges are omitted for simplicity. In other words, we can use the canonical ensemble to evaluate ionization constants of oligomers (i.e., pK values). Generalized Potts Model and Metal Binding The development is rather analogous if two types of ions are binding to the macromolecule. One simple possibility would be that each metal ion binds to the same site as the protons, and this situation can be described with si ) 0 (empty site), si ) 1 (proton bound), and si ) 2 (metal bound). As will be discussed below, metal binding is more complicated, and additional states must be introduced. For the moment, however, we merely assume that all binding states can be uniquely characterized by a proper choice of the (multivalued) discrete state variables. Because the variables are multivalued, we are dealing with a (generalized) Potts model.26
6668 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Koper and Borkovec
Again we can discuss two types of averages. In the canonicallike average, we now have to constrain not only the number of bound protons n′ but also the number of bound metal ions m′. The expectation value of any function of state variables can be written as -1 〈 f 〉n′m′ ) K h n′m′
∑ f ({si})e-βF({s })δn′,n({s })δm′,m({s }) i
i
{si}
i
(10)
where we have introduced the number of bound metal ions m({si}) and n({si}) is the number of bound protons, as introduced above. The partition function now reads
∑e-βF({s })δn′,n({s })δm′,m({s })
K h n′m′ )
i
i
{si}
(11)
i
}) m({s }) -βF({s }) aM e ∑ f ({si})an({s H i
i
i
(12)
{si}
where the (semi)grand partition function reads
Ξ)
θM )
}) m({s }) -βF({s }) aM e ∑an({s H i
i
i
(13)
{si}
∑n,m mKh nmanH amM ∑n,m Kh nmanH amM
(19)
The latter relations are well-known in solution chemistry, and K h nm corresponds to the to the equilibrium constant of the reaction
A + nH + mM h HnMmA
(20)
The charges are omitted as above. The canonical ensemble can be again used to evaluate binding constants of metal ions to ligands. For m ) 0, the case of proton binding is recovered.
Let us briefly review the action of the transfer matrix for a linear chain26,27 albeit in a slightly more general way to accommodate the extension to metal binding. Our formulation is rather similar in spirit to the transfer matrix formalism introduced by Flory to model conformational isomerism in polymer chains.28 Consider a subchain consisting of the sites i ) 1, 2, ..., N′ (N′ e N). The state variables of such a subchain s1, s2, ..., sN′ will be abbreviated as {si}′. For such a subchain with N′ sites, we now define a partition (column) vector Ξ(N′), whose elements are the partition functions of the subchain with the last state variable sN′ constrained to a particular value s. Thus, we have
By making use of eq 11 we obtain
Ξ(N′)(s) )
N
Ξ)
∑ n,m
K h nmanH
amM
(14)
This expression generalizes the binding polynomial for a system containing protons and metal ions. In the absence of the metal ions (aM ) 0), the above-discussed case of proton binding is recovered. In solution chemistry, it is customary to parametrize the activities as pH ) -log10 aH and pH ) -log10 aM. The average degree of protonation is again given by
θH )
aH ∂ ln Ξ N ∂aH
(15)
whereas the average degree metal complexation obtained from
θM )
aM ∂ ln Ξ N ∂aM
(16)
From eqs 15 and 16 it becomes clear that, for all equilibrium systems, the isotherms must obey
aM
1 N
Transfer Matrix Formulation for Linear Chains
In the (semi)grand canonical ensemble, two activities are kept constant, namely, the one of the protons aH and of the metal ions aM. The average becomes
〈 f 〉 ) Ξ-1
and
∂θH ∂θM ) aH ∂aM ∂aH
(17)
In the language of thermodynamics, this identity is called a Maxwell relation, whereas in the adsorption literature it is referred to as a thermodynamic consistency relation. Inserting the binding polynomial eq 14 into eqs 15 and 16, we recover the isotherms
1 θH ) N
∑n,m nKh nmanH amM ∑n,m Kh nmanH amM
(18)
}′) m({s }′) aM δs,s ∑e-βF({s }′)an({s H i
i
i
N′
{si}′
(21)
Knowing the partition vector of a subchain, we can find the partition vector of a subchain that is extended (to the left) by one site with standard matrix algebra
Ξ(N′+1) ) TΞ(N′)
(22)
where the expression for the transfer matrix T can be found by inspection of eq 21. As we shall demonstrate in the following examples, the transfer matrices assume simple forms when the interactions are short ranged. Moreover, if the sites and their interactions are the same along the chain, there is just a single transfer matrix describing the whole chain. This situation will be assumed for simplicity here, but the formalism approach is easily extended beyond. Because the sites within a chain are numbered from the left to the right in an increasing fashion, the transfer matrix must be defined such that it steps through the chain in the same direction. The recursion relation eq 22 can be initiated by using the partition vector of a single site. More practical, however, is to define a generating vector Vg, which generates the first partition vector by application of the transfer matrix, namely, Ξ(1) ) TVg. The advantage of introducing this generating vector is its very simple structure, as will be exemplified below. The partition function of the entire chain with N sites is found by summing the elements of the partition vector of the final (Nth) subchain over all possible values of the site variable
Ξ)
∑s Ξ(N) (s)
(23)
This operation is also easily formulated in a matrix notation by defining a terminating vector Vt. In its simplest form, this vector
Binding of Metal Ions to Polyelectrolytes
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6669
contains only unit entries and allows us to write eq 23 as a scalar product
Ξ)V ˜ tΞ(N)
(24)
where V ˜ t denotes the transposed (row) vector. With the generating and terminating vectors Vg and Vt, the partition function can be written in a compact fashion as
Ξ)V ˜ tTNVg
(25)
This formulation is most useful in conjunction with numerical and symbolic computation packages, and various properties of finite chains are easily evaluated. The treatment of long chains is discussed in most statistical mechanics textbooks.26,27 The long chain limit is typically dominated by the largest eigenvalue λ of the transfer matrix T, and one can show that the partition function is asymptotically given by27
for N f ∞
Ξ ∼ λN
(26)
This long-chain approximation is usually satisfactory for chain lengths exceeding 20-30. If the eigenvalues of the transfer matrix cannot be evaluated analytically, they are easily determined numerically. The binding isotherms follow by taking the appropriate derivative of the largest eigenvalue (see eqs 7, 15, and 16). Proton Binding: A Simple Illustration Before we apply the presented formalism to metal binding by linear polyelectrolytes, first recall the simple case of proton binding and recapitulate the classical discussion of the linear Ising chain with nearest neighbor pairwise interactions. This discussion is given here to reiterate on the close correspondence between the linear Ising chain and proton binding to linear polyelectrolytes and its oligomers. This simple model captures quantitatively the ionization behavior of linear weak polyacids and polybases to a good degree of approximation. By introducing a few unimportant modifications, various types of low and high molecular weight linear polyamines can be modeled to a surprising degree of accuracy; the details are given elsewhere.19,20,24 In the simplest case, the free energy of a linear chain with identical sites and nearest neighbor interactions is given by N
∑ i)1
F({si}) ) -µ1
N-1
si + E11
sisi+1 ∑ i)1
(27)
where µ1 is a chemical potential term and E11 is the pair interaction energy between the neighboring sites. (We have introduced a subscript 1 to symbolize the monodentate character of proton binding. This notation will be extended to include multidentate binding of metal ions in later sections.) Because of Coulombic forces between protons, the interactions are repulsive and E11 > 0. The chemical potential term can be related to the microscopic binding constant K ˆ 1 of the proton to a site when all other sites are deprotonated by eβµ1 ) K ˆ 1. The transfer matrix is particularly simple in this situation and is given by
[
1 1 T) z z u 1 1 11
]
(28)
Figure 2. Microstates of a linear chain with four sites (N ) 4) for (monodentate) proton binding with the macroscopic binding constant K h n or K h n0. For symmetrical microstates, only one species is shown.
where we have introduced the reduced activity
z 1 ) a HK ˆ1
(29)
and parametrized the pair interaction energy by u11 ) e-βE11. The two supplementary (transposed) partition vectors are given by V ˜ g ) (1, 0) and V ˜ t ) (1, 1). The partition functions of finite Ising chains are easily evaluated by inserting these quantities into eq 25. For example, for a chain with four sites (N ) 4), one obtains
Ξ ) 1 + 4z1 + 3(1 + u11)z21 + 2u11(1 + u11)z31 + u311z41 (30) By equating the coefficients with eq 5 and using eq 29, the macroscopic binding constants K h n follow. Their values are summarized in Figure 2. Each individual term corresponds to a particular microscopic state of the chain, and they are grouped according to the number of protons bound. For example, there are three different possibilities to bind two protons without invoking any pair interactions, and again three possibilities having one interaction of this kind. In the long chain limit, the largest eigenvalue can be evaluated analytically. The result is
λ ) (1 + z1u11)/2 + xz1 + (1 - z1u11)2/4
(31)
With eqs 7 and 26 an analytical expression of the titration curve of a linear polyelectrolyte follows:18,19,21
θH ) [2 + (λ/z1)(1 - z1u11)/(1 - u11 + λu11)]-1 (32) In the absence of interactions (E11 ) 0 or u11 ) 1), this isotherm reduces to the familiar Langmuir isotherm
θH )
z1 θH or z1 ) 1 + z1 1 - θH
(33)
The latter isotherm immediately follows from a simple mass action law argument. Competitive Metal Binding: Simplest Case of Bidentate Complexes We shall now extend the above formalism to metal binding. The peculiar aspect of interactions of metal ions with a linear
6670 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Koper and Borkovec
TABLE 1: Transfer Matrix for a Linear Chain with Monodentate Binding of Protons and Bidentate Complexing of a Metal Ion 0 1 2-1 2-2
0
1
2-1
2-2
1 z1 z2 0
1 z1u11 z2u12 0
0 0 0 1
1 z1u12 z2u22 0
chain molecules is that a metal ion typically binds to several consecutive sites. Because of the chelate effect, a metal ion will be bound at least to two sites (ligands) simultaneously (see Figure 1). How to handle this situation with transfer matrix techniques is not entirely obvious and will be illustrated in the following. Consider first a simple situation where sites are either involved in protonation or in a bidentate complexation of a metal ion. The model is defined by the transfer matrix given in Table 1. The abbreviations are similar as before. We introduce the ˆ 1 and with z2 ) aMK ˆ 2 for monoreduced activities z1 ) aHK dentate and bidentate species. Thereby, we have introduced K ˆ1 and K ˆ 2 as the microscopic binding constants for protons and metal ions. The interactions are expressed as uνµ ) exp(-βEνµ), where the pair interaction energies are denoted by E11 for proton-proton interactions, E12 proton-metal interactions, and E22 for metal-metal interactions. Note that the pair interaction matrix is symmetric, namely, Eνµ ) Eµν. When considering the pH scale, it is useful to parametrize the interactions in terms of dimensionless interaction parameters νµ ) βEνµ/ln 10. There are two important modifications with respect to the Ising chain. First, the number of states being increased to four. However, rather than using a state variable running from 1 to 4, we choose the following symbols to represent these states. As before, we symbolize an empty site with 0 and a protonated site with 1. However, sites involved in a bidentate complex can be in two states, as it can have a metal ion bound either on its right or on its left. A site with right-bound metal will be denoted as 2-1, and with a left-bound metal 2-2. When a site is bound to a metal ion, it can no longer protonate. The second modification lies in introducing zero elements within the transfer matrix. To appreciate this modification, let us dwell on the statistical interpretation of this matrix in more detail. An element T(s, s′) of the transfer matrix is the conditional probability of finding a site in state s given that the preceding site is in state s′. This observation can be used to eliminate certain combinations of succeeding sites. For example, because the succession of two left-bound bidentate sites is forbidden, we choose the corresponding element of the transfer matrix to be zero. The other vanishing entries of the transfer matrix follow by similar arguments. These observations have also implications for the structure of the two partition vectors. In this example, V ˜ g ) (1, 0, 0, 0) as expected but V ˜ t ) (1, 1, 0, 1) because we cannot have a halfway bound metal ion at the (right) end of the chain. With this information at hand, the complexation constants for finite chains and binding isotherms for long chains can be evaluated. Returning to our example of a chain with four sites (N ) 4), with eq 25 we find
Ξ ) 1 + 4z1 + 3(1 + u11)z21 + 2u11(1 + u11)z31 + u311 z41 + 3z2 + 2(1 + 2u12)z1z2 + u12(2u11 + u12)z21z2 + u22z22 (34) The macroscopic binding constants K h nm follow from eq 14 by comparing with the coefficients of the corresponding powers of the activities aH and aM. The constants involving metal ions
Figure 3. Microstates of a linear chain with four sites (N ) 4) for (monodentate) proton binding and bidentate metal binding with the macroscopic binding constant K h nm. Only the species involving metal ions are shown (m g 1); species without the metal (m ) 0) are given in Figure 2.
Figure 4. Binding isotherms for long chains involving protonation and bidentate metal complexes. The microscopic binding constant for protons is log10 K ˆ 1 ) 9, and the metal binding constant is log10 K ˆ2 ) 14. The pair interaction parameters are 11 ) 2 (proton-proton), 12 ) 1 (proton-metal), and 22 ) 3 (metal-metal).
(m g 1) are shown in Figure 3. The first five terms in eq 34 are the same as for the protonation only (cf. eq 30); the corresponding protonation constants (m ) 0) are already summarized in Figure 2. The binding isotherms in the long chain limit can be evaluated by differentiating the largest eigenvalue of the transfer matrix (cf. eqs 15, 16, and 26). This calculation is easily carried out numerically. We will always report the degree of protonation θH as a function of pH at constant pM, and the degree of complexation θM as a function of pM at constant pH. Exemplary results are shown in Figures 4 and 5. The following parameters are chosen. We take the microscopic binding constant for protons log10 K ˆ 1 ) 9 and for metal ions log10 K ˆ 2 ) 14. The
Binding of Metal Ions to Polyelectrolytes
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6671 the configuration where bidentate bound metal alternates with deprotonated sites, see in Figure 6b. The first hump (on the ˆ 2 ) 14, and the splitting is given by 222 ) right) lies at log10 K 6. When the chain protonates in the presence of a metal, another ordered state may become stable, where bound metal ions alternate with protonated sites (see in Figure 6c). This state leads to the same intermediate isotherm plateaus for metal ions and protons, namely, θM ) θH ) 1/3. This state only develops if the proton-metal interactions are sufficiently weak; for stronger interactions, this characteristic feature of the competitive isotherm disappears (see Figure 5). It is worth noting that in the absence of interactions the dominant eigenvalue can be evaluated explicitly and reads
λ ) (1 + z1)/2 + xz2 + (1 + z1)2/4
(35)
With eqs 15, 16 and 26, analytical expressions for both isotherms follow. For the protons, one obtains
θH )
z1
(36)
x4z2 + (1 + z1)2
whereas for the bound metal ions (bidentate complexes)
θM ) Figure 5. Binding isotherms for long chains involving protonation and bidentate metal complexes. The microscopic binding constant for protons is log10 K ˆ 1 ) 9, and the metal binding constant is log10 K ˆ2 ) 14. The pair interaction parameters are 11 ) 2 (proton-proton), 12 ) 2 (proton-metal), and 22 ) 3 (metal-metal).
pair interaction parameters are assumed to be 11 ) 2 (protonproton) and 22 ) 3 (metal-metal). In Figure 4, we choose 12 ) 1, and in Figure 5, 12 ) 2. In the latter case, the protonmetal repulsion is stronger. Let us first discuss Figure 4, as this set of parameters illustrates the characteristic regimes clearly. In the absence of metal ions, we recover the proton titration curve (see Figure 4a). With increasing pH, the protonation increases in a first hump, reaches an intermediate plateau at θH ) 1/2, increases again in a second hump, and finally levels off at θH ) 1 because all sites protonate. The intermediate plateau is related to the stability of alternating protonated and deprotonated sites, as depicted in Figure 6a. The first hump (on the right) is located at log10 K ˆ 1 ) 9, whereas the splitting between both humps is 211 ) 4, because two pair interaction have to be overcome to protonate the intermediate state. In the absence of protons, the metal binding isotherm shows two humps as well. However, the intermediate plateau lies at θM ) 1/3, and the isotherm levels off at θM ) 1/2 because of the bidentate nature of the complexes (see Figure 4b). The intermediate plateau at θM ) 1/3 originates from the stability of
(37)
λx4z2 + (1 + z1)2
In the absence of metal (z2 ) 0), the Langmuir isotherm is recovered (cf. eq 33), whereas in the absence of protons (z1 ) 0), the known isotherm for adsorption of dimers on a linear chain follows30,31
θM )
Figure 6. Schematic representations of stable configurations for long chains involving protons and bidentate metal binding. (a) Protons only, (b) metal ions only, and a mixed state involving protons and metal ions.
z2
2z2 1 + 4z2 + x1 + 4z2
or z2 )
θM (1 - 2θM)2
(1 - θM) (38)
Although the shape of these isotherms is sigmoidal, their form is different from what one would expect on the basis of simple mass action law consideration. The latter leads to the so-called Flory approximation.29 For the isotherm given in eq 38, the corresponding approximate expression is obtained by replacing the term (1 - θM) in the second part of eq 38 by unity. Competitive Metal Binding: Toward a Realistic Description with Multidentate Complexes We now discuss a generalization of this model, which we believe to represent a realistic model for the binding of metal ions to linear polyelectrolytes. Studies of metal binding with linear poly(ethylene imine) have shown that complexes with various coordination geometries are being formed.16 We shall now demonstrate that such situations can be treated with a straightforward extension of the present formalism. However, we shall only illustrate a few generic features of the model here; a detailed comparison with experimental data is beyond the scope of the paper and will be discussed elsewhere. Different complexes can be considered as follows. The model is again defined by its transfer matrix, which is shown in Table 2 including up to tetradentate complexes. We have parametrized the normalized activities as zν, where the index ν ) 1, 2, 3, 4, ... is the coordination number of the complex (i.e., monodentate, bidentate, tridentate, tetradentate, etc.). For the protons, ˆ 1, whereas for the metal ions, zν ) aMK ˆ ν for we have z1 ) aHK ν ) 2, 3, 4, .... The interactions are specified with uνµ ) exp-
6672 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Koper and Borkovec
TABLE 2: Transfer Matrix for a Linear Chain for Competitive Binding of Protons and of Bidentate, Tridentate, and Tetradentate Metal Complexes 0 1 2-1 2-2 3-1 3-2 3-3 4-1 4-2 4-3 4-4
0
1
2-1 2-2 3-1 3-2 3-3 4-1 4-2 4-3 4-4
1 z1 z2 0 z3 0 0 z4 0 0 0
1 z1u11 z2u12 0 z3u13 0 0 z4u14 0 0 0
0 0 0 1 0 0 0 0 0 0 0
1 z1u12 z2u22 0 z3u23 0 0 z4u24 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
1 z1u13 z2u23 0 z3u33 0 0 z4u34 0 0 0
0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 1
1 z1u14 z2u24 0 z3u34 0 0 z4u44 0 0 0
(-βEνµ), where Eνµ are the corresponding pair interaction energies. Again we use pair interaction parameters νµ ) βEνµ/ ln 10 as above. The structure of the problem is the same as before, just the number of states has increased. In the case of a tridentate complex, a site can be in three different states, namely, a site can have a left-bound metal (3-1), a middle-bound metal (32), and a right-bound metal (3-3). Naturally, a site with a leftbound metal must be followed by a site with a middle-bound metal and then by a site with a right-bound metal. For a tetradentate complex, a site can be in four states, namely, with an outer left-bound metal (4-1), an inner left-bound metal (42), an inner right-bound metal (4-3), and an outer right-bound metal (4-4). Again, these states can only occur in succession. These simple rules define the zero entries in the corresponding submatrixes. With the auxiliary vectors V ˜ g ) (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) and V ˜ t ) (1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1), all binding properties of interest can be evaluated. An inspection of Table 2 shows that the transfer matrix is easily generalized to complexes with arbitrary coordination numbers. From the practical point of view, complexes up to hexadentate must be considered, which leads to a 22 × 22 matrix. This size is still easily handled numerically as well as symbolically by the same techniques as before. For example, one can again evaluate the partition function of the four-site linear oligomer
Figure 7. Microstates of a linear chain with four sites (N ) 4) for (monodentate) proton binding and all possible complexes with any coordination number with the macroscopic binding constant K h nm. Only the species involving metal ions are shown (m g 1), the species without the metal (m ) 0) are the same as in Figure 2.
Ξ ) 1 + 4z1 + 3(1 + u11)z21 + 2u11(1 + u11)z3 + u311 z41 + 3z2 + 2(1 + 2u12)z1z2 + u12(2u11 + u12)z21 z2 + u22z22 + 2z3 + z4 + 2u13z1z3 (39) Comparing with eq 34, one finds that this partition function differs from the bidentate case in the last three terms proportional to z3 and z4. These terms correspond to the tridentate and tetradentate complexes, as shown in Figure 7. They also lead to modifications of the corresponding binding constants as can be seen by comparing with Figure 3. Obviously, no complexes with a higher coordination number can be formed with a tetradentate ligand. In the long chain limit, however, all types of complexes can coexist. Although our implementation can handle any combination of complexes, we show some typical results for the simultaneous presence of tridentate, tetradentate, and hexadentate complexes (see Figures 8 and 9). We choose these complexes for illustration, because they were suggested to dominate the binding of nickel (II) to poly(ethylene imine).16 The proton binding is described with the microscopic binding ˆ 1 ) 9 and with pair interactions characterized constant log10 K by 11 ) 2. At this point, we have little information about the corresponding parameters describing the metal binding. The
Figure 8. Binding isotherms for long chains involving protonation and bidentate metal complexes. All interactions involving metals are set to zero. The microscopic binding constant for protons is log10 K ˆ1 ) 9, and the metal binding constants are log10 K ˆ 3 ) 13, log10 K ˆ 4 ) 16, and log10 K ˆ 3 ) 18 for tridentate, tetradentate, and hexadentate complexes, respectively. The only nonzero pair interaction parameter is 11 ) 2 (proton-proton).
ˆ 3 ) 13, log10 binding constants are arbitrarily chosen as log10 K K ˆ 4 ) 16, and log10 K ˆ 3 ) 18. These values are intended for illustration only and will require modification in order to describe real systems. The results shown in Figure 8 assume no interactions involving bound metals, whereas in Figure 9, the proton-metal interactions were taken into account. First, we discuss the metal binding isotherm without protons in the absence of metal-metal interactions (see Figure 8b). The isotherm is extremely wide and dominated by successive formation of metal complexes. It shows three shallow humps,
Binding of Metal Ions to Polyelectrolytes
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6673
Figure 9. Binding isotherms for long chains involving protonation and bidentate metal complexes with interactions. The microscopic binding constant for protons is log10 K ˆ 1 ) 9, and the metal binding constants are log10 K ˆ 3 ) 13, log10 K ˆ 4 ) 16, and log10 K ˆ 3 ) 18 for tridentate, tetradentate, and hexadentate complexes, respectively. The pair interaction parameters Vµ are 2 for proton-proton and protonmetal interactions and 3 for all metal-metal interactions.
which indicate the formation of hexadentate, tetradentate, and tridentate complexes (from right to left). ˆ 6 ) 18 and corresponds to The first hump occurs at log10 K the formation of hexadentate complexes, as evidenced by the intermediate plateau at θM ) 1/6. The second hump occurs because the hexadentate complexes are displaced by tetradentate complexes. When a complex with coordination number µ is displaced by a complex with coordination number ν (ν < µ), one finds from simple mass action law considerations that both complexes occupy approximately a comparable fraction of sites at
pM )
1 ˆ ν - log10 K ˆ µ) (µ log10 K µ-ν
(40)
For our example, this equation predicts that hexadentate complexes are displaced by tetradentate complexes at pM ) 12, whereas tetradentate complexes are displaced by tridenate complexes at pM ) 4. Because the model does not involve any bidentate complexes, tridentate complexes dominate at high metal loading and the isotherm levels off at θM ) 1/3. With increasing proton concentration, the protons increasingly compete for binding sites with the metal ions, and thus, the metal ions are being forced into lower coordination numbers at low pH. Figure 9 shows the same situation in the presence of interactions. For illustration, we choose all proton-metal interactions to be the same 1µ ) 2 with µ ) 3, 4, 6, whereas all metal-metal interactions are taken to be the same with νµ ) 3 with ν, µ ) 3, 4, 6. In reality, these interactions will depend on the particular coordination of the metal complex. Comparison with Figure 9 shows that their main effect is to lower the amount of metal bound because empty sites are being favored. By the same token, the degree of protonation of the polymer decreases. To illustrate the competition between various complexes of various coordination numbers, we have considered the site
Figure 10. Degree of coordination for the different metal complexes as a function of pM for the same parameters as given in Figure 9: (a) pH 6 and (b) pH 10.
occupancy with a complex of given coordination number or, briefly, the degree of coordination. These quantities are given by
θµ )
zµ ∂ ln Ξ N ∂zµ
(41)
and represent the fraction of the sites involved binding a complex of a the coordination number µ. Figure 10 shows these quantities for the tridentate, tetradentate, and hexadentate complexes at two different pH values. The same parameter set is being used as that in Figure 9. One observes that at low metal concentration and high pH complexes with a high coordination number are formed, whereas smaller coordination numbers are being favored at lower metal loading and/or lower pH. A final point worth mentioning is that even in the absence of interactions the metal binding isotherm invoking complexes of various coordination numbers is nontrivial. When a single complex of coordination number µ binds to a linear arrangement of sites, in the absence of interactions, the exact adsorption isotherm is known analytically.31 In our notation it reads
zµ )
θM (1 - µθM)µ
[1 - (µ - 1)θM]µ-1
(42)
Note that this equation reduces to the Langmuir isotherm (µ ) 1, cf. eq 33) and to the case of bidentate binding (µ ) 2, cf. eq 38). The Flory approximation29 is recovered by deleting the term in square brackets. The exact isotherms for mixtures of various noninteracting complexes are not generally known explicitly. Conclusion and Outlook In this paper, we have studied theoretically the competitive binding of metal ions and protons with polyelectrolytes and the corresponding oligomers. The process is modeled in a statistical mechanics framework with a generalized Potts model. The different states are specified in terms of appropriate state variables. The interactions are assumed to be short ranged and parametrized in terms of nearest neighbor pair interactions. We
6674 J. Phys. Chem. B, Vol. 105, No. 28, 2001 focus on linear chains, which are treated with transfer matrix techniques in a straightforward fashion. This approach opens a new way to treat competitive interactions between metal ions and polyelectrolytes. An extension, which might be needed to model metal ion binding data quantitatively, calls for consideration of different coordination geometries for a given coordination number. For example, in a tetradentate coordination geometry, the ligands may or may not lie within a plane, thus yielding two different tetradentate complexes. If necessary, the present formalism can be easily extended to incorporate these effects by extending the number of states per site. By introducing additional states, one can similarly include interactions of longer range or treat mixed systems containing various types of metal ions. The challenge is to derive a consistent parameter set capable to quantify the binding of protons and various metal ions to linear polyelectrolytes and their oligomeric analogs. The present transfer matrix procedure can be also generalized to branched (tree-like) structures. We have shown the applicability of this approach to model proton binding to various branched polyamines, including dendrimers.21,22 The same generalization should be feasible for metal binding. This extension will enable us to treat various branched polyelectrolytes, which are known to complex metal ions more strongly than their linear counterparts. Back folding of the polyelectrolyte chain is not considered. It is conceivable that various parts of the polymer chain may participate in a multidentate complexation of a metal ion. This effect is analogous to the excluded volume problem in polymers, as it involves long-range interactions between the polymer segments. It is not clear, however, whether it will be important in the present situation. It can be only relevant at low metal concentrations, and even in this case, it may turn out that the complexes form predominantly locally along the chain. Another restriction of this approach is its validity only at sufficiently low ligand concentrations. The consideration of ligand-ligand interactions seems difficult at this point. Our treatment focuses on the case of a single ligand, and any complexes involving more than one ligand are neglected. Although such complexes are well-know for low molecular weight ligands, their existence is rather uncertain for polyelectrolytes. For that reason, the present description will be useful to model interactions between metal ions and polyelectrolytes. Acknowledgment. This paper is dedicated to Bruce J. Berne on the occasion of his 60th birthday. We thank Jo¨rg Kleimann for discussions and comments on the manuscript. References and Notes (1) Cantor, R. C.; Schimmel, P. R. Biophysical Chemistry, Part III: The BehaVior of Biological Macromolecules; Freeman: New York, 1980.
Koper and Borkovec (2) Svensson, B.; Jo¨nsson, Bo; Thulin, E. Biochemistry 1993, 32, 2828-2834. (3) Kesvatera, T.; Jo¨nsson, Bo; Thulin, E.; Linse, S. Biochemistry 1994, 33, 14170-14176. (4) Kim, J.; Tripathy, S. K.; Kumar, J.; Chittibabu, K. G. Mater. Sci. Eng. C 1999, 7, 11-18. (5) Yoshida, N.; Oshio, H.; Ito. T. J. Chem. Soc., Perkin Trans. 2 1999, 975-983. (6) Kurth, D. G.; Lehmann, P.; Schutte, M. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 5704-5707. (7) Smith, B. F.; Gibson, R. R.; Jarvinen, G. D.; Robison, T. W.; Schroeder, N. C.; Stalnaker, N. D. J. Radioanal. Nucl. Chem. 1998, 234, 225-229. (8) Diallo, M. S.; Balogh, L.; Shafagati, A.; Johnson, J. H.; Goddard, W. A.; Tomalia, D. A. EnViron. Sci. Technol. 1999, 33, 820-824. (9) Rampley, C. G.; Ogden, K. L. EnViron. Sci. Technol. 1998, 32, 987-993. (10) Wilkinson, G. ComprehensiVe Coordination Chemistry; Pergamon Press: Oxford, U.K., 1987. (11) Arago, J.; Bencini, A.; Bianchi, A.; Garcia-Espana, E.; Micheloni, M.; Paoletti, P.; Ramirez, J. A.; Paoli, P. Inorg. Chem. 1991, 30, 18431849. (12) Bianchi, A.; Mangani, S.; Micheloni, M.; Nanini, V.; Orioli, P.; Paoletti, P.; Seghi, B. Inorg. Chem. 1985, 24, 1182-1187. (13) Zimmer, A.; Mu¨ller, I.; Reiss, G. J.; Caneschi, A.; Gatteschi, D.; Hegetschweiler, K. Eur. J. Inorg. Chem. 1998, 2079-2086. (14) Kobayashi, S.; Hiroishi, K.; Tokunoh, M.; Saegusa, T. Macromolecules 1987, 20, 1496-1500. (15) Rivas, B. L.; Moreno-Villoslada, I. J. Phys. Chem. B 1998, 102, 11024-11028. (16) Spuck, J. Ph.D. Thesis, University of Fribourg, Switzerland, 1999. (17) Kirkwood, J. G.; Shumaker, J. B. Proc. Natl. Acad. Sci. 1952, 38, 855-871. (18) Steiner, R. F. J. Chem. Phys. 1954, 22, 1458-1459. (19) For a review, see: Borkovec, M.; Jo¨nsson, B.; Koper, G. J. M. In Surface and Colloid Science; Matijevic´, E., Ed.; Plenum Press: New York, 2001; Vol. 16, pp 99-339. (20) Borkovec, M.; Koper, G. J. M. Langmuir 1994, 10, 2863-2865. (21) Borkovec, M.; Koper, G. J. M. Macromolecules 1997, 30, 21512158. (22) van Duijvenbode, R. C.; Borkovec, M.; Koper, G. J. M. Polymer 1997, 39, 2657-2664. (23) Borkovec, M.; Daicic, J.; Koper, G. J. M. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 3499-3503. (24) Borkovec, M.; Koper, G. J. M. J. Phys. Chem. 1994, 98, 60386045. (25) Wyman, J. J. Mol. Biol. 1965, 11, 631-638. (26) Baxter, R. J. Exactly SolVed Models in Statistical Mechanics; Academic Press: New York, 1982. (27) Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Press: New York, 1987. (28) Flory, P. Statistical Mechanics of Chain Molecules; Interscience Publishers: New York, 1969. (29) Flory, P. J. Chem. Phys. 1942, 10, 51. (30) Maltz, A.; Mola, E. E. J. Math. Phys. 1981, 22, 1746. (31) Ramirez-Pastor, A. J.; Eggarter, T. P.; Pereyra, V. D.; Riccardo, J. L. Phys. ReV. B 1999, 59, 11027-11036.
