pubs.acs.org/Langmuir © 2010 American Chemical Society
Modeling Ion Binding to Humic Substances: Elastic Polyelectrolyte Network Model Silvia Orsetti, Estela M. Andrade, and Fernando V. Molina* INQUIMAE, Departamento de Quı´mica Inorg anica, Analı´tica y Quı´mica Fı´sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, pabellon II, C1428EHA Buenos Aires, Argentina Received August 18, 2009. Revised Manuscript Received December 16, 2009 A new model for the electrostatic contribution to ion binding to humic substances is proposed and applied to published data for proton binding to fulvic and humic acids. The elastic polyelectrolyte network model treats humic substance particles as composed by two parts, an external one directly in contact with the solution, and an internal part or gel fraction which is considered, from a statistical point of view, as a charged polymer network swelled by the electrolyte solution, in the framework of the Flory polymer network theory. The electrostatic effect is given by a Donnan-like potential, which can be regarded as an average value over the gel fraction of the humic particle. The gel fraction expands as the pH and humic charge are increased, determining the Donnan potential and consequently the ion activity inside the gel. The model was fitted to published experimental data with good agreement. The model predictions are discussed, and the behavior suggests, for some cases, the presence of a transition between closed and open structures attributed to the presence, at low pH, of intramolecular hydrogen bonds which are removed as the carboxylic sites become deprotonated.
Introduction Humic substances (HS) are important components of natural organic matter in groundwaters and in soils, where they have a fundamental role in nutrient availability for plants, among several other important properties.1,2 HS show colloidal behavior and interact with other chemical species in the natural environment, including binding of protons and inorganic cations; this ability, in particular for binding heavy metals, has an important role in the fate of such pollutants in the environment. Metal binding occurs always in competence with protons, and besides the acid-base properties of HS are important themselves; thus, this topic has attracted a high number of researchers over the last decades.3-14 Among the different experimental studies, equilibrium binding data, usually in the form of titration curves, have been reported in the literature; a number of studies have been reviewed by Milne et al.15 The interpretation of such curves in terms of equilibrium *Corresponding author. Telephone: þ54-11-4576-3378/80 ext 230. Fax: þ54-11-4576-3341. E-mail:
[email protected].
(1) Senesi, N.; Loffredo, E. In Soil Physical Chemistry; Sparks, D. L., Ed.; CRC: Boca Raton, FL, 1998; pp 239-370. (2) Baldock, J. A.; Nelson, P. N. In Handbook of Soil Science; Sumner, M. L., Ed.; CRC: Boca Raton, FL, 1999; pp B75-B84. (3) Tipping, E. Aquat. Geochem. 1998, 4, 3–48. (4) Kinniburgh, D.; Van Riemsdijk, W.; Koopal, L.; Borkovec, M.; Benedetti, M.; Avena, M. Colloids Surf., A 1999, 151, 147–166. (5) Avena, M. J.; Koopal, L. K.; van Riemsdijk, W. H. J. Colloid Interface Sci. 1999, 217, 37–48. (6) Perdue, E.; Lytle, C. Environ. Sci. Technol. 1983, 17, 654–660. (7) Rhea, J.; Young, T. Environ. Geol. 1987, 10, 169–173. (8) Ritchie, J.; Michael Perdue, E. Geochim. Cosmochim. Acta 2003, 67, 85–93. (9) Gustafsson, J.; Kleja, D. Environ. Sci. Technol. 2005, 39, 5372–5377. (10) Cooke, J.; Hamilton-Taylor, J.; Tipping, E. Environ. Sci. Technol. 2007, 41, 465–470. (11) Lopez, R.; Gondar, D.; Iglesias, A.; Fiol, S.; Antelo, J.; Arce, F. Eur. J. Soil Sci. 2008, 59, 892–899. (12) Atalay, Y.; Carbonaro, R.; Di Toro, D. Environ. Sci. Technol. 2009, 43, 3626–3631. (13) Van Zomeren, A.; Costa, A.; Pinheiro, J.; Comans, R. Environ. Sci. Technol. 2009, 43, 1393–1399. (14) Gustafsson, J. P. J. Colloid Interface Sci. 2001, 244, 102–112. (15) Milne, C. J.; Kinniburgh, D. G.; Tipping, E. Environ. Sci. Technol. 2001, 35, 2049–2059.
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binding reactions is complicated by the fact that HS are heterogeneous in nature, showing a high number of binding chemical groups (sites) with varying equilibrium constant (or affinity for the bound species). A fully detailed molecular description of HS is, at best, very difficult to achieve, so that one way to tackle the problem is the use of simplified models to interpret the titration curves. These models generally include a description of the intrinsic binding constants in terms of a distribution, the affinity distribution or affinity spectrum, and an electrostatic contribution to account for the colloidal behavior of HS. The intrinsic affinity distribution has been assumed to be either discrete (effectively similar to a mixture of weak acids)3,14,16 or continuous, with some assumed distribution function,4,7,17,18 but in both cases two main types of groups or sites have been included: carboxylic sites, with pKa in the range 3-5, and phenolic ones, with pKa about 8-10. The electrostatic contribution modeling has also followed two basic lines:19 one treated the humic substances as rigid particles,14,18,20-22 applying standard colloid chemistry models, whereas other authors applied, albeit sometimes in a restricted and approximate way, the concept of Donnan equilibrium.3,23,20 In Tipping’s WHAM models V and VI3,16 a Donnan phase is assumed, surrounding the humic particles and containing only positively charged counterions, whereas the electrostatic effect on binding is accounted by an empirical (16) Tipping, E.; Hurley, M. Geochim. Cosmochim. Acta 1992, 56, 3627–3641. (17) Orsetti, S.; Andrade, E. M.; Molina, F. V. J. Colloid Interface Sci. 2009, 336, 377–387. (18) de Wit, J. C. M.; van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1993, 27, 2015–2022. (19) Saito, T.; Nagasaki, S.; Tanaka, S.; Koopal, L. K. Colloids Surf., A 2005, 265, 104–113. (20) Companys, E.; Garces, J. L.; Salvador, J.; Galceran, J.; Puy, J.; Mas, F. Colloids Surf., A 2007, 306, 2–13. (21) Bartschat, B. M.; Cabaniss, S. E.; Morel, F. M. M. Environ. Sci. Technol. 1992, 26, 284–294. (22) Milne, C. J.; Kinniburgh, D. G.; De Wit, J. C. M.; Van Riemsdijk, W. H.; Koopal, L. K. Geochim. Cosmochim. Acta 1995, 59, 1101–1112. (23) Benedetti, M. F.; Van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1996, 30, 1805–1813.
Published on Web 01/07/2010
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The subject of modeling the ion binding equilibrium of humic substances has interested researchers for several decades. However, as said above the heterogeneity and complexity of these substances have made very difficult to formulate models on physicochemical grounds. Bartschat et al.21 presented a simple
thermodynamic model, treating the humic particles as spheres and considering the opposite cases of permeable and impenetrable spheres. In the permeable sphere model (PSM), the humic particle is considered a separate phase or gel where the humic molecules are penetrated (swelled) by water and ions, and surrounded by a diffuse layer region. In the impenetrable sphere model (ISM), the humic molecules are treated similarly to solid colloidal particles (such as oxides), having a surface charge and a diffuse layer. Bartschat et al. solved numerically the Poisson-Boltzmann equation and found that both PSM and ISM models were in principle applicable, but chose the ISM for convenience. Bryan et al.32 formulated a physicochemical model for metal cationhumic binding based on consideration of energies and entropies of binding; however, the lack of structural knowledge prevented quantitative comparison with experimental results. The electrostatic contribution was modeled through the PSM. More recently, Saito et al.19 analyzed several electrostatic models, separating them in permeable and impenetrable sphere types, and used the master curve approach to evaluate the models considered. They found that, under particular assumptions, both types were applicable under appropriate conditions. The ISM requires knowledge of the particle size; Saito et al.19 used a radius optimized to make the individual curves merge into a master curve. For the PSM, testing was done using only hydrodynamic radii estimated from viscosimetry, and an averaged internal electrostatic potential, finding it not satisfactory. Further variants of the PSM, based on assumption of Donnan equilibrium were studied, finding that if a part of the solution is included in the “Donnan phase” (i. e., subject to the Donnan potential) a satisfactory master curve is found. Ohshima and co-workers33 introduced a PSM model with a hard, impenetrable core which could be either charged or uncharged (but considered uncharged in the model developed there), termed “soft particle model” (SPM). See Appendix A (Supporting Information). A number of recent studies on HS show that humic molecules are associated and/or entangled forming aggregates in most of the pH range,26-28,34-37 and behave as soft, permeable particles.25 In particular, Baigorri et al.27 proposed recently the presence of three different types of entities in HS: relatively small molecules, macromolecules and supramolecular assemblies of the other two types. Thus, the PSM appears to be a realistic view of humic particles. However, a detailed electrostatic modeling would require knowledge of the particle size and shape, which is difficult to obtain; furthermore, as shown by Duval et al.25 and other authors (see for example Avena et al.38 and Siripinyanond et al.39), HS undergo size changes due pH- and compositiondependent aggregation as the medium changes. With the aim of proposing a physically more realistic model, and in view of the above considerations, the humic particles will be considered here as an elastic polyelectrolyte network, in the sense given by Flory:40 a set of polymer chains cross-linked
(24) Christensen, J.; Tipping, E.; Kinniburgh, D.; Groen, C.; Christensen, T. Environ. Sci. Technol. 1998, 32, 3346–3355. (25) Duval, J. F. L.; Wilkinson, K. J.; Van Leeuwen, H. P.; Buffle, J. Environ. Sci. Technol. 2005, 39, 6435–6445. (26) Simpson, A.; Kingery, W.; Hayes, M.; Spraul, M.; Humpfer, E.; Dvortsak, P.; Kerssebaum, R.; Godejohann, M.; Hofmann, M. Naturwissenschaften 2002, 89, 84–88. (27) Baigorri, R.; Fuentes, M.; Gonzalez-Gaitano, G.; Garcı´ a-Mina, J. Colloids Surf., A 2007, 302, 301–306. (28) Sutton, R.; Sposito, G. Environ. Sci. Technol. 2005, 39, 9009–9015. (29) Wershaw, R. L. Evaluation of conceptual models of natural organic matter (humus) from a consideration of the chemical and biochemical processes of humification; USGS: Denver, CO, 2004. (30) Chilom, G.; Rice, J. A. Langmuir 2009, 25, 9012–9015. (31) Kinniburgh, D. G.; Milne, C. J.; Benedetti, M. F.; Pinheiro, J. P.; Filius, J.; Koopal, L. K.; Van Riemsdijk, W. H. Environ. Sci. Technol. 1996, 30, 1687–1698.
(32) Bryan, N. D.; Jones, D. M.; Appleton, M.; Livens, F. R.; Jones, M. N.; Warwick, P.; King, S.; Hall, A. Phys. Chem. Chem. Phys. 2000, 2, 1291–1300. (33) Ohshima, H. Theory of colloid and interfacial electric phenomena; Interface Science and Technology; Academic Press: Amsterdam, 2006; Vol. 12, pp 39-55. (34) Rizzi, F.; Stoll, S.; Senesi, N.; Buffle, J. Soil Sci. 2004, 169, 765–775. (35) Kawahigashi, M.; Sumida, H.; Yamamoto, K. J. Colloid Interface Sci. 2005, 284, 463–469. (36) Kucerı´ k, J.; Smejkalov a, D.; Cechlovska, H.; Pekar, M. Org. Geochem. 2007, 38, 2098–2110. (37) Baigorri, R.; Fuentes, M.; Gonzalez-Gaitano, G.; Garcia-Mina, J. M. J. Phys. Chem. B 2007, 111, 10577–10582. (38) Avena, M. J.; Vermeer, A. W. P.; Koopal, L. K. Colloids Surf., A 1999, 151, 213–224. (39) Siripinyanond, A.; Worapanyanond, S.; Shiowatana, J. Environ. Sci. Technol. 2005, 39, 3295–3301. (40) Flory, P. J. J. Chem. Phys. 1950, 18, 108–111.
Boltzmann factor; here the humic particles are assumed to be of fixed size (one value for fulvic acids, FA, and another one for humic acids, HA). In the NICA-Donnan model4,15 a “Donnan volume” of solution around the humic particle is assumed to contain enough ions so as to be (the whole system of particle plus solution) electrically neutral, the volume being adjustable as a function of ionic strength. Both approaches lead to good fit of experimental data;24 the Donnan volume model, however, has been criticized as it leads in some situations to unreasonably larger volume values.5 Besides being chemically heterogeneous, humic substances have been show to have a soft and permeable nature, as deduced from electrophoretic mobilities by Duval et al.25 Their results are interpreted considering HS as permeable particles, with a pHdependent hydrodynamic size which increases as pH decreases in most of the range, and (for some cases) departs from this behavior at the low and high pH extremes. The fact that the size increases as pH decreases is explained in terms of molecular aggregation, being less important at high pH values due to intermolecular electrostatic repulsion. At the high pH range (9-11) a size increase is observed attributed to electrostatically driven molecular expansion; that is, aggregation is prevented and the observed changes are due to molecular swelling caused by the increasing charge. At the low range (pH 4-5) the hydrodynamic radius levels or decreases slightly, which is interpreted as the aggregates becoming more compact due to increased hydrogen bonding. Humic aggregation has also been observed and discussed by several authors.26-30 The most commonly used humic models, NICA-Donnan4,31 and WHAM,3 show good fitting and prediction ability for many cases examined so far. However, in view of the recent research mentioned above, some of the assumptions involved appear to be not very realistic. With the aim of introduce physically more realistic modeling, in this work a new electrostatic model for humic substances, the elastic polyelectrolyte network (EPN) model is proposed. This electrostatic model will be combined with a continuous affinity spectrum, in principle obtained from conditional spectrum calculations,17 and afterward adjusted as part of the procedure. In the following sections, the electrostatic model is introduced and combined with an intrinsic affinity spectrum, the numerical procedures are presented and fitting to literature data is performed. Finally, model predictions and comparisons are discussed.
Elastic Polyelectrolyte Network Model for Humic Substances
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through the ends with the functionality, fP, giving the number of bonds at one chain end (see Appendix B, Supporting Information, for details). This network is assumed to be composed of N2 polymer segments, cross-linked or entangled to form a network, with a pH dependent charge evolved through proton bindingdissociation processes; the electrostatic repulsion arising causes expansion and thus swelling of the network. These N2 polymer segments are, as a whole, statistically equivalent to the humic particle, and not necessarily bear a direct relationship to the actual molecules or molecular segments forming the particle. The process of swelling of this network is caused by the incorporation of a charge-dependent number, N1, of solvent molecules; these solvent molecules are water molecules carrying electrolyte ions so as to satisfy Donnan equilibrium with the external medium. As a simplifying assumption, the electrolyte solution is assumed to have the bulk fluid properties of water. The swelling process is assumed to be isotropic and composed of several contributions, namely humic-electrolyte mixing, network deformation and humic charging through proton dissociation; these contributions are assumed to be independent. The condition for swelling equilibrium, when a network composed of N2 statistical polymer segments incorporates N1 molecules of solvent, can be written as41 DF DV
T , N2
DΔFsw ¼ DV
T , N2
D ¼ ½ΔFmix þ ΔFdef þ ΔFel DV
¼0
T , N2
ð1Þ where F is the Helmholtz free energy, T the temperature and V the volume; sw indicates swelling, and mix, def, and el indicate the different contributions. In an actual humic particle, result of the aggregation of several (probably different) molecules, the electrostatic potential arising from the charge of ionized groups will have a higher value at the center and will decay toward the bulk solution in a complex way due to the distribution of individual charges (due to the humic and medium ions) over the interior and surroundings of the particle. Even with a detailed knowledge of the HS structure, this is a difficult scenario to model, thus some simplifying assumptions are made. Modeling the HS (and other polyelectrolytes) as a separate phase in Donnan equilibrium with the bulk solutions has been proposed by a number of authors, including among others Marinsky-Miyajima and co-workers,42-44 Koopal-van Riemsdijk and co-workers4,5,23,38 and Companys et al.20 The humic particles are here treated as consisting of two different parts (similar in some aspects to Gustaffson’s SHM model14): an internal gel in Donnan equilibrium with the external medium, which is the part modeled as an elastic network, and an external part which is assumed to be bathed by the bulk solution. The gel fraction, gf, is the mole fraction of the humic forming the gel; see Figure 1 for a schematic depiction. The total acid-base sites, Qmax (as moles of charge per mass unit), are assumed to be uniformly distributed on all the humic volume, thus gfQmax is the number of sites per unity of mass inside the gel whereas (1 - gf)Qmax sites are outside. Proton and metal binding equilibrium are assumed to be essentially similar in both parts, only affected in the internal gel by the Donnan equilibrium. Figure 2 (41) (42) (43) 5307. (44) 266.
Hill, T. L. J. Chem. Phys. 1952, 20, 1259–1273. Marinsky, J. A. J. Phys. Chem. 1985, 89, 5294–5302. Marinsky, J. A.; Baldwin, R.; Reddy, M. M. J. Phys. Chem. 1985, 89, 5303– Miyajima, T.; Mori, M.; Ishiguro, S. J. Colloid Interface Sci. 1997, 187, 259–
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Figure 1. Schematic representation of the humic electrostatic model: a fraction gf of the total humic particle is modeled as a gel-like region in Donnan equilibrium with the bulk solution, defining a Donnan potential ψD which affects the ion activity inside the gel. The small circles represent negatively charged sites. Ion charges are not shown for clarity.
Figure 2. Schematic representation of the potential profile at the gel border, x = 0, according to Ohshima33 (continuous line) for a negatively charged gel; ψD is the Donnan potential and ψ0 the surface potential. d is penetrable layer thickness, if an impenetrable core is present, or the particle radius, if there is no core. It is assumed that d . 1/κ0 (see Appendix A, Supporting Information). The dashed line shows the approximation used in this work.
shows the profile of the electrostatic potential, ψ, following Ohshima33 for a negatively charged particle in the SPM. Deep inside the gel ψ reaches a limiting value (assuming that the particle size is greater than the Debye length), being the Donnan potential ψD; near the particle border the potential decays in magnitude, being ψ0 at the border and decaying in a nearly exponential way in the bulk solution (see Appendix A, Supporting Information, for details). As observed in Figure 2, there is a diffuse layer outside the particle, resulting in a contribution to the positive charge bound. If the particle radius is much larger than 1/κ, the border effect can be neglected; that is the potential profile can be assumed to be that given by the dashed line in Figure 2. The diffuse layer charge neglected is small compared with the charge inside the particle, and furthermore it is partially compensated by the excess resulting from assuming ψ = ψD from x = -d to x = 0. According to Ohshima,33 the electrostatic potential reaches the ψD value at a distance of the order of 2/κ, which is within the range of humic particle sizes, specially for soil HS, except at very low I. Furthermore, aggregation is expected to occur to some extent. Thus, here the Donnan potential is assumed to be constant; i.e., border effects are neglected. The assumptions made result in a simplified form of the PSM. In the following, the different contributions to the free energy are introduced. The swelling equilibrium is treated in the framework of the Flory-Huggins polymer solution theory45-47 where the solution is considered a reticule with solvent molecules occupying single sites and polymer molecules (segments forming a network in this case) occupying multiple sites. The solvent and polymer volume (45) Flory, P. J. Chem. Phys. 1942, 10, 51–61. (46) Huggins, M. J. Phys. Chem. 1942, 46, 151–158. (47) Hill, T. L. An introduction to statistical thermodynamics; Courier Dover Publications: Dover, 1986.
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results in
fractions are, respectively: j1 ¼
V1 N 1 v1 ¼ V N1 v1 þ N2 Mv1
ð2Þ
and j2 ¼
V2 N2 Mv1 ¼ V N1 v1 þ N2 Mv1
ð3Þ
where V1 and V2 are the volumes of electrolyte and dry humic solid, respectively, forming the gel and v1 is the partial molecular volume of the electrolyte. The humic solid volume is related to the humic mass mh by V2 = mh/δ where δ is the dry humic substance density. Here, the statistical polymer “molecules” are assumed to fill, on average, M sites of volume v1; thus, the humic statistical average molecular volume is v2 = Mv1. Mixing Contribution. The free energy of mixing is given, following the Flory-Huggins theory and considering the N2 polymer molecules as forming a network, by41 ΔFmix ¼ kT½N1 ln j1 þ χðN2 M þ N1 Þj1 j2 V -V2 V -V2 χðV -V2 ÞV2 þ ln ¼ kT v1 V v1 V
ð4Þ
where k is the Boltzmann constant, T the absolute temperature and χ is an interaction parameter related to the “Van Laar heat of mixing”, ΔUm, by ΔUmix ¼ kTχðN2 M þ N1 Þj1 j2
ð5Þ
Deformation Contribution. The deformation contribution is purely entropic, and is given, following Flory,40 by (see Appendix B, Supporting Information) 3 3 2 ð6Þ ΔFdef ¼ N2 kT j2 -2=3 - þ ln j2 2 2 fP where fP is the Flory functionality giving the degree of crosslinking of the polymer segments; fP = 2 means linear bonds (i.e., effectively no cross-links), whereas fP = 4 means tetrahedrical bonding. Electrostatic Contribution. The electrostatic contribution to the free energy will be treated following Hill.41 The PoissonBoltzmann equation inside the gel can be written nion X 1 zi eci ð1 -j2 Þe -zi eψ=kT ð7Þ r2 ψ ¼ - 0 ½F þ εr ε0 i ¼1 where ε0 r is the relative permittivity of the internal medium, zi the charge of ion i and ci its concentration, e is the elementary charge and F is the HS charge density, given by Z gf Qmh ¼ gf Qδj2 F ¼ ¼ V V
ð8Þ
where Z is the total charge and Q is the humic specific charge. The factors (1 - j2) in eq 7 and j2 in 8 account for the charge density decrease due to swelling (mixing of HS and electrolyte). Under the assumption of constant ψ = ψD inside the gel, eq 7 becomes X 0 ¼ Fþe zi ci ð1 -j2 Þ expð -zi eψD =kTÞ ð9Þ i
which is the electroneutrality condition inside the gel fraction. For a 1-1 electrolyte solution of ionic strength I, eq 9 Langmuir 2010, 26(5), 3134–3144
ψD ¼
kT F arcsinh e 2Ið1 - j2 Þ
ð10Þ
The electrostatic contribution to the free energy change is thus found to be (see Appendix C, Supporting Information) " F ΔFel ¼ ZkT arcsinh 2Ið1 - j2 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2Ið1 - j2 Þ 2Ið1 - j2 Þ ð11Þ - 1þ2 þ F F The Swelling Equilibrium Condition. Considering the different contributions above derived in eq 1, the following condition is obtained: 1 1 2j ½χj2 2 þ j2 þ lnð1 - j2 Þ þ j2 1=3 - 2 v1 v2 fP 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 gf Qδj2 5 þ 2I 41 - 1 þ ¼0 ð12Þ 2Ið1 -j2 Þ From this expression j2 can be obtained, then using eq 3 the volume can be computed and, in turn, the Donnan potential can be calculated using eq 10. It should be noted here that the humic particle size and shape are not needed; this fact is a consequence of assuming a constant Donnan potential, which implies neglecting diffuse layer effects. The introduction of the gel fraction approximation has the effect of averaging the potential profile, so that the potential obtained from eq 10 is in fact an averaged value corresponding to a potential difference across a hypothetical gel/solution interface. Proton Binding and Intrinsic Affinity Spectrum. This model can in principle be applied to various intrinsic affinity schemes. The proton binding equilibrium is written, for a continuous affinity spectrum,17,48-50 as Z QH ¼ Qmax
"
¥ -¥
KH aint KH aH H gf þ ð1 - gf Þ int 1 þ KH aH 1 þ KH a H
f ðlog KH Þ d½log KH þ QH, el
#
ð13Þ
where KH is the local binding constant, f(log KH) is the constant distribution (affinity spectrum), aH is the proton activity (aH = 10-pH), and aHint is the proton activity in the interior of the gel, given in electrochemical equilibrium by aH int ¼ aH e -FψD =RT
ð14Þ
where R is the molar gas constant and F is Faraday’s constant. QH,el is the specific charge of electrostatically bound Hþ ions, given by the difference between the internal and bulk proton concentration (cHint and cH, respectively) integrated over the gel volume Vg: Z 1 ðcint -cH Þ dV ð15Þ QH, el ¼ mh Vg H
(48) Benedetti, M. F.; Milne, C. J.; Kinniburgh, D. G.; Van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1995, 29, 446–457. (49) Cernı´ k, M.; Borkovec, M.; Westall, J. Langmuir 1996, 12, 6127–6137. (50) Bersillon, J.; Villieras, F.; Bardot, F.; Gorner, T.; Cases, J. J. Colloid Interface Sci. 2001, 240, 400–411.
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Orsetti et al. Table 1. Data Sets Interpreted with the Electrostatic Polyelectrolyte Network Modela data set
humic substance
ionic strength values analyzed
ref
FH-09 FH-15þFH-11þFH-03 FH-23 FH-24 TS-FA HH-09 HH-11 HH-17 HH-18 HH-19 HH-23 TS-HA
Lake Drummond FA Suwannee River FA Laurentian soil FA PUFA Toledo soil FA Purified peat HA Eliot silt loam HA Shitara Black HA Purified Aldrich HA Tongbersven Forest HA PUHA Toledo soil HA
0.001, 0.010, 0.10, 0.50 0.001, 0.002, 0.010, 0.020, 0.050, 0.10, 0.20 0.009, 0.030, 0.10 0.020, 0.034, 0.10, 0.30 0.010, 0.050, 0.10, 0.30 0.002, 0.008, 0.014, 0.087, 0.093, 0.31, 0.35 0.015, 0.016, 0.031, 0.10, 0.11 0.003, 0.020, 0.11 0.005, 0.016, 0.10 0.004, 0.020, 0.12 0.014, 0.032, 0.10, 0.30 0.010, 0.050, 0.10, 0.30
59 55-57 60 61 62 63 64 5 5 5 61 62
a
Except for TS-FA and TS-HA, all data and data set designations are taken from Milne et al.15,53
Because of the constant potential in the gel, the internal concentration is also constant, thus QH, el ¼ ðcint H -cH Þ ¼ ðcint H -cH Þ
gf V g f Vh ¼ ðcint H -cH Þ mh mh j2 gf δj2
ð16Þ
and, clearly aH = cHγH, where γH is the corresponding activity coefficient. To apply the EPN model in the prediction of QH, eq 13, first j2 should be found from eq 12, then F from 8, ψD from 10, finally aHint from eq 14 and cHint using the appropriate activity coefficient. The affinity spectrum f in eq 13 can in principle be a discrete one as in WHAM 3 and SHM14 models or continuous. Here, a continuous affinity spectrum described by Gaussian functions, as reported before, 17 will be mainly applied, as f ðlog KH Þ ¼
ng X
2 qi pffiffiffiffiffiffiffiffie -2ð½log KH -log Ki =wi Þ i ¼1 wi π=2
ð17Þ
Numerical Calculations The model is applied to literature data; the data sets employed are presented in Table 1, mostly taken from Milne et al.53 The numerical procedure was as follows: 1 A conditional affinity distribution was first obtained with a constrained regularization method, applying the CONTIN package to an Hþ binding curve of high ionic strength, as described previously.17 2 The conditional affinity distribution was fitted to a sum of Gaussian functions given by eq 17. Two Gaussians were employed initially, but in some cases a satisfactory fit only was found using three Gaussian peaks.17 3 With a conditional affinity distribution found, all the data sets available for a given HS where fitted to the model through eq 13 using a Levemberg-Marquardt algorithm. Note that the intrinsic affinity spectra and the electrostatic model are coupled through aHint in eq 13. The experimental data is actually, in several cases, ΔQA vs pH, with ΔQA ¼ -Qmax þ QH -Q0 ð20Þ with Q0 the humic charge at the initial pH of the experiment. Equation 20 includes the excess Hþ retained in the gel fraction QH,el; however, it was found to be a quite small correction, even at the lowest I values, and neglecting it does not result in significative differences in the fitting parameters; the activity coefficients were calculated using the Davies equation. The experimental Q values (ΔQA) were used in eq 12 to compute an initial j2 value, which was used in turn to compute F and ψD (eqs 8 and 10). With ψD, aHint was calculated from eq 14 and then a new Q value was obtained from eqs 13 and 20, and the whole process was iterated until Q converged within a specified accuracy. In the case of curve simulation, an arbitrary initial Q value was used. Upon convergence, the resulting value was independent of the initial guess.
where ng ranges between 2 and 3 depending on the case, qi is the fraction of total sites corresponding to sites of type i, K i is the location of the Gaussian function center, and wi is the width parameter. However, other forms of continuous spectra can be considered, such as that of Sips 51,52 f ðlog KH Þ ¼
ng X i ¼1
qi ½ln 10 sinðπni Þ π½10ni ðlog KH -log Ki Þ þ 10ni ðlog Ki
0
log KH Þ
þ 2cosðπni Þ
ð18Þ where ni is the width parameter of the Sips equation. As it is well-known, the Sips distribution can be integrated with a Langmuir local isotherm to give the Langmuir-Freundlich isotherm; thus, eq 13 results in this case QH ¼
ng X
"
ni ðKi aint H Þ Qmax qi gf int 1 þ ðKi aH Þni i ¼1
ni
4
#
ðKi aH Þ þ QH, el þ ð1 -gf Þ 1 þ ðKi aH Þni
ð19Þ
(51) Sips, R. J. Chem. Phys. 1948, 16, 490–495. (52) Garces, J. L.; Mas, F.; Cecı´ lia, J.; Companys, E.; Galceran, J.; Salvador, J.; Puy, J. Phys. Chem. Chem. Phys. 2002, 4, 3764–3773.
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The adjusted parameters were χ, gf, v2, Qmax, and Q0, which was adjusted for each curve individually. The dry humic density was taken as δ = 1.5 kg L-1, following Dinar et al.54 and the functionality fP was taken as 3.0, which is an intermediate value between 2, corresponding to a linear polymer (non cross-linked),
(53) Milne, C. J.; Kinniburgh, D. G.; van Riemsdijk, W. H.; Tipping, E. Environ. Sci. Technol. 2003, 37, 958–971. (54) Dinar, E.; Mentel, T.; Rudich, Y. Atmos. Chem. Phys. 2006, 6, 5213–5224.
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Table 2. Parameter Values Obtained from Model Fitting to the Data Sets Enumerated in Table 1 Using Gaussian Affinity Spectra
FH-09 SR-FAd FH-23 FH-24 TS-FA HH-09 HH-11 HH-17 HH-18 HH-19 HH-23 TS-HA
χ
v2, L mol-1
gf
Qmax, mol kg-1
log K1
w1
q1
log K2
w2
q2
0.71 0.67 0.18 0.65 0.55 0.55 0.07 0.66 0.67 0.63 0.84 0.60
2.08 2.39 0.40 1.06 5.1 2.81 6.60 3.90 4.55 7.32 3.0 11.2
0.32 0.11 0.27 0.32 0.50 0.56 0.24 0.34 0.35 0.33 0.45 0.87
8.16 8.60 7.94 7.25 7.83 6.02 6.79 6.02 5.18 4.11 5.16 4.63
3.08 4.27 3.65 3.84 3.90 4.18 4.22 4.23 4.54 4.55 3.47 4.77
3.29 1.90 2.29 1.58 1.72 2.95 3.37 3.28 2.76 3.06 2.42 3.11
0.66 0.35 0.41 0.45 0.48 0.45 0.62 0.84 0.64 0.65 0.56 0.75
10.4 6.65 5.64 6.46 5.23 10.3 10.1 9.10 9.64 9.87 7.93 9.07
2.35 3.30 3.16 8.19 1.84 4.16 3.81 1.92 4.21 2.66 5.17 1.36
0.34 0.16 0.24 0.55 0.19 0.55 0.38 0.18 0.36 0.35 0.44 0.25
log K3
w3
q3
na
R2b
RMSEc
223 0.997 54 0.0522 227 0.998 31 0.0604 252 0.999 33 0.0423 117 0.999 67 0.0249 9.99 2.60 0.33 282 0.999 86 0.0186 980 0.999 58 0.0191 262 0.996 70 0.0710 252 0.999 77 0.0213 254 0.999 92 0.0105 251 0.999 67 0.0159 150 0.999 55 0.0217 268 0.999 60 0.0226 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P P 2 P 2 2 a b 2 c Number of points considered in the fitting. R ¼ 1 - ðΔQi -ΔQi, calc Þ =½ ΔQi 2 -ð ΔQi Þ =n. RMSE ¼ ðΔQi -ΔQi, calc Þ =n. d Data condensed from data sets FH-03, FH-11, and FH-15. 10.4 9.14
1.54 3.01
0.49 0.35
Table 3. Parameter Values Obtained from Model Fitting to the Data Sets Enumerated in Table 1 Using Sips Affinity Spectra (Langmuir-Freundlich Isotherms) data set FH-23 FH-24 TS-FA HH-09 HH-11 HH-17 HH-18 HH-19 HH-23
χ
v2, L mol-1
0.17 0.64 0.47 0.56 0.07 0.45 0.53 0.60 0.84 qffiffiffiffiffiffiffiffiffiffi
0.44 1.07 6.54 5.35 6.97 7.03 4.50 9.29 2.11
gf
Qmax, mol kg-1
log K1
n1
w1a
q1
log K2
n2
w2a
q2
nb
R2c
RMSEd
0.30 0.33 0.34 0.48 0.48 0.39 0.36 0.37 0.45
7.91 7.25 8.25 6.01 6.71 6.38 5.18 4.10 5.16
4.23 3.84 3.99 4.23 4.42 4.31 4.63 4.45 3.53
0.40 0.71 0.48 0.44 0.40 0.43 0.48 0.45 0.53
3.62 1.56 2.92 3.80 3.94 3.34 2.92 3.08 2.56
0.63 0.46 0.79 0.45 0.61 0.79 0.65 0.63 0.57
8.98 6.40 9.52 10.16 10.03 9.51 9.52 9.61 7.94
0.43 0.16 0.71 0.31 0.40 0.39 0.32 0.38 0.26
3.34 9.54 1.56 4.88 3.60 3.72 4.66 3.90 5.82
0.37 0.56 0.21 0.55 0.38 0.23 0.34 0.37 0.44
255 117 285 980 262 252 254 251 150
0.999 30 0.999 56 0.999 21 0.999 36 0.996 46 0.999 42 0.9997 0.999 34 0.999 61
0.0435 0.0287 0.0432 0.0237 0.0764 0.0339 0.02 0.0224 0.0204
-ni es et al.52 b Number of points considered in the fitting. c R2 ¼ 1 wi ¼ ln2π10 13n 2 following Garc i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P d RMSE ¼ ðΔQi -ΔQi, calc Þ2 =n. a
2
and 4 for a tetrahedral croslinking40 (see also Appendix B in the Supporting Information). The value of 3 would correspond to a partially cross-linked network, which is expected to reflect better the structure of the humic particles. Tests conducted varying fP shown that its value had little influence on the results.
P
ðΔQi -ΔQi,
2 P ΔQi 2 calc Þ =½
P -ð ΔQi Þ2 =n.
Results and Discussion
Figures 3 and 4 shows examples of fittings to FA and HA titration data, respectively. For the other data sets presented in Table 1, Figures S1 and S2 (Supporting Information) show the fitting results. All the fittings are very good. The data sets selected are those with a relatively high number of data points and/or a wide range of pH and ionic strength measured. As discussed previously,17 a reliable application of the regularization methodology to extract conditional affinity distributions requires data of high quality and density (number of points); the same can be said for the extraction of model parameter values with a fitting procedure. For the case of the Suwannee River FA (SR-FA), the Milne et al. data contains three data sets, but each one covered a relatively restricted range, so that we fitted data condensed from all data sets, FH-03 (originally from Ephraim55), FH-11 (from Ephraim et al.56) and FH-15 (originally from Glaus et al.57) simultaneously, also to have a better comparison with other literature data (see below). However, it should be noted that, even when the fitting looks fair, it was a difficult case, and some parameters had to be restricted in order to get physically reasonable results. If a Sips function-based affinity distribution is employed, a good fitting is also obtained, as shown in Table 3 for selected cases; the fitting quality parameters are slightly better using Gaussian functions. On the other hand, no significant improvement was found considering three peaks instead of two. The model parameters are in most cases similar for both types of
Fitting Results. Table 1 enumerates the data sets used whereas Table 2 presents the parameter results found in the fitting. The affinity distribution parameters given in columns 6-15 of Table 2 correspond, as specified above, to a multi-Gaussian function, eq 17. Table 3 shows the results found using Sips instead of Gaussian distributions.
(55) Ephraim, J. H. Ph.D. Dissertation, Buffalo, NY, 1986. (56) Ephraim, J.; Alegret, S.; Mathuthu, A.; Bicking, M.; Malcolm, R. L.; Marinsky, J. A. Environ. Sci. Technol. 1986, 20, 354–366. (57) Glaus, M.; Hummel, W.; Van Loon, L. Experimental determination and modelling of trace metal-humate interactions: A pragmatic approach for applications in groundwater; Paul Scherrer Institut: Villigen, 1997.
5 The fitting was continued by adjusting the Gaussian parameters of eq 17, and then the model parameters detailed in point 4 again, and so on until no further improvement was found. 6 In a number of cases the data were fit using the Langmuir-Freundlich isotherm, eq 19, preferentially starting from the results found in point 5. It was observed that this fitting procedure was more difficult, and the EPN parameters in some cases could not be all adjusted simultaneously. All the fittings were performed extensively starting from different initial values in order to test the uniqueness of the fit. No significant differences in the resulting fitting values were found. Also, the whole procedure was tested previously with synthetic data.
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Figure 3. Model fitting result for two fulvic acids: Lake Drummond (a) and Suwannee River (b). Symbols indicate experimental results; lines are fitted curves. Ionic strengths are in molar units.
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affinity spectra, so that in the following we will refer to those in Table 2. Considering the fitting parameter values collected in Table 2, it is observed that χ has positive values, most of them falling in the range 0.55-0.71. That implies a positive heat of mixing (or, in other words, that water is not a good solvent for these substances), which is to be expected because HS have a significant fraction of non polar groups. There are no apparent differences between fulvic and humic acids, however. The average molar volume v2 shows, on the other hand, lower values for FA than for HA, with the exception of TS-FA; nevertheless, v2 for this substance is below the HA from the same origin, TS-HA, which show the highest average molar volume of all the substances considered here. The values found correspond to between 60 and 600 times the water molar volume. The gf values fall in most cases in the range 0.3-0.5; this represents only the part of the humic particle where the electrostatic potential will have the value given by eq 10 when the true potential profile is averaged over the particle to the profile depicted in Figure 1. If the particle is assumed to have a uniform charge distribution, the true potential profile will have a maximum value at its center which will decay in a relatively smooth way to a border value and beyond that fall more rapidly to the bulk value. Duval et al.25 solved that profile with the aid of electrophoretic mobility data showing the features mentioned. However, as pointed out by Bartschat et al.21 in a typical humic molecule the number of charges could fall between 5 and 10 approximately, thus being below what can be considered a polyelectrolytic particle (over about 60). In this case, the humic particle (which can be composed by a single molecule or a few bound molecules) would be more appropriately seen as an oligoelectrolyte, and as such a discrete charge distribution model would be required. The solution of the Poisson-Boltzmann equation for this system is more complex and will require extensive numerical evaluation, but it can be anticipated that the potential profile will be of oscillating nature, with maxima (in absolute value) located on the individual charges. Thus, the averaged profile proposed here, even when it can appear as a quite strong approximation, could be a reasonable compromise to avoid excessive complexity due to extensive numerical evaluation and a higher number of adjustable parameters to model the discrete charge distribution. The Qmax values have essentially the same meaning as Qtot from Milne et al.15 In fact, most of the values found here are fairly coincident with those reported there: all the HA (excepting TSHA, not present there) are coincident but for the FA only for SRFA the Qmax value found here is coincident with that of Milne et al. for data set FH-15. The remainder parameters describe the normalized intrinsic affinity distributions which are shown in Figure 5. The general shape is similar to those reported by Avena et al.5 and Milne et al.,15 among others. Model Predictions. The model allows prediction of the volume expansion (through j2) of humic particles upon pH (and consequently, Q) increase, and also the electrostatic potential. Recalling that j2 = V0/V, where V0 is the “dry” HS gel volume, the total humic volume in solution VH can be estimated as VH ¼
Figure 4. Model fitting result for two humic acids: purified peat HA (a) and purified Aldrich HA (b). Symbols indicate experimental results, lines are fitted curves. Ionic strengths are in molar units. 3140 DOI: 10.1021/la903086s
gf ð1 -gf Þ þ δ δj2
ð21Þ
where j2-1 gives the volume increase of the gel fraction due to swelling, whereas the external fraction 1 - gf is assumed to not swell. Figure 6 shows results of VH for two different HAs, Langmuir 2010, 26(5), 3134–3144
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Figure 5. Normalized intrinsic affinity distributions found for (a) fulvic acids and (b) humic acids.
Figure 6. Total humic volume as a function of pH: EPN model results (lines) and calculated from viscosity measurements of Avena et al.38 (points) for different ionic strengths: (a) purified Aldrich HA; (b) Tongbersven Forest HA. Ionic strengths are in molar units. Langmuir 2010, 26(5), 3134–3144
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comparing with values calculated from viscosity measurements (assuming spherical shape) of Avena et al.;38 in that study, it was found that an increase in solution pH leads to swelling of the HS, whereas increase in ionic strength leads to shrinkage, as it is also found in our results. There is qualitative agreement in Figure 6 in order of magnitude and general tendency to increase with pH and decrease with increasing I. In comparing those results, it should be kept in mind that the EPN model predicts the overall humic volume, irrespective of the individual particle size, whereas viscosity measurements essentially probe the individual particle size, and can be affected by factors such as aggregation. Note that aggregation, being a phenomenon which involves essentially the outer part of the particles, should not alter significantly the inner Donnan potential, as long as the model assumptions remain fulfilled. Also, it should be noted that the predictions of the EPN model are obtained assuming that the HA is a gel with a solid/dry part and an hydrated one, whereas in the other models (such as NICA-Donnan and Companys et al.) this volume has non physical existence and only is a means to ensure the electroneutrality condition necessary to use the Donnan equilibrium approximation. Figure S3 (Supporting Information) shows results for other HS; note the different scale for fulvic and humic acids. The overall tendency is for VH to increase with pH and decrease with I, as expected. Appendix D (Supporting Information) includes a comparison with present Donnan-Volume models. Alternatively, j2-1/3 = (V/V0)1/3 can be regarded as a measure of the expansion in one dimension. Pinheiro et al.58 studied the mobility of PPHA by light dispersion as a function of pH, adjusting the pH by two different methods. In method A, the pH was directly increased from the just dissolved HA by NaOH additions; in method B, the pH was first increased to 10.0 and then adjusted backward by appropriate HNO3 additions. From the measured mobilities, the particle diameter was estimated through the Stokes-Einstein equation; because this equation is valid for a rigid, impermeable sphere in a continuous medium, the comparison can only be considered in a qualitative way. The EPN model results for PPHA are shown in Figure 7, with the inset presenting Pinheiro et al.58 data. Whereas by method A a sharp size decrease was observed around pH = 5, by method B a slight diameter increase is found as pH increases. The change observed in method A is attributed to particle disaggregation, whereas the increase observed by the second method is qualitatively consistent with the results presented here. Other results for j2-1/3 are presented in Figure S4 (Supporting Information). Duval et al.25 analyzed theoretically electrophoretic mobility measurements, being able to calculate electrostatic potential profiles in the permeable sphere model framework and to estimate sizes through the Stokes-Einstein equation. Their size results for SR-FA (Figure 6 of ref 25) show, as pH increases, a size increase in the range 4-5, then a steady decrease up to pH ∼ 8-9 and finally a small increase for higher pH values; the decrease in the mid pH range was attributed to disaggregation induced by the charge increase, the small increase at higher pHs to swelling of the individual molecules caused by the further charge increase, and (58) Pinheiro, J. P.; Mota, A. M.; d’Oliveira, J. M. R.; Martinho, J. M. G. Anal. Chim. Acta 1996, 329, 15–24. (59) Cabaniss, S. Anal. Chim. Acta 1991, 255, 23–30. (60) Pinheiro, J.; Mota, A.; Benedetti, M. Environ. Sci. Technol. 1999, 33, 3398– 3404. (61) Christl, I. Ph.D. Dissertation, ETH, Zurich, Switzerland, 2000. (62) Fernandez, J. M.; Plaza, C.; Senesi, N.; Polo, A. Chemosphere 2007, 69, 630–635. (63) Milne, C.; Kinniburgh, D.; de Wit, J.; van Riemsdijk, W.; Koopal, L. Geochim. Cosmochim. Acta 1995, 59, 1101–1112. (64) Robertson, A. P. Ph.D. Dissertation, Stanford University: Palo Alto, CA, 1996.
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Figure 7. Model results for linear expansion of purified peat HA (PPHA) as a function of pH at different ionic strengths: inset, PPHA particle diameter, d, as a function of pH adapted from Pinheiro et al.;58 squares, method A; triangles, method B; see text for details. Ionic strengths are in molar units.
the increase in the low range (4-5) to hydrogen bond breaking as the carboxylic groups are deprotonated. The model presented here does not account for aggregation-disaggregation, but the increase observed at high pH by Duval et al. is in qualitative agreement with our results for SR-FA (Figure S3a, Supporting Information). The particle (gel fraction) electrostatic Donnan potential results, presented in dimensionless form (Ψ = FψD/RT) are shown in Figure 8 for selected cases, and in Figure S5, Supporting Information, for the remainder. Whereas for some cases (Figure 8a for example) the potential becomes monotonically more negative as the pH increases, as expected, in others (such as HH-18, Figure 8c) a minimum (maximum in absolute value) is observed at pH 4-5 approximately for low I values. This is accompanied by a marked change in slope of j2-1 (Figure 6a) observed at about the same pH where the minimum is located. This behavior is observed in a number of cases, as can be seen in Figures 8 and S5 (Supporting Information); in some cases the minimum is rather shallow (e.g., HH-09), and in others clearly marked. This feature indicates that, in the minimum region, an electrostatic effect should be found: a close inspection of the binding curves (Figure 4b for HH-18) reveals that in this pH region the separation between these curves is higher. In the absence of electrostatic effect, the curves for all I should merge (master curve); as the electrostatic effect appears (here given essentially by ψD) the curves apart each other. Thus, the extreme in ψD reflects the higher separation between curves in pH region 4-6, which in turn indicates, for low I, that a higher electrostatic Hþ binding is present (less negative QA). In the Flory-Huggins polymer solution theory, a positive enough χ value will lead to poor interaction and phase separation (see for example Hill,47 chapter 21). Briefly, the chemical potential of water inside the gel, μ1, can be found from ΔFsw (eq 1) DΔFsw μ1 ðj2 Þ -μ1 ð0Þ ¼ DN1
ð22Þ T , N2
which, assuming that water is an incompressible fluid, is directly related to the osmotic pressure π: πv1 ¼ μ1 ðj2 Þ -μ1 ð0Þ
ð23Þ
Thus, the osmotic pressure can be obtained and a plot of π vs j2-1 is essentially a diagram akin to a P-V plot for liquid-gas system or one of spreading pressure vs area for a surface phase: for 3142 DOI: 10.1021/la903086s
Figure 8. Model results for dimensionless electrostatic potential of humic particles as a function of pH for different ionic strengths: (a) Suwannee River FA; (b) purified peat HA; (c) purified Aldrich HA. Ionic strengths are in molar units.
certain χ values critical phenomena is predicted, leading for polymer solutions to phase separation.47 In the present case, the situation is more complex, because of the electrostatic contribution to the free energy. A full analysis is beyond the scope of the present work; however, eq 26 can be evaluated numerically, which is shown in Figure 9, as dimensionless pressure Π = πv1/kT as a function of j2-1 (and also as a function of pH in the inset). For PAHA, which has χ = 0.67 (Table 2) it is observed that as j2-1 (volume) increases, the pressure decreases, as expected, but around j2-1 ∼ 6 (pH ∼ 5.5) the slope changes markedly until the curve becomes, for the lowest I, almost horizontal. In a P-V plot, a horizontal line is indicative of a phase transition. In the case of a network, phase separation is not possible, but some sort of transition, as the relatively sharp size increase observed around pH 5-6 in Figure 6, can be present. That can be interpreted as a transition from a relatively close structure to a more open one. In the case of Laurentian Soil FA, having χ = 0.18, no change of slope is observed, and no minimum is found in the Ψ vs pH plot. Langmuir 2010, 26(5), 3134–3144
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Figure 10. Comparison of electrostatic potential predictions
(lines) with data from Duval et al.25 (points) for SR-FA at pH = 10.7 as a function of ionic strength: squares, border potential, ΨB; circles, center potential, ΨC; continuous line, this work; dotted line, Donnan potential predicted by the NICA-Donnan model.
Figure 9. Model results for the dimensionless osmotic pressure of humic gel phase as a function of j2-1 for different ionic strengths: (a) purified Aldrich HA; (b) Laurentian Soil FA. Inset: the same but as a function of pH. Ionic strengths are in molar units.
From a molecular point of view, this behavior can be rationalized observing that the pH where the Ψ minimum and the change of slope in j2-1 is located at approximately the same pH than log K1, suggesting that the transition is found when the carboxylic sites become deprotonated. Thus, the behavior can be attributed to the presence of hydrogen bonding: as the pH is increased from the acidic side of the curve, the humic particle volume is held approximately constant by hydrogen bonding, and consequently increasing the charge density, the Donnan potential and the electrostatic contribution to Hþ binding; when the sites become deprotonated, hydrogen bonds are broken and so the HS expands rapidly, decreasing F, Ψ, and the electrostatic binding. Even when hydrogen bonding is not explicitly accounted for in the EPN model, its influence is included in χ, which is an average interaction parameter. In order to compare the present results with literature data, to the best of our knowledge there are almost no reports of electrostatic potentials (either Donnan-like or surface) as a function of pH or I in HS available. As said before, Duval et al.25 were able to derive potential profiles from mobility measurements, giving some results for the potential at the center, ΨC, and at he border, ΨB, as a function of ionic strength and pH. Figure 10 shows the results for the dependence of both potentials with I for SR-FA at pH = 10.7. Although this pH value is outside the range of experimental data employed here, using the parameters found (Table 2) the Donnan potential predicted by the EPN model was calculated and is shown as a continuous line in Figure 10; the same behavior is observed. For comparison, it was also computed the potential predicted by the NICA-Donnan model, using the optimized parameters for SR-FA (FH-15) given by Milne et al.,15 with the aid of Gustafsson’s program Vminteq.14 Langmuir 2010, 26(5), 3134–3144
Figure 11. Comparison of electrostatic potential predictions
(lines) with data from Duval et al.25 (points) for SR-FA at I = 5 mM as a function of pH: squares, center potential, ΨC; continuous line, this work; dotted line, Donnan potential predicted by the NICA-Donnan model.
The results are shown as a dotted line in Figure 10, where it is seen that this model predicts much lower absolute values. In Figure 11, on the other hand, the pH dependence found by Duval et al. is compared with the EPN and NICA-Donnan predictions; the behavior of EPN is only qualitatively similar to the results of Duval; again, NICA-Donnan predictions are much lower in magnitude. In doing these comparisons, it should be taken into account that Duval et al. used in their derivation humic particle radii calculated with the Stokes-Einstein equation, which precludes quantitative comparisons. It is interesting to note that in the experimental data sets which show this particular behavior, it was reported that the titration data was collected after several cycles of forward (pH increasing) and backward titration, until the curves did not show any hysteresis. Thus, the presence of absence of the minimum in ψD could be related to the aggregation state of the HS. In the analysis of electrostatic binding effects in colloid science, and in the case of HS in particular, a check of validity for a theory is the ability to predict a master curve, that is, correction of the experimental binding curves for the electrostatic effects, should make them to merge on a single curve, which is the master curve. Saito et al.,19 among others, have considered this question and studied several models from this point of view. In the EPN model presented here, due to the procedure involving all the data at different ionic strengths simultaneously, the fact that a good DOI: 10.1021/la903086s
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fitting is achieved implies merging into a master curve (the intrinsic binding curve). Nevertheless, the merging can be verified as follows. Usually, the master curve is constructed by plotting the experimental charges against the “surface” or “internal” pH (depending on the model considered). In the present model, however, part of the HS (gf) is considered to be in equilibrium with the internal gel pH whereas the remainder (1 - gf) is in equilibrium with the bulk solution pH. Because of the convolute nature of eq 13, it is difficult to apply that method. Instead we used an alternative approach consisting in correcting the experimental charges for the amount of bound Hþ due to electrostatic effect; the intrinsic Hþ bound charge would be Z QH, intr ¼ Qmax
¥
KH a H f ðlog KH Þ dðlog K H Þ 1 þ KH a H -¥
ð24Þ
and the difference between the experimental QH and QH,intr will be
Z
¥ -¥
"
QH, exc ¼ QH - QH, intr ¼ gf Qmax # KH aint KH a H H f ðlog KH Þ dðlog KH Þ þ QH, el 1 þ KH a H 1 þ KH aint H ð25Þ
so that the corrected experimental charge is given by Qcorr ¼ QH -QH, exc
ð26Þ
The Qcorr vs pH curves are found to merge at all I values, as is shown in Figure S6 (Supporting Information). Saito et al.19 found that PSM models with an external double layer explicitly considered did not predict master curves, whereas Donnan type models do, such as the Donnan volume model used in combination with NICA isotherm and the Donnan extended volume model. The model presented here is similar to these, but with the added consideration of that a part of the HS interacts with the bulk solution, not with the internal gel.
Concluding Remarks The EPN model is proposed. It is found that it gives good agreement to experimental data with physically reasonable parameter values. The experimental curves are shown to merge into a master curve. The model predicts the humic particle expansion as pH increases and the electrostatic effect through a Donnan potential. This potential shows in several cases a maximum in
3144 DOI: 10.1021/la903086s
absolute value, which is attributed to a transition from a closed to an open gel structure as the carboxylic groups are deprotonated. This is interpreted as being due to presence, at low pH, of hydrogen bonds which are broken as deprotonation proceeds. The main electrostatic effect on Hþ binding is the increase of HS gel sites protonation, rather than free proton accumulation in the gel fraction, which is a negligible contribution in most cases, as found in the calculations. The EPN model gives fitting quality similar to other models for HS, such as WHAM, NICA-Donnan or SHM; however, it provides an improvement in the physical insight on the behavior of HS. The model could be conceivably improved in a number of ways, but at the expense of introducing additional parameters which, in the absence of additional experimental information, would lead to an unreasonable number of adjustable parameters. The EPN model can straightforwardly be applied to metalhumic interaction, including Hþ and other metals competence, by application of eq 14 with due consideration to the respective cation charges, once the Donnan potential is obtained. Acknowledgment. The authors gratefully acknowledge financial support from the Universidad de Buenos Aires (UBACYT 2004-2007 X105 and 2008-2010 X148), the Consejo Nacional de Investigaciones Cientı´ ficas y Tecnicas (CONICET, PIP 05216) and the Agencia Nacional de Promocion Cientı´ fica y Tecnologica (Grant No. 06-12467). F.V.M. is a member of the Carrera del Investigador Cientı´ fico of CONICET. The comments of two anonymous reviewers are also acknowledged. Supporting Information Available: Figures S1 and S2, showing fittings to the proposed model of acid-base titration data for several fulvic and humic acids, detailed in Tables 1 and 2, Figure S3, showing predicted results for humic volume as a function of pH and ionic strength, Figure S4, showing predicted linear expansion (j-1/3) results for several fulvic and humic acids, Figure S5, showing dimensionless Donnan potential (Ψ) results for all fulvic and humic acids not shown in Figure 8, Figure S6, showing results of master curve merging for several fulvic and humic acids, Appendix A, giving a brief summary of Ohshima’s model, Appendix B, giving a brief account of Flory polymer network statistics, Appendix C, giving details of Hill’s derivation of the electrostatic contribution to the free energy, and Appendix D, presenting comparisons with Donnan volume calculations in present models. This material is available free of charge via the Internet at http://pubs.acs.org.
Langmuir 2010, 26(5), 3134–3144
