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Langmuir 2002, 18, 2506-2514
Effect of Counterions on the Adsorption of Ionic Surfactants at Fluid-Fluid Interfaces P. Warszyn´ski,*,† K. Lunkenheimer,‡ and G. Czichocki‡ Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Krako´ w, Poland, and Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Potsdam, Germany Received March 13, 2001. In Final Form: January 14, 2002 The equilibrium surface tension of anionic surfactant n-decyl sulfate (DS-) for various monovalent (alkali) counterions was investigated. It was found that surface activity of surface chemically pure DSsignificantly increases with decreasing hydrated size of the counterion. We describe our experimental results in terms of the previously developed adsorption model, which assumes that the counterions may penetrate the Stern layer where the surfactant headgroups are adsorbed. The headgroups and counterions have a finite size that leads to the surface exclusion effects in the adsorption isotherm. The model is improved by explicitly taking into account the electric interactions between adsorbed ions in the adsorbed layer. We obtain a good correlation between the relative counterion size in the Stern layer, the measure of the area excluded by the adsorbed counterions, and the effective radius of the hydrated counterion in the solution. The limiting areas per molecule at the critical micelle concentration for the adsorbed decyl sulfate for various counterions are in good agreement with those measured by neutron scattering.
Introduction The understanding of the interfacial behavior of ionic surfactants at fluid-fluid interfaces is very important for a variety of colloidal phenomena like surface tension, thin film stability, micellization, and so forth. Counterions play a significant role in the interfacial properties of ionic surfactants. The formation of ionic micelles, the critical micelle concentration (cmc), and the micellar size are strongly dependent on the counterion.1-4 For example, taking into consideration only the simple monovalent alkali counterions, the cmc of dodecyl sulfate follows the rule cmcCs < cmcK < cmcNa < cmcLi, that is, it increases with the effective radius of the hydrated counterion. The effect of counterions on the adsorption of ionic surfactants at the air/solution interface was studied most extensively for inorganic salts of sodium dodecyl sulfate (SDDS) using surface tension5,6 and neutron scattering.7,8 The neutron scattering technique not only can give information about the amount of the surfactant adsorbed at the interface but, using the appropriate contrasting techniques, can provide some insight about the structure of the adsorbed layer.9 Similar information concerning the structure of the adsorbed layer can also be obtained from X-ray reflectometry;9 however, the interpretation of the results is rather ambiguous, heavily dependent on * Corresponding author. Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30239 Krako´w, Poland. E-mail:
[email protected]. † Polish Academy of Sciences. ‡ Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. (1) Mukerjee, K.; Mysels, J.; Kapauen, P. J. Phys. Chem. 1967, 71, 4166. (2) Chen, S. H. Annu. Rev. Phys. Chem. 1986, 37, 351. (3) Clint, J. Surfactant Aggregation; Chapman & Hall: London, 1992. (4) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1989. (5) Weil, I. J. Phys. Chem. 1969, 70, 133. (6) Oh, S. G.; Shah, D. O. J. Phys. Chem. 1993, 97, 284. (7) Lu, J. R.; Marocco, A.; Su, T. J.; Thomas, R. K.; Penfold, J. J. Colloid Interface Sci. 1993, 158, 303. (8) Su, T. J.; Lu, J. R.; Thomas, R. K.; Penfold, J. J. Phys. Chem. B 1997, 101, 937. (9) Thomas, R. K.; Penfold, J. Curr. Opin. Colloid Interface Sci. 1996, 1, 23.
the model assumed. In all experimental works reported, it was found that the surface tension of aqueous SDDS was lowered more effectively for smaller, less hydrated counterions. The effect of counterions on the adsorption of cationic surfactants has also been studied.10,11 Theoretical efforts concerning the influence of the counterions on the adsorption of ionic surfactants are rather scarce due to the relative complexity of the problem. When the ionic surfactant molecules are adsorbed at the fluid-fluid interface, an electric double layer is built and its electric potential is determined by the charge of surfactant molecules. As a result, an energetic barrier of electrostatic origin appears for the adsorption of surfactant. Therefore, the buildup of the double layer hinders the adsorption of subsequent surfactant molecules and facilitates the adsorption of oppositely charged counterions. First Davies and Rideal12 and then Borwankar and Wasan13 employed the Gouy-Chapmann theory to describe the electric double layer formed when the ionic surfactant adsorbs at the fluid/fluid interface. They considered the surfactant adsorption in terms of the Frumkin isotherm (Langmuir in the case of Davies and Rideal). They assumed that the surfactant ions were adsorbed in the Stern layer14 and the counterions remained in the diffuse part of the electric double layer. They obtained the dependence of the surface tension on the ionic surfactant concentration in solution by applying the Gibbs adsorption equation. However, their model could not predict any counterion-specific behavior. Recently, Kalinin and Radke15 developed a new model of ionic surfactant adsorption at the fluid-fluid interface. Their approach allows counterion binding to the surfactant ions adsorbed in the Stern layer, and therefore the adsorption of the ionic surfactant may depend on counterion type. (10) Goralczyk, D. J. Colloid Interface Sci. 1996, 184, 139. (11) Adamczyk, Z.; Para, G.; Warszyn´ski, P. Langmuir 1999, 15, 8383. (12) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1963. (13) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1986, 1, 199. (14) Stern, O. Z. Elektrochem. 1924, 30, 508. (15) Kalinin, V. V.; Radke, C. J. Colloids Surf., A 1996, 114, 337.
10.1021/la010381+ CCC: $22.00 © 2002 American Chemical Society Published on Web 03/07/2002
Effect of Counterions on Adsorption
They adopted Grahame’s16 concept of the triple layer with counterions located at a plane different from that of the adsorbed surfactant headgroups. Although their model seems to be more suitable for charged insoluble monolayers, they successfully described data concerning the dependence of the surface tension at the air/aqueous SDDS and oil/aqueous SDDS interfaces on the SDDS and NaCl concentration in the aqueous phase. The diffuse layer potential values predicted by their model were also more realistic and better agreed with the experimentally measured values than those obtained using the model of Borwankar and Wasan. They did not apply their model to study any counterion-specific effects. In our previous paper,17 we proposed a model of the adsorption of ionic surfactants at the fluid-fluid interface which is alternative to the one developed by Kalinin and Radke. The model is based on the assumption of adsorption of ionic surfactants and counterions in the same region of the Stern layer. The counterion located in that layer is not associated with a single headgroup but can be shared by several headgroups. Therefore, the Stern layer can be regarded as a quasi-two-dimensional electrolyte in which the electroneutrality condition is not fulfilled (the overall charge in the Stern layer is not zero). This model explicitly takes into account the counterion-specific effects. We applied our model to describe the surface tension and surface potential isotherms of SDDS in aqueous NaCl solution in a broad range of concentrations (0-1 M).17 Recently, Kralchevski et al.18 formulated an alternative model of ionic surfactant adsorption. They also postulated independent adsorption of counterions in the Stern layer. They considered the thermodynamic consistency criteria for the adsorption isotherms of surfactant ions and counterions. The models of ionic surfactant adsorption were recently reviewed by Prosser and Franses.19 Most of the results reported in the literature indicate strong binding of the counterions at the charged liquidliquid interface,8,20-22 but there exists controversy concerning the location of the counterions adsorbed in the Stern layer. There is experimental evidence from neutron reflectivity of strong counterion penetration in the region of the headgroup location for soluble8 as well for insoluble23 monolayers. The counterions are located in the surface layer together with the headgroups, and a single counterion can be shared by several headgroups. The molecular dynamics simulations24,25 and X-ray reflectivity measurements of the thin films26 seem to support a high degree of penetration of the surfactant adsorption region by counterions. Also, from the experimentally observed limiting area per molecule at cmc (0.35-0.50 nm2) one can conclude that there is enough space for the counterions to penetrate the adsorption layer. On the other hand, the developed theoretical models of ionic micelles20,21 as well as electron spin-echo modulation spectroscopy results concerning the structure of the Stern layer around SDDS (16) Grahame, D. C. Chem. Rev. 1947, 41, 441. (17) Warszyn´ski, P.; Barzyk, W.; Lunkenheimer, K.; Fruhner, H. J. Phys. Chem. B 1998, 102, 10948. (18) Kralchevsky, P. A.; Danov, K. D.; Broze, G.; Mehreteab, A. Langmuir 1999, 15, 2351. (19) Prosser, A. J.; Franses, E. I. Colloids Surf., A 2001, 178, 1. (20) Stigter, D. J. Phys. Chem. 1975, 79, 1008. (21) Buenen, J. A.; Ruckenstein, E. J. Colloid Interface Sci. 1983, 96, 469. (22) Yoshida, T.; Taga, K.; Okabayashi, H.; Matsushita, H.; Kamaya, H.; Ueda, I. J. Colloid Interface Sci. 1986, 109, 336. (23) Su, T. J.; Thomas, R. K.; Penfold, J. Langmuir 1997, 13, 2133. (24) Gamba, Z.; Hautman, J.; Sheley, J. C.; Klein, M. L. Langmuir 1992, 8, 3155. (25) Schweighofer, K. J.; Essmann, U.; Berkowitz, M. J. Phys. Chem. B 1997, 101, 3793. (26) Belorgey, O.; Benattar, J. J. Phys. Rev. Lett. 1991, 66, 313.
Langmuir, Vol. 18, No. 7, 2002 2507
Figure 1. Model of the adsorption of ionic surfactant at the air-aqueous solution interface.
micelles27 suggest a compact layer of headgroups with a low counterion penetration. In this paper, we modified the theoretical model developed previously17 in order to consider the effect of the excluded volume (in the interfacial layer) due to the finite size of surfactant headgroups and counterions in the Stern layer as well as strong lateral electric interaction of adsorbed ions in the surface layer. We use our improved model to describe the equilibrium surface tension data of the aqueous, surface chemically pure28,29 solution of ndecyl sulfate (DS) in the presence of various alkali counterions. Model of the Adsorption of Ionic Surfactants The main principle of the model is presented in Figure 1. Adsorption of the ionic surfactant molecules leads to the buildup of the electric double layer (EDL) at the airsolution interface. Surfactant headgroups, assumed to be negatively charged, are adsorbed at the interface in the Stern layer. The counterions can adsorb also in the Stern layer at the same Helmholtz plane as the headgroups. Both ions in the Stern layer preserve their freedom, there is no complex formation, and an adsorbed counterion can be shared by several headgroups. The Stern layer can be considered as a two-dimensional electrolyte, which does not fulfill the electroneutrality condition. The total charge in the Stern layer determining the diffuse layer potential is the sum of charges of the adsorbed negative surfactant headgroups and positive counterions:
σ ) zsFΓs + zcFΓc
(1)
where σ is the surface charge density, Γs is the surface concentration of the surfactant in the Stern layer, Γc is (27) Szajdzinska-Pietek, E.; Maldonato, R.; Kevan, L.; Berr, S. S.; Jones, R. M. J. Phys. Chem. 1985, 89, 1547. (28) Miller, R.; Lunkenheimer, K. Colloid Polym. Sci. 1986, 264, 273. (29) Lunkenheimer, K.; Miller, R. J. Colloid Interface Sci. 1987, 120, 176.
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the surface concentration of the counterions in the Stern layer, F is the Faraday constant, and zs and zc are the valencies of surfactant ions and counterions, respectively. For the sake of simplicity, we consider further a system with z+ ) |z-| ) 1. The diffuse layer potential at the boundary between the Stern layer and the diffuse part of EDL can be determined from the formula
ψd )
(
)
σe 2kT sinh-1 e 20kTκ
σδ 0s
(3)
where δ is the thickness of the Stern layer and s is the dielectric constant in the Stern layer. The adsorption isotherms for the surfactant ions and counterions describing relationships between ionic concentration in the bulk and in the surface layer can be derived from the equilibrium condition of equal electrochemical potentials for the bulk phase and the Stern layer and by applying the Flory-Huggins statistics30,31 for the concentrated twodimensional electrolyte in the Stern layer. The electrochemical potential in the bulk is given by
µ( ) µ0( + kT ln a(
(4)
where µ0( is the standard chemical potential and a( is the activity of counterions and surfactant ions, respectively, which can be conveniently calculated using the RobinsonStokes formula.32 In the Stern layer, the electrochemical potential can be written in the form33
µσs ) µσ,0 s + kT ln
[
]
θs
( )
and
( )
( )
eψs φc a+ exp (1 - θs - θc)gsc ) θc exp Rc kT kT
(8)
0 where Rs ∼ exp((µR,0 s - µ-)/kT) is the “surface activity” of the surfactant ion, being a measure of the free energy of the adsorption after separating the contribution of the - µ0+)/kT) is the electric component, and Rc ∼ exp((µR,0 c “surface activity” of the counterion which is a measure of the binding in the Stern layer and the image forces acting on the ionic charge at the interface between the media with different dielectric properties.34 For the sake of simplicity, we assumed that gs ) 1 so gsc ) Γs∞/Γc∞. For the dilute solutions (200 mol/dm3) means that the alkali ions by themselves are not surface active. In fact, the small increase of the surface tension (∼2 ÷ 3 mN/m) upon addition of l M of electrolyte to water can be interpreted in terms of negative adsorption of simple ions.45 However, this phenomenon cannot be accounted for in our present theory. Figure 5 presents the correlation between asc, the relative size of the alkali cation (with respect to the size of the DS- headgroup) obtained from the fit to the theoretical model, and the hydrated ion radii in the bulk electrolyte rh. The values of the hydrated ion radii were taken from ref 46. Figure 5 also shows the dependence between the effective ionic radii in the surface layer aS, obtained from our calculation, and its effective size in the bulk electrolyte ab. The latter values were taken for alkali chlorides32 as we could not find the respective data for (44) Sundell, S. Acta Chem. Scand., Ser. A 1977, 31, 799. (45) Weisenborn, P. K.; Pugh, R. J. J. Colloid Interface Sci. 1996, 184, 550. (46) Koryta, J.; Dvorak, J.; Bohackova, V. Lehrbuch der Electrochemie; Springer-Verlag: Wien, 1975.
Effect of Counterions on Adsorption
Langmuir, Vol. 18, No. 7, 2002 2511
Figure 5. The correlation between the relative size asc of the counterion (with respect to the size of the DS- headgroup) and its hydrated radius rh (circles) and the effective ionic radius aS in the Stern layer and its effective radius in electrolyte for alkali chlorides. The relative size of the counterions and their effective ionic radius in the Stern layer were obtained from the fit of experimental data to the theoretical model. Lines are drawn to guide the eye. Table 2. Limiting Area per Decyl Sulfate Ahg and Limiting Area per Molecule in the Stern Layer Acmc (Decyl Sulfate and Counterion, at cmc)a Ahg [nm2] Li+
Na+ NH4+ K+ Cs+ a
0.49 0.46 0.44 0.41 0.40
Ahg [nm2] SDDS from neutron reflectivity (ref 7) 0.50 0.44 0.38
Acmc [nm2] 0.27 0.25 0.24 0.22 0.215
Calculated for Γs∞ ) 6.4 × 10-10 mol/cm2.
alkali decyl sulfate solutions. It can be concluded that the effective ionic radii in the surface layer are comparable to the ones in the bulk electrolyte. This suggests that counterions are transferred to the adsorption layer together with most of their hydration shell. Results presented in Figure 5 clearly indicate that there exists a strict correlation between the relative size of the counterion which is the measure of the surface excluded by the counterion adsorbed in the Stern layer and its hydrated size in the solution. Adsorption of the counterion of the finite size in the Stern layer makes that part of the interface inaccessible for further adsorption of surfactant molecules or other counterions. The larger the size of the counterion, the more effectively it blocks the surface. Simultaneously, presence of less hydrated cations in the Stern layer leads to the larger correction accounting for the lateral electric interaction. The activity coefficient for the quasi-two-dimensional electrolyte formed by surfactant ions and counterions adsorbed in the Stern layer assumes larger (negative) values for smaller counterions (cf. eq 9). Table 2 presents the dependence of the area per DSmolecule adsorbed Ahg and the area per both DS- and counterions adsorbed in the Stern layer Acmc on the cation obtained from our fitting procedure at the cmc. The area per DS molecule adsorbed at cmc agrees qualitatively with the ones obtained from neutron reflection for dodecyl
Figure 6. The dependence of the surface tension of LiDS (black circles), CsDS (black diamonds), and decanol (white diamonds) on their concentration in solution. Solid lines represent fits of the theoretical model of adsorption of ionic surfactants to the experimental data; the dashed line represents the prediction of our model for the nonionic compound with a hydrophobic part identical to that of n-decyl sulfate.
sulfate.7 Dodecyl sulfate has two more CH2 groups in the hydrocarbon chain than decyl sulfate. However, the crosssectional area of the chain is much smaller (ca. 0.20 nm2) than the respective area per sulfate headgroup. Additionally, at the surface concentration corresponding to cmc the hydrophobic chains should preferentially be directed out of the aqueous phase, and since they are quite flexible, they can adjust their conformation to the available surface area. Therefore, it seems to be reasonable to assume that the area per molecule close to cmc is determined predominantly by the headgroup and counterion size, and consequently we can compare the predictions of our model for n-alkali decyl sulfate and the results of neutron scattering for dodecyl sulfate. Data presented in Table 2 suggest that for smaller counterions a more compact structure of the adsorbed layer is formed. That may be the reason for increased gas permeability observed in thin films formed from the dodecyl sulfate solutions if the sodium counterion is replaced by lithium.47 Figure 6 presents the comparison between the surface tension isotherms of lithium decyl sulfate, cesium decyl sulfate, and their parent compound, n-decanol, which has the same number of carbon atoms in the hydrophobic chain. There is a difference of two concentration decades in the surface activity between the ionic surfactant and the nonionic one with the identical hydrophobic part. That is due to repulsive electric interactions between the surfactant ions adsorbed in the surface layer and in the bulk. This repulsion can be effectively reduced by addition of inorganic electrolyte, and therefore in the presence of an excessive amount of salt the surface activity of the ionic surfactant strongly increases.11,16,17 Solid lines in Figure 6 represent fits of our model of ionic surfactants to the experimental data, while the dashed line shows the prediction of the model when all charges (potentials) are set to zero, that is, the Frumkin isotherm for the single nonionic compound having a hydrophobic part identical to that of DS-. The dashed line describes the experimental (47) Krustev, R.; Platikanov, D.; Nedyalkov, M. Colloids Surf., A 1997, 123, 383.
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dependence of surface tension versus concentration of n-decanol quite accurately. This indicates that our model can correctly account for the effect of electric interactions on the adsorption of ionic surfactant. Conclusions We have presented an improved model of adsorption of ionic surfactants at the fluid-fluid interface. The model is based on the assumption of adsorption of surfactant ions and counterions in the Stern layer. Both ions in the layer preserve their freedom; therefore, the layer can be considered as a quasi-two-dimensional electrolyte. Headgroups and counterions adsorbed in the Stern layer have a finite size. That leads to the surface exclusion effects accounted for in our model by Flory-Huggins statistics. The electric interaction between adsorbing molecule and charged interface are described in terms of the SternDebye-Hu¨ckel model of the double layer. The model also takes into account, in the approximate manner, the lateral electric interactions in the Stern layer. We have applied our model to interpret the dependence of the surface tension of decyl sulfate on its concentration in solution for various monovalent alkali counterions. The model describes very well the experimentally observed changes in the surface activity depending on the type of counterion. At the same bulk concentration of decyl sulfate, the surface tension is more reduced for hydrated cations of smaller size. This observation agrees with other experimental findings reported in the literature concerning adsorption behavior of ionic surfactants. We have been able to describe the dependence of the surface tension of decyl sulfate on its concentration in solution for five types of counterions in terms of only two parameters, the relative size of the counterion to the headgroup and the effective ionic radius in the surface layer. The resulting values of those parameters are well correlated with the hydrated radii of counterions. The limiting area per molecule of decyl sulfate adsorbed in the Stern layer at cmc for various counterions is in good agreement with one determined by neutron reflectivity. The structure of the adsorbed layer is more compact for smaller, less hydrated counterions, and that can result for instance in different permeabilities of the thin film formed from the solution of the ionic surfactant with various counterions. Acknowledgment. The authors thank Max-PlanckGesellschaft for the support of this work. Appendix Consider an ion (surfactant or counterion) adsorbed in the Stern layer at the air/solution interface. This ion induces a “hemispherical” ionic cloud (self-atmosphere) in the neighboring diffuse layer, a “circular” ionic cloud in the Stern layer, and electrostatic image charges above the interface. We assume that centers of all adsorbed ions lie on the same plane in the Stern layer and the distance of closest approach between any ions is as. The electrostatic potential at any point (F,z) in the vicinity of an adsorbed ion may be expressed as
ψ1(F,z) ) ψ(z) + φ1(F,z)
(A1)
where ψ(z) is the mean electrostatic potential, φ1(F,z) is the fluctuation (self-atmosphere) potential, z is the distance normal to the adsorption plane, and F is the radial distance from the center of the adsorbed ion. The mean electrostatic potential in the diffuse layer satisfies the
Poisson-Boltzmann equation,
d2ψ(z) 2
dz
)
[
]
e ψ(z) kT 2 κ sinh e kT
(A2)
with the solution
ψ(z) )
( [ ]
)
eψd 4kT tanh-1 tanh exp(-κ(z - δs)) e 4kT z g δs (A3)
In the Stern layer and in air, that is, for z < δs, the mean potential satisfies the Laplace equation,
d2ψ )0 dz2 The boundary conditions at the respective planes (see Figure 1) are
dψ dψ a |z)-δa-0 - 1 |z)-δa+0 ) 0 dz dz
(A4)
σ dψ dψ 1 |z)-0 - 2 |z)+0 ) dz dz 0
(A5)
dψ dψ 2 |z)δs-0 - |z)δs+0 ) 0 dz dz
(A6)
The total electrostatic potential in the diffuse layer also satisfies Poisson-Boltzmann equation:
∇2ψ1 )
d2ψ(z) dz
2
+ ∇2 φ1(F,z) )
[
]
e ψ1(z) kT 2 κ sinh e kT z > δs (A7)
Subtracting (A2) from (A7) and linearizing the result with respect to the self-atmosphere potential, we obtain
∇2 φ1(F,z) )
kT 2 κ (z) φ1(F,z) e
z > δs
(A8)
where κ(z) is a local Debye-Hu¨ckel parameter:
κ2(z) ) κ2 cosh
[
]
e ψ(z) kT
(A9)
For z < δs, the self-atmosphere potential satisfies the Laplace equation ∇2 φ1(F,z) ) 0; φ1(F,z) f 0 as z or F tends to infinity and the other boundary conditions are
dφ1 dφ1 - 1 )0 a dz |z)-δa-0 dz |z)-δa+0
(A10)
dφ1 dφ1 ∆σ(F) ) 1 |z)-0 - 2 dz dz |z)+0 0
(A11)
dφ1 dφ1 - )0 2 dz |z)δs-0 dz |z)δs+0
(A12)
where ∆(F) is the surface density of fluctuation charge in the Stern layer connected with the presence of an adsorbed ion and its self-atmosphere. For F > as it can be found, if we express the surface density of surfactant ions at given distance F from the center as35
Effect of Counterions on Adsorption
Langmuir, Vol. 18, No. 7, 2002 2513
[
ν-(F) ) ν0- exp
]
e ψ1(F,0) kT
(A13)
φ1(F,z) )
∫0∞ k J0(kF) [A(k) exp(kz) + B(k) exp(-kz) + C(k) exp(-k|z|)] dk +
whereas for F f ∞, it becomes the mean surface density of adsorbed ions:
[
ν-(∞) ) ν0- exp
]
[ ]
eψs e ψ(0) ) ν0- exp ) NAΓs kT kT
(A14)
Therefore, it follows from (A1), (A13), and (A14) that
[
ν-(F) ) NAΓs exp
]
e φ1(F,0) kT
]
(A16)
Therefore, the local charge density can be expressed as
[
]
e φ1(F,0) kT e φ1(F,0) eNAΓs exp (A17) kT
[
]
After linearizing (A17), we obtain for the fluctuation charge
∆σ(F) ) -κs0s φ1(F,0)
φ1(F,0) ) φ0(F,0) + φa(F,0) + φd(F,0)
F > as
(A18)
φ0(F,0) )
e 4π0s
e 1 w ∫0∞ J0(kF) H(k) dk Ff0 4π0s F
φa(F,0) ) -
asσ 4π0s
φd(F,0) ) 1 4π0s
∫0∞ J0(kF) H(k) ∫a∞F′ ∆σ(F′) J0(kF′) dF′ dk s
(A27) is the potential due to the two-dimensional ionic cloud.
F < as
(A19)
Using eq A18 for the fluctuation charge and following the procedure of Levine et al.,35 we obtain a Fredholm integral equation of the second kind:
where
φ1(F,z) )
Φ(F,0) ) -4π0sκs
F(k) Q(z - δs) exp(-zxκ2 + k2) J0(kF) k dk z > δs (A20)
where
φd(F,0) ) Φ(F,0) - 4π0sκs
2R - 1 1ζ exp(-2κz) 2R + 1 Q(z) ) 1 - ζ exp(-2κz)
)
]
R)
[ ]
ζ ) tanh2
[ ]
1 k2 1+ 2 2 κ
eψd 4kT
1/2
(A21)
and J0(kF) is the zero-order Bessel function of the first kind. The solution of the Laplace equation for z < δa is35
∫0∞ k J0(kF) E(k) exp(kz) dk
z < -δa (A22)
(48) Williams, W. E. Proc. Phys. Soc., London, Sect. A 1953, 66, 372.
∫a∞ F′ M(F,F′) φd(F′,0) dF′ s
(A29)
∫a∞ F′ M(F,F′) [φ0(F′,0) + s
φa(F′,0)] dF′ (A30) and
M(F,F′) )
φ1(F,z) )
(A26)
is the potential due to the excluded charge, and
where σ is the mean charge in the Stern layer (cf. eq 1), since upon placing an ion the disk of radius as is depleted of charge. We can find φ1(F,z) by applying Fourier-Bessel transforms. The solution for eq A8 was obtained by Williams:48
[ (
∫0∞ k1 J1(kF) J0(kF) H(k) dk
H(k) ) 1 + 4π0sk[A(k) + B(k) + C(k)] (A28)
∆σ(F) ) -σ
∫0
(A25)
is the potential due to the point charge at (0,0),
On the other hand,
∞
(A24)
where
e φ1(F,0) ν+(F) ) NAΓc exp kT
σ1(F) ) σ + ∆σ(F) ) eNAΓc exp -
The functions A(k), B(k), C(k), D(k), E(k), and F(k) have to be determined from the boundary conditions (A10)(A12). Applying a procedure analogous to one used by Levine et al.35 for the single ion adsorbed at the electrode surface, one can find that in the Stern layer
(A15)
The same applies for the surface density of counterions:
[
1 1 s x 2 F + z2 -δa < z < δs (A23)
1 4π0s
∫0∞ H(k) J0(kF) J0(kF′) dk
(A31)
The contribution of the self-atmosphere potential to the chemical potential of the ion (surfactant or counterion) can be found by applying the Guntelberg charging process:33
φ)
∫0e
1
φd(0,0) de′
(A32)
where e1 is the charge of an adsorbed ion. This contribution calculated separately for surfactant ion φs and counterion φc has to be used in eqs 5 and 6 respectively. Therefore, an effective algorithm for solving eq A29 and then for integrating eq A32 is desired if we would like to formulate the correct model of ionic surfactant adsorption. However, it is a complicated numerical task and the resulting
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algorithm may be difficult to implement for fitting to experimental data. Therefore, some crude approximations of (A32) could initially be used to assess the implications of the theoretical model. Since the derivation described above is analogous to one which leads to the calculation of the activity coefficient of strong electrolytes, on the basis of the scaling arguments (κs-1 is the correlation length for the two-dimensional electrolyte as κ-1 is for the threedimensional one) we propose eq 9 as such an initial approximation. It should correctly describe the contribution of the self-potential in the Stern layer provided that κs . κ Equation 9 can also be substantiated using the following arguments. If the screening occurs predominantly in the Stern layer, that is, for κs . κ ≈ 0, one can write set of equations eq A8 to eq A12 in the approximate form
( )
∂2φ1 e1 1 ∂ ∂φ1 F + 2 )δ(x) δ(y) δ(z) F ∂F ∂F ∂z 0s σ δ(z) (1 - θ(F - as)) + κsφ1 δ(z) θ(F - as) (A33) 0s where δ(x) and θ(F) are the Dirac delta and step functions, respectively. The three terms at the right-hand side correspond to the point charge, the excluded charge, and the two-dimensional ionic cloud. Considering the equation for the third term only and approximating the delta function by a thin slab (of the order of 1/κs) of the infinite cylinder of radius as, one obtains
( )
1 ∂ ∂φd F ) κs2φdθ(F - as) F ∂F ∂F
(A34)
balance the unit charge at the origin, we obtain the solution of eq A34 in the form49
φd(F) ) -
e1 K0(κsF) 4π0sas K1(κsas)
(A35)
where K0 and K1 are the zero- and first-order modified Bessel function of the second kind. Therefore, the potential of the ionic cloud at the center of point charge can be found as
φd(0,0) = -
K0(κsas) e1 4π0sas K1(κsas)
(A36)
Taking into account the asymptotic behavior of K0,
K0(r) )
x2rπ exp(-r)
for r . 1
(A37)
and the dependence K1(r) ) -dK0(r)/dr, one finds that if κsas . 1,
e1 e1 2κsas κs )) 4π0sas 1 + 2κsas 2π0s 1 + 2κsas e1 e1 κs κs ≈(A38) 4π0s 1 4π0s 1 + κsas + κsas 2
φd(0,0) = -
so the approximate formula for the potential of the ionic cloud at the center of point charge is analogous to that for the three-dimensional electrolyte with κ being replaced by κs. Applying the Guntelberg charging according to eq A32, one obtains eq 9. LA010381+
Taking into account the boundary condition that the charge of the two-dimensional ionic cloud must counter-
(49) White, L. R. J. Chem. Soc., Faraday Trans. 2 1977, 73, 577.
