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Notes Influence of the Compressibility of Adsorbed Layers on the Surface Dilational Elasticity V. B. Fainerman,† R. Miller,*,‡ and V. I. Kovalchuk§ International Medical Physicochemical Centre, Donetsk Medical University, 16 Ilych Avenue, Donetsk 83003, Ukraine, Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Am Mu¨ hlenberg 1, 14424 Potsdam, Germany, and Institute of Biocolloid Chemistry, 42 Vernadsky Avenue, 03680, Kyiv (Kiev), Ukraine
Here, γ is the surface tension, A is the surface area, and Γ is the adsorption. To express the elasticity only by measurable quantities, an additional relationship is needed. Various equations of state exist which can be used for this purpose. In the present conceptual model, we will use the SzyszkowskiLangmuir model, given by
(γ0 - γ)ω ) -ln(1 - θ) RT θ)
Received January 8, 2002. In Final Form: May 13, 2002
Introduction The dependence of the surface dilational elasticity of surfactant solutions on the surfactant concentration shows the following features: (i) at high oscillation frequencies, the elasticity levels off at a plateau; (ii) at higher surfactant concentration, that is, adsorption layer saturation, a limiting value of the elasticity is reached or the elasticity even passes through a maximum. The first point is due to a negligible exchange between surface layer and solution bulk at sufficiently high frequencies. The second fact has been observed so far for all surfactant systems;1-5 however, there is no satisfactory explanation in the framework of existing concepts of surface elasticity. The aim of this paper is to demonstrate the limitation of the existing theoretical models and to suggest an additional effect that is able to explain the experimentally observed concentration dependence of the dilational elasticity of surfactant adsorption layers.
bc 1 + bc
(3) (4)
The expression for the dilational elasticity of the surface layer in the limiting case of high oscillation frequencies reads
)-
RTΓ RTbc dγ ) ) d ln Γ 1 - θ ω
(5)
* Corresponding author. † Donetsk Medical University. ‡ Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. § Institute of Biocolloid Chemistry.
γ0 is the surface tension of the solvent, ω is the molar area, θ ) Γω is the surface coverage, R is the gas constant, T is the temperature, b is the adsorption constant, and c is the surfactant bulk concentration. With other equations of state (adsorption isotherms), the equation for the elasticity is changed; however, the general picture to be discussed here will remain valid, so that we restrict ourselves to this adsorption model here. Experimentally determined elasticities of four normal alcohols obtained from oscillating bubble measurements4 are shown in Figure 1 as a function of the dimensionless concentration bc. In addition, the theoretical values calculated from eq 5 are shown for comparison. The experimental data refer to the limiting oscillation frequency of 500 Hz; that is, the elasticities have reached the respective plateau values.4 The calculations with eq 5 have been performed using values of bc and ω determined from experimental surface tension isotherms. Such data from various authors were summarized recently in ref 6 showing that the Langmuir isotherm allows a sufficiently accurate description of the equilibrium adsorption state. Using the more accurate Frumkin isotherm would yield only little differences in the calculated curve. One can see that in the range of low alcohol concentration, the experimental results are in good agreement with the calculated values. However, at bc > 1 significant discrepancies arise between theory and experiment. The experimental elasticities reach a certain limit at bc . 1, which increases with the alkyl chain length. Similar results were obtained in refs 5 and 7 for the homologous series of alkyl dimethyl phosphine oxides (ADMPO). Experimental data for C12DMPO solutions taken from ref 7 are shown in Figure 2. The isotherm
(1) Lucassen, J.; Hansen, R. S. J. Colloid Interface Sci. 1967, 23, 319. (2) Bonfillon, A.; Langevin, D. Langmuir 1993, 9, 2172. (3) Tian, Y.; Holt, R. G.; Apfel, R. E. Phys. Fluids 1995, 7, 2938. (4) Wantke, K.-D.; Fruhner, H.; Lunkenheimer, K. J. Colloid Interface Sci. 1998, 208, 34. (5) Wantke, K.-D.; Fruhner, H. J. Colloid Interface Sci. 2001, 237, 185.
(6) Fainerman, V. B.; Miller, R.; Aksenenko, E. V.; Makievski, A. V. In Surfactants-Chemistry, Interfacial Properties and Application; Fainerman, V. B., Mo¨bius, D., Miller, R., Eds.; Studies in Interface Science, Vol. 13; Elsevier: New York, 2001; p 189-286. (7) Kovalchuk, V. I.; Kra¨gel, J.; Makievski, A. V.; Loglio, G.; Liggieri, L.; Ravera, F.; Miller, R. J. Colloid Interface Sci., in press.
Theoretical Concept Starting from the definition of the surface dilational elasticity ,
)
dγ d ln A
(1)
and using the conservation of the mass of adsorbed species in the monolayer (AΓ ) const), which is true at a sufficiently high oscillation frequency, one obtains1
)-
dγ d ln Γ
(2)
10.1021/la020024e CCC: $22.00 © 2002 American Chemical Society Published on Web 08/30/2002
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Figure 1. Experimental values of surface elasticity at limiting oscillation frequencies obtained using the oscillating bubble method (ref 4): (9) nonanol, (0) octanol, (2) heptanol, and (4) hexanol; theoretical curves were calculated from eq 5.
Figure 2. The same as in Figure 1 for C12DMPO solutions, with data obtained in ref 7 using the oscillating bubble method: (0, 9) ground conditions and (4, 2) microgravity conditions; filled and open symbols refer to different methods of calculation.
parameters for this surfactant given in ref 6 show that C12DMPO follows the Langmuir isotherm almost perfectly. Wantke et al.4 attempted to explain the deviations of the experimental elasticity values in the range bc > 1 from those predicted by eq 5. They propose that the dilational elasticity is influenced not only by adsorbed molecules but also by surfactant molecules present in the sublayer adjacent to the surface. Here, we want to propose another explanation for this phenomenon; however, the model given will be only on a rather qualitative basis and requires further future refinement. Effect of Surface Layer Compressibility on Dilational Elasticity All known theoretical approaches for the analysis of the elasticity of adsorption layers disregard the fact that the molar (molecular) area ω of adsorbed surfactant molecules can depend on the surface pressure. This means that a certain compressibility of the molecules is not assumed so far. In contrast, there are experimental data published in the literature showing the dependence of the
molecular area of insoluble amphiphilic molecules in the condensed state on surface pressure. This was illustrated, in particular, by synchrotron X-ray diffraction (grazing incidence X-ray diffraction) in refs 8-15. As an example, the dependencies of the area per molecule on surface pressure in the condensed monolayer are shown for 1-stearylamine monoglycerol in S(-) chiral form (S-AMD18)8,15 and 1-monosteroyl-rac-glycerol in racemic R(+) (8) Gehlert, U. Ph.D. Thesis, Technischen Universita¨t, Berlin, 1996. (9) Brezesinski, G.; Scalas, E.; Struth, B.; Mo¨hwald, H.; Bringezu, F.; Gehlert, U.; Weidemann, G.; Vollhardt, D. J. Phys. Chem. 1995, 99, 8758. (10) Gehlert, U.; Vollhardt, D.; Brezesinski, G.; Mo¨hwald, H. Langmuir 1996, 12, 4892. (11) Gehlert, U.; Weidemann, G.; Vollhardt, D.; Brezesinski, G.; Wagner, R.; Mo¨hwald, H. Langmuir 1998, 14, 2112. (12) Melzer, V.; Vollhardt, D.; Weidemann, G.; Brezesinski, G.; Wagner, R.; Mo¨hwald, H. Phys. Rev. E 1998, 57, 901. (13) Melzer, V.; Weidemann, G.; Vollhardt, D.; Brezesinski, G.; Wagner, R.; Struth, B.; Mo¨hwald, H. Supramol. Sci. 1997, 4, 391. (14) Vollhardt, D.; Brezesinski, G.; Siegel, S.; Emrich, G. J. Phys. Chem. B 2001, 105, 12061. (15) Gehlert, U.; Vollhardt, D. Langmuir 2002, 18, 688.
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Notes
Figure 3. Dependencies of the area per molecule as a function of the surface pressure in the condensed monolayer: curve 1, S-AMD-18, (refs 8 and 15); curve 2, Rac-ESD-18 (refs 8 and 10).
chiral form (Rac-ESD-18)8,10 in Figure 3. One can see that with increasing surface pressure, the area per molecule decreases. This decrease is almost linear and can be described by the simple equation
ω ) ω0 - k(γ0 - γ) ) ω0 - kΠ
(6)
Here k is an experimental parameter analogous to the compressibility coefficient, and Π is the surface pressure. Similar dependencies were found for other amphiphiles in the condensed state. The constant k is usually in the range between 0.0005 and 0.003 nm2/(mN/m) per molecule, or 300-2000 m2/(mN/m) per mol.8-15 Therefore, for condensed monolayers formed by molecules with changing molar area, the monolayer coverage can be assumed to be constant: θ ) Γω ) 1 ) const, which in turn yields d ln ω ) -d ln Γ. If we introduce this condition into eq 2, we obtain an expression for the dilational elasticity of the surface layer,
)
dγ d ln ω
(7)
From eq 6 it follows that dω ) kdγ; hence, eq 7 transforms into
)
ω dγ ) d ln ω k
(8)
For condensed monolayers, = const, because 1/ ) k/ω is the relative two-dimensional compressibility coefficient of surfactant molecules. The above conclusion is valid for a “two-dimensional compressibility”, because a decrease in ω can be accompanied by an increase in the thickness of the condensed monolayer. The idea of compressibility of adsorbed molecules has been already mentioned 20 years ago by de Feijter and Benjamins16 assuming the (16) de Feijter, J.; Benjamins, J. J. Colloid Interface Sci. 1982, 90, 289.
Table 1. Values of E ) ω/k for Condensed Monolayers of Some Insoluble Amphiphiles, According to the Given References surfactant
T [°C]
) ω/k [mN/m]
ref
Rac-ESD-16 S-ESD-16 Rac-ESD-18 S-ESD-18 Rac-ETD-16 S-ETD-16 Rac-AMD-18 S-AMD-18 THBAA THPAA
20 20 20 20 20 20 20 20 15 15
280 196 300 256 169 199 142 110 295 220
8, 9 8 8, 10 8, 10 8 8 8, 15 8, 15 12 13
molecules to be a kind of soft spheres; however, a thorough model was not presented. To understand the effects of the molecular geometry on the compressibility coefficient k, we can analyze the experimentally determined values obtained for insoluble monolayers. The values of ) ω/k for condensed monolayers of the racemic R(+) and S(-) forms of monoglycerol ethers (ETD),8,9 esters (ESD), and amines (AMD)8,10,15 and of N-tetradecyl-γ-hydroxybuturic acid amine (THBAA)12 and N-tetradecyl-γ-hydroxypropionic acid amine (THPAA)13 are listed in Table 1. The limiting value of the monolayer elasticity depends on the type of the surfactant, while in the same homologous series there is a significant increase with the hydrocarbon chain length. Generally, the larger the polar (alkyl chain) tilt angle with respect to the normal at low pressures, the higher the value of k and the lower the value of . During the compression of the monolayer, the tilt angle decreases; therefore the area per molecule becomes smaller, and the monolayer becomes thicker. For low Π, the initial angle between the azimuthal tilt direction and the (1.0) axis and the angle between the (1.0) and (0.1) directions also strongly influence the value of k.8-15 If the surface coverage and the compression rate are sufficiently large, we can assume that eq 6 remains valid also for adsorption layers. Thus, we can derive an equation for the dilational elasticity taking into account the two-
Notes
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Figure 4. Dependence of dilational elasticity on bc calculated from eq 11 at ω ) 2 × 105 m2/mol: curve 1, k ) 0; curve 2, k ) 500; curve 3, k ) 1000; curve 4, k ) 2000; curve 5, k ) 5000; and curve 6, k ) 10 000 m2/(mN/m) per mol.
dimensional compressibility of the surfactant molecules at the surface. Only as an example, we use the equation of state (3) again. When we replace ω by the expression given in eq 6,
(γ0 - γ)[ω0 - k(γ0 - γ)] ) -ln[1 - Γ(ω0 - k(γ0 - γ))] RT (9) we obtain
dγ ) d ln Γ -
RTΓ(ω0 - k(γ0 - γ))
[ω0 - 2k(γ0 - γ)][1 - Γ(ω0 - k(γ0 - γ))] + RTΓk (10)
For ω . kΠ, together with eqs 4 and 6, one can transform eq 10 into
)-
RTΓ RTbc dγ ) ) d ln Γ (1 - θ) + RTΓ(k/ω) ω + RTbc(k/ω) (11)
This equation has two limiting cases, which are just those obtained above. For k ) 0 (absolutely two-dimensionally incompressible molecules), eq 11 becomes identical to eq 5, while for k * 0 and bc . 1 (densely packed monolayer of two-dimensionally compressible molecules) we obtain eq 8. When we use other equations of state, relationships analogous to eq 11 will be obtained. However, due to the complexity of equations such as the Frumkin isotherm, the results would obviously be not so easy to discuss. Discussion The results of model calculations with eq 11 with values for k between 500 and 10 000 m2/(mN/m) per mol and ω ) 2 × 105 m2/mol are presented in Figure 4. The value of 2 × 105 m2/mol for the molar area ω is average between the values valid for alkyl dimethyl phosphine oxides (ω ) 2.5 × 105 m2/mol) and alcohols (ω ) 1.7 × 105 m2/mol).6 Note that the upper limit for k used here is about 5 times larger than the maximum value of the compressibility coefficient found experimentally for the insoluble condensed monolayer formed by S-AMD-18 (cf. Figure 3). This extension of the parameter range can be justified by
the fact that the molecules in adsorption layers do not exist in a condensed state (i.e., are less closely packed). Moreover, the length of the hydrocarbon chain in soluble surfactants is shorter than that characteristic for insoluble ones. The theoretical curves for bc > 1 shown in Figure 4 deviate from a straight line which corresponds to the so-called Gibbs elasticity (eq 5). This agrees well with the experimental data shown in Figures 1 and 2. For bc g 100, the dilational elasticity becomes independent of bc and is determined only by the two-dimensional compressibility coefficient k. For k ) 2000 m2/(mN/m) per mol, which was found experimentally as the maximum for a condensed monolayer, the corresponding theoretical curve in Figure 4 is quite close to the experimental data of C12DMPO shown in Figure 2. To achieve agreement between the proposed model and the experimental data for normal alcohols (see Figure 1), the k values should be in the range between 5000 and 10 000 m2/(mN/m) per mol. It is also very important to note that the derivative dc/dΓ which follows from eq 9 is not as large as for bc . 1 obtained from eq 3. This means that the problem connected with the linearity condition at high surfactant concentrations, that is, high surface coverage, as discussed in ref 7 is not crucial. The underestimation of the diffusional exchange of matter is another possible explanation for the discrepancy between theory and experiment. However, at sufficiently large frequencies this contribution is negligible. It is controlled by the diffusion relaxation frequency ωD ) D(dc/dΓ)2, where D is the diffusion coefficient. As D does not change too much with the surfactant molecule and the derivative dc/dΓ reflects essentially the adsorption isotherm, there is no free parameter which can be tuned to increase the exchange of matter contribution significantly, that is, to better match the experimental data by a theoretical model. Thus, the proposed effects given in refs 4 and 5 are unable to explain the huge discrepancy between theory and experiment. In ref 17, it was tried to find agreement between experimental data for C10EO5 solutions and theoretical models via combinations of the equation of state and Gibbs equation. The Frumkin equation of state was used over the entire concentration interval or at each experimental concentration, reflected by local values of the nonideality parameter H, neither of which ended up in a satisfactory description of the experiments. Moreover, the nonideality parameter was negative which actually decreases the resulting elasticity as compared to the Langmuir isotherm as a special case. This however is physically not reasonable for nonionic surfactants, as for these surfactants the reorientation model of ref 18 is optimum. In the analysis of the second derivative of γ(c), the authors of ref 17 used a higher polynomial to describe the experimental points, which leads to a better agreement between experiment and theory. In our opinion, such a procedure cannot be seen as universal for the following reasons: (i) For example, the surface tension isotherms of normal alcohols or ADMPO, in contrast to CnEOm, are very well described by the Frumkin isotherm with a positive parameter H,6 even though this equation does not yield a satisfactory description of the elasticity. The deviation from the experiments is even larger than that obtained for the simpler Langmuir model. (ii) The application of higher polynomials for the description of γ(c) does not change the situation for normal (17) Jayalakshmi, Y.; Ozanne, L.; Langevin, D. J. Colloid Interface Sci. 1995, 170, 358. (18) Fainerman, V. B.; Miller, R.; Mo¨hwald, H. J. Colloid Interface Sci. 2002, 247, 193.
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alcohols or ADMPO and in addition increases the number of model parameters (coefficient of the polynomial) without any physical sense. (iii) The calculation of the second derivative in respect to concentration c increases the role of errors of each of the experimental points, which is particularly sensitive close to the critical micelle concentration (cmc). On the contrary, the compressibility coefficient k introduced in the present model has a clear physical sense and its value has been well determined for a number of insoluble monolayers. Note that the coefficient k (cf. eq 9) has approximately the same influence as a negative parameter H in the Frumkin equation: for a given surface pressure, the surface coverage increases with decreasing k. Conclusions The proposed model for the description of a limiting dilational elasticity of adsorption layers, essentially based on the additional parameter of two-dimensional compressibility of densely packed adsorption layers, allows a
Notes
satisfactory explanation of existing experimental results. The order of magnitude of this compressibility coefficient is in good agreement with data obtained for condensed monolayers using the synchrotron X-ray diffraction method. This new model, being applied to experimental data obtained at high oscillation frequencies, provides new information concerning the structure and mechanical properties of surfactant adsorption layers. For the sake of simplicity, the given model is based on the LangmuirSzyszkowski adsorption model; however, the general physical idea can be easily transferred to other adsorption models. An analysis of the consequences of the introduced compressibility coefficient k on the adsorption isotherm and equation of state for surfactant adsorption layers is under way. Acknowledgment. The work was financially supported by the Max Planck Society, the DFG (Mi 418/11), and the Ukraine SFFR (03.07/00227). LA020024E
