Langmuir 1993,9, 3174-3190
3174
A New Theoretical Approach to Adsorption of Ionic Surfactants at Water/Oxide Interfaces: Effects of Oxide Surface Heterogeneity L.Lajtar, J. Narkiewicz-Michalek,and W. Rudzinski' Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska University, PI. Marii Curie-Sklodowskiej 3, 20-031 Lublin, Poland
S . Partyka Laboratoire de Physico-Chimie des Systemes Polyphases, LA 330, USTL, Place Eugene Bataillon, 34060 Montpellier Cedex, France Received April 13, 1 9 9 9 On the basis of the scaled particle theory, a new theoretical approach is developed for adsorption of ionic surfactantsat watedoxide interfaces. The adsorbed phase is treated as a mixture of oblate aggregates of various dimensions, interacting via excluded volume interactions. A three-parameter equation for the adsorption isotherm is developed, and then successfully applied to fit experimental isotherms above the cac (criticalaggregationconcentration). By adjustingjust onemore parameterin the correapondingequation for the heat of adsorption,very good agreement with calorimetric experimental data is achieved. Having determined the adsorptionparameters, one can calculate the distributionof the surface aggregates among their correspondingaggregationnumbers. The calculatedaverageaggregationnumber treated as a function of the degree of surface coverageexhibits behavior found in fluorescencedecay spectroscopyexperiments. The adsorption proceeds both via the increasing number of surface aggregates and via the increasingsize of the aggregates. It is concluded that the surface heterogeneity of oxides may be a leading factor in the process of surface aggregation.
Introduction In the past, researchers dealing with ionic Surfactant adsorption onto polar surfaces followed two opposite approaches. (a) Some of the researchers were looking for selectivity improvementin flotation processes. Thus, they were using adsorbate-adsorbent systems in which the surfactant has a strong affinity for the surface, i.e., a strong adsorbateadsorbent bond, mostly of electrostatic origin. These systems lead to the formation of a monolayer for high values of the undersaturation [low equilibrium concentrations with regard to the critical micelle concentration]. Thus, the particle becomes hydrophobic. The monolayer is reached long before the air-liquid interface is saturated by the surfactant. It is possible then to get a good flotation for equilibrium concentration values much lower than the cmc. These systems are c d e d strong normal adsorbateadsorbent bond systems. (b) Other researchers, dealing with studies aiming at enhanced oil recovery using microemulsions, employed ionic or nonionic surfactanta showing little affinity toward mineral surfaces (quartz or kaolinites) in order to keep the detergent properties of microemulsionsafter injection. Such adsorbate-adsorbent systems are called weak normal adsorbate-adsorbent bond systems. As flotation-related studies started much earlier than enhanced oil recovery-related ones, the researchersat first were studying strong adsorbate-adsorbent bond systems leading to hydrophobic particles. Thus, the following models of adsorption were successively worked out. (a)Surface Micellization Hypothe~sis.~*~ Thistheory
* To whom correspondence should be addressed.
Abstract published in Advance ACS Abstracts, October 1,1993. (1) Caees, J. M.; Villieras, F. Langmuir 1992,8, 1251. (2) Gaudin, A. M.; Fueratenau, D. W. Tram. AIME 1966,202,958. (3) Fuenrtenau, D. W.; Raphvan, S. In Flotation; Furetenau, M. C., Ed.; AIME: New York, 1976; p 21. @
,
assumed that growth of the adsorbed layer proceeds through the formation of admolecule aggregates, formed due to strong lateral hydrophobic bonds. As the strong normal adsorbate-adsorbent bond systems were most frequently studied at that time, experimental fiidings advocated the monolayer character of these surface aggregates. So,this is why they were called "hemimicelles". No theoretical description existed at that time of the structure of these surface aggregates, and the origin of their limited size. The first, and practically the only, two advanced theoretical works on the nature of the surface aggregates of ionic surfactants were published by Harwell et al. in 1986: and then by Yeskie and Harwell in 1988: Becauseof their importance,they will be discussedin more detail in the next section. (b) Theory of Two-DimensionalCondensation on a Heterogeneous Surface.61oFirst developed by Cases and Mutaftachiev," it was presented next in the series of papers, including the most recent review.' This theory takes into account the whole potential energy (normaland lateral bonds), and the energetic heterogeneity of the real adsorbent surfaces. According to that theory the origins of the formation of the two dimensional aggregates are (i) the lateral bonds which lead to two-dimensional condensation and (ii) the surface heterogeneity which controls the size of the aggregates, and different surface domains (4)Harwell, J. H.; Hoskina, J. C.; Schechter, R. S.; Wade, W. H. Langmuir 1985,1, 261. (5) Yeskie, M. A.; Harwell, J. H. J. Phys. Chem. 1988,92, 2346. (6) Cases, J. M.; Mutaftschiev, B. Surf. Sci. 1968, 9, 57. (7) Cases, J. M. Bull. Mineral. 1979,102, 684. (8) Cases, J. M.; Canet, D.; Doerler, N.; Poirier, J. E. In Adeorption at the Gau-Solid and Solid-Liquidlnterface;Roquerol, J., Sing, K. S. W., Eds.; Elsevier: Amsterdam, 1982; p 21. (9) Cases,J. M.; Levitz, P.; Poirier,J. E.; Van Damme, H. In Advances in Mineral Processing; Somasundaran, P., Ed.;SME: Littleton, 1986; p 171. (10) Cases, J. M.;Poirier,J. E.; Canet, D. In Solid-LiquidInteractiom in Porous Media; Cases, J. M., Ed.; Technip: Paris, 1985; p 336.
0 1993 American Chemical Society
Adsorption of Ionic Surfactants
are filled in the sequenceof their decreasing energies. The numerous flotation systems studied by Cases and coworkers provided impressive support for this theory. The Bragg-Williams approximation was accepted to describe the lateral interactions in the adsorbed monolayer. This model was elaborated further by Scamehorn et al.” They considered also the possibility of a bilayered adsorption, and allowed for the possibility of first-layer phase transitions on homogeneous domainsto occur, with subsequent adsorption on the top of the first layer to complete the bilayered structure. Michalek12J3has proposed recently still another generalization of the theory of two-dimensional condensation on a heterogeneoussurface,taking into account the obvious fact that the conformation of adsorbed molecules must change during the adsorption process. Some recent achievementsalong these lines have been presented in the series of papers published by BBhmer et al.lPl6 In the next section we will refer to these papers in more detail. From the experimental point of view, adsorption studies involve application of a variety of adsorption techniques, such as adsorption isotherm, surface charge, zeta potential, and particle wettability measurements. The use of other techniques such as infrared spectroscopy or ellipsometry, for instance, is less popular since it involves ex situ procedures such as freezing, drying, etc. However, the most essential discoveries concerning the mechanism of surfactant adsorption onto ionic surfaces have come more recently from fluorescence decay spectroscopy and calorimetric studies. The fluorescence decay technique, applied first by Levit~”-~lto investigate the adsorption of nonionic surfactants onto hydrophilic surfaces, was used next by Somasundaran and co-workers to study the structure of the adsorbed layer of dodecyl sulfate at the alumina-water have found that, except interface. Thus, Chandar et al.22 for the region of very small initial surface coverages, the adsorption of dodecyl sulfate proceeds via the formation of two-dimensional surface compact aggregates. The average aggregation number increases as the total surface coverage increases. The surface aggregation phenomena are demonstrated also clearly in the heat effects accompanying surfactant adsorption, as demonstrated in the studies published by Partyka and co-worker~.~~ ~~
(11)Scamehom,J.F.;Schechter,R. S.;Wade, W.H.J.ColloidInterface Sci. 1982,85, 463. (12)Narkiewin-Michalek, J. Ber. Bunsen-Ges. Phys. Chem. 1991,95, 85. (13)Narkiewicz-Michalek, J. Langmuir 1992,8, 7. (14)Bdhmer, M. R.; Koopal, L. K.; Janssen, R.; Lee, E. M.; Rennie, A. R. Langmuir 1992,8, 2228. (15) BBhmer, M. R.; Koopal, L. K. Langmuir 1992,8, 1594. (16)BBhmer, M. R.; Koopal, L. K. Langmuir 1992,8, 2649. (17)Levitz,P.; Azouz el Miri; Keravis, D.; Van Damme, H. J. Colloid Interface Sei. 1984, 99, 484. (18)Levitz, P.; Van Damme, H.; Keravis, D. J.Phys. Chem. 1984,88, 2228. (19)Levitz, P. Doctoral Thesis, University of Orleans, 1985. (20)Levitz, P.; Van Damme, H.; Keravis, D. In Solid-Liquid Interactions in Porous Media; Cases, J. M., Ed.; Technip: Paris, 1985;p 473. (21)Levitz,P. Langmuir 1991, 7, 1595. (22)Chandar, P.; Somasundaran, P.; T w o , N. J. J. Colloid Interface Sci. 1987, 117, 31. (23)Amin-Alami, A.; Kamenka, N.; Partyka, S. Thermochim. Acta 1987,122, 171. (24)Partyka, S.; Rudzinski, W.; Brun, B.; Clint, J. H. Langmuir 1989,
__
5. _,297. ..
(25)Thomas, F.;Bottero, J. Y.;Partyka, S.; Cot, D. Thermochim. Acta
1987,122,197.
(26)Lindheimer,M.; Keh, E.; Zaini, S.;Partyka, S. J. Colloidlnterface Sei. 1990, 138, 83.
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The abovediscussedfluorescencedecay and calorimetric studies, combined with the results obtained in the earlier adsorption studies, yield the following picture of the adsorption mechanism. At very low surface coverages the adsorption runs via monomers lying flatly on the adsorbent surface. Then, at a certain concentration a quite sudden change in the orientation of the adsorbed monomers takes place to a more vertical one. The vertically oriented monomers start to associate into two-dimensional aggregates. As the surfacecoverageincreases,the average aggregationnumber of the aggregates also increases. The initially hydrophilic surface becomes hydrophobicat small surface Coverages, and then again hydrophilic at higher surface coverages. At the plateau, the adsorbed amount and solid surface area measurements suggest that at least bilayered condensed structures must exist on the solid surface. The fluorescencedecay spectroscopy shows that, except for the region of very small coverages, the adsorption proceeds via the formation of surface aggregates. However, the fluorescence decay spectroscopy detecta only the existence of the compact hydrophobic core in these aggregatesand their average aggregation number, but does not provide any information about their shape and structure. So, various and sometimes controversial views have been expressed here. Thus, Scamehorn et al. and Harwell et al. argue that, in the systems which they studied, only bilayered aggregates called “admicelles”were practically formed. So, the initial hydrophobization of the surface would be due to the flatly adsorbed monomers. Yeskie and Harwell conclude that “low counterions bindings, low surface charge densities and low dielectric constants all favour the formation of a hemimicelle at a lower surfactant concentration than necessary for an admicelle to form”. This would suggest that at such conditions the adsorption may proceed also via the formation of hemimicelles on the solid surface. BBhmer and Koopal argue that “for uncharged surfaces the first step, (i.e. formation of monolayered condensed phase) in the isotherm is missing and once the adsorption starts a bilayer is formed almost immediately”. There is some controversy concerning the role of the surface energetic heterogeneity. While Harwell et al. believe that in surfactant adsorption the ”primary importance is the role of surface heterogeneities”, BBhmer and Koopal argue that the behavior of these adsorption systems “is more likely due to the character of the surfactant and the variable surface charge than to surface heterogeneity”. Thus, in spite of the large body of existing experimental data our understanding of these adsorption phenomena is far from being satisfactory and deserves further extensive theoretical study. One of the most intriguing and fundamental questions is the origin of the limited size of the formed surface aggregates. That fundamental question was the main inspiration for the present work. Theory I. Discussion of the Previous Approaches. Before showing our approach to the problem, we will review briefly the existing most recent theories to see to which extent they reproduce the behavior of the studied experimental systems. (27)Partyka,S.;Lindheimer,M.;Zaini,S.;Keh,E.;Brun,B.Lagmuir 1986, 2, 101.
3176 Langmuir, Vol. 9, No. 11,1993
Lajtar et al.
As far as the "surface micellization" approach is concerned, advanced theoretical studies were initiated by Harwell et al. in 1985.4 Their theory is based on the noncontroversial assumption that 'if the surface is homogeneous, an adsorption isotherm will consist of a low-concentration region, region I, in which the adsorption will increase slowly with concentration, and a vertical step to complete bilayer coverage at the cac (Critical Admicelle Concentration). Adsorption will then remain constant at increased solution concentrations because bilayer coverage represents complete saturation of the surface". Harwell et al. believe that this is the surface charge density 6, which is the most essential characteristic determining the cac. They also assume that the existing oxide surfaces have patchwise topography. Differentpatches exhibit different binding energies with respect to protons and other simple ions of the basic electrolyte, including counterions. As an effect, at a given pH and basic electrolyte concentration, different patches are characterized by different charge densities 6,. Thus, according to Harwell et aL4"for each solution of given surfactant concentration there corresponds a surface charge density sufficient to stabilize an admicelle". The total adsorption of surfactant at a given surfactant and counterion concentration I't is therefore given by the sum of two contributions:
10-9
L lo-' 1 0 - ~ 11 2
10-~
10-1
where 6,* is the minimum value of 6, which stabilizes the existence of admicelles,f @ , ) is the differential distribution of the number of patches into corresponding values of 6, (normalized to unity), I'I(p8,6,) is the isotherm of monomer adsorption at the surfactant bulk concentration pa, and r, is the maximum amount adsorbed (whenthe adsorbed phase is a dense bilayered structure). Although the Coulombic molecule-surface interactions seem to play a crucial role in the adsorption of ionic surfactants on polar surfaces, the dispersion of the nonCoulombic molecule-surface bonds may be still another possible source of the surface energetic heterogeneity effects in adsorption. There is strong experimental evidence that the dispersion of the non-Coulombic interactions affects adsorption strongly at low surface coverages where not only the polar head but also the alkyl chain (hydrophobicmoiety) is in direct contact with the solid surface. First of all, the log-log plots of the experimental isotherms of ionic surfactants tend to be linear at very small surface coverages, but the tangent of that limiting slope is equal to unity only in some rare cases. For instance, this is the case for the anionic surfactant adsorption on kaolinite investigated by Scamehorn et Generally,the tangent of the log-logplots of the isotherms of ionic surfactants tends to values of about 112 at very small surface coverages. This would indicate adsorption on a strongly heterogeneous solid surface, characterized usually by the Freundlich isotherm equation at low surface coverages. As an example we compare in Figure 1 the adsorption of anionic surfactantson alumina investigated by Wakamatsu and Furstenau,= and by Scamehorn et al.11
Figure 2 shows the same Freundlich behavior of the log-log plots for the adsorption of two cationic surfactants adsorbed on silica, studied in the CNRS laboratory in Montpellier. (28) Wakamatsu, T.;Furstenau, D. W. Adu. Chem. Ser. 1968,79,161.
io-' C imol.dm-'I
.,
Figure 1. (A)Adsorption isotherms of sodium alkane sulfonates of varying alkyl chains on alumina reported by Wakamatsu and Fmtenaumfor pH 7.2 and 2 X 1 V moEdm4 ionic strength: 0, C16; A, Clr; A, (212; Clo; 0, Ca. (B)Adsorption isotherms for alkylbemnesulfonateaof varying hydrocarbonchainson alumina reported by Scamehom et al." for pH 7.7 and 0.171 M NaCk A, cl2;0 , ClO; v, C8.
-
2.2
-
2.0
-
1.8
-
r
0,
2
E
3
-
L-
-
1.6 -
m
1.4 -
1.2
-
1.0
-
0.8 I -4.5
I
-4.0
I
-3.5
I
I
-3.0
-2.5
-2.0
-1.5
log lC/mol.dm-31
Figure 2. Adsorption isotherms of DDAB ( 0 )and TTAB (0) on silica RP 63-876 at 25 "C and pH 8.2. The experimental details are described in the Ph.D. Thesis by M. Bouzerda, CNRS Laboratory 330, Montpellier University, 1991.
The "patchwise" model seems to be a very convincing picture of the surface heterogeneities of the ionic solid surfaces studied by Cases and co-workers. The oxide surfaces studied by Schechterand co-workersmust exhibit
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Langmuir, Vol. 9, No. 11, 1993 3177
a much smaller degree of surface organization, so the patchwise model may not be a perfect picture of the surface heterogeneity in this case. However, as the patchwise model is so commonly accepted it seems worthwhile to explore the extent to which it can be applied in the description of adsorption of ionic surfactants on oxide surfaces. The obtained results will be very useful in our attempts to describe the surfactant adsorption on solid surfaces exhibiting departures from the perfect patchwise topography. We start our analysis by accepting eq 1proposed by Harwell et al. Because at low surface coveragts not only polar heads but also hydrophobicsegments are adsorbed on the solid surface, I’I should be an isotherm expression developed for multi-site-occupancyadsorption. Nevertheless, in the limit of very low surface concentrations every such isotherm equation developed for adsorption on a homogeneous surface must reduce to Henry’s isotherm. Thus, until the patchwise topography is assumed, I’I in eq 1may practically be replaced by Henry’s isotherm, provided that the admicelles are formed already at low surface concentrations. Every Henry’s isotherm will, in general, have the following form:
rI= Kx
(2)
where x is the bulk surfactant concentration and the theoretical expression for the constant K will result from a particular theoretical treatment. It is to be expected that it will be a function of the surface charge density 6,, the concentration of the basic electrolyte, temperature, etc. Harwell et al. emphasize that fitting an experimental isotherm by eq 1creates, for the first time, a chance to determine the dispersion of the surface charge 6, on different patches, characterizedby the functionf(6,). Next they have performed such a best fit, based on their theoretical expressionsdeveloped to describethe monomer and aggregate adsorption. Assuming, thus, that the dispersion of the surface charge f(6,) is an important adsorption characteristic, we would like to focus our attention on still another possibility of its determination from an experimental isotherm I’t. This is a kind of standard procedure which allows in addition some further interesting conclusions to be drawn. As adsorption on different patches will be characterized by different values of K,the first step is to calculate the differential distribution of ln K,f ~ ( l Kn ) . That problem has been known for a long time in the theories of gas adsorption on heterogeneous surfaces, so we refer only to the existing theoretical solutions.29 In eq 1 the “local” isotherm equation describing the adsorption on a given patch is the linear combination of Henry’s and step isotherms. In such a case the differential distribution f d l n K ) can, to a good approximation, be calculated by using the assymptotically correct condensation appr~ximation:~~
a2e
ae 0.4343-a l n z (3) fKClnK) ==where 8 = I’tJI’-. The second term is a kind of correction to the first leading term (aela ln x ) corresponding to the condensation approximation. Rather than introducing a (29)Rudziki, W.; Everett, D. H. Adsorption of Gmes on Heterogeneous Surfuces; Academic Press: New York, 1992. (30)Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L.K. J. Colloid Interface Sei. 1990, 135, 410.
meaningful correction,that second term introduces usually a serious “noise”to the calculated functionf d n K),caused by a second differentiation of the discrete values of the function I’t(ln x ) , comprising experimental errors. We will neglect it, therefore, in our further considerations. The next step is to look for the meaning of K arising from a particular theoretical treatment of monomer adsorption. The approach proposed by Harwell et al. belongs to the category of the Stern-Grahame-Gouy approaches. It starta with Ben Naim’s statistical treatment of mixtures, according to which the chemical potential of a solute molecule in a bulk mixture pb is given by the expression4
W + kT ln(p,A3q-’) (4) where W is the work of putting a surfactant molecule at a fixed position within the bulk. This work is dependent upon the composition of the bulk phase {pi). Further, q is the internal partition function for a surfactant molecule (assumed to be independent of the local environment), and the translational contribution is defined by pb =
A = (h2/2?rMkT)’f2 (5) One can apply a similar treatment to determine the chemical potential of a molecule in the admicelle phase. Thus, pa=
ii’ + kT ln(4?r2&i2q[’)
(6)
where i!’ is the work of placing a surfactant molecule at a fixed position within the admicelle in its most favorable orientation. That work is dependentupon the composition of the admicelle {pi). Equations 4 and 6, while similar in appearance, differ. The ps is the surface density of surfactant molecules in admicelles whereas p , is the bulk concentration. Further qa is the internal partition function of a surfactant molecule in the admicelle. At equilibrium, the chemical potentials in the bulk phase pb and in the surface phase pamust be equal, so
When only a single type of surfactant molecule is present in the aggregates, that is, for pure componentadsorption, p s is the critical admicelle concentration (cac) of the local patch on which the admicelle has formed. When a surfactant monomer adsorbs directly in the Stern plane in region I, its chemical potential pm is given by where qm # qa for obvious reasons, and
The calculation of the work term (W- W) represents a very complex problem. So, to arrive at tractable expressions, Harwell et al. have decided to “sacrificesome of the thermodynamic precision” and assume that this work term is given by the following sum:
W - w = E”’- FAG
(10) where Eel is the electrical work and FAG is the free energy change of removing the F methyl groups of the hydrocarbon moiety from the bulk solution to the admicelle. The same assumption is made for the monomer adsorption, except that F is replaced by f i I 7. denoting the effective number of methyl groups removed from the solution to the inner
Lajtar et al.
3178 Langmuir, Vol. 9, No. 11, 1993
Helmholtz plane (IHP). Also AG is expected to be different, and Eel is given by a simplified form of the expression Eel developed for the admicelle. Most of the work is devotedto developing the theoretical expressions for Eel as a function of pH, the concentration of the basic electrolyte, and the concentration of counterions. That advanced theory is able to explain most of the fundamental features of the adsorption of ionic surfactants, except the finite aggregation number. This is because it considers surface aggregates infinite in two dimensions. Now let us consider certain consequences of assuming that the heterogeneity effects in surfactant adsorption are due solely to the dispersion of the surface charge values across the surface. The equation for monomer adsorption, developed by Harwell et al., takes the form
-2.0
1
-4.0
-4.5
1
-3.5
-2.5 log (C/mol. dm-'I
-3.0
1
where co is a constant,
6, + 61
y=l+4kTepo
0.0 -2.0
-1.0
0.0
1.0 -In
and PO = h a + + PS-
(14)
For small values of adsorption 61 can be neglected since it only serves toreduce y, which then is practically constant and equal to about 10.56. With these approximations,
Harwell et al. conclude that the second term under the exponent sign (6,kTleco) should be small compared to the first one. Thus, monomer adsorption should not be very sensitive to the surface charge, and should always reduce to Henry's isotherm. Now, let us assume that the surface heterogeneity is due to different values of 6, on different surface patches. Provided, thus, that AAG is not a function of 6,, Le., A is constant in the course of adsorption, the function aela In x plotted against the function -In x should have the same shape as the function f(6,) plotted against 6,kTleco. It means also that if two adsorption systems are characterized by the same charge dispersion f(S,), their plots of arJaIn x normalized to unity should match after shifting one of them from right to left on the axis -In x . In particular, that conclusion should be valid in the case of two surfactants having the same structure, but different hydrophobic moieties. Our analysis of appropriate experimental data does not support that conclusion. It is shown in Figure 3. Thus, the surface heterogeneity effects in adsorption of ionic surfactants on oxide surfaces must be a more complicated phenomenon. It should, however, be emphasized that the abovediscusseddiscrepancy between the predicted and observed behavior does not oppose the theory by Harwell et al. Its most essential part related to the description of the surface aggregates is still a milestone in the theory. However, as far as the monomer adsorption is concerned, equations developed by adopting a lattice description formalism will be more useful for our further consideration.
2.0 3.0 it/mmol.d~
Figure 3. (A) log-log plots of the experimental adsorption isotherms of alkylbetaineson spherosilXOBO15 measured at 25 "C and pH 6 0 , CIZ;0,Clr. The experimental details are described in the Ph.D. Thesis by A. Amin-Alami, CNRS Laboratory330,Montpellier University,1989. (B)Functionarda In C calculated from the experimentaldata reported in (A). The
broken line is the approximation by eq 29 of the experimental plot of arJa ~n c.
For this purpose, we adopt the theoretical approach to monomer adsorption developed by Koopal and co-worke r ~ That . ~ ~approach is based on the following assumptions: (1) The surfactant molecule is a chain containing one charged group (valency TI and r - 1apolar segments. As the first approximation each CH2 group of the aliphatic chain can be considered as one segment. (2) In the adsorbed state each chain of, in total, r segments has a sequence of m segments adsorbed in the first layer as a train, and the remaining r - m segments protrude into the solution as a tail. (3) Due to Coulombic attraction the charged head segment has a strong electrostatic affinity to the surface, so that all adsorbed head segments are present in the first adsorbed layer (closest to the surface). (4) In a poor solvent, as water is for surfactants, segments tend to cluster together. The segment density distribution is therefore assumed to be homogeneous (step function), and extends over rlm layers each with a volume fraction &equal to 41. Only for such a simplified segment density distribution (4i)the calculation of the freeenergy of mixing is relatively simple. (5) The surfactant chain is flexible both in the solution and in the adsorbed state. With these assumptions Koopal et arrived at the following isotherm equation:
where the subscript b refers to the equilibrium bulk (31)Koopal, L. K.;Wilkinson, G. T.;Ralston, J. J . Colloid Interjace Sci. 1988, 126, 493.
Adsorption of Ionic Surfactants
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solution and Q is a combinatorial factor, the simplest form of which is the following:
n = Aom-lA,
(17) In eq 17 Xo is the fraction of the nearest neighbors in the same lattice layer and A1 is that in each of the adjacent layers: hence, 2x1 + Xo = 1. The last exponential term on the right-hand side of eq 16 describes the interactions between the surfactant and surface (mi,'j, then between the adsorbed surfactant molecules [2rx'(41- &)I, and between the polar head and the surface (-TY,). Above 2; is the weighed average net adsorption energy per segment:
+
mi: = (x,P + A1xPo) (m - l)(x," + A1xao) (18) where x,P is the polar segment (p)-surface ( 8 ) interaction energy, x p is the "exchange energy" parameter for the polar segment-water (PO) interaction, and the m_eanings of xaoand x a P are now obvious. The meaning of x' is the following:
(A,
+ A, - -)Aol XBP (19)
m r Finally Y1 is the reduced potential e$l/kT, where $1 is the potential at the plane through the centers of the adsorbed polar heads. The potential $l(x) can be interpreted as a sum of two contributions, the first one originating from the plane charge itself and the second one from the image charge:
where a0 is the area of a unit cell (the molecular cross section of a segment), 6, is the surface charge density sign included, E = ere, with eo the dielectric permittivity of a vacuum and e, that of water, T is the valency of the adsorbing ion, and K is the reciprocal Debye length, which in the presence of a symmetrical indifferent electrolyte can be given as = e(2n+2/ckT)1/2 (21) where no is the number of ions per unit volume, z is the valency of the indifferent electrolyte, and k is the Boltzmann constant. Of course, such a simple expression for the lateral Coulombic interactions between adsorbed ions, like that in eq 20, can be accepted only for low ion densities. However, as we apply these expressions to describe monomer adsorption at low surface coverages, the application of that simple expression is fully justified. For most surfactants, water is a poor solvent and the maximum value of (bb is given by the bulk critical micelle concentration. Hence, for most practical situations f#Jb
