Chemical Modeling of Aqueous Systems II - American Chemical Society

Chapter 20

Numerical Simulation of Coadsorption of Ionic Surfactants with Inorganic Ions on Quartz

Downloaded by UNIV LAVAL on October 9, 2014 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch020

Rebecca L. Rea and George A. Parks Department of Applied Earth Science, Stanford University, Stanford, CA 94305-2225

When little organic carbon is present in a sediment or aquifer, sorption of organic solutes must be controlled by mineral surfaces. Adsorption of these solutes depends on pH, electrolyte concentration, and adsorption of unrelated inorganic ions. Interactions among adsorbing species can significantly enhance or inhibit adsorption. The adsorption of anionic and cationic surfactants on quartz and corundum in the presence of a background electrolyte, and the coadsorption of an anionic surfactant enhanced by Ca have been simulated successfully using HYDRAQL, a chemical speciation program including a triple-layer adsorption model. Surface ionization constants, electrolyte binding constants, and surfactant binding constants were determined from five independent experimental investigations. Surfactants are assumed to adsorb as single ions at low solution concentration and as clusters resulting from hydrophobic interactions between molecules at high concentration. The simulation accounts for experimental adsorption densities, zeta potentials, and for Ca coadsorption, froth flotation recovery. ++

++

Adsorption of organic solutes is commonly controlled by the solid organic matter in soils, sediments, and aquifers ( 1 - 3 ) . However, when organic matter is absent, mineral surfaces play an important role and interactions with inorganic solutes may significantly influence adsorption of the organic species. Tipping and Heaton (4) and Davis (5), for example, found that adsorption of natural organic solutes is sensitive to calcium concentration and pH. Gerstl and Mingelgrin (6) found that pre-adsorption of a long-chain cationic surfactant on the clay mineral, attapulgite significantly enhanced the adsorption capacity of the clay for parathion. Ainsworth et al. (7) showed that the adsorption of PCBs on goethite is enhanced in the presence of surfactants and that cationic and anionic surfactants introduced opposite pH dependence. Finally, Gaudin and Chang (8) in one of the few direct measurements of simultaneous or co-adsorption of two species, found that adsorption of B a enhances adsorption of laurate ion (C H 3COO ) on quartz. As a first step in the development of methods for dealing with these synergistic effects in hydrogeochemical transport programs, this paper demonstrates that conceptual models developed for adsorption of surfactants on oxides, including enhancement by Ca can be incorporated into an existing surface complexation model, HYDRAQL (9). HYDRAQL is a version of MINEQL, a chemical speciation program originated by Westall et al. (JO) and expanded to deal with adsorption in a triple layer model by Davis et al. (11). ++

-

n

2

++

0097-6156/90/0416-Ο26Ό$0ό.0Ο/0 c 1990 American Chemical Society

In Chemical Modeling of Aqueous Systems II; Melchior, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

20. REA & PARKS

Numerical Simulation of Coadsorption of Ionic Surfactants 261

Adsorption of Surfactants Conceptual models for adsorption of surfactants are described in detail by Somasundaran and Fuerstenau (12) and Parks (13). Illustrative experimental data for adsorption of dodecylamine ion (C H25NH ) on quartz (a-SiO^ are provided by DeBruyn (14) and for adsorption of dodecylsulfate ion (C H S0 ~) on corundum (α-Α1 0 ) by Chandar et al. (15). At low concentrations, ionic surfactants adsorb chiefly as counterions in response to surface charge. Anionic species adsorb strongly when pH < pH , where surface charge is positive, and weakly, if at all, when pH > p H ^ . Cations adsorb weakly when pH < \>H and strongly when pH > pH . In this range, electrokinetic behavior is identical to that of simple monovalent indifferent electrolytes, i.e., the adsorbed surfactant reduces, but does not reverse the sign of the zeta potential, ζ (16). In keeping with the behavior of indifferent counterions, the surfactant is assumed to adsorb in the outer Helmholtz plane (OHP) and the diffuse layer. At higher ionic surfactant concentrations, Van der Waals interaction between hydrocarbon chains and hydrophobic bonding results in aggregation, forming clusters called "hemi-micelles" (12.17-21). The aggregation numbers of hemi-micelles are not well established; estimates range from 2 to -250 (15.22.23). Surface charge is reduced more rapidly than at lower solution concentrations and is ultimately reversed as solution concentration and adsorption increase so the adsorption bond includes a "chemical" or "specific" contribution. +

12

3

12

25

4

2

3

pzc

pzc


pzc

Cation Enhancement of Surfactant Adsorption The presence of multivalent inorganic cations can initiate or enhance adsorption of otherwise weakly or non-adsorbing anionic surfactants (8.24). Multivalent anions inhibit adsorption of anionic surfactants, but may enhance adsorption of cationic surfactants. There are two conceptual models for cation enhancement (24.25). When a surfactant fails to adsorb because its charge is the same as that of the surface, then: (1) the enhancing ion may adsorb onto the surface, reversing surface charge, with the surfactant adsorbing in response to that charge, or (2) the surfactant and coadsorbing ion may form an aqueous complex with charge opposite that of the surface, and this complex may adsorb. In either case, adsorption is possible because the surfactant and surface are oppositely charged in the presence of the enhancing ion. These mechanisms are thermodynamically indistinguishable. ADSORPTION MODEL HYDRAQL (10) treats adsorption as surface complexation with bound hydroxide functional groups, SOH, and their ionization products, SO" and SOH . The calculations in this paper use HYDRAQL in its triple layer mode. Surface charge and countercharge accumulate in three "layers": (1) at the surface itself, i.e., in the plane of the SOH groups where the surface potential is Ψ ; (2) in the outer Helmholtz plane (OHP), where adsorbed ions retain their inner hydration sheaths (26) and the potential is Ψβ; and (3) in the diffuse layer. The triple layer model is ideal for our purposes because of its ability to compute an estimate of Ψβ. The computed Ψβ can be compared with experimental measurements of the zeta potential, providing an additional means of constraining models. In triple layer approximations the location of each adsorbate with respect to the surface must be specified. Protons and all ions assumed bound as inner-sphere complexes (specifically or chemically adsorbed species) are assumed to lose part of their hydration sheaths, bonding directly to sites in the surface itself. Adsorbates assumed to remain hydrated, forming outer-sphere surface complexes, are assigned to the OHP. In the intrinsic equilibrium constants for adsorption reactions, K ' ' , the activities of ions transferred from solution to the surface are corrected for the electrical potential they experience, Ψ or Ψ (27). A variety of parameters must be evaluated in any surface complexation model. Common to all surface reactions are the density of SOH sites, N , the specific surface area, A , the solid/liquid ratio, and the near-surface capacities, Q inside the OHP, and C outside the OHP. N is a property of the solid; for both quartz and corundum, N was taken as 5 sites/nm (11.28). A and solid/liquid ratio are experimental variables; values reported with published adsorption data were used in each case. The value, 0.20 F/m (Farads per m ) (27), was adopted for C throughout this work. Every surface +

2

0

n

0

β

s

s

2

s

2

s

2

s

2

2


262

CHEMICAL MODELING OF AQUEOUS SYSTEMS II

ionization and adsorption reaction requires selection of the stoichiometric reaction itself, the location of the adsorption complex, and an equilibrium constant. As a first approximation in formulating any reaction, the aqueous species predominant under the conditions of the experiment and the oppositely charged surface species were selected as reactants. These reactants were combined in one-to-one stoichiometry and treated as outer sphere complexes. If the resulting reaction proved inadequate for simulating experimental data, reactants and/or stoichiometry and/or location were adjusted. The complex systems of interest involve adsorption of H , Na , CF, a surfactant ion, and the enhancing ion, if any. For each solid, ionization and electrolyte binding reactions and equilibrium constants were estimated for H , Na , and C F using an appropriate set of experimental adsorption data. The value of Q is determined as part of this process. A l l of these parameters were subsequently held constant and used when the appropriate ions were present in other systems. Having fixed these parameters, only adsorption reactions and their equilibrium constants were used to describe adsorption of surfactants and enhancing ions. Using the same strategy, these were determined using data from the simplest possible experimental systems, then applied without further adjustment in the final effort to simulate the most complex systems containing all interacting solutes. The only new parameters needed to simulate enhanced adsorption were a single reaction describing interaction of the enhancing ion and the surfactant, and its equilibrium constant. +


+

+

+

+

Surface Ionization and Na and CF Binding Constants Reactions and selected intrinsic equilibrium constants describing ionization of SOH and adsorption of Na and C F on quartz and corundum are listed in Table I. The formulas of surface complexes indicate the assumed location of constituents of the complex in the electrical triple layer. The entire surface complex is located directly on the surface unless a dash is present in the formula. If a dash is present, everything to the left of the dash is located on the surface, and everything to the right of the dash is located in the OHP. +

Table I. Reactions and Selected Intrinsic Equilibrium Constants for Surface Ionization and Electrolyte Adsorption

Reaction

Quartz

Q

1.295 F/m

SOH

+ 2

SOH SOH + Na SOH -Cl 2

+

SOH + H SO" + H

+

pK^

+

SO-Na + H

+

7

Corundum 1.4 F/m

2

pK"

= 3.2

a 7

2

= 5.2

p K ' ^ = 7.2

p K ' ^ = 12.3

p ' K ' V = 6.7

p ' K ' V = 9.7 P * K ' V = 7.9

+

SOH + H + CF

+

As first approximations, equilibrium constants for ionization of SOH and for Na and C F binding on quartz and corundum were assumed equal to values selected by Davis et al. (27) for amorphous silica and by James and Parks (28) for γ-Α1 0 , respectively. If necessary, the equilibrium constants were adjusted to fit experimental data for systems more closely approximating those in which surfactant adsorption data were available. The adjusted equilibrium constants are summarized in Table I. Sources of data and details of fitting procedure are described below. Li and DeBruyn (19) measured the adsorption density of Na on ground Brazilian optical quartz (using radio-tracer methods) and zeta potential, both as functions of pH and NaCl concentration. Tabular data were obtained directly from Li's SM thesis (29). Davis et al. provide only pK" '^. A 2

3

+

1


20.

REA & PARKS

Numerical Simulation of Coadsorption of Ionic Surfactants 263 +

reaction forming a positive surface site, SOH , was included, and its equilibrium constant selected so that the model p H ^ matched Li's experimental value of p H ^ = 2.0. The values of pK + and Q , only, were then adjusted to optimize simulation of the sodium adsorption and zeta-potential data (29). Results are shown in Figure 1. The fits are typical of results obtained by others (ϋ,28). No direct adsorption data were available for Na and C F on corundum. Fuerstenau and Modi (18), however, computed the zeta potential, ζ, as a function of pH from experimental measurements of streaming potential in NaCl solutions. The value of Ψβ computed by the triple layer model in HYDRAQL is an approximation of ζ (Π). Because ζ depends on H , Na , and C F adsorption densities, it was possible to use these data to constrain the equilibrium constants and Q . The value of pK" ^ was adjusted to force p H ^ = 9.1 as observed by Fuerstenau and Modi (18), and v*¥L + was adjusted to fit the zeta-potential for pH < pH . Results are shown in Figure 2. 2

mi

Na

+

+

+

1

tnt

Na

pzc


Surfactant adsorption Dodecvlamine on Quartz DeBruyn (14) measured the adsorption density of dodecylamine on ground Brazilian optical quartz as a function of concentration and pH, using radio-tracer methods. Two adsorption reactions are needed to represent the conceptual model for surfactant adsorption, one describing outer-sphere adsorption of R N H and one describing hemi-micelle formation. A third reaction, producing an adsorbed RNH C1 complex, is suggested by evidence that both adsorbed surfactants (30) and micelles (31) bind counterions. These three reactions and the p K ' values derived by fitting deBruyn's data in the context of the appropriate surface ionization and electrolyte binding reactions (Table I), accounting for aqueous hydrolysis and dimerization of the amine (32) are: +

3

3

m

SOH + R N H SOH + R N H

+ 3

+

SOH-RNH

3

+ CF

SOH + 5RNH

+ 4

+

SOH-RNH Cl

P^RNHSCI

3

+

^> SO(RNH ) (RNH ) + 4H 4

2

3

= ·

V^RNNI*

3

3

3

+

P*K^

=

4

·

3

8

= 12.5

M +

W 2

()

(3)

The resulting simulation is compared with experimental data in Figure 3. Monomeric surface species were assumed to be outer-sphere complexes located in the OHP. The hemi-micelle was assumed to be an inner-sphere complex located on the surface. These assumptions are supported by the work of Chander, et al (22). They observed significant ionic strength dependence of the adsorption of dodecylsulfonate on alumina in NaCl solutions at low surfactant concentrations and no ionic strength dependence in the higher concentration region corresponding to hemi-micelle formation. High ionic strength dependence suggests outer-sphere complexes, while lack of ionic strength dependence suggests inner-sphere adsorption complexes (33). The composition of the hemi-micelle in Reaction 3 was determined by optimizing the fit to the adsorption density at high concentration and high pH, recognizing an experimentally observed reversal in zeta potential (16). and the high-density packing of molecules in hemi-micelles observed by Waterman et ai (34). Dodecvlsulfate on Corundum Chandar et al. Q5) measured the adsorption density of dodecylsulfate on corundum at pH = 6.5 in 0.1 M NaCl. The experimental data are adequately described by a model closely analogous to that used for dodecylamine on quartz. In addition to the surface ionization and NaCl binding reactions (Table I), the reactions and equilibrium constants used were: +

SOH -RS0 ^ SOH + H + RS0 2

3

3

P*K

+

SOH (RS0 H) (RS0 ) - ^ SOH + H + 5RS0 2

3

3

3

2

3

ρ

I

N

T R

S

O

K"V

R

= 1.1

(4)

= 41.1

(5)


264



+

Figure 1. Na on Quartz: Comparison of experimental (29) and simulated adsorption and zetapotential data. Modeling parameters for surface ionization and sodium adsorption are given in Table I. Surface site concentration: Σ SOH = 7.06 χ 10" M . 4


REA & PARKS



20.

+

Figure 2. Na and C F on Corundum: Comparison of experimental Q8) and simulated zetapotential data. Modeling parameters for surface ionization and electrolyte adsorption are given in Table I. Surface site concentration: Σ SOH = 2.34 χ 10" M. 4



266


PH Figure 3. Dodecylamine on Quartz: Comparison of experimental (14) and simulated adsorption data. Modeling parameters for surface ionization and NaCl electrolyte adsorption are given in Table I. Surface site concentration: Σ SOH = 1.27 χ ΙΟ" M . Dodecylamine adsorption Reactions 1, 2, and 3 were assumed with V*K. + = 3.3, V*K = 4.8, and ρ Κ + = 12.5, respectively. 4

INT

INT

RNI{3

RNH}CL

ίΛί

//Μ


20. REA & PARKS


Equilibrium constants for Reactions 4 and 5 were estimated by trial and error to fit the experimental dodecylsulfate adsorption data. No counterion binding reaction, equivalent to Reaction 2, was needed. The experimental data and simulation are compared in Figure 4. Coadsorption


++

C a improves froth flotation recovery of quartz with dodecylsulfate, implying that calcium enhances adsorption of dodecylsulfate on the mineral, probably by coadsorption (35). Direct simultaneous adsorption measurements of both surfactant and enhancing ion are not presently available, but adsorption behavior can be inferred from flotation studies. Froth flotation is a technique used to separate one kind of particulate solid from another through selective attachment to air bubbles in an aqueous suspension. Successful flotation is a complex process, dependent upon both equilibrium chemistry and bubble attachment kinetics. However, there are good empirical correlations between flotation recovery and adsorption density (36.37). Flotation requires adsorption of an appropriate surfactant at a density equivalent to at least 5% surface coverage (14.36). As reported by Fuerstenau and Healy (35) and Fuerstenau el al. (37), calcium-enhanced flotation of quartz by dodecylsulfate commences near pH = 10 (implying surface coverage of at least 5% at that point), peaks near pH = 12, and drops off at higher pH (Figure 5). To simulate coadsorption of calcium and surfactant, it is first necessary to define reactions and equilibrium constants for adsorption of calcium on quartz in the absence of surfactants. The same two reactions used by other investigators to describe adsorption of divalent cations (38) were adequate to match the limited Ca adsorption data (Figure 5) published by Cooke and Digre (39). The reactions and equilibrium constants used in addition to those listed in Table I were: ++

++

SOH + Ca

+

^> SO-Ca + H

++

+

SOH + Ca + H 0 ^=> SO-CaOH + 2H

K

P* ""c +

++ a

V^caon*

2

= ·

3

8

1 3

·

=

6

() 5

7

()

Calcium-enhanced surfactant adsorption can be simulated by addition of only one additional reaction, formation of a mixed Ca -surfactant surface complex: ++

SOH + Ca + RS0 - £5 SOCa-RS0 + H ++

3

3

+

Ρ K

I N T C A R S 0 3

+

=

8.7

(8)

++

Accepting the predetermined surface ionization and NaCl binding reactions (Table I), Ca binding Reactions 6 and 7, and surfactant adsorption Reactions 4 and 5, it was possible to simulate the necessary 5% surface coverage at pH = 10.3 and adsorption maximum at pH « 12 by setting ρ K = 8.7. Results are shown in Figure 6. I

N

T

C

A

R

S

0

3

+

CONCLUSIONS These results demonstrate the feasibility of simulating the adsorption of monofunctional anionic and cationic alkyl surfactants on mineral surfaces, and the cation enhancement of surfactant adsorption with the electrical triple layer, surface complexation model in HYDRAQL. No changes in the basic model were needed. No adjustment of surface ionization or electrolyte binding constants or of capacities were needed, once these parameters had been derived from independent data obtained from surfactant free systems; in fact, these parameters proved nearly identical to those derived by others for a different silica and a different alumina (27.28). Surfactants were accommodated by adding a set of aqueous solution reactions to account for known solution behavior and a small set of adsorption reactions based on conceptual models for surfactant adsorption promulgated largely by the flotation community. The fact that it was possible to apply reactions and equilibrium constants derived from diverse simple systems, without change, in increasingly complex systems suggests that this modeling approach has wide general application. The simulation models are assuredly not unique. Adsorption density measurements, electrokinctic data, and even froth flotation data constrain the problem, and more extensive data in these areas would be useful, but direct observation of the compositions and aggregation numbers of


268



-7

-6

-5

-4

-3

log equil. amine cone.

Figure 4. Dodecylsulfate on Corundum: Comparison of experimental (15) and simulated adsorption data. Modeling parameters for surface ionization and NaCl electrolyte adsorption are given in Table I. Surface site concentration: Σ SOH = 2.34 χ 10" M . Dodecylsulfate adsorption Reactions 4 and 5 were assumed with p*K = 1.1 and p*K! - = 41.1, respectively. 4

int

nt

RSor

HM



20. REA & PARKS

Numerical Simulation ofCoadsorption ofIonic Surfactants 269

log equilibrium sulfate cone. (M/l) ++

Figure 5. C a on Quartz: Comparison of experimental (39) and simulated adsorption data. No background electrolyte was present; Σ CaCl = 7.2 χ 10~ M . Modeling parameters for surface ionization and C F adsorption are given in Table I. Surface site concentration: Σ SOH = 2 χ 10" M . C a adsorption Reactions 6 and 7 were assumed with p*K ' ++ = 3.8 and ν*Κ οΗ - 5 , respectively. 3

2

4

++

m

Ca

ιηί

+

=

l3

€α


270


100 h

>


Ε ο û. Β ω Ν

10" Μ NaCl 10' Μ NaCl 10" Μ NaCl 3

4

5

-100

10

12

PH ++

Figure 6. Comparison of simulated Ca enhanced adsorption of dodecylsulfate on quartz with experimental, Ca -enhanced dodecylsulfate flotation of quartz (3537). In the simulation, ionic strength was set at 0.01 M with NaCl. Modeling parameters for surface ionization and electrolyte adsorption are given in Table I. Surface site concentration: Σ SOH = 2.34 χ 10 M. Adsorption of Ca was simulated with Reactions 4 and 5 assuming Σ C a = 7.2 χ ÎO^ M , p*K"" = 3.8, and p * K ' ' = 13.5. Dodecylsulfate adsorption was simulated with Reactions 6 and 7 assuming Σ R S 0 = 2.3 χ \0* Μ, p~K' ρ K. - = 4\A. Ca -dodecylsulfate coadsorption Reaction 8 was assumed with p K " CaRSOi + = 8.7. ++

-4

++

++

4

n

Cû++

Cû0//+

3

int

++

HM

adsorption complexes and their locations relative to the surface are needed to define reactions uniquely. LITERATURE CITED 1. 2. 3. 4. 5. 6. 7.

Goring. C. Α. I. Ann. Rev. Phytopathol. 1967, 5, 285-318. Karickhoff, S. W. Chemosphere 1981, 10, 833-846. Schwarzenbach, R. P.; Westall, J. T. Environ. Sci. Technol. 1981, 15, 1360-1367. Tipping, E.; Heaton, M. J. Geochim. Cosmochim. Acta 1983, 47, 1393-1397. Davis. J. A. Geochim. Cosmochim. Acta 1982, 46, 2381-2393. Gerstl, Z.; Mingelgrin, U. Clays and Clay Minerals 1979, 27, 285-290. Ainsworth, C. C.; Chou, S. F. J.; Griffin, R. A. 24th Hanford Life Sciences Symposium October 1985. 8. Gaudin, A. M.; Chang, M. C. AIME Trans. 1952, 193, 193-201. 9. Papelis, C.; Hayes, K. F.; Leckie, J. O. Technical Report No. 306, Environmental Engineering and Science, Dept. Civil Engineering, Stanford University, 1988.



20. REA & PARKS


10. Westall, J. C.; Zachary, J. L.; Morel, F. M. M. Technical Note 18, Ralph M. Parsons Laboratory, M. I. T., Cambridge, MA 1976. 11. Davis, J. Α.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480-499. 12. Somasundaran, P.; Fuerstenau, D. W. J. Phys. Chem. 1966, 70, 90-96. 13. Parks, G. A. In Marine Geochemistry; Riberg, J. P.; Skirrow, G., Eds.; Academic: New York, 1976,p241-308. 14. deBryun, P. L. Trans. AIME 1955, 202, 291-296. 15. Chandar, S.; Somasundaran, P.; Turro, N. J. J. Colloid Interface Sci. 1987, 117, 31-46. 16. Fuerstenau, D. W. J. Phys. Chem. 1956, 60, 981-985. 17. Gaudin, A. M.; Fuerstenau, D. W. AIME Trans. 1955, 202, 958-962. 18. Fuerstenau, D. W.; Modi, H. J. J. Electrochemical Soc. 1959, 106, 336-341. 19. Li, H. C.; deBruyn, P. L. Surface Sci. 1966,5,203-220. 20. Wakamatsu, T.; Fuerstenau, D. W. In Advances in Chemistry Series 79, Adsorption from Aqueous Solution; Gould, R. F., Ed.; American Chemical Society: Washington, DC, 1968; p 161-172. 21. Dick, S. G.; Fuerstenau, D. W.; Healy, T. W. J. Colloid Interface Sci. 1971, 37, 595-602. 22. Chander, S.; Fuerstenau, D. W.; Stigter, D. In Adsorption From Solution; Ottewill, R. H.; Rochester, C. H.; Smith, A. L., Eds.; Academic: London, 1983; p 197-210. 23. Cases, J. M.; Levitz, P.; Poirier, J. E.; Van Damme, H. In Advances in Mineral Processing; Somasundaran, P., Ed.; S.M.E. Pub.: Littleton, CO, 1986;p171-188. 24. Fuerstenau, M. C., Miller, J. D., Kuhn, M. C. Chemistry of Flotation; AIME: New York, 1964. 25. Davis, J. Α.; Leckie, J. O. Environ. Sci. and Tech. 1978, 12, 1309-1315. 26. Bockris, J. O'M.; Reddy, Α. Κ. N. Modern Electrochemistry; Plenum: New York, 1970. 27. Davis, J. Α.; James, R. O.; Leckie, J. O. J. Colloid Interface Sci. 1978, 63, 480-499. 28. James, R. O.; Parks, G. A. Surface and Colloid Sci. 1982, 12, 119-216. 29. Li, H. C. S. M. Thesis, M. I. T., Cambridge, MA, 1955. 30. Bitting, D.; Harwell, J. H. Langmuir 1987, 3, 500-511. 31. Adamson, A. W. Physical Chemistry of Surfaces; John Wiley and Sons: New York, 1976. 32. Anamanthpadmabhan, K.; Somasundaran, P.; Healy, T. W. AIME Trans. 1979, 266, 20032009. 33. Hayes, K. F.; Leckie, J. O. J. Colloid Interface Sci. 1987, 115, 564-572. 34. Waterman, K. C.; Turro, N. J.; Chandar, P.; Somasundaran, P. J. Phys. Chem. 1986, 90, 68286830. 35. Fuerstenau, M. C.; Healy, T. W. In Adsorptive Bubble Separation Techniques; R. Lemlich, Ed.; Wiley: New York, 1972;p91-131. 36. Fuerstenau, D. W. Trans. AIME 1957, 208, 1365-1367. 37. Fuerstenau, M. C.; Martin, C. C.; Bhappu, R. B. Trans AIME 1963, 226, 449-454. 38. Davis, J. Α.; Leckie, J. O. J. Colloid Interface Sci. 1949, 67, 90-107. 39. Cooke, S. R. B.; Digre, Marcus AIME Trans. 1949, 184, 299-305. RECEIVED August 18, 1989


Chemical Modeling of Aqueous Systems II - American Chemical Society

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