On the Binding of Calcium by Micelles Composed ... - ACS Publications

Nov 2, 2006 - Department of Chemistry and Biochemistry, UniVersity of the AlgarVe,. Gambelas Campus, 8005-139 Faro, Portugal. J. P. A. Custers, L. J. ...
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Langmuir 2006, 22, 10932-10941

Articles On the Binding of Calcium by Micelles Composed of Carboxy-Modified Pluronics Measured by Means of Differential Potentiometric Titration and Modeled with a Self-Consistent-Field Theory Y. Lauw,* F. A. M. Leermakers, and M. A. Cohen Stuart Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, Wageningen 6700 EK, The Netherlands

J. P. Pinheiro Department of Chemistry and Biochemistry, UniVersity of the AlgarVe, Gambelas Campus, 8005-139 Faro, Portugal

J. P. A. Custers, L. J. P. van den Broeke, and J. T. F. Keurentjes Process DeVelopment Group, EindhoVen UniVersity of Technology, P.O. Box 513, EindhoVen 5600 MB, The Netherlands ReceiVed May 9, 2006. In Final Form: August 22, 2006

We perform differential potentiometric titration measurements for the binding of Ca2+ ions to micelles composed of the carboxylic acid end-standing Pluronic P85 block copolymer (i.e., CAE-85 (COOH-(EO)26-(PO)39-(EO)26COOH)). Two different ion-selective electrodes (ISEs) are used to detect the free calcium concentration; the first ISE is an indicator electrode, and the second is a reference electrode. The titration is done by adding the block copolymers to a known solution of Ca2+ at neutral pH and high enough temperature (above the critical micellization temperature CMT) and various amount of added monovalent salt. By measuring the difference in the electromotive force between the two ISEs, the amount of Ca2+ that is bound by the micelles is calculated. This is then used to determine the binding constant of Ca2+ with the micelles, which is a missing parameter needed to perform molecular realistic self-consistentfield (SCF) calculations. It turns out that the micelles from block copolymer CAE-85 bind Ca2+ ions both electrostatically and specifically. The specific binding between Ca2+ and carboxylic groups in the corona of the micelles is modeled through the reaction equilibrium -COOCa+ h -COO- + Ca2+ with pKCa ) 1.7 ( 0.06.

I. Introduction Studies of triblock copolymer poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide), also known as Pluronic block copolymer, have intensified during the last two decades.1-3 The micellization of this block copolymer is adjustable with respect to both the Pluronic concentration and the temperature. This behavior makes it favorable in various applications, such as in drug delivery processes and as detergents. The introduction of charges, by means of carboxylic groups, into the Pluronic block copolymer enhances the complexity of the system yet widens its applicability. In this case, the block copolymer selfassembly also becomes dependent on the ionic strength I and the pH. The study of modified Pluronics has a relatively short history; for example, see Huang et al.4 and Li et al.5 (1) Alexandridis, P.; Hatton, T. A. Colloids Surf., A 1995, 96, 1-46. (2) Goldmints, I.; Yu, G. E.; Booth, C.; Smith, K. A.; Hatton, T. A. Langmuir 1999, 15, 1651-1656. (3) Su, Y. L.; Wang, J.; Liu, H. Z. Langmuir 2002, 18, 5370-5374. (4) Huang, K.; Lee, B. P.; Ingram, D. R.; Messersmith, P. B. Biomacromolecules 2002, 3, 397-406. (5) Li, J. T.; Carlsson, J.; Lin, J. N.; Caldwell, K. D. Bioconjugate Chem. 1996, 7, 592-599.

The micellization of Pluronic P85 (i.e., (EO)26(PO)39(EO)26,where EO is ethylene oxide and PO is propylene oxide) was studied by Mortensen et al.6 A complete temperature-polymer concentration phase diagram was obtained from small-angle neutron-scattering measurements. In the present article, the main focus is on the binding of metal ions to the micelles composed of modified Pluronic P85 block copolymers. The modified block copolymer is based on P85 with additional carboxylic groups COOH, denoted by X, at both free ends of the PEO groups. It is called CAE-85 (carboxylic acid end-standing P85) with configuration X(EO)26(PO)39(EO)26X. Recently, by means of calorimetric measurements Custers et al.7 has proven the thermoreversibility of both the micellization and the ability of the block copolymer micelle to bind bivalent metal ions. It was shown for temperatures lower than the critical micellization temperature (CMT) that there is no binding of metal ions to the CAE-85 block copolymers, whereas at temperature higher than the CMT metal binding does occur. This is attributed to the (6) Mortensen, K.; Pedersen, J. S. Macromolecules 1993, 26, 805-812. (7) Custers, J. P. A.; Kelemen, P.; Van den Broeke, L. J. P.; Cohen Stuart, M. A.; Keurentjes, J. T. F. J. Am. Chem. Soc. 2005, 127, 1594-1595.

10.1021/la061300i CCC: $33.50 © 2006 American Chemical Society Published on Web 11/02/2006

Binding of Ca by Micelles Composed of Pluronics

micellization, which implies a cooperative effect that eventually generates a significant negative electrostatic potential Ψ in the corona of the micelles. A detailed self-consistent-field analysis has been performed for a large set of Pluronic molecules, concentrating either on the critical micellization temperature CMT or the cloud-point temperature CPT.8 From this previous analysis, many parameters for the self-assembly of CAE-85 are already available. The intrinsic value of the proton dissociation constant of the carboxylic groups is also well known. The only missing parameter is the binding constant between Ca2+ ions and the negatively charged carboxylic groups. For this reason, it is necessary to measure the Ca2+ binding accurately. The binding of calcium in various aqueous systems, such as protein or polyelectrolyte solutions, has been widely studied experimentally by applying photometric, calorimetric, or potentiometric measurement.9-11 In this article, the Ca2+ binding behavior of the micelles from block copolymers CAE-85 is studied further by using direct differential potentiometric measurements with Ca2+ ion-selective electrodes (ISEs). The function of the calcium-selective electrode in surfactant solutions such as Pluronics is very poor because the Pluronics interfere with the electrode by an adsorption (binding) process. Our solution to this problem is to cover the electrode with a membrane that prevents Pluronics from reaching the electrode but still allows for the free transport of calcium. In this article, we want to show that it is possible to quantify precisely the binding of Ca2+ with anionic micelles composed of CAE-85. It is performed by first analyzing the experimental results with Scheutjens-Fleer self-consistent-field (SF-SCF) theory. The result of this analysis is a useful estimation for the intrinsic binding constant of Ca2+ with the carboxylic groups. As a result, all necessary SCF parameters are available. It will be shown below that the electrostatic interaction is mainly responsible for the variation of the micellar Ca2+ uptake, whereas the short-range specific interaction is considered to be a secondary effect. The rest of the article is arranged as follows. The Experimental Section is followed by an explanation of SF-SCF theory and the model parameters used in the numerical calculation, closed by the Results and Discussion and Conclusions sections. II. Experiments A. Materials. Block copolymer carboxylic acid end-standing P85 (CAE-85) with architecture X(EO)26(PO)39(EO)26X, where X is the additional carboxylic group, is synthesized as described by Custers et al.7 The final product is in a paste form with an acid concentration of 5 × 10-4 mol/g and pKa 4.51 for the carboxylic group protonation. This block copolymer is then diluted with deionized water from a Barnstead water purifier and is used as the titrant in the differential potentiometric titration with an active carboxylic concentration of [-COO-] ) 1.4 × 10-3 M. The pH is kept close to 7 by adding KOH, which corresponds to the fully deprotonated state of the carboxylic groups. In all measurements except the ITC experiments, 10-4 M Ca2+ from Ca(NO3)2 (pro analysi, Merck) is used as the initial solution, and KNO3 (pro analysi, Merck) with a known concentration is used as the additional electrolyte. In total, we use four different total ionic strengths I (i.e., I ) 7 × 10-4, 1.2 × 10-3, 1.8 × 10-3, and 5.8 × 10-3 M). All of the solutions are prepared (8) De Bruijn, V. G.; Van den Broeke, L. J. P.; Leermakers F. A. M.; Keurentjes J. T. F. Langmuir 2002, 18, 10467-10474. (9) Kinosian, H. J.; Newman, J.; Lincoln, B.; Selden, L. A.; Gershman, L. C.; Estes, J. E. Biophys. J. 1998, 75, 3101-3109. (10) Hendrix, T.; Griko, Y. V.; Privalov, P. L. Biophys. Chem. 2000, 84, 27-34. (11) Sinn, C. G.; Dimova, R.; Antonietti, M. Macromolecules 2004, 37, 34443450.

Langmuir, Vol. 22, No. 26, 2006 10933 at room temperature. EDTA is obtained from Sigma-Aldrich (cell culture tested, minimum 99.5% titration) in the form of EDTAtetrasodium salt dihydrate (C10H12N2Na4O8‚2H2O). The EDTA solutions are prepared according to the instructions from the ISE manufacturer.12 Together with PMAA (20% stock solutions, BDH Chemicals Ltd.), they were used to validate our potentiometric setup and measurements. For the isothermal titration calorimetry (ITC) experiments, a 9.96 × 10-3 M CaCl2 solution (pro analysi, Merck) is titrated into a CAE-85 solution with a carboxylate group concentration of 6.19 × 10-3 M, and a 10-2 M MgCl2 solution (pro analysi, Merck) is titrated into a CAE-85 solution with an initial [-COO-] ) 5 × 10-3 M. The pH of both CAE-85 solutions was adjusted to 6 with NaOH. B. Differential Scanning Calorimetry (DSC). Differential scanning calorimetry (Setaram DSC III) is used to determine the CMT of the CAE-85 block copolymer at different polymer concentration and total ionic strength. The temperature scan from 15 to 70 °C and back to the original temperature value is performed at a scan rate of 0.5 K/min. From the measurements, the heat capacity Cp as a function temperature T is obtained. By subtracting the baseline value, one gets the effective heat capacity Cpeff(T). The CMT is defined as the temperature at which there is a first-order-like transition in the Cpeff(T) curve.13 C. Isothermal Titration Calorimetry (ITC). The ITC experiments are performed using a MicroCal VP-ITC apparatus with a cell volume of 1.4431 mL. The experimental procedure consists of adding 70 injections of a 4 µL metal chloride salt solution to the sample cell that contains a solution of the CAE-85 block copolymer. In all of the experiments, the reference cell is filled with demineralized water. Before each experiment, the temperature is equilibrated at 50 °C, which is well above the CMT of CAE-85 solution in the sample cell. During the titration, the addition of a small, known amount of reactant to a well-stirred sample cell will cause small changes in the temperature. The resulting absorbed or evolved heat is measured accurately, and the difference in heat between the sample and the reference cell that keeps both cells at the same temperature is obtained. This is a measure of the cumulative enthalpic change in the sample cell. Generally, the observed heat effect Qobs is a summation of various enthalpic contributions that complicate the interpretation of the data, especially as one considers the titration of divalent cation d into an initial solution of block copolymer CAE-85 that contains background monovalent salt s. Suppose the carboxylic groups in the initial solution are either in the undissociated form, -COOH, or bound form, -COOs, such that the observed heat change Qobs of the titration can be interpreted as Qobs ) Qmix + Qdiss + Qhyd,s + Qhyd,PEO-X + Qdehyd,PEO-X + Qdehyd,d + Qbind,d + Qmic (1) where Qmix is the mixing heat , Qdiss is the dissociation heat of the bound acid groups -COOs, Qhyd,s is the hydration heat of the monovalent cation, Qhyd,PEO-X and Qdehyd, PEO-X are the heats of hydration and dehydration of the polymer headgroup, respectively, Qdehyd,d is the dehydration heat of the divalent cation, Qbind,d is the binding heat of the divalent cation, and Qmic is the micelle conformation heat. Quantities Qdehyd,PEO-X and Qhyd,PEO-X have opposite signs and cancel each other such that eq 1 becomes Qobs ) Qmix + Qdiss + Qhyd,s + Qdehyd,d + Qbind,d + Qmic

(2)

The hydration enthalpies are always exothermic. Hence, the dehydration enthalpies are always endothermic. In general, the valence and the ionic radius of the cation mainly determine the value (12) Thermo-Orion. Calcium Electrode Instruction Manual, Orion 93-20 and 97-20; Thermo Electron Corporation: Waltham, MA, 2003. (13) Lau, B. K.; Wang, Q.; Sun, W.; Li, L. J. Polym. Sci., Part B 2004, 42, 2014-2025.

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Lauw et al.

Table 1. Unhydrated and Hydrated Ionic Radii (Ri and Rh)14,15 and Enthalpies of Hydration16 of Ca2+ and Mg2+ at Room Temperature cation

Ri (Å)

Rh (Å)

∆Hhyd (kJ/mol)

Ca2+

1.00 0.72

2.42 2.09

-1602 -1949

Mg2+

of the hydration enthalpy. The higher the valence and the smaller the ionic radius, the more exothermic the hydration enthalpy. The radii and hydration enthalpy for Ca2+ and Mg2+ are listed in Table 1. The mixing heat Qmix is the result of mixing the divalent cationic solution with the CAE-85 solution. This quantity is usually exothermic and considerably smaller than the hydration heat, so Qmix can be neglected. The mixing heat becomes experimentally important where the interaction heat, Qint, is small. The interaction heat is defined as the observed heat without the dissociation and binding heat contributions (cf. eq 5). The binding heat Qbind,d is the heat due to the binding of divalent cation d to CAE-85 molecules in the micelle. The binding of two opposite charges is energetically favorable, which makes this term exothermic. In general, the binding enthalpy is likely to become more exothermic when the ionic radius of the cation decreases.17 During the titration, the binding heat Qbind,d and the dehydration heat Qdehyd,d depend on the concentration of the bound divalent [dbound] cations. They are defined as follows: Qbind,d ) ∆Hbind,d[dbound]

(3)

Qdehyd,d ) ∆Hdehyd,d[dbound]

(4)

In contrast to the binding heat, the dissociation heat Qdiss is endothermic because energy is required to split a bound ion pair into separate ions. As mentioned above, the total interaction heat Qint, which is often referred to as the displacement energy, can then be defined as Qint ) Qdiss + Qhyd, s + Qdehyd,d + Qbind,d

(5)

Combining eqs 2 and 5 then leads to Qobs ) Qmix + Qint + Qmic

(6)

where again Qmic is the micellization heat that includes all of the structural changes in the aggregate taking place during the titration. The changes in the micellar structure are always driven by a gain in free energy to obtain the most favorable conformation. In general, the enthalpic contribution Qmic is smaller than the interaction energy Qint. In an ideal situation, Qmic and Qmix are much smaller than Qint, and the observed heat change Qobs value approaches Qint according to eq 6. D. Dynamic Light Scattering. To obtain the critical micellization concentration (CMC) of block copolymer CAE-85 at fixed temperature T ) 40 °C, dynamic light scattering (DLS) measurements are performed at a scattering angle of 90°. An Ar laser source with λ ) 514.5 nm (1W Spectra Physics series 2000 laser) and ALV5000 digital correlator software are used for the DLS measurement. The scattering intensity IS of each CAE-85 solution is normalized with the standard scattered intensity I0 of toluene. This ratio is then depicted at different active group concentrations [-COO-] to obtain the CMC of block copolymer CAE-85 at given total ionic strength I. E. Differential Potentiometric Measurement. The setup of the differential potentiometric cell is depicted in Figure 1. The temperature of the cell is controlled during the measurement at 40 (14) Marcus, Y. Chem. ReV. 1988, 88, 1475-1498. (15) Shannon, R. D. Acta Crystallogr., Sect. A 1976, 32, 751-767. (16) Marcus, Y. Ion SolVation; John Wiley & Sons: New York, 1985. (17) Pochard, I.; Foissy, A.; Couchot, P. Colloid Polym. Sci. 1999, 277, 818826.

Figure 1. Setup for the differential potentiometric measurement. Two Ca2+-ion-selective electrodes are used as an indicator (ISE-1) and as a reference (ISE-2). Both ISEs are connected to the standard Ag/AgCl reference electrode, and a Microlink 3000 module is used for data collection. The sensing membrane of the ISE-1 is covered by a dialysis membrane to protect it from block copolymer adsorption. The ISE-2 is placed in a salt-bridge junction that contains an identical initial Ca2+ solution within the main cell. ( 0.2 °C by circulating warm water that encloses the cell. The constant temperature of the water is regulated by a temperature bath. Nitrogen is used to degas the cell. Two Ca2+-ion-selective electrodes (liquid membrane Thermo Orion 93-20 ISE) are used. The first ISE (ISE-1) is used as an indicator electrode, whereas the second ISE (ISE-2) is the reference of the ISE-1 and is placed in a salt-bridge junction (Schott-Gerate B522). As a reference electrode, the standard Ag/AgCl electrode is used (Mettler Toledo 363-S7/120 with 4M KCl filling solution). During the measurement, the pH is measured by using a combined pH electrode (SenTix Mic) that is linked to a pH meter (Knick). The potentiometric measurement with the ISE is a useful and straightforward method of detecting the activity of ions under study. However, its main disadvantage is the possible absorption of other compounds, such as other types of ions, on the sensing membrane, which eventually interferes with the signal readings. Sinn et al.11 documented such a disturbance in their measurement using a Ca2+ISE and noted that this is caused by the penetration of low-molecularweight polymers into the sensing membrane. During our measurements, we also encountered similar problems, which in this case were caused by the adsorption of block copolymer CAE-85 onto the membrane electrode. We solved this problem by covering the sensing membrane of the ISE-1 with a Spectra/Por-6 RC dialysis membrane with a 1 kDa MWCO (132 640) so that such adsorption was prevented and the signal interference disappeared (Figure 1). The differential potentiometric measurements are made continuously with subsequent additions of block copolymer to the sample solution. Preceding the titration, the cell and the salt-bridge junction are filled with an initial solution of 10-4 M Ca2+ from Ca(NO3)2 and additional salt KNO3 with a known concentration. A PC is connected to the Microlink 3000 module for data collection and control of the polymer additions. During the titration, the electromotive force (EMF) of both Ca2+-ISEs are measured and recorded. Because the solution in the glass junction does not change during the titration, the difference of the EMF of the two ISEs is proportional to the logarithm of the ratio of Ca2+ activity at the cell and glass junction as follows dEMF )

( )

2+ RT aCaISE-1 ln nF aCaISE-2 2+

(7)

where dEMF is the difference between the electromotive force of ISE-1 and ISE-2 after being normalized by each of its standard cell potentials. Throughout this article, it is assumed that the activity coefficients in both cells are always the same, thus dEMF is

Binding of Ca by Micelles Composed of Pluronics

Langmuir, Vol. 22, No. 26, 2006 10935

proportional to the logarithm of the ratio of Ca2+ concentrations between the two cells: dEMF )

(

)

2+ RT [CaISE-1] ln nF [Ca2+ ] ISE-2

(8)

All measurements are performed at constant temperature T ) 40 ( 0.2 °C, which is above the CMT when the concentration ratio of the total number of carboxylic groups (ligand) with Ca2+ is beyond 2:1 (the ligand is well in excess with respect to Ca2+). The pH electrode and ISEs are calibrated before the measurements. The calibration is performed by titrations of EDTA and poly(methacrylic acid) (PMAA, Mw ) 86.1 g/mol) to a similar Ca2+ solution according to the instruction from the manufacturer (ThermoOrion) and relevant publications.18,19

dF can be expressed in terms of the change in temperature dT, the volume of the system dV, the number of molecules in the ith component dNi, and the number of micelles dN, at a given entropy of the system S, pressure p, chemical potential of the ith component µi, and excess free energy per micelle ,22,23 c

2+ + 2+ 2K+ bound + Cafree h 2Kfree + Cabound

(9)

2 with an apparent equilibrium binding constant of K ) [K+ free] [ 2+ + 2+ 2 Cabound]/[Kbound] [Cafree]. This can be rewritten, giving the binding isotherm as

θ ) K0[Ca2+ free] 1-θ

(10)

+ 2 where θ/(1 - θ) ) [Ca2+ bound]/[Kbound] is the relative number of 2 bound Ca2+ and K+ ions and K0 ) K/[K+ free] is the effective ] binding constant. Both θ/(1 - θ) and [Ca2+ free are measurable quantities and thus can be used to obtain K0. All of these parameters can also be extracted easily from the SF-SCF calculation such that the experiments can be mimicked. Comparison between experimental and theoretical K0 values are used to optimize the parameters for the SCF model.

IV. Self-Consistent-Field Theory of Weakly Charged Amphiphiles’ Self-Assembly The thermodynamic study of self-assembly is based on the classical thermodynamic work of Gibbs.21 In particular, the overall system is seen as an ensemble of many small systems in equilibrium with identical (macroscopic) thermodynamic characteristic as described by Hill.22 Hall and Pethica then applied this method to micelle systems.23 In the canonical ensemble (N, V, T), thermodynamic properties are identical in each small system. For a macroscopic system consisting of c components, the change in the Helmholtz energy (18) Tackett, S. L. Anal. Chem. 1969, 41, 12, 1703-1705. (19) Christensen, T.; Gooden, D. M.; Kung, J. E.; Toone, E. J. J. Am. Chem. Soc. 2003, 125, 7357-7366. (20) Gerstner, J. A.; Bell, J. A.; Cramer, S. M. Biophys. Chem. 1994, 52, 97-106. (21) Gibbs, J. W. The scientific papers of J. Willard Gibbs; Ox Bow Press: Woodbridge, CT, 1993; Vol. 1. (22) Hill, T. L. Thermodynamics of Small Systems; Dover Publications: New York, 1994; Parts 1 and 2. (23) Hall, D. G.; Pethica, B. A. In Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1967; Chapter 16.

(11)

The thermodynamic equilibrium and micellar stability conditions lead to the following constraints:

|

∂F ))0 ∂N T,V,{Ni} ∂ 2F ∂N2

III. Ion-Exchange Model of Metal Binding To obtain parameters that describe the binding behavior of metal ions to the acid groups of the micelles, a simple ionexchange model is implemented first to treat the experimental results.20 In short, the binding of Ca2+ ions to the micelles composed of CAE-85 block copolymers is described as competitive binding between Ca2+ and K+ ions. The latter is the additional electrolyte present in the solution. Thus, monovalent ions bound to the acid groups of the micelles are exchanged with free calcium ions according to the reaction equilibrium

∑i µidNi + dN

dF ) -S dT - p dV +

|

)

T,V,{Ni}

|

∂ ∂N

T,V,{Ni}

(12) >0

(13)

In an isothermic system, the primitive of eq 11 can be written as the following: c

F + pV -

∑i µiNi ) N

(14)

In the SF-SCF model, we consider only a single average micelle in one small system in the canonical ensemble. The center of this micelle is positioned at the center of the coordinate system that is used. Thus, by defining a new variable, the so-called translationally restricted grand potential Ω, the subdivision potential  can be expressed as

Ω - TStrans ) 

(15)

where Strans is a translational entropic penalty for placing the micelle in the center of the coordinate system for which Strans = -kB ln φm where kB is the Boltzmann constant and φm is the volume fraction of micelles in the system.24 At equilibrium, eq 15 leads to

Ω ) -kBT ln φm

(16)

φm ) e-Ω/kBT

(17)

or equivalently

where the translationally restricted grand potential value Ω is obtainable from the SF-SCF calculation. SF-SCF theory is applicable to the study of the self-assembly of copolymers in selective solvents. Here, we give a very short overview and refer to the literature for more details.25-30 The numerical SF-SCF scheme is a combination of a first-order Markov approximation for the chain statistics with a local meanfield approach or the Bragg-Williams approximation for the local energetic interactions. It is implemented in a lattice with spherical geometry while enforcing an incompressibility con(24) Leermakers, F. A. M.; Eriksson, J. C.; Lyklema, J. In Fundamentals of Interface and Colloid Science; Lyklema, J., Ed.; Elsevier: Amsterdam, 2005; Volume 5, Chapter 4. (25) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 12, 16191635. (26) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178-190. (27) Hurter, P. N.; Scheutjens, J. M. H. M.; Hatton, T. A. Macromolecules 1993, 26, 5592-5601. (28) Bohmer, M. R.; Evers, O. A.; Scheutjens, J. M. H. M. Macromolecules 1990, 23, 2288-2301. (29) Bohmer, M. R.; Koopal, L. K. Langmuir 1992, 8, 1594-1602. (30) Lauw, Y.; Leermakers, F. A. M.; Cohen Stuart, M. A. J. Phys. Chem. B 2006, 110, 465-477.

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Lauw et al.

straint. The method employs Flory-Huggins interaction parameters χ for unlike contacts. To determine the spatial distribution of the molecules in the system, the molecules are considered to be composed of various segments. For example, in our case, the block copolymers consist of a linear sequence of segments. In spherical geometry, space is discretized into a number of lattice layers r ) 1,..., M where the number of lattice sites in layer r, L(r), grows quadratically with r (i.e., L(r) ≈ r2). Each lattice site can be occupied by only one segment, an ion or a solvent molecule. Homogeneity of the density of molecular components within a given layer is assumed, and a density gradient is allowed in the normal direction. As mentioned above, by applying spherical geometry, the center of a micelle, which is formed by the self-assembled block copolymers, coincides with the origin of the coordinate system. In SF-SCF theory, the probability of finding a particular segment type in a certain lattice layer r depends in a complex way (sum over all possible and allowed conformations) on the set of potential fields {ux(r)} where x indicates each segment type. Every potential field itself depends on the set of distribution {φx(r)} of all segments in the lattices. Here, electrostatic and short-range interactions contribute to the segment-type-dependent potentials ux(r). The electrostatic contribution is obtained by solving the discrete form of the Poisson equation, whereas the short-range interactions between neighboring segments are evaluated by an inhomogeneous variant of the Flory-Huggins theory. A numerical procedure is used that varies the potentials and evaluates the resulting segment distributions until a so-called self-consistent solution is found. It fulfills all governing equations and reflects the thermodynamic equilibrium condition of all present entities in the small system. With these self-consistent segment distributions and potentials at each layer r, thermodynamic properties such as the chemical potential µ and the translationally restricted grand potential Ω can be obtained. The SF-SCF model requires us to work in terms of discretized (dimensionless) variable quantities at a particular layer r. As an example, the dimensionless volume fraction φ(r) is used instead of the (molar) concentration. To obtain this experimentally accessible variable from the dimensionless quantity, a conversion unit is needed (further details discussed in the next section). Throughout this article, except for the volume fraction notation, we use the tilde symbol for the dimensionless variables to distinguish them from the standard experimental variables. Our main interest is to discuss how the self-consistent-field scheme accounts for the weakly charged segments (i.e., with the carboxylic units). In principle, we follow the multiple-state approximation initiated by Bjorling et al.31 Here, we consider the carboxylic groups X in the block copolymers CAE-85 to exist in three possible states: X1, X2, and X3. Segment X1 represents the uncharged state -COOH, X2 stands for the negatively charged state -COO-, and X3 represents the positively charged state -COOCa+. This means that we consider the competition between H+ and Ca2+ to bind to -COO-. More specifically, the two first states are in equilibrium through the reaction

carboxylic groups is formed by the specific binding of Ca2+ ions with the negative state -COO-, and it is modeled through the reaction equilibrium

-COOCa+ h -COO- + Ca2+ K ˜ Ca )

(φ-COO-)(φCa2+) φ-COOCa+

(19)

In addition, the water equilibrium is given by

2H2O h H3O+ + OH(φH3O+)(φOH-)

K ˜w )

(φH2O)2

(20)

Here, water has also three states, H2O, H3O+, and OH- denoted by W1, W2, and W3, respectively. The K ˜ w is the corresponding dissociation constant of the water molecule. There is a constraint on segment X that can exist in one of the three different states (i.e., the total fraction of all states is unity):

R-COO-(r) + R-COOH(r) + R-COOCa+(r) ) 1, ∀r (21) Rb-COO- + Rb-COOH + Rb-COOCa+ ) 1, in the bulk (22) All R(r) differ from Rb in the region where the electrostatic potential Ψ(r) deviates from zero. In the bulk solution, by using eq 22, eq 18 can be rewritten as

Rb-COO˜ 2O ) pH ˜ - log b pK ˜ a + pH R-COOH

(23)

where pK ˜ a ) -log K ˜ a, pH ˜ 2O ) -log φH2Ob, and pH ˜ ) -log b φH3O+ . The superscript b denotes the corresponding quantity in the bulk solution. Analogously, eq 19 can be rewritten as

˜ a - log pK ˜ Ca ) pC

Rb-COORb-COOCa+

(24)

where pK ˜ Ca ) -log K ˜ Ca and pC ˜ a ) -log φCa2+b. By using eqs 22-24, given the values of pK ˜ Ca, pK ˜ a, and pC ˜a at a certain proton concentration in the bulk as input parameters, all values of the fraction of states RbX can be computed. These values are needed to compute the overall statistical weight of state-dependent segments X in the SF-SCF calculation at arbitrary layer r. All further details of the self-consistent-field scheme for weakly charged system that is used in the calculation can be found elsewhere.32 In the next section, we specify and explain the use of the remaining parameters in the SF-SCF theory.

V. Model Parameters -

+

-COOH + H2O h -COO + H3O K ˜a )

(φ-COO-)(φH3O+) (φ-COOH)(φH2O)

(18)

where K ˜ a is the dissociation constant of segment-type X and φi denotes the volume fraction of segment i. The third state of the (31) Bjorling, M.; Linse, P.; Karlstrom, G. J. Phys. Chem. 1990, 94, 471-481.

There are four sets of input parameters that have to be predefined before starting the SCF calculation. The first set determines the total number of molecules in the system, the architecture of the chains, and the possible internal states of each segment. The second set defines the Flory-Huggins interaction parameters between all segments. The third set consists of the dissociation constants and electrostatic properties of each segment (32) Israels, R.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1994, 27, 3087-3093.

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Table 2. List of the Flory-Huggins Interaction Parameters Used in the Model at 40 °C χ

X

C

O

Cbr

W

s

Ca2+

X C O Cbr W s Ca2+

0 1.6 -0.7 2.3 0 0 0

1.6 0 1 0.6 1.6 1.6 1.6

-0.7 1 0 1.6 -0.7 -0.7 -0.7

2.3 0.6 1.6 0 2.3 2.3 2.3

0 1.6 -0.7 2.3 0 0 0

0 1.6 -0.7 2.3 0 0 0

0 1.6 -0.7 2.3 0 0 0

in the internal states level, and the fourth set specifies the lattice coordinate system. Most of the parameters are taken to be identical to those used in the previous study.30 The modified Pluronic CAE-85 used in the model has the segment sequence X(C2O)26(C2CbrO)39(C2O)26X. The carboxylic group is indicated as X. As previously mentioned, X exists as a neutral segment or a negatively or positively charged species. C refers to the CH2 group, O represents the oxygen group, and Cbr refers to a CH3 group (which in PO is a side group but in this model is located in the linear chain). Water is modeled as a monomer, and its weak dissociation depends on the pH. In addition to the calcium ions, a 1:1 electrolyte is used. It is denoted by s. It is important to note that the total ionic strength I is defined as (1/2)Σi cizi2 where ci is the total concentration of electrolyte i and zi is its valence. As mentioned in the previous section, in our model the concentrations are expressed in a dimensionless unit (volume fraction). To convert the volume fraction of monomeric species to a molar value, a conversion factor of 55.4 M is used following the concentration value given from the monomeric solvent. For example, φ ) 10-5 corresponds to the concentration 5.54 × 10-4 M. In the calculation, the total electrolyte concentration is computed from the volume fraction of the electrolyte both in the micellar region and in the bulk (i.e., φti ) θσi /Vss + φbi where θσi is the excess amount of electrolyte i in the micellar region, Vss is the volume of the small system, and φbi is the volume fraction of electrolyte i in the bulk solution). The volume of the small system Vss is expressed as σ Vss ) Vm/φm ) θCAE-85 /e-Ω/kBT where Vm is the volume of the σ micelle, i.e., θCAE-85, the amount of block copolymer CAE-85 in the micellar region (in number of occupied lattice sites); φm is the volume fraction of micelles in the small system for which φm ) e-Ω/kBT, where Ω is the grand potential per micelle (cf. eq 17). This expression is valid only in a very dilute regime where the steric and electrostatic interactions between micelles are negligible. This is ensured by choosing the volume V of the lattice model such that V . Vss. In an earlier publication by Custers et al.,7 the amount of calcium uptake is presented as fractional counterion binding θ with respect to the concentration ratio of Ca2+ and carboxylic groups. Here, we use a different symbol Φ for fractional Ca2+ binding (i.e., the fraction of carboxylic groups that are bound electrostatically and specifically by Ca2+). From the SCF calculation, the fractional binding is calculated by Φ ) θσCa2+ + θσ-COOCa+/θσX*, the ratio between the amount of bound Ca2+ and charged carboxylic group X* in the micellar region. The Flory-Huggins parameters used in the model are given in Table 2. Here, s indicates salt ions from the 1:1 electrolyte. The hydrophobicity of the PPO block is reflected in the relatively high value of χCbrW compared with χCW. The repulsion between CH2 or CH3 groups and water drives the self-assembly process. In this set of interaction parameters, the interaction between oxygen and solvent segments is attractive, thus χOW < 0. This parameter is used to stop the aggregation of Pluronics such that the micelles have a well-defined aggregation number.

Pluronics are temperature-sensitive polymeric surfactants. It is known that around room temperature the solubility of, for example, alkanes in water is not a strong function of the temperature.33 Therefore, we fix χCW and χCbrW to be constant, irrespective to the temperature. The thermosensitivity of the micellization of the Pluronic P85, and consequently the CAE-85 block copolymers, is thus governed only by the parameter χOW. As the temperature increases, the hydrophobicity of EO and PO groups increases, and water molecules are expelled from the micelles (increasing micellar dehydration). In our model, this is mimicked by tuning the χOW parameter, as already done in a previous publication.8 In general, a Flory-Huggins χij parameter can be divided into enthalpic and entropic terms (i.e., χij ) Aij -Bij/T where Aij and Bij are constants). For the case of χOW, the constant BOW is a definite positive. For significant values of BOW, χOW is a decreasing function of T. Below, we will discuss the quantitative effect of changing the temperature by varying χOW. The set of χ parameters that is used in the model (Table 2) follows the parameter set used by De Bruijn et al. in the modeling of Pluronic surfactants self-assembly.8 Because of the lack of experimental data to be fitted, the χ parameters of De Bruijn are not necessary the optimum ones. Therefore, they are used only as sensible indicators. We follow the χ parameters from the dimeric water model of De Bruijn with an exception in the interaction parameter between hydrocarbon and water molecules. This is because the water molecule dissociation and the charge regulation of the carboxylic groups require a monomeric water component. For simplicity, the χ parameters for the monovalent salt s, calcium ions Ca2+, and carboxylic groups X follow the values of the water molecule. We consider that the chosen χ parameters (cf. Table 2) are reasonable albeit not necessarily the most optimal ones. The carboxylic group X dissociates with an intrinsic dissociation constant of pK ˜ a ) 6.25 (i.e., pKa ) 4.51.)7 It is assumed that this pKa value does not depend on whether the block copolymers exist as micelles or as unimers. The carboxylic group exists in three possible states (i.e., as a negative, neutral, or positive charge group with corresponding valence values υCOO- ) - 1, υCOOH ) 0, and υCOOCa+ ) 1, respectively). The relative permittivity r of this group is set equal to 10. Each salt ion of the 1:1 electrolyte is modeled as a monomer with valences υs+ ) 1 and υs- ) -1. The relative permittivities for both ions are set equal to 10. Ca2+ is modeled as a monomer with valence 2. Water is modeled as a monomer with pK ˜ w ) 17.5 (thus pKw ) 14). It exists in three possible states with corresponding valences υOH) - 1, υH2O ) 0, and υH3O+ ) 1. The relative permittivity of water is equal to 80. It is assumed that the electrostatic properties can be kept constant while changing the temperature. More specifically, the product T is fixed, implying a fixed Debye length. Note that in the model the dimensionless pH ˜ is used. It is defined as pH ˜ ) -log φHb 3O+ ) - log(RHb 3O+ φbW) where RHb 3O+ is the degree of dissociation of the proton in the bulk solution and φbW is the volume fraction of water in the bulk. Again, the pH ˜ corresponds to the experimental pH by using the conversion factor 55.4 M such that pH ˜ ) pH + log 55.4. This conversion is also valid for the case of the equilibrium constant of specific Ca2+ binding (i.e., pK ˜ Ca ) pKCa + log 55.4). In this article, the experimental variables are used in describing the results with exceptions in the use of the (dimensionless) volume fraction of calcium in the bulk φbCa2+ and added electrolyte volume fraction φs. The spherical coordinate system is used in all calculations. The total number of lattice layers M ) 150 ensures that the (33) Tanford C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; John Wiley & Sons: New York, 1973.

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Lauw et al.

- 1602 kJ/mol), eq 26 can be rewritten as

[Ca2+ bound] [Mg2+ bound]

Figure 2. Plot of the heat change Qobs from ITC measurements of block copolymer CAE-85 solutions for the titration of two different divalent cationic solutions (Ca2+ and Mg2+) as a function of the concentration ratio between the divalent cation d and ligand concentration [-COO-] at a total ionic strength of I ) 10-2 M, pH ) 6, and T ) 50 °C.

volume of the small system is exceeded such that micelles, each in its small system, are located far apart. The number of layers M is considerably larger than the Debye length, so the electrostatic potential vanishes at the system boundary. The 1D bond-weighting factor λ in all calculations is set equal to 1/3, and the characteristic dimension of a lattice site a is equal to 0.3 nm.

VI. Results and Discussions This section consists of two parts. Subsection A describes a calorimetric study of Ca2+ and Mg2+ binding. In subsection B, the search for an appropriate Ca2+ binding parameter of block copolymer CAE-85 is performed by correlating differential potentiometric titration results with SF-SCF predictions. A. Calorimetric Study to Compare Calcium and Magnesium Binding to CAE-85. In the ITC experiment, solutions of Ca2+ or Mg2+ ions are titrated into an initial CAE-85 solutions with corresponding carboxylate concentrations 6.19 × 10-3and 5 × 10-3 M, respectively, with total ionic strength I ) 10-2 M, pH ) 6, and T ) 50 °C. At this temperature, the CAE-85 solution is above the CMC. The characterization of the CMC and CMT of CAE-85 solutions will be discussed further in the next subsection. The observed heat change from the ITC measurements (i.e., Qobs,Ca2+ and Qobs,Mg2+) are plotted in Figure 2 as a function of the concentration ratio between the corresponding divalent cation d and ligand concentration [-COO-]. In general, from eqs 5 and 6 the difference in the observed heat change Qobs from the titration of two types of divalent cations is solely due to the difference in the dehydration heat Qdehyd,d and the binding heat Qbind,d of each divalent cation. From Table 1, the values of the molar enthalpy of dehydration for Ca2+ and Mg2+ ions are ∆Hdehyd,Ca2+ ) 1602 kJ/mol and ∆Hdehyd,Mg2+ ) 1949 kJ/mol. From Figure 2, at large [d]/[ligand], Qobs,Ca2+ > Qobs,Mg2+. According to eq 6, the difference in the observed heat between the Ca2+ and Mg2+ titrations can be estimated to be

Qobs,Ca2+ - Qobs, Mg2+ ≈ Qint,Ca2+ - Qint, Mg2+ > 0 (25) where Qint,Ca2+ and Qint,Mg2+ are the interaction heats of Ca2+ and Mg2+, respectively. By applying eqs 3 and 4, the interaction heat difference can be expressed as follows:

Qint,Ca2+ - Qint,Mg2+ ) [Ca2+](∆Hdehyd,Ca2+ + ∆Hbind,Ca2+) [Mg2 +](∆Hdehyd,Mg2+ + ∆Hbind,Mg2+) > 0 (26) Provided that ∆Hdehyd,Ca2+ + ∆Hbind,Ca2+ > 0 (i.e., ∆Hbind,Ca2+ >

>

∆Hdehyd,Mg2+ + ∆Hbind, Mg2+ ∆Hdehyd,Ca2+ + ∆Hbind, Ca2+

(27)

From eq 27, it is known that the term on the right-hand side is positive because the absolute value of the binding enthalpy is usually smaller than the enthalpy of dehydration for each corresponding cation. In general, the binding enthalpy ∆Hbind,d can be divided into an electrostatic term and a specific binding term. It is known that Mg2+ binds the carboxylic group electrostatically without any specific interactions34,35 so that the binding enthalpy ∆Hbind,Mg2+ between Mg2+ and -COO- is due to only the electrostatic interaction. For the case of Ca2+ binding to carboxylic groups in CAE-85, the main contribution to the binding enthalpy is from the electrostatic interaction. This will be shown below by potentiometric (DPT) measurement. For this reason, the ratio of bound calcium and bound magnesium on the left-hand side of eq 27 may be larger than 1 as more calcium is bound to the carboxylic groups of CAE-85 compared to magnesium. This suggests that there is a specific binding of Ca2+ to -COO-. More quantitative proof of this specific binding is given below by combining the results from experiments and SF-SCF calculations. B. SF-SCF Parametrization from Experimental Results. The CMT of block copolymer CAE-85 at given carboxylic (ligand) concentration [-COO-] and total ionic strength I is determined from DSC measurements. The CMT is the temperature at which there is a sudden change in the slope of the effective heat capacity Cpeff. The Cpeff(T) values for two different ligand concentrations, [-COO-] ) 4.7 × 10-4 and 4.7 × 10-3 M, both at I ) 5 × 10-3 M, are depicted in Figure 3a. Throughout this section, the ligand concentration [-COO-] is calculated by taking into account the degree of dissociation of carboxylic groups at the corresponding pH with pKa ) 4.51. From Figure 3a, the CMTs are about 20 and 30 °C for [-COO-] ) 4.7 × 10-3 and 4.7 × 10-4 M, respectively. At sufficiently low copolymer concentration (i.e., [-COO-] < 4.7 × 10-4 M), we consider that micelles are not formed even at high temperature. The CMTs of the CAE-85 solution for different total ionic strength, I ) 2 × 10-3, 5 × 10-3, and 10-2 M at [-COO-] ) 4.7 × 10-3 M, are approximately 24 °C (Figure 3b). We conclude that the CMT of block copolymer CAE-85 is indifferent at low ionic strength. In summary, the CMT is predominantly a function of the polymer concentration. The CMCs of CAE-85 solutions at constant temperature T ) 40 °C are obtained from DLS measurements. The intersection point of two linear curve fits of the ratio of scattering intensity IS/I0 vs total concentration [-COO-] indicates the corresponding CMC. In Figure 4, two sets of results from DLS measurements for the CAE-85 solution at I ) 3 × 10-3 and 10-2 M are depicted. The corresponding CMCs are 2.9 × 10-4 and 1.6 × 10-4 M, respectively (as indicated by the arrows). By considering the CMC and CMT values of CAE-85 solutions, the differential potentiometric titration (DPT) is performed in order to measure directly the Ca2+ uptake by the micelles composed of CAE-85. The polymer concentration cp and thus the ligand concentration [-COO-] change during titration. An initial [Ca2+] ) 10-4 M is chosen for each measurement. It corresponds to a ligand concentration at a “bidentate” binding (34) Malovikova, A.; Rinaudo, M.; Milas, M. Biopolymers 1994, 34, 10591064. (35) Tamura, T.; Kawabata, N.; Kawauchi, S.; Satoh, M.; Komiyama, J. Polym. Int. 1998, 46, 353-356.

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Figure 5. Plot of dEMF vs the concentration ratio of ligand to Ca2+ from differential potentiometric titrations of EDTA, PMAA, and CAE-85 into the initial Ca2+ solution. The titration of a block copolymer CAE-85 solution is performed at I ) 7 × 10-4 M and T ) 40 °C.

Figure 3. (a) Effective heat capacity Cpeff(T) of block copolymer CAE-85 solutions with a ligand concentrations [-COO-] ) 4.7 × 10-4 and 4.7 × 10-3 M at fixed total ionic strength of I ) 5 × 10-3 M. (b) Similar plot at a constant [-COO-] ) 4.7 × 10-3 M for various ionic strengths of I ) 2 × 10-3, 5 × 10-3 M, and 10-2 M, as indicated.

Figure 4. Plot of the ratio of scattering intensity IS/I0 from DLS measurements of block copolymer CAE-85 solutions as a function of the ligand concentration [-COO-] at two different total ionic strengths, I ) 3 × 10-3 and 10-2 M, at T ) 40 °C. Each of the CMCs is obtained from the intersection between two bilinear curve fits as indicated by the arrows.

of [-COO-] ) 2 × 10-4 M. On the basis of the DLS measurements (cf. Figure 4), it is known that at T ) 40 °C and at low ionic strength I < 10-2 M micelles exist only beyond the value [-COO-] ) 2 × 10-4 M, which is larger than the CMC of the CAE-85 solution. The DPT is validated by reference systems (EDTA and PMAA) that bind Ca2+ ions in solution. Both measurements are performed at room temperature.12,18,19 The collection of these results is presented in Figure 5. Here, the titration of block copolymer CAE-85 solution at I ) 7 × 10-4 M is included as a comparison. The deprotonated carboxylic groups are denoted as ligands. The

Figure 6. Plot of dEMF vs the concentration ratio of ligand to Ca2+ from differential potentiometric titrations of CAE-85 at different total ionic strengths, I ) 7 × 10-4, 1.8 × 10-3, and 5.8 × 10-3 M, at constant temperature T ) 40 °C.

decrement of about 30 mV in dEMF means that [Ca2+] is decreased a decade from its initial concentration. From Figure 5, at any [ligand]/[Ca2+] the dEMF of the CAE-85 titration curve is less than that of EDTA or PMAA. This indicates that the Ca2+ uptake by CAE-85 is relatively weak. The result of the titration of CAE-85 at three different total ionic strengths I ) 7 × 10-4, 1.8 × 10-3, and 5.8 × 10-3 M are given in Figure 6. As before, the initial [Ca2+] for every titration curve is fixed at 10-4 M. From Figure 6, it follows that the Ca2+ ions are bound more weakly at higher ionic strength. Apparently the 1:1 electrolytes compete with the Ca2+ ions to screen the -COO- charges, which is indeed in line with the applied ion-exchange model. 2+ + As dEMF is converted to [Ca2+ bound] and [Cafree], [Kbound] is calculated from the remaining ligands that are not occupied by Ca2+. The number of remaining ligands is obtained by subtracting the 2[Ca2+ bound] value from the total ligand concentration [-COO-] (bidentate complex). This is based on the assumption that all deprotonated carboxylic groups collect counterions in their neighborhood (local electroneutrality approximation). The resulting plot of θ/(1 - θ) versus [Ca2+ free] for the case of a CAE85 titration at I ) 7 × 10-4 M is depicted in Figure 7. A bilinear curve can be fit to the result, exemplifying the presence of block copolymer CAE-85 in two different regimessmicellar and unimeric. At first, at several titration points at a constant measurement temperature of T ) 40 °C, the CMC has not been reached, implying that only unimers exist. Further along in the titration as the block copolymer concentration cp becomes larger

10940 Langmuir, Vol. 22, No. 26, 2006

Figure 7. Plot of the relative amount of bound Ca2+ and K+, θ/(1 - θ), as a function of free Ca2+ concentration for the titration of CAE-85 into a 10-4 M Ca2+ solution at I ) 7 × 10-4 M and T ) 40 °C. Micellar and unimeric regimes correspond to a bilinear curve fit of the result, as indicated. The slope of the linear fit in the micellar region gives the effective binding constant K0.

Figure 8. Plot of the relative amount of bound Ca2+ and K+, θ/(1 - θ), as a function of free Ca2+ concentration for the theoretical titration of CAE-85 into a 10-4 M Ca2+ solution at I ) 1.1 × 10-3 M and T ) 40 °C. As an addition to the electrostatic interaction, the specific binding between Ca2+ and carboxylic groups is given with the indicated pKCa values. The slopes of the linear fits in the micellar regions give the K0 values.

than the CMC, micelles are formed. In this regime, the slope of the bilinear fit gives the K0 value of interest, which in this case is K0 ) 2.42 × 107 M-2. SF-SCF calculations are performed to mimic experimental details. Here, we do not include all of the details of the calculation but rather refer the reader to a previous publication on this matter.30 As in the experiments, where the total concentration of Ca2+ ions is practically the same throughout the titration, the total amount of Ca2+ in the calculation is also fixed. It is achieved by imposing the same total amount of Ca2+ in the small system (i.e., φCa2+ ) θCa2+σ/Vss + φCa2+b ) 1.8 × 10-6, which corresponds to the initial [Ca2+] ) 10-4 M in the experiment). In the initial setting of the parameters, only electrostatic contributions are taken into account, omitting any specific interaction between Ca2+ and acid groups. In all other parameters settings, a specific interaction is included through the reaction equilibrium -COOCa+ h -COO- + Ca2+. As an example, in Figure 8, analogous to the experimental plot in Figure 7, θ/(1 - θ) is plotted against -3 M and [Ca2+ free] for the case of CAE-85 titration at I ) 1.1 × 10 T ) 40 °C. The bilinear curve fit corresponds to unimer and micellar regimes of block copolymers. Gradients of the linear fit in the micellar regime are the effective binding constants K0 with values of 8.52 × 107, 2.29 × 107, and 1.84 × 107 M-2 (and subsequently for the binding with only an electrostatic contribution

Lauw et al.

Figure 9. Plot of the effective binding constant K0 at different total ionic strength I. The experimental results are indicated by the dots. Results from the SF-SCF calculation are indicated by dashed lines. When only electrostatic interactions are taken into account, numerical calculations deviate from experimental results, particularly at low ionic strength. The other two SF-SCF results are obtained by including the specific interaction with corresponding pKCa ) 1.66 and 1.76. These values are selected to give a good fit.

Figure 10. Plot of the fractional Ca2+ binding Φ as a function of the concentration ratio of total Ca2+ to ligand from the titration of CAE-85 into a 10-4 M Ca2+ solution at I ) 7 × 10-4 M and T ) 40 °C. The result from direct potentiometric measurement (DPT) is compared with the SF-SCF theoretical titration by taking into account the electrostatic interaction and specific binding with pKCa ) 1.71 as indicated by the solid line. The dashed SF-SCF isotherm curve results by considering only the electrostatic binding.

and for the binding that includes specific interaction with pKCa ) 1.66 and 1.76). On the basis of a similar treatment of the experiment and the model as described above, the experimental and theoretical K0 values are obtained as a function of ionic strength. K0 values at four different ionic strengths (I ) 7 × 10-4, 1.2 × 10-3, 1.8 × 10-3, and 5.8 × 10-3 M) are attained experimentally. The theoretical K0 values from the SF-SCF model of Ca2+ binding (with electrostatic and specific interactions) are used in order to fit the experimental result. The results are presented in Figure 9. Here, the dots are the experimental values. They are plotted together with three series of SF-SCF predictions. When only the electrostatic interactions are taken into account, the K0 fit deviates considerably from the experimental result, especially at low ionic strength. The inclusion of specific binding, as given by eq 19, results in a much better fit. In particular, for pKCa values of 1.66 and 1.76, the value of K0 is fit very well over the whole range of ionic strength. We implement pKCa ) 1.71 in the following SCF predictions. As an example, results from the theoretical titrations are compared with the experimental results from the direct potentiometric

Binding of Ca by Micelles Composed of Pluronics

measurement of CAE-85 in a Ca2+ solution at I ) 7 × 10-4 M and T ) 40 °C. This is presented in Figure 10. Here, the fractional Ca2+ binding Φ is plotted as a function of the concentration ratio of Ca2+ to ligand. Again, good agreement is found at a low [Ca2+]/[ligand] ratio because the K0 fitting is performed within this region (where CAE-85 block copolymers exist as micelles). For comparison, the dashed line in Figure 10 is the result for which there is only an electrostatic contribution in the Ca2+ binding. Indeed, the conclusion is that Ca2+ is bound predominantly by an electrostatic binding mechanism and that the specific contribution is a secondary effect. In a forthcoming article, we will elaborate on the SCF predictions for carboxy-modified Pluronics with special attention paid to the Ca2+ binding efficiency for which the pKCa is a central quantity.

VII. Conclusions In this article, a differential potentiometric titration measurement is performed in order to obtain important information about the binding behavior of Ca2+ ions to novel block copolymer CAE-85. The metal complexes are governed by the temperature, polymer (ligand) concentration, total ionic strength, and pH. At a given temperature above the CMT, the effective binding constants of the ion-exchange equilibrium are obtained. The results were used subsequently to find the intrinsic pKCa value needed in the SF-SCF calculations. It turns out that micelles composed of CAE-85 bind Ca2+ ions both electrostatically and specifically. The specific binding between Ca2+ and carboxylic groups on the surface of the micelles is modeled through the reaction equilibrium

Langmuir, Vol. 22, No. 26, 2006 10941

-COOCa+ h -COO- + Ca2+ with pKCa ) 1.7 ( 0.06. In general, the Ca2+ uptake occurs mostly as a result of electrostatic interactions, leading to controllable Ca2+ binding via micellization. Because of its binding characteristics, the novel block copolymer CAE-85 has wide potential applications, especially when a controlled binding process is preferred. The fact that the electrostatic interaction contributes the most to Ca2+ uptake makes the binding uniquely adjustable through the changing of temperature, pH, or added salt concentration. By changing these parameters, one can control the micellization or demicellization process of CAE-85 that causes the uptake or release of Ca2+ ions. In a future publication, we will investigate the binding capacity of the CAE-85 variants by using the full set of parameters in a more thorough SF-SCF analysis. This will enable us to predict the Ca2+ binding efficiency of similar modified Pluronic copolymers with different numbers and positions of carboxylic groups. Acknowledgment. The Dutch Science and Technology Foundation (STW), Aquacare, GWA, KIWA, Witteveen&Bos, and ETD&C are greatly acknowledged for financial support. Part of this work is supported by grant EPC.5516. We thank A. Korteweg in helping with the calorimetric measurement and W. Threels and R. Wegh for their great help with the DPT setup. We thank Professor J. Lyklema and P. Iakovlev for very helpful discussions and suggestions. LA061300I