Langmuir 1996, 12, 6127-6137
6127
Affinity Distribution Description of Competitive Ion Binding to Heterogeneous Materials Miroslav C ˇ ernı´k and Michal Borkovec* Swiss Federal Institute of Technology, ETH-ITO, Grabenstrasse 3, CH-8952 Schlieren, Switzerland
John C. Westall Department of Chemistry, Oregon State University, Corvallis, Oregon 97331 Received January 2, 1996. In Final Form: September 11, 1996X Semiempirical equilibrium models of competitive, two-component binding of simple ions to heterogeneous materials are based on a combination of independent binding isotherms, such as two-component Langmuir isotherms or exchange isotherms. Such models are usually sufficiently flexible to rationalize various experimental data sets quantitatively. Since such models are thermodynamically consistent, they have substantial predictive capabilities. Furthermore, such models are rather simple, may depend on a relatively small number of parameters, and provide an intuitive framework for comparison of binding properties of different materials. We solve the ill-posed data inversion problem with constrained, regularized leastsquares techniques. This approach to the interpretation of two-component competitive ion binding to heterogeneous materials is illustrated with two data sets; the binding of Sr2+ and H+ to ferrihydrite (amorphous FeOOH) and that of Cu2+ and H+ to a humic acid.
Introduction Binding of simple ions to complex, heterogeneous materials (e.g. oxides, clays, soils, polyelectrolytes, humics, proteins, nucleic acids, cell walls) plays an essential role in biochemistry, polymer chemistry, chromatography, material science, and environmental chemistry.1-13 Within these disciplines, many applications require multicomponent equilibrium models that are able to describe quantitatively the partitioning of one or more ions as a function of solution composition (e.g. pH, salt concentration, competing ion concentrations). The classical approach to the development of a binding model builds on the knowledge of the underlying molecular binding mechanism. On the basis of this knowledge one can construct a mechanistic binding model. This approach usually involves an elaborate process, which requires detailed information about the molecular structure of the material and the geometrical arrangement of the individual binding sites and their chemical environments. A * To whom correspondence should be addressed. Phone: (+41) 1 633 6003. Fax: (+41) 1 633 1118. E-mail: borkovec@wawona. vmsmail.ethz.ch. X Abstract published in Advance ACS Abstracts, November 15, 1996. (1) Stumm, W.; Morgan, J. J. Aquatic Chemistry; John Wiley: New York, 1996. (2) Brinker, C. J.; Scherer, G. W. Sol-Gel Science; Academic Press: New York, 1990. (3) Bashford, D.; Karplus, M. Biochemistry 1990, 29, 10219. (4) Yang, A.-S.; Gunner, M. R.; Sampogna, R.; Sharp, K.; Honig, B. Proteins 1993, 15, 252. (5) Honig, B.; Nicholls, A. Science 1995, 268, 1144. (6) Smits, R. G.; Koper, G. J. M.; Mandel, M. J. Phys. Chem. 1993, 97, 5745. (7) Borkovec, M.; Koper, G. J. M. Langmuir 1994, 10, 2863. (8) James, R. O.; Parks, G. A. Surf. Colloid Sci. 1982, 12, 119. (9) Dzombak, D. A.; Morel, F. M. M. Surface Complexation Modeling: Hydrous Ferric Oxide; John Wiley: New York, 1990. (10) Hiemstra, T.; Riemsdijk, W. H.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91. (11) Sverjensky, D. A. Nature 1993, 364, 776. (12) Rudzinski, W.; Charmas, R.; Partyka, S.; Bottero, J. Y. Langmuir 1993, 9, 2641. (13) Westall, J. C. FITEQL: A Program for the Determination of Chemical Equilibrium Constants from Experimental Data. Technical Report; Department of Chemistry, Oregon State University: Corvallis, OR, 1982.
S0743-7463(96)00008-X CCC: $12.00
high degree of sophistication has been reached in this regard in the description of proton binding to proteins.3-5 Related developments can be cited for polyelectrolytes6,7 and, with respect to multicomponent competitive ion binding, for the water-oxide interface.8-13 Indeed, such a mechanistic approach has several advantages: (i) the model reflects actual molecular mechanisms and is based on the proper statistical mechanical description of the systemsthus, the same formalism can be applied, in principle, to all systems of interest; (ii) such models allow predictions of ion partitioning to be made in the absence of the experimental partitioning data themselves;4,7 and (iii) one can also predict the binding characteristics of individual binding sites, for example, the protonation behavior of individual ionizable amino acid residues in a protein.4,5 There are also disadvantages to this mechanistic approach, however: (i) such models may employ a rather involved computational apparatus,4 and (ii) they are based on very detailed structural information at the molecular level. Particularly, the second point represents a serious obstacle for the development of binding models for natural, heterogeneous materials (e.g. soil particles, humic substances, cell walls). At present, structural information is virtually impossible to obtain for such materials. In many cases, however, one is indeed interested in such natural, heterogeneous materials, but one is concerned only with ion partitioning between the aqueous phase and the separable phase. This partitioning can be measured in the laboratory in a straightforward fashion, and such laboratory data can be used to construct a suitable empirical binding model that allows interpolation of the binding data over the solution composition range spanned by the experimental data window. The advantage of such an empirical approach is immediately obvious: the model is constructed solely on the basis of experimentally accessible partitioning data; detailed structural information, which is very hard to come by, is no longer necessary. Partly for this reason, various authors have investigated various empirical models of ion binding.14-20 A promising class of simple empirical models is based on a linear © 1996 American Chemical Society
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C ˇ ernı´k et al.
combination of independent binding isotherms, an idea that dates back to Langmuir.21 Such empirical models are uniquely defined by an affinity distribution, which describes the site concentration as a function of the binding constant of the site. However, it is important to recognize that the determination of such affinity distributions from experimental data is an ill-posed problem.22-24 A given set of experimental data can often be compatible with rather different affinity distributions, and the resulting distribution can strongly depend on the nature of the data errors. This problem can be partly circumvented by standard regularization techniques, which select affinity distributions consistent with some a priori hypothesis about the structure of the distribution (e.g., smooth distribution, small number of discrete sites). This rather promising affinity distribution approach has been applied primarily to one-component binding.14-17 Generalization of this approach to two or more components has been discussed in the context of gas adsorption,25 and very recently, such ideas were also used to examine ion binding in solution.18-20 In this article we explore the use of regularized and constrained least-squares techniques for the calculation of affinity distributions for binding data in aqueous suspensions of heterogeneous materials (e.g., humic substances, oxides) where concentrations of two solution components are varied. Mathematical Description Consider equilibrium binding of two components A and B from a liquid to a separable phase. Later we shall discuss competitive binding of Sr2+ and H+ to ferrihydrite as an example. Let us represent the component activities (or free concentrations at constant ionic strength) by cA and cB (both in mol/L). The corresponding bound amounts, which are denoted by qA(cA,cB) and qB(cA,cB) (in mol/g), are functions of both component concentrations if competitive binding processes are considered. Applying Langmuir’s idea that adsorption can be interpreted as the result of binding to independent sites,21 we write the binding isotherms as M
qA(cA,cB) )
siφA,i(cA,cB) ∑ i)1
(1)
∑ i)1
siφB,i(cA,cB)
abbreviationa L11
(2)
where si (in mol/g) denotes the total concentration of site i (i ) 1, ..., M), while φA,i(cA,cB) and φB,i(cA,cB) are local isotherms in units of the total site concentration for (14) van Riemsdijk, W. H.; Koopal, L. K. In Environmental Particles; Buffle, J., van Leeuwen, H. P., Eds.; Lewis Publishers: Chelsea, 1992. (15) Stanley, B. J.; Bialkowski S. E.; Marshall D. B. Anal. Chem. 1993, 65, 259. (16) Stanley, B. J.; Guiochon, G. J. Phys. Chem. 1993, 97, 8098. (17) C ˇ ernı´k, M.; Borkovec, M.; Westall, J. C. Environ. Sci. Technol. 1995, 29, 413. (18) Koopal, L. K.; van Riemsdijk, W. H.; de Wit, J. C. M.; Benedetti, M. F. J. Colloid Interface Sci. 1994, 166, 51. (19) Benedetti, M. F.; Milne, C. J.; Kinniburgh, D. G.; van Riemsdijk, W. H.; Koopal, L. K. Environ. Sci. Technol. 1995, 29, 446. (20) Westall, J. C.; Jones, J. D.; Turner, G. D.; Zachara, J. M. Environ. Sci. Technol. 1995, 29, 951. (21) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (22) Tikhonov, A. N.; Goncharsky, A. V. Ill-posed problems in the Natural Sciences; MIR: Moscow, 1987. (23) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213. (24) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in FORTRAN, The art of scientific computing, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992. (25) Jaroniec, M. Adv. Colloid Interface Sci. 1983, 18, 149.
name
reaction
competitive Langmuir site A + X h AX B + X h BX specific Langmuir site A + X h AX specific Langmuir site B + X h BX exchange site (one-to-one) B + AX h A + BX competitive Langmuir site A + X h AX 2B + X h B2X specific Langmuir site 2B + X h B2X exchange site (one-to-two) 2B + AX h A + B2X
L10 L01 E11 L12 L02 E12
a The letter “E” refers to an exchange site, while “L” refers to a Langmuir site. The first and second number represent the stoichiometric coefficients in the reaction of the first and second components, respectively.
component A and B, respectively.14,17,25 Each local isotherm φA,i(cA,cB) or φB,i(cA,cB) additionally depends on one or two equilibrium constants. The set of all site concentrations then defines the affinity distributions and, together with the local isotherms, the binding model. (If one of the components is H+, the corresponding isotherm may also involve an additive constant which is related to the initial protonation state and the total number of ionizable sites of the material.17) Once we have decided on particular types of local isotherms, we evaluate the corresponding affinity distribution from experimental data. A two-component data set is of the form (j) (j) q(j) for j ) 1, ..., NA A ,cA ,cB
(3)
(j) (j) for j ) 1, ..., NB q(j) B ,cA ,cB
(4)
The model parameters are determined with a leastsquares technique by minimizing a suitable objective function χ2(x1,...,xL) which depends on several variables x1, ..., xL such as the total site concentrations si and, in general, the corresponding equilibrium constants. The objective function is usually taken to be the sum of squares of the difference between the experimental and calculated bound amounts13,24
χ2(x1,...,xL) ) NA
M
qB(cA,cB) )
Table 1. Summary of the Various Sites Considered
∑ j)1
[
] [
(j) (j) q(j) A - qA(cA ,cB )
2
NB
+
σ(j) A
∑ j)1
]
(j) (j) q(j) B - qB(cA ,cB )
σ(j) B
2
(5)
(j) where σ(j) A and σB are estimates of the standard deviations (j) (j) of qA and qB , respectively. Due to the ill-posed nature of this optimization problem, this scheme to determine the affinity distributions often leads to numerical instabilities.17,24 The better approach is to introduce regularization and bounds. The application of such techniques to one-component situations, where the local isotherm is represented by the Langmuir isotherm, has been discussed recently.17 Here we discuss the two-component case, where the choice of local isotherms is more complicated. Local Isotherms. In order to construct the set of local isotherms φA,i(cA,cB) and φB,i(cA,cB) (cf. eqs 1 and 2), we shall rely on the same intuitive approach that leads to the Langmuir isotherm in one-component situations. We picture each site as a real, physical site which obeys an underlying binding reaction. Various possibilities are discussed below and summarized in Table 1. Competitive Langmuir Sites (L11). Focus first on the simplest picture of competitive Langmuir sites. Binding of two components leads to the reactions
Competitive Ion Binding
Langmuir, Vol. 12, No. 25, 1996 6129
A + X h AX
(6)
B + X h BX
(7)
where X represents a competitive Langmuir site (L11, see Table 1). For brevity, we have dropped the index i which labels all different sites X. Reactions 6 and 7 lead to two-component, competitive Langmuir isotherms
φA(cA,cB) ) φB(cA,cB) )
KAcA 1 + KAcA + KBcB
(8)
KBcB 1 + KAcA + KBcB
(9)
where KA and KB (both in L/mol) are the equilibrium constants of eqs 6 and 7, respectively. To construct the mass action laws, we always represent the activities of bound species by mole fractions, as is the convention used in ion-exchange modeling. The denominator in eqs 8 and 9 is a sum of three terms. If one of these terms is negligible compared to the other two, we obtain one of three limiting forms discussed below. Specific Langmuir Sites (L01, L10). The first limiting form of eqs 8 and 9 is recovered by considering the case where the bound amount of B is negligible (KBcB , 1 and KBcB , KAcA, or KB f 0). In this case, we can consider solely the reaction
A + X h AX
(10)
where X now represents a specific Langmuir site which binds component A only (L10). (Recall that we have dropped the index; the symbol X in eq 10 refers to a different type of site than the symbol X in eqs 6 and 7. This notation will be used throughout.) For component A, the corresponding local isotherm is given by the classical one-component Langmuir isotherm
φA(cA,cB) )
K A cA 1 + KAcA
(11)
where KA (in L/mol) is the equilibrium constant of eq 10. For component B
φB(cA,cB) ) 0
(12)
An analogous argument applies to the second limiting case where the roles of A and B are interchanged (L01). Exchange Sites (E11). The last limiting case is obtained by considering the case where the fraction of unoccupied binding sites is negligible (KAcA . 1 and KBcB . 1, or KA f ∞ and KB f ∞). In this case, one recovers the one-toone exchange reaction
B + AX h A + BX
A + X h AX
(16)
2B + X h B2X
(17)
where X is a competitive Langmuir site (L12). Note that, in this reaction, two units of the monovalent component B are bound simultaneously and not stepwise. From these reactions we obtain the following local isotherms
φA(cA,cB) )
φB(cA,cB) )
KAcA 1 + KAcA + K′Bc2B 2K′Bc2B 1 + KAcA + K′Bc2B
cA φA(cA,cB) ) cA + KBAcB
(14)
KBAcB cA + KBAcB
(15)
where KBA ) KB/KA (dimensionless) is the equilibrium constant of eq 13.
(18)
(19)
where KA (in L/mol) and K′B (in L2/mol2 ) are the equilibrium constants of eqs 16 and 17, respectively. Again, three limiting forms can be considered. Specific Langmuir Sites (L02). In the first case, where the bound amount of component B is negligible, we recover specific Langmuir sites (L10) as discussed further above (cf. eqs 10-12). In the second case, where the bound amount of component A is negligible (KAcA , 1 and KAcA , K′Bc2B, or KA f 0), one obtains the reaction
2B + X h B2X
(20)
where X are specific Langmuir sites for component B (L02). The corresponding local isotherms are given by
φA(cA,cB) ) 0 φB(cA,cB) )
2K′Bc2B 1 + K′Bc2B
(21) (22)
where K′B (in L2/mol2) is the equilibrium constant of eq 20. Exchange Sites (E12). In the third limit, where the fraction of unoccupied binding sites is negligible (KAcA . 1 and K′Bc2B . 1, or KA f ∞ and K′B f ∞), one recovers a one-to-two exchange reaction
2B + AX h A + B2X
(23)
where X represents an exchange site (E12). The corresponding local isotherms read
(13)
where X now represents an exchange site (E11). The corresponding local isotherms read
φB(cA,cB) )
Competitive Langmuir Sites (L12). Besides the simplicity of the model, there is no particular reason to assume that sites bind both components only in a monodentate fashion. To illustrate this point, let us consider a site that binds one unit of the (divalent) component A but two units of the (monovalent) component B. For example, this site binds one Cu2+ or two H+. This scenario leads to the reactions
φA(cA,cB) )
φB(cA,cB) )
cA cA + K′BAc2B 2K′BAc2B cA + K′BAc2B
(24)
(25)
where K′BA ) K′B/KA (in L/mol) is the equilibrium constant of eq 23. While the isotherms given in eqs 24 and 25 are very similar to the ion-exchange isotherms which are derived from the classical Gaines-Thomas convention, we note that their mathematical form is different and that the reaction given in eq 23 is not the same as the
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reaction on which the classical Gaines-Thomas convention is based.1 Determination of Affinity Distributions. We have found that two different models can be used to interpret most experimental data in a satisfactory fashion. (i) Models based on competitive Langmuir sites (L11, L12), which result in competitive Langmuir models, are characterized by affinity distributions which depend on two affinity constants. (ii) Models based on various types of exchange sites (E11, E12) and specific Langmuir sites (L01, L02) are referred to as exchange models, since binding to specific Langmuir sites can be viewed as exchange with the solvent. The exchange model is characterized by one or more affinity distributions which depend on a single affinity constant. These sites do not exhibit simultaneous three-way competition among an unoccupied site, occupancy by the first component, and occupancy by the second component but can be viewed as exhibiting twoway competitions between two of these three states. As we shall see further below, this observation can be used to simplify the description of the binding process. Once we decide on the appropriate combination of local isotherms, we use constrained and regularized leastsquares methods to minimize eq 5 for the determination of the corresponding affinity distributions. In all cases, we employ non-negativity constraints (bounds) on the total site concentrations si. Stabilization of the resulting affinity distributions is achieved by introducing further regularization. We use two types of regularization methods: (i) regularization for a small number of sites, which leads to discrete distributions consisting of a small number of isolated peaks, and (ii) regularization for smoothness, which leads to gradually varying distributions. With a sufficiently fine grid, the regularization for smoothness is usually used to generate smooth and continuous distributions.17,24 We have applied two different types of solution schemes, which employ either a fixed grid or a variable grid for the equilibrium constants. In both cases, these grids can be one- or two-dimensional. One-dimensional grids are used for sites which are characterized by a single equilibrium constant (exchange model), while two-dimensional grids are necessary if the local isotherms depend on two equilibrium constants (competitive Langmuir model). The limits of these grids are finite and define the computational window which is calculated on the basis of the experimental data. As in the case of binding of one-component, a fixed grid spacing of 1.0 on a logarithmic scale (base 10) is sufficient for accurate representation of most data sets. Additional details are given in the Appendix and ref 17. Thermodynamic Consistency. One important facet of a pair of two-component equilibrium binding isotherms qA(cA,cB) and qB(cA,cB), which is sometimes overlooked, is that these isotherms are not independent but satisfy the thermodynamic consistency relation (Maxwell relation or Gibbs-Duhem relation)26,27
( ) ( ) ∂qB ∂ ln cA
)
cB
∂qA ∂ ln cB
(26)
cA
Equation 26 implies that complete experimental records of qA(cA,cB) and qB(cA,cB) contain redundant information. This redundancy can be seen in the following fashion. Suppose that the entire isotherm qA(cA,cB) is known as a function of cA and cB. Suppose further that the isotherm (26) Kemball, C.; Rideal, E. K.; Guggenheim, E. A. Trans. Faraday Soc. 1948, 44, 948. (27) Franses, E. I.; Siddiqui, F. A.; Ahn, D. J.; Chang, C.-H.; Wang, N.-H. L., Langmuir 1995, 11, 3177.
qB(c(0) A ,cB) of the other component is known as a function of cB but only at one fixed concentration c(0) A . This information is sufficient to construct the rest of the isotherm. Integration of eq 26 shows that
qB(cA,cB) ) qB(c(0) A ,cB) +
∫cc
A (0) A
( )( ) cB ∂qA c′A ∂cB
dc′A (27)
cA)c′A
For example, this relation says that all necessary information about the binding of Sr2+ and H+ is contained in experimental records of bound Sr2+ as a function of Sr2+ and H+ in solution, and a single acid-base titration curve (bound H+ as a function of H+ in solution) in the absence of Sr2+. Titration curves with additions of Sr2+ are thus not really necessary but, of course, extremely useful as a check of the thermodynamic consistency of the entire data set. One should note, however, that the thermodynamic consistency relation eq 26 applies only if cA and cB represent the activities (or free concentrations at constant ionic strength) of the species in question. If these quantities are interpreted as total solution concentrations, this relation no longer applies. This fact is particularly important if complexation in solution takes place. Derivation of local isotherms from an underlying chemical reaction, as we have done above, has its advantages. In doing so, we automatically ensure that the resulting isotherms always satisfy the thermodynamic consistency relation eq 26. This equation is readily verified for pairs of competitive isotherms derived in the previous section (cf. eqs 8 and 9, eqs 11 and 12, eqs 14 and 15, eqs 18 and 19, eqs 21 and 22, eqs 24 and 25). If one is able to fit the experimental binding data where amounts adsorbed were measured for both components by the proposed models, the data set is also thermodynamically consistent. Due to the thermodynamic consistency relation, such models are no longer “mere data fitting exercises” but can be used to make predictions within a given data set. Because of these predictive capabilities, these models are superior to an entirely “empirical” model, and we shall thus refer to a “semi-empirical” model. Analysis of Experimental Data We have successfully applied these affinity distribution methods to interpret two-component experimental data for binding of ions to a wide variety materials including oxides, fulvic and humic substances, polyelectrolytes, proteins, and soil particles; a selection of these results will be given elsewhere.28 Here, we focus on techniques for the derivation of affinity distributions for these systems and base our discussion on two experimental data sets only, namely binding of Sr2+ and H+ to ferrihydrite29,30 and of Cu2+ and H+ to a humic acid.19 These examples are used to illustrate various isotherms and regularization methods, as well as to demonstrate the flexibility of the present class of binding models. No attempt was made to present a detailed comparison to other modeling approaches.9,19 Binding of Sr2+ and H+ by Ferrihydrite.29,30 Focus now on the binding of Sr2+ and H+ to ferrihydrite (amorphous FeOOH) in 1.0 mol/L NaNO3 as reported by Kolarˇ´ık.29 Experimental data are shown in Figure 1a as they were originally published, namely as fraction of Sr2+ bound as a function of pH for different total Sr2+ (28) Borkovec, M.; Rusch, U.; Westall, J. C. In Sorption of Metals by Earth Materials; Jenne, E. A., Ed.; Academic Press: New York, in press. (29) Kolarˇ´ık, Z. Collect. Czech. Chem. Commun. 1962, 25, 938. (30) Pyman, M. A. F.; Posner, A. M. J. Colloid Interface Sci. 1978, 66, 85.
Competitive Ion Binding
Langmuir, Vol. 12, No. 25, 1996 6131 Table 2. Semiempirical Models for Sr2+ and H+ Binding Data to Ferrihydrite goodness of fit
model no.
sitea
gridb
1 2 3 4 5 6
L11 L11 L11 E11, L01 E11, L01 E11, L01
fixed at 1.0 variable fixed at 0.4 fixed at 1.0 variable fixed at 2.0
regc
Figure Table (Sr2+)d
2a, 3a SNS 2b, 3b SMO 3c 2c SNS 2d, 4 SMO 5
3 4
0.065 0.081 0.075 0.095 0.106 0.119
(H+)e 0.0023 0.0070 0.0034 0.0013 0.0040 0.0085
a Types of sites as defined in Table 1. b Grid spacing on a base 10 logarithmic scale. c Regularization toward smoothness (SMO) and a small number of sites (SNS). d Mean of the absolute values of the relative deviations. e Mean of the absolute values of the absolute deviations normalized by the experimental window.
considered, we can identify the total concentrations in solution with the activities and understand the obtained binding constants as conditional at the ionic strength considered. Competitive Langmuir Model. Let us first interpret this data set in terms of the following reactions (cf. eqs 6 and 7)
Sr2+
Figure 1. Experimental data of competitive binding of and H+ to ferrihydrite in 1.0 mol/L NaNO3. (a) Relative amount of Sr2+ bound as a function of pH for different total Sr2+ concentrations at an Fe concentration of 0.10 mol/L.29 (b) Acidbase titration in the same medium.30 Lines serve to guide the eye.
concentrations. Note that the data cover a very wide concentration range. To complete this data set, we need information about H+ binding. We use the acid-base titration in the same medium in the absence of Sr2+ as determined by Pyman and Posner.30 Their data are shown in Figure 1b. While the ferrihydrite used by Pyman and Posner is not exactly the same ferrihydrite as used by Kolarˇ´ık, the binding properties of ferrihydrite are known not to vary widely from sample to sample.9 Note that we have no information about the acid-base behavior in the presence of Sr2+, but due to the thermodynamic consistency relation eq 26, this data set contains all the information necessary. However, the binding isotherms must be interpreted as a function of solution activities. As the data in this particular system were all recorded at constant ionic strength and Sr2+ does not hydrolyze in the pH range
Sr + X h SrX
(28)
H + X h HX
(29)
where X is a competitive Langmuir site (L11). In our notation, we have dropped the ionic charges for simplicity, as charges are not included explicitly in the present description of ion binding. To start, we analyze the data with a nonregularized fit on a fixed grid (model 1 in Table 2). As shown in Table 2 and Figure 2a, the fit is excellent. The resulting affinity distribution, which depends on two affinity constants, is displayed in Figure 3a and consists of 23 sites. Only 9 sites represent true competitive Langmuir sites (L11), as they lie completely inside the computational window (solid rectangle in Figure 3a). Most of these sites, however, lie on the border of the window and thus correspond to limiting cases discussed further above. We shall return to this key point below, but for the moment, let us proceed with the analysis without paying any attention to this fact. Nonregularized affinity distributions are very unstable and sensitive to small changes in input data.17 The solution can be stabilized by introducing further regularization techniques, as described in the Appendix. The first possibility is regularization toward a small number of sites (model 2 in Table 2). As evident from Table 2 and
Figure 2. Experimental data for binding of Sr2+ and H+ to ferrihydrite with best fits of the different models. Data are from Figure 1. While the individual models are rather different, the fit of the data is satisfactory in all casessthis feature is typical of an ill-posed problem. (a) Competitive Langmuir model (L11) without regularization (model 1 in Table 2; for the affinity distribution, see Figure 3a). (b) Competitive Langmuir model (L11) regularized for a small number of sites (model 2 in Table 2; for the affinity distribution, see Figure 3b or Table 3). (c) Exchange model (L01, E11) without regularization (model 4 in Table 2). (d) Exchange model (L01, E11) regularized for a small number of sites (model 5 in Table 2; for the affinity distribution see Figure 4 or Table 4).
6132 Langmuir, Vol. 12, No. 25, 1996
C ˇ ernı´k et al. Table 3. Competitive Langmuir Model of Sr2+ and H+ Binding to Ferrihydrite with a Small Number of Sites (Model 2 in Table 2)
Sr + X h SrX
KSr
H + X h HX
KH
log10 KSra
log10 KHa
sb (mmol/g of Fe)
-1.00 0.00 1.00 3.18 4.18 5.18 7.16 8.16
6.31 10.07 7.85 12.90 9.25 5.47 10.17 15.00
5.46 × 10-1 5.71 × 10-1 4.77 × 10-1 2.48 3.60 × 10-1 1.00 × 10-4 1.48 × 10-2 1.31
a In L/mol. b Additive constant of -4.98 mmol/g of Fe was added to the H+ binding isotherm.
Mathematical Description. Thus, one may try to represent the data just in terms of the following reactions (cf. eqs 10 and 13)
H + X h HX
(30)
where X represents a specific Langmuir site (L01), and
H + SrX h Sr + HX
Figure 3. Two-dimensional affinity distributions of the competitive Langmuir model (L11) of Sr2+ and H+ binding to ferrihydrite. The dashed rectangle represents the experimental window, and the solid rectangle the computational window (as discussed in the Appendix). (a) No regularization (model 1 in Table 2). (b) Regularization for a small number of sites (model 2 in Table 2; the same affinity distribution is also given in Table 3). Black circles refer to individual sites; their size is proportional to the logarithm of the site concentration. (c) Regularization for smoothness (model 3 in Table 2). The continuous affinity distribution is normalized to unity; the contour interval is 0.005.
Figure 2b the fit remains very good. The resulting affinity distribution, which is given in Figure 3b and Table 3, consists of 8 sites only. The advantage of this description is that it is based on a simple binding mechanism and involves a rather small number of parameters. The second possibility is to regularize toward smooth affinity distributions (model 3 in Table 2). As apparent from Table 2, the fit also remains very good. The resulting distribution, which is shown in Figure 3c, is characterized by two broad peaks. While graphically appealing, such distributions do not seem to us to be very useful, mainly because of the large number of parameters involved. Exchange Model. As already mentioned above, the nonregularized fit yields many sites on the borders of the computational window (see Figure 3a). These sites represent two kinds of either specific Langmuir sites (left and bottom border) or exchange sites (top and right border), corresponding to the three limiting cases discussed in the
(31)
where X represents an exchange site (E11). We have also considered specific Langmuir sites for Sr2+ (L10), but their concentrations turn out to be zero. This result is consistent with Figure 3; no sites appear at the bottom of the window. First we have performed a nonregularized fit on fixed grids for the exchange model (model 4 in Table 2). As evident from Table 2 and Figure 2c, the fit remains excellent. The fit is essentially as good as the nonregularized fit based on the competitive Langmuir model (model 1 in Table 2, Figure 2a). This surprising result seems to apply rather generally, and most experimental binding data can be rationalized in terms of the exchange model. Another example will be discussed further below; a few additional examples are presented elsewhere.28 As discussed above, the resulting affinity distributions can be regularized in various ways. The first possibility is regularization toward a small number of sites (model 5 in Table 2). As shown in Table 2 and Figure 2d the fit remains very good. The resulting distributions, which are given in Figure 4 or Table 4, consist of 8 sites. This semiempirical model is now very attractive, as it uses a simple mechanism and a small number of parameters and is easily displayed graphically. The second possibility is to regularize for smoothness. This regularization could be done on a fine grid, and one would obtain two smooth, continuous affinity distributions. Here, we examine another interesting possibility; regularization for smoothness on a coarse grid (model 6 in Table 2). As indicated in Table 2, the fit also remains very good. The resulting affinity distributions, which are shown in Figure 5, now vary only gradually. This model still involves a moderate number of parameters, but since the logarithms of the equilibrium constants are equally spaced, the description is more suitable for comparisons of binding properties of different materials. Binding of Cu2+ and H+ by Humic Acid.19 Consider now the experimental data set of Cu2+ and H+ binding to a purified peat humic acid in 0.1 mol/L KNO3 by Benedetti et al.19 This substance is a natural, heterogeneous polyelectrolyte. These authors have measured the activities of the free Cu2+ ion directly with a Cu2+-selective electrode over a wide range; the amount bound to the
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Langmuir, Vol. 12, No. 25, 1996 6133
Figure 4. Small number of sites affinity distributions for the exchange model (L01, E11) for Sr2+ and H+ binding to ferrihydrite (model 5 in Table 2; the same affinity distribution is also given in Table 4). Sr2+
Table 4. Exchange Model of and Binding to Ferrihydrite with a Small Number of Sites (Model 5 in Table 2) H + X h HXb
H + SrX h Sr + HXc
H+
log10 K
sa (mmol/g of Fe)
4.12 6.55 8.03 9.79 2.91 5.29 7.27 10.0
2.25 4.60 × 10-1 5.24 × 10-1 7.92 × 10-1 1.40 × 10-2 5.35 × 10-1 1.45 2.62
a Additive constant of -5.75 mmol/g of Fe was added to the H+ binding isotherm. b Specific Langmuir sites for H+; equilibrium constant in L/mol. c Exchange site; equilibrium constant is dimensionless.
humic acid was calculated by difference, taking into account copper complexation in solution at fixed ionic strength. In Figure 6 the experimental data points from Benedetti et al.19 are compared with various semiempirical models which will be discussed further below. Each column in Figure 6 corresponds to a different model; each row (labeled as a-c), to one subset of the experimental data. In Figure 6a we show Cu2+ bound as a function of free Cu2+ activity at three different pH values, in Figure 6b we show the acid-base titration curve in the absence of Cu2+ (selected, equidistant data points), and in Figure 6c we show differential H+/Cu2+ exchange ratios as a function of Cu2+-free activity for two different pH values.
Figure 5. Smooth affinity distributions on coarse grids for the exchange model (L01, E11) for Sr2+ and H+ binding to ferrihydrite (model 6 in Table 2).
We have constructed all models solely on the basis of the Cu2+ binding data and the acid-base titration curve in the absence of Cu2+ (i.e. data points shown in Figure 6a and b). Later we shall discuss the predictions of the exchange ratios based on these models. The experimental exchange ratio data shown in Figure 6c were not used for the construction of the models. Competitive Langmuir Model. In contrast to the previous Sr2+ data set, a nonregularized fit with the competitive Langmuir model, which is based on sites which bind one Cu2+ and one H+ (L11), leads to an extremely poor description of the Cu2+-binding data (model 1 in Table 5). We conclude that the experimental Cu2+-binding data are incompatible with this model. An alternative competitive Langmuir model is based on sites which bind one Cu2+ and two H+ (L12), with corresponding reactions (cf. eqs 16 and 17)
Cu + X h CuX
(32)
2H + X h H2X
(33)
where X is a competitive Langmuir site (L12). On the basis of this mechanism, we perform a nonregularized fit (model 2 in Table 5). As evident from Table 5 and Figure 6a and b (left), the resulting fit is excellent. It is also possible to regularize this solution toward a smooth distribution or towards small number of sites. These results are not described here in detail. In the latter case, it is possible to rationalize the experimental data with 12 competitive Langmuir sites. However, this model
Table 5. Semiempirical Models for Cu2+ and H+ Binding Data to Peat Humic Acid goodness of fit
model no.
sitea
gridb
1 2 3 4 5
L11 L12 E11, E12, L01 E11, E12, L01 E11, E12, L01
fixed at 1.0 fixed at 1.0 fixed at 1.0 variable fixed at 1.0
regc
Figure
Table
6 (left) SNS SMO
6 (middle) 6 (right), 7
6
(Cu2+)d
(H+)e
0.386 0.053 0.117 0.124 0.138
0.0015 0.0029 0.0012 0.0050 0.0012
a Types of sites as defined in Table 1. b Grid spacing on a base 10 logarithmic scale. c Regularization toward smoothness (SMO) and a small number of sites (SNS). d Mean of the absolute values of the relative deviations. e Mean of the absolute values of the absolute deviations normalized by the experimental window.
6134 Langmuir, Vol. 12, No. 25, 1996
C ˇ ernı´k et al.
Figure 6. Binding of H+ and Cu2+ by peat humic acid at 0.1 mol/L KNO3. Symbols are experimental data points from Benedetti et al.19 Different subsets of the experimental data set are arranged in rows. (a) Binding of Cu2+ as a function of free Cu2+ activity at various pH values. (b) Acid-base titration curve in the absence of Cu2+. (c) Differential H+/Cu2+ exchange ratios as a function of free Cu2+ activity at various pH values. All models are constructed on the basis of Cu2+ binding isotherms at various pH values and acid-base titration curves (experimental data shown in parts a and b). Exchange ratio data (shown in c) were not used. Lines are calculated and represent (a and b) the best fit or (c) independent predictions for the following three models: the nonregularized competitive Langmuir model (left column, model 2 in Table 5), the exchange model regularized for a small number of sites (middle column, model 4 in Table 5), and the exchange model regularized for smoothness (right column, model 5 in Table 5).
does not provide an optimally smooth interpolation of the data. The problem arises mainly due to the large pH gap between individual Cu2+-binding isotherms. Exchange Model. As in the previous Sr2+ data set, we can rationalize the experimental data in terms of independent exchange sites and specific Langmuir sites. However, we now also have to consider one-to-two exchange reactions. The reaction mechanism involves (cf. eqs 10, 13, and 20)
H + X h HX
H + CuX h Cu + HXc
2H + CuX h Cu + H2Xd
(35)
where X represents an exchange site (E11), and
2H + CuX h Cu + H2X
H+Xh
HXb
(34)
where X represents a specific Langmuir site (L01),
H + CuX h Cu + HX
Table 6. Exchange Model of H+ and Cu2+ Binding to Peat Humic Acid with a Small Number of Sites (Model 4 in Table 5)
(36)
where X represents an exchange site (E12). On the basis of this mechanism, we perform a nonregularized fit on fixed grids (model 3 in Table 5). As indicated in Table 5 the fit is satisfactory. Within the exchange model the fit cannot be improved significantly, either by including other types of exchange sites or by decreasing the grid spacing. Again, this model was regularized toward various directions. First, we have performed regularization for a small number of sites (model 4 in Table 5). As shown in Table 5 and Figure 6a and b (middle), the resulting fit
log10 K
sa (mmol/g)
3.59 5.24 7.26 9.04 -4.81 -2.37 -0.85 1.12 0.27 2.56 4.44 6.15
1.22 8.76 × 10-1 7.01 × 10-1 9.57 × 10-1 2.35 × 10-3 3.30 × 10-2 1.28 × 10-1 1.30 4.29 × 10-3 6.07 × 10-3 3.67 × 10-2 9.22 × 10-2
a Additive constant of 5.60 mmol/g was added to the H+ binding isotherm. b Specific Langmuir site for H+ (L01); equilibrium constant in L/mol. c One-to-one exchange site (E11); equilibrium constant is dimensionless. d One-to-two exchange site (E12); equilibrium constant in L/mol.
remains satisfactory. The resulting affinity distributions, which are given in Table 6, consist of 12 sites. Second, we have regularized for smoothness on a coarse grid (model 5 in Table 5). As shown in Table 5 and Figure 6a and b (right), the resulting fit does not change substantially. The resulting affinity distributions are shown in Figure 7. Note that the fit based on the competitive Langmuir model is somewhat better than the fit based on the exchange models, as evident from Figure 6a and b (left and middle).
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Langmuir, Vol. 12, No. 25, 1996 6135
shown in Figure 6c (right), yields the best prediction of the exchange ratios. The differences between the predictions of various models can be interpreted as follows. Let us hypothesize that the experimental data contain some minor experimental errors (originating from electrode drifts, kinetic effects, etc.) and the exchange model is sufficient to represent the binding process. The larger flexibility of the competitive Langmuir model may then allow a good fit of the experimental data including the minor experimental errors. The exchange model, being less flexible, cannot fit the data perfectly. In a sense, this model “filters” out the experimental errors and thus yields a better prediction of the exchange ratio. The minor differences in the prediction of both exchange models are easily understood. Regularization for a small number of sites will result in less smooth isotherms, especially in the present case of wide pH gaps between the experimental Cu2+ isotherms. Regularization for smoothness leads, in spite of these pH gaps, to very smooth isotherms, and the exchange ratio also becomes a smooth function. Discussion
Figure 7. Affinity distribution of the exchange model regularized for smoothness of Cu2+ and H+ binding to peat humic acid (model 5 in Table 5). (a) Specific Langmuir sites for H+ (L01). (b) One-to-one exchange sites (E11). (c) One-to-two exchange sites (E12).
Prediction of H+/Cu2+ Exchange Ratio. Consider the differential H+/Cu2+ exchange ratio at constant pH
( ) ∂qH ∂qCu
cH
(∂qH/∂cCu)cH )
(∂qCu/∂cCu)cH
(37)
which is a function of free Cu2+ activity. Benedetti et al.19 have estimated these exchange ratios from their experimental data based on a first-order difference approximation to eq 37. Their data points are shown in Figure 6c and compared with predictions of our models which are based on direct evaluation of eq 37. Recall that the experimental exchange ratio data were not used to construct these models. Inserting eq 26 into eq 37, we observe that the exchange ratio is uniquely given by the Cu2+-binding isotherm. If the experimental data would be sufficiently dense and accurate, any thermodynamically consistent model must be able to predict the exchange ratios precisely. Indeed, from Figure 6c we also conclude that all models are able to predict the correct range of the exchange ratios. However, there are substantial differences between the three models. The prediction of the exchange ratio by the nonregularized competitive Langmuir model (L12), which is shown in Figure 6c (left), is the poorest for all discussed models. Recall that this model yields an almost perfect fit of the Cu2+-binding data. On the other hand, the exchange models, which cannot fit the Cu2+-binding data perfectly, predict the exchange ratios very well. The prediction of the exchange ratio by the exchange model with the small number of sites, which is shown in Figure 6c (middle), exhibits some oscillations. The prediction of the exchange model regularized for smoothness, which is
In this paper we have discussed how two-component, competitive ion binding processes to heterogeneous materials (e.g., humic substances, oxides) can be modeled with a linear combination of independent binding isotherms. The affinity distributions associated with these isotherms are determined by regularized least-squares techniques. Various types of local isotherms can be used. One approach is based on a combination of competitive Langmuir isotherms. In this case, for two-component binding data, a binding site will be in one of three states: occupied by the first component, occupied by the second component, or unoccupied. Because each competitive Langmuir site involves two affinity constants, the affinity distribution is a function of two variables. Models based on such two-variable distributions generally do represent the experimental data satisfactorily, but such distribution functions are somewhat unwieldy and inconvenient for practical applications. An alternative approach is based on a combination of exchange sites and specific Langmuir sites. In this case, sites do not exhibit simultaneous three-way competition among occupancy by the first component, occupancy by the second component, and being unoccupied, but only two-way competitions between two of these three states. As a result, each affinity distribution is a function of only a single variable; the corresponding affinity distributions are practically more convenient than two-variable distributions. Furthermore, specific Langmuir sites are often necessary only for H+. This result reflects the fact that most metal binding sites are also weak acids or bases, but some of the weak acids and bases are not also metal binding sitessthis behavior is observed for complexing agents in solution as well. This observation applies to a wide variety of heterogeneous materials and reflects the classical interpretation of binding processes in terms of ion exchange. In this article, we have illustrated this observation with two examples; a few additional examples will be discussed elsewhere.28 Because of the ill-posed nature of the inversion problem, the application of regularization techniques is necessary. Two techniques that appear to be most promising for the applications discussed here are (i) regularization for a small number of sites and (ii) regularization for smoothness on a coarse grid. These approaches may have substantial potential for modeling and for phenomeno-
6136 Langmuir, Vol. 12, No. 25, 1996
logical characterization of binding properties of heterogeneous materials (see Figures 4, 5, and 7). Nonetheless, the ill-posedness implies that experimental binding data can be described by rather different models and affinity distributions (see Figure 2). The wide variety is rather striking, and for this reason, one remains skeptical whether conclusions about the actual molecular mechanism can be inferred only from a good fit of experimental binding data. Nevertheless, a few indications about such mechanisms appear to be clear-cut, for example, the necessity to use one-to-two exchange sites or the absence of specific Langmuir sites for metal cations. It is essential to stress, however, that sites derived from such semiempirical models do not correspond to any real physical sites on the surface; they should be rather viewed as hypothetical sites, the binding properties of which are essentially indistinguishable from the binding properties of the material under investigation (in the window of concentrations in which experimental data are available). The same caveat applies to distributions derived from consideration of one-component binding.7 The thermodynamic consistency26,27 of the semiempirical models described here incorporates substantial predictive power within a given data range. If the bound amount of one component is known as a function of the activity of both components in solution, we have considerable information about the bound amount of the other component due to the thermodynamic consistency relation eq 26. One argues similarly in solution chemistry as well.31 Particularly, if we can neglect specific Langmuir sites for this second component, then the bound amount of the second component follows. This feature was illustrated here with the successful prediction of the H+/Cu2+ exchange ratio shown in Figure 6c (right); other examples will be given elsewhere.28 In this paper we have shown that it is often possible to represent two-component, competitive binding data to heterogeneous materials in terms of exchange isotherms and specific (noncompetitive) Langmuir isotherms; full competitive Langmuir isotherms are not necessary. That is, it is possible to represent data sets, in which two concentrations are being varied, through affinity distributions which depend on a single affinity constant only. While this reduction in dimensionality by one is of great practical significance for two-component data sets, the practical problems of dealing with data sets, in which three or more concentrations are being varied, remain to be solved. Acknowledgment. We thank D. Kinniburgh and C. Milne for several enlightening discussions and for making the actual experimental data points from ref 19 available to us. This research has been supported by the Swiss National Science Foundation and the Swiss part of the EU-project EV5V-CT94-0536. J.C.W. acknowledges support from the U.S. Department of Energy (DOE) under Contract DE-AC06-76RLO 1830 as part of DOE’s Subsurface Science Program and the Co-Contaminant Chemistry Subprogram. Appendix Experimental and Computational Window. Before the analysis we must define the limits of the logarithmic grids for the equilibrium constants. Suppose that the maximum and minimum values of the measured concen(min) and c(max) . These values are trations c(i) A are given by cA A used to define the window of equilibrium constants by (31) Motekaitis, R. J.; Martell, A. E. Inorg. Chem. 1984, 23, 18.
C ˇ ernı´k et al.
log10 K(min) ) -log10 c(max) -γ A A
(A.1)
log10 K(max) ) -log10 c(min) +γ A A
(A.2)
where γ is a suitably chosen constant. Similarly, c(min) B and c(max) denote the extreme values of the concentraB tions of component B and similar relations define K(min) and K(max) . If we set γ ) 0, we refer to the B B experimental window, while for γ > 0 we obtain the window of the equilibrium constants used for computations. We always use γ ) 2. Such limits are applied directly as borders for windows of equilibrium constants for specific Langmuir sites (L10, L01) and competitive Langmuir sites (L11). For other types of sites we use similar expressions involving the appropriate stoichiometric coefficients. Fixed Grids. The first solution scheme represents a straightforward generalization of the methods discussed to analyze for binding of a single component.17,23,24 In the case of a single component,17 local isotherms depend on a single equilibrium constant only. Values considered for the equilibrium constants are then uniformly spaced on a logarithmic grid. The limits of the grid are calculated by the procedure described above. This procedure applies to all types of specific Langmuir sites or exchange sites, namely to L10, L01, E11, L02, and E12 and the corresponding eqs 11, 14, 15, 22, and 24 and 25. In the case of the competitive Langmuir sites L11 and L12, the local isotherms depend on pairs of equilibrium constants, as can be seen from eqs 8 and 9 and eqs 18 and 19. Such isotherms are predefined in a similar fashion on a rectangular, uniformly spaced logarithmic grid, where all combinations of constants are considered. This approach is common in image analysis.24 Since we keep the grid of equilibrium constants fixed, the only unknowns are the site concentrations. These site concentrations are determined by minimizing17,24
χ2(s1,...,sM) + λR(s1,...,sM) subject to si g 0 for i ) 1, ..., M
(A.3)
The regularization parameter λ defines the weight of the regularizing function R(s1,...,sM), which is included in order to bias the solution toward some a priori defined features of the distribution. A useful class of regularizing functions are24 M
R(s1,...,sM) )
∑ Aijsisj
(A.4)
i,j)1
where the structure of the symmetric matrix Aij depends on the type of distributions for which one attempts to bias. Choosing Aij ) 1 for i * j and Aii ) 0 favors affinity distributions involving a small number of sites.17 Smooth distributions can be selected by choosing Aij such that it represents the square of a first-order difference approximation of the curvature of the affinity distribution integrated over the region of interest; for a linear grid, the curvature is approximated by the second derivatives, while for a rectangular grid, it is approximated by the Laplacian in two dimensions.24 The optimization problem of the minimization of a nonlinear function subject to bounds has received substantial attention in the numerical mathematics literature, and robust algorithms for its solution are avail-
Competitive Ion Binding
able.32,33 The final number of unknowns depends on the grid dimension, the range of experimental values, and the number of sites considered per decade. If only onedimensional grids are involved, the number of unknowns is rather small (M ∼ 10-100 ), but this number is much larger (M ∼ 100-1000 ) as soon as competitive Langmuir sites based on a two-dimensional, rectangular grid are taken into consideration. Further details are given in ref 17. Variable Grids. The following, alternative leastsquares method proved superior for generating stable solutions with a small number of discrete sites and was used in all such examples discussed in the text. The advantage to the variable grid technique is that the optimization scheme must handle a rather limited number of variables. Consider first the case of a single linear, one-dimensional grid. We minimize
χ2(s1,...,sM;K1,...,KM) subject to si g 0 for i ) 1, ..., M and log10 K(min) e log10 Ki + ∆ e log10 Ki+1 e log10 K(max) for i ) 1, ..., M - 1 (A.5) The minimal spacing between two successive equilibrium constants is defined by ∆, which is suitably chosen (we (32) Gill, P. E.; Murray, W.; Wright, M. H. Practical Optimization; Academic Press: New York, 1981. (33) NAG Fortran Library, Mark 15, Numerical Algorithm Group Ltd, Wilkinson House, Jordan Hill Road, Oxford OX2 8DR, U.K.
Langmuir, Vol. 12, No. 25, 1996 6137
use ∆ ) 1.0). The solution is “regularized” by increasing M progressively until a good fit is reached. Usually, quite a small number of sites suffice. This method represents an interesting alternative for generating solutions which involve a small number of sites in one-component situations where a simple Langmuir isotherm is chosen as a local isotherm. However, it is easily generalized to the case of several linear grids and represents a more stable and more reliable approach than the regularization method discussed above. This optimization problem involves the minimization of a nonlinear function subject to bounds and linear constraints. As usual the number of unknowns is not too large (say < 30) and several robust algorithms are available.32,33 The same method can also be applied to two-dimensional grids. Here we concentrate on a simple extension of the above situation where the KA,i values are constrained the same way as in the one-component situation and the KB,i values are bound in a corresponding window; namely, as one minimizes
χ2(s1,...,sM;KA,1,...,KA,M;KB,1,...,KB,M) subject to si g 0 for i ) 1, ..., M, e log10 KA,i + ∆ e log10 KA,i+1 e log10 K(min) A for i ) 1, ..., M - 1, and log10 K(max) A e log10 KB,i e log10 K(max) log10 K(min) B B for i ) 1, ..., M (A.6) LA960008F
