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Adsorption Layer Characteristics of Mixed Sodium Dodecyl Sulfate/CnEOm Solutions 1. Dynamic and Equilibrium Surface Tension V. B. Fainerman,† E. V. Aksenenko,‡ S. V. Lylyk,† J. T. Petkov,§ J. Yorke,§ and R. Miller*,
Donetsk Medical University, 16 Ilych Avenue, 83003 Donetsk, Ukraine, ‡Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky Avenue, 03680 Kyiv (Kiev), Ukraine, §Unilever R&D Port Sunlight, Quarry Road East, Bebington, CH63 3JW, U.K., and Max-Planck-Institut f€ ur Kolloid- und Grenzfl€ achenforschung, Am M€ uhlenberg 1, 14424 Potsdam, Germany )
†
Received June 10, 2009. Revised Manuscript Received July 7, 2009 Bubble profile analysis tensiometry is used to study the dynamic and equilibrium surface tensions of mixed sodium dodecyl sulfate (SDS)/C12EO5 and SDS/C14EO8 solutions. For the data analysis, a new theoretical model was employed, which assumes different adsorption mechanisms for each type of surfactants. In particular, the adsorption behavior of oxyethylated surfactants was described by the so-called reorientation model, which assumes two states of surfactant molecules with different molar areas in the surface layer, and additionally an intrinsic compressibility of the adsorbed layer. For the anionic surfactant SDS, a modified Frumkin adsorption model was assumed, which also accounts for the intrinsic compressibility. For the theoretical analysis of the dynamic surface tensions, the theoretical model was based on the numerical solution of Fick’s diffusion equation for a spherical geometry of the bubble. The proposed set of theoretical models describes accurately and consistently the experimental results of the equilibrium and dynamic surface tensions of the studied mixed solutions.
1. Introduction The widespread use of surfactants in the huge variety of applications is typically based on mixtures. In modern technologies (such as detergency, foam and emulsion stabilization, painting and coating, textile finishing, etc.), surfactant mixtures are composed such that special effects are reached that cannot be achieved by any single compound. These compositions are usually mixtures of homologues or surfactants of different nature, for example, nonionic surfactants mixed with ionics, ionic surfactants with zwitterionics, or anionic with cationic surfactants. As a result of this growing importance, mixed surfactants systems have gained increasing attention from both a theoretical *Corresponding author. (1) Lucassen-Reynders, E. H. In Anionic Surfactants: Physical Chemistry of Surfactant Action; Lucassen-Reynders, E. H., Ed.; Marcel Dekker, Inc.: New York/ Basel, 1981; p 1. (2) Damaskin, B. B.; Frumkin, A. N.; Borovaja, N. A. Elektrokhimija 1972, 8, 807. (3) Rosen, M. J.; Hua, X. Y. J. Colloid Interface Sci. 1982, 86, 164. (4) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 87, 469. (5) Joos, P. Dynamic Surface Phenomena; VSP: Utrecht, The Netherlands, 1999. (6) Franses, E. I.; Siddiqui, F. A.; Ahn, D. J.; Chang, C.-H.; Wang, N. H. L. Langmuir 1995, 11, 3177. (7) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Colloids Surf., A 1998, 143, 141. (8) Rodakiewicz-Nowak, J. J. Colloid Interface Sci. 1982, 85, 586. (9) Karolczak, M.; Mohilner, D. M. J. Phys. Chem. 1982, 86, 2840. (10) Diamant, H.; Andelman, D. J. Phys. Chem. 1996, 100, 13732. (11) Siddiqui, F. A.; Franses, E. I. AIChE J. 1997, 43, 1569. (12) Mulqueen, M.; Blankschtein, D. Langmuir 1999, 15, 8832. (13) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2680. (14) Lu, J. R.; Thomas, R. K.; Penford, J. Adv. Colloid Interface Sci. 2000, 84, 143. (15) Danov, K. D.; Vlahovska, P. M.; Kralchevsky, P. A.; Mehreteab, A.; Broze, G. Colloids Surfaces A 1999, 156, 389. (16) Kralchevsky, P. A.; Danov, K. D.; Broze, G.; Mehreteab, A. Langmuir 1999, 15, 2351. (17) Fainerman, V. B.; W€ustneck, R.; Miller, R. Tenside Surfactants Detergents 2001, 38, 224. (18) Fainerman, V. B.; Miller, R. J. Phys. Chem. B 2001, 105, 11432. (19) Fainerman, V. B.; Miller, R.; Aksenenko, E. V.; Makievski, A. V. In Surfactants: Chemistry, Interfacial Properties and Application; Fainerman, V.B., M€obius, D., Miller, R., Eds.; Studies in Interface Science; Elsevier: New York, 2001; Vol. 13, pp 189-286.
284 DOI: 10.1021/la902072v
and experimental viewpoint.1-26 All theoretical models proposed so far dealt with the predictions of surface or interfacial tensions of mixed solutions from known characteristics of the individual compounds. In addition, to match the experimental data with theoretical predictions, an additional parameter was introduced to account for the mutual influence of the monolayer components. A simple equation was derived in refs 18-20, which is able to predict the surface tension of a surfactant mixture from the surface tension values or isotherm parameters of individual solutions. It should be noted that significant progress has been made recently in the understanding of the adsorption behavior of individual solutions of surfactants by considering the reorientation or intrinsic compressibility of adsorbed molecules depending on the surface pressure.7,19,26-28 These advances have yet to be incorporated into theoretical models to describe the adsorption behavior of mixtures. The influence of intrinsic compressibility on the adsorption and rheological behavior of surfactant mixtures was taken into account (in the framework of Frumkin’s model) in ref 29, which led to a significantly improved agreement between the theory and experimental data. (20) Fainerman, V. B.; Miller, R.; Aksenenko, E. V. Adv. Colloid Interface Sci. 2002, 96, 339. (21) Fainerman, V. B.; Lucassen-Reynders, E. H. Adv. Colloid Interface Sci. 2002, 96, 295. (22) Kralchevsky, P. A.; Danov, K. D.; Kolev, V. L.; Broze, G.; Mehreteab, A. Langmuir 2003, 19, 5004. (23) Danov, K. D.; Kralchevska, S. D.; Kralchevsky, P. A.; Ananthapadmanabhan, K. P.; Lips, A. Langmuir 2004, 20, 5445. (24) Lopez-Dı´ az, D.; Garcı´ a-Mateos, I.; Mercedes Velazquez, M. J. Colloid Interface Sci. 2006, 299, 858. (25) Penfold, J.; Thomas, R. K.; Dong, C. C.; Tucker, I.; Metcalfe, K.; Golding, S.; Grillo, I. Langmuir 2007, 23, 10140. (26) Miller, R.; Fainerman, V. B.; M€ohwald, H. J. Colloid Interface Sci. 2002, 247, 193. (27) Fainerman, V. B.; Zholob, S. A.; Lucassen-Reynders, E. H.; Miller, R. J. Colloid Interface Sci. 2003, 261, 180. (28) Fainerman, V. B.; Miller, R.; Kovalchuk, V. I. J. Phys. Chem. B 2003, 107, 6119. (29) Aksenenko, E. V.; Kovalchuk, V. I.; Fainerman, V. B.; Miller, R. J. Phys. Chem. C 2007, 111, 14713.
Published on Web 07/21/2009
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In this series of publications, a comprehensive experimental study of the adsorption behavior of the ionic surfactant sodium dodecyl sulfate (SDS) mixed with oxyethylated alcohols (C14EO8 and C12EO5) is presented along with a theoretical analysis of the experimental data. It was shown recently26,27,30-32 that the adsorption characteristics of solutions of oxyethylated surfactants are best described by the reorientation model, which assumes that the surfactant molecule can exist at the surface in two states (orientations) with different molar areas. As an example, it was shown in ref 33 that the whole set of experimental data available for C14EO8 solutions, including dynamic and equilibrium surface tensions, adsorption values, and dilational rheology, is described in the best possible way by a combined reorientation/compressibility model. This model assumes the ability of the oxyethylene groups to be adsorbed at the solution/air interface at low surface coverage, and to be partially desorbed with increasing surface pressure, i.e., reorientation of the EO groups, and also accounts for the intrinsic compressibility of the surface layer in the state with minimal molar area. However, neither of the theoretical models for the surfactant mixtures developed so far used the two-state reorientation model. Therefore, one of the theoretical problems that we will deal with here is the extension of the model given in ref 29 to the case of mixtures of two surfactants, one of which is performing a reorientation in the surface layer at sufficiently high surface pressure. In this first part of our series, bubble profile analysis tensiometry is used to study the dynamic and equilibrium surface tensions of mixed SDS/CnEOm solutions. For the analysis of the experimental data, a new theoretical model was developed, which assumes different adsorption mechanisms for mixed surfactants. In further work, studies of the dilational rheology are presented using the oscillating bubble method. A third part is dedicated to the dynamic surface tensions and dilational rheology, using the maximum bubble pressure and bubble profile analysis tensiometry, of micellar solutions of mixed SDS/CnEOm solutions. The comparison of experimental results with theoretical models shows that, for mixed SDS/CnEOm solutions, the model considering the reorientation of CnEOm and the twodimensional compressibility for both surfactants allows a good description of the complete set of equilibrium and dynamic experimental results.
2. Experimental Section The experiments were performed with the bubble/drop profile analysis tensiometers PAT-1 and PAT-2P (SINTERFACE Technologies, Germany), the principle of which was described in detail elsewhere.34,35 The temperature of the measuring cell with a volume of V = 20 mL was kept constant at 25 °C. In this study we used a buoyant bubble formed at the tip of a Teflon capillary with a diameter of 3 mm. It should be noted that, for studies of the mixtures of surfactants with different adsorption activity, one has to employ the bubble profile rather than the drop profile method in order to obtain a large volumeto-surface area ratio, so that the adsorption of the surfactants (30) Valenzuela, M. A.; Garate, M. P.; Olea, A. F. Colloids Surf., A 2007, 307, 28. (31) Ritacco, H. A.; Busch, J. Langmuir 2004, 20, 3648. (32) Lee, Y.-C.; Liu, H.-S.; Lin, S.-Y. Colloids Surf., A 2003, 212, 123. (33) Fainerman, V. B.; Zholob, S. A.; Petkov, J. T.; Miller, R. Colloids Surf., A 2008, 323, 56. (34) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Ravera, F.; Ferrari, M.; Liggieri, L. In Novel Methods to Study Interfacial Layers; M€obius, D., Miller, R., Eds.; Elsevier: Amsterdam, 2001; pp 439-484. (35) Zholob, S. A.; Makievski, A. V.; Miller, R.; Fainerman, V. B. Adv. Colloid Interface Sci. 2007, 134-135, 322.
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at the interface does not deplete the concentration in the bulk.36 The substances studied were SDS, with a purity higher than 99% (SigmaUltra, Sigma-Aldrich) and used without further purification, and the oxyethylated surfactants C14EO8 and C12EO5, which were purchased from Sigma Chemical and also used without purification. The SDS/CnEOm mixed solutions were prepared with Milli-Q water in the presence of 0.01 M NaCl, having a surface tension of 72.0 ( 0.2 mN/m at 25 °C over a time interval of up to 105 s (about 28 h). The presence of the NaCl concentration partially diminishes the adsorption of dodecanol in the SDS solutions37 as a result of a significant increase in the SDS adsorption activity for low SDS concentrations.
3. Theory 3.1. Adsorption Equilibrium. The equations of state and adsorption isotherm for the mixture of two surfactants and for the surfactant that can assume two adsorption states (with different molar areas) at the surface were derived from Butler’s equation for the chemical potentials in the surface layer, as explained in detail elsewhere.21,27,38 The relations accounting for the intrinsic compressibility of surfactant molecules for solutions of individual surfactants and mixtures were presented in ref 29. It should be noted that several versions of the reorientation model were proposed in ref 27. Here we employ the one in which the nonideality of enthalpy and entropy of mixing in the surface layer is accounted for in the most rigorous way. In what follows, for the sake of simplicity, we refer to surfactants that adsorb according to the reorientation model as “reorientable surfactants” and to the surfactant that obeys Frumkin’s model as the “Frumkin surfactant”. The subscripts R and F refer to the reorientable and Frumkin component of the surface layer, respectively. Using the approximation ωRO = ωFO for surfactants R (reorientation model) and F (Frumkin’s model), the following equation of state for surfactants mixture results:27,38
-
Πω0 ¼ lnð1 -θR -θF Þ þ θR ð1 -ωR0 =ωR Þ þ aR θR 2 RT þ aF θF 2 þ 2aRF θR θF
ð1Þ
with
ω0 ¼
ωR0 θR þ ωF0 θF θR þ θF
ð2Þ
where Π = γ0 - γ is the surface pressure, γ is the surface tension, γ0 is the surface tension of the solvent, T is the temperature, R is the gas constant, θi = ωi 3 Γi is the surface coverage by surfactant molecules of component i (i = R or F), Γi is the adsorption, ωi is the molar area, ωi0 is the molar area at zero surface pressure, and ai and aRF are the interaction constants. The adsorption of the reorientable surfactant consists of the amounts of surfactant adsorbed in two states: ΓR = Γ1 + Γ2. The molar areas of the reorientable surfactant adsorbed in states 1 and 2 are denoted by ω1 and ω2 (for definiteness, we assume ω2 > ω1), hence the coverage of the monolayer by this surfactant is θR = ωRΓR = ω1Γ1 + ω2Γ2. Note that, in refs 27 and 33, the mixtures of reorientable and Frumkin surfactants were discussed on the basis (36) Makievski, A. V.; Loglio, G.; Kr€agel, J.; Miller, R.; Fainerman, V. B.; Neumann, A. W. J. Phys. Chem. 1999, 103, 9557. (37) Gurkov, T. D.; Dimitrova, D. T.; Marinova, K. G.; Bilke-Krause, C.; Gerber, C.; Ivanov, I. B. Colloids Surf., A 2005, 261, 29. (38) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Adv. Colloid Interface Sci. 2003, 106, 237.
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of a less rigorous theory: in eq 1 the second term on the right-hand side (which accounts for the entropy nonideality) was disregarded, and in eq 2 the parameter ωR was used instead of ωR0. However, as the difference between ωR and ωF0 values is quite significant, especially at low surface layer coverage when the molar area of the reorientable surfactant is maximum and ωR = ω2, this simplified form of eq 1 can lead to larger deviations between theory and experiment. The molar areas of the two surfactants (ωF for the Frumkin surfactant and ω1 for the reorientable surfactant in the state of minimum area) can be approximated by a linear dependence on surface pressure Π and total surface coverage θ = θR + θF: 29 ω1 ¼ ωR0 ð1 -εR ΠθÞ
ð3Þ
ωF ¼ ωF0 ð1 -εF ΠθÞ
ð4Þ
where εi is the two-dimensional relative surface layer compressibility coefficient which characterizes the intrinsic compressibility of molecules in the surface layer. The intrinsic compressibility reflects the change of the tilt angle of the adsorbed molecules upon surface layer compression. The adsorption isotherms for states 1 and 2 of the reorientable surfactant are bR cR ¼
b R cR ¼
Γ1 ωR0 ð1 -θÞω1 =ωR0
ω1 exp ð2aR θR Þ -2aRF θF ωR0
Γ2 ωR0 RR
ðω2 =ω1 Þ ð1 -θÞ
ω2 =ωR0
exp -
ð5Þ
ω2 ð2aR θR Þ -2aRF θF ωR0
ð6Þ where bR is the adsorption equilibrium constant, and cR is the surfactant bulk concentration. Equation 6 for state 2 with the maximum surface molar area was derived under the assumption that the adsorption constant increases with increasing molar area, according to a power law with the constant exponent RR.27 Therefore, the ratio of the adsorptions in states 1 and 2 for the reorientable surfactant is given by
Γ1 1 ðω2 -ω1 Þ ¼ exp ð2aR θR Þ ωR0 Γ2 ðω2 =ω1 ÞRR ð1 -θÞðω2 -ω1 Þ=ωR0 ð7Þ The adsorption isotherm for the Frumkin component reads bF cF ¼
θF exp½ -2aF θF -2aRF θR ð1 -θÞ
ð8Þ
molecules. Similarly, for cR = 0 (and hence θR = 0), eqs 1-3, 4, and 7 turn into the equation of state and adsorption isotherm derived in refs 28 and 29 for the Frumkin surfactant with intrinsic compressibility of adsorbed molecules. The procedure used for the numerical calculations was quite straightforward: given the known values of T, ωR0, ω2, ωF0, RR, aR, aF, aRF, εR, εF, bR and bF, a computation procedure was developed that, for any given values of the surfactant concentrations cR and cF, determines θR and θF from eqs 5-8. These θ1 and θ2 values are then used to calculate the values of adsorption ΓR = Γ1 + Γ2 and ΓF, and, finally, the surface pressure value Π is obtained via eqs 1 and 2. 3.2. Dynamic of Adsorption and Surface Tension. The theoretical description of the concurrent dynamic adsorption on the bubble interface follows the lines described in ref 40, however, it is now generalized onto the case when the system contains two surfactants in the solution. Let us consider a spherical “cell” of radius RE, which consists of an internal spherical region (bubble) of radius RI, the center of which coincides with the center of the cell, and an external region (solution) with the radius RE. The diffusion of the surfactants in the solution is governed by Fick’s law, which, in spherical coordinates (obviously corresponding to the geometry of the cell), reads Dci D2 ci 2 Dci ¼ Di þ r Dr Dt Dr2
! for RI < r < RE
ð9Þ
where ci = ci(r,t) is the ith surfactant (i = R or F) concentration at time t and distance r from the center of the cell (chosen as the origin of the coordinate system), and Di are the corresponding diffusion coefficients of the surfactants. In this study it was assumed that there is a homogeneous initial distribution of the surfactants in the solution, although the numerical procedure developed for the solution of the problem allows for an inhomogeneous radially symmetrical initial distribution. Therefore, the initial conditions for the set of eqs 9 are ð0Þ
ci ðr, 0ÞÞ ¼ ci
for RI < r < RE
ð10Þ
Now the boundary conditions for the concentrations should be defined. These conditions at the external boundary of the system r = RE follow from the symmetry of the system and the fact that the system is closed: Dci r ¼RE- ¼ 0: Dr
ð11Þ
Assuming a diffusion controlled adsorption at the interface, the change of adsorptions Γi with time caused by the diffusive fluxes is dΓi Dci ¼ Di r ¼RIþ dt Dr
ð12Þ
where here bF is the adsorption equilibrium constant, and cF is the surfactant bulk concentration. If only one of the two components of the mixture is present in the solution, the equations listed above become identical to those developed earlier for the individual solutions. In particular, for cF = 0 (and hence θF = 0), eqs 1-3, 5, and 6 are transformed into the equations of state and adsorption isotherm derived in ref 39 for the reorientable surfactant regarding the nonideality of the surface layer and the intrinsic compressibility of the surfactant
These equations are the boundary conditions at the interface located at r = RI. Note that, as the dependencies of Γi on ci(r = RI) involve the equation of state of the surface layer and the adsorption isotherms of the surfactants defined by eqs 1-8 above, eqs 12 couple the set of eqs 9. Also, for nonzero initial adsorptions, the subsurface concentrations ci(0,0) can be determined via eqs 1-8. This completes the formulation of the problem.
(39) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Makievski, A. V.; Petkov, J. T.; Yorke, J.; Miller, R. Colloids Surf., A 2009, 334, 1.
(40) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Liggieri, L.; Makievski, A. V.; Petkov, J. T.; Yorke, J.; Miller, R. Colloids Surf., A 2009, 334, 8.
286 DOI: 10.1021/la902072v
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Figure 1. Dynamic surface tension for individual solutions of C12EO5 and SDS and of mixed SDS/C12EO5 solutions with molar ratios of 3:1, 10:1, and 30:1, all at the same total concentration of 10 μmol/L; symbols = experimental data; lines = calculated dependencies.
Figure 2. The same as in Figure 1 for a total concentration of 50 μmol/L.
The equations above were implemented in a Windows-based software package designed in a modular structure: the “core” of the package contains the procedures used to calculate the solution of Fick’s equations with the respective boundary and initial conditions, the input/output modules, and the visual interface, which serves to control the calculations. Additional modules can be incorporated if required to perform the calculations for any other particular isotherm. Details of the numerical procedure are presented in the Appendix.
4. Results and Discussion The dynamic surface tensions of the SDS/CnEOm mixtures in the presence of 0.01 M NaCl for various molar ratios of the components (3:1, 10:1, and 30:1 for SDS/C12EO5, and 5:1, 30:1, and 200:1 for SDS/C14EO8 mixtures) were measured using the buoyant bubble profile analysis tensiometry. Figures 1 and 2 illustrate present examples of the experimental dependencies of γ(t) for SDS/C12EO5 mixed solutions for total concentrations of 10 and 50 μmol/L, respectively. In addition, the dynamic surface tensions of the individual C12EO5 and SDS solutions at the same concentrations are shown. It was verified in the experiments that the addition of the inorganic electrolyte (0.01 M NaCl) does not affect the surface tension of the CnEOm solutions due to its Langmuir 2010, 26(1), 284–292
nonionic nature. A similar result was obtained for Tritons (polyethylene glycol octylphenyl ethers).39 Figure 3 shows the dynamic surface tensions of SDS/C14EO8 mixtures for various molar ratios at a total concentration of 50 μmol/L, along with the dynamic surface tensions for the individual solutions of the same concentration. It is seen from Figures 1-3 that the increase of the relative amount of CnEOm in mixtures with SDS results in a significant decrease of the surface tensions and in a decrease in the time necessary to attain the adsorption equilibrium. For the mixtures and individual solutions of CnEOm, the equilibrium state is attained within 2000-6000 s after bubble formation. However, the SDS solution and the SDS/C14EO8 mixtures with a large SDS fraction require even more than 10 000 s to reach the adsorption equilibrium. This fact is ascribed to the SDS hydrolysis, which leads to the formation of dodecanol, which was discussed in detail in ref 41. Figures 1-3 also contain the curves calculated from model equations. The equilibrium surface tension isotherms of the individual components SDS and C12EO5, and the isotherms of SDS/C12EO5 (41) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Petkov, J. T.; Yorke, J.; Miller, R. Colloids Surf., A [Online early access]. DOI: 10.1016/j.colsurfa.2009.02.022.
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Figure 3. Dynamic surface tension for individual solutions of C14EO8 and SDS and of mixed SDS/C14EO8 solutions with molar ratios of the components of 5:1, 30:1 and 200:1, all at a total concentration of 50 μmol/L; symbols = experimental data; lines = calculated dependencies.
Figure 4. Equilibrium surface tension isotherms for individual solutions of C12EO5 (() and SDS (9), and for SDS/C12EO5 mixtures at molar ratios of 3:1 (4), 10:1 ()) and 30:1 (O). Theoretical curves were calculated with the proposed model using the parameters listed in the text.
mixtures at various ratios (in the presence of 0.01 M NaCl for all systems) are presented in Figure 4 as dependencies on the total concentration of the surfactants. The isotherm for the individual SDS solution in the presence of 0.01 M NaCl is reproduced from ref 41. This isotherm was theoretically described by Frumkin’s model with intrinsic compressibility, i.e., by eqs 1, 2, 4, and 8 at θR = 0. The isotherm parameters found in ref 41 are given in Table 1. The experimental isotherm for individual C12EO5 solutions shown in Figure 4 agrees well with the results reported by other authors.19,42-44 This isotherm was theoretically described by the model for reorientable surfactant assuming the nonideality of entropy of the surface layer and intrinsic compressibility of the surfactant molecules, i.e., eqs 1-3 and 5-7 with θF = 0. The model parameters are summarized in Table 1. One can see good agreement between the theoretical curve and the experimental (42) Binks, B. P.; Fletcher, P. D. I.; Paunov, V. N.; Segal, D. Langmuir 2000, 16, 8926. (43) Kjellin, U. R. M.; Claesson, P. M.; Linse, P. Langmuir 2002, 18, 6745. (44) Tyrode, E.; Johnson, C. M.; Rutland, M. W.; Claesson, P. M. J. Phys. Chem. C 2007, 111, 11642.
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values for C12EO5. The comparison between the adsorption equilibrium constants bi for the substances studied shows that the surface activity of C12EO5 is by a factor of 200 higher than that of SDS. It will be shown below that the experimental values of dynamic surface tensions and dilational rheological characteristics of C12EO5 solutions are described by this model with the same values of the model parameters. The experimental isotherms of SDS/C12EO5 mixtures for various molar ratios are also shown in Figure 4. Note that the theoretical dependencies for mixtures were calculated from eqs 1-8 with the same values of model parameters as those found for the individual SDS and C12EO5 solutions. The only additional parameter for mixtures was the intermolecular interaction coefficient aRF, for which the optimum fitting value is 1.3 for all studied mixing ratios. It is clearly seen that the proposed theoretical model agrees well with the experimental data for SDS/C12EO5 mixtures. In a similar way, Figure 5 illustrates the experimental and theoretical isotherms of individual SDS and C14EO8 solutions and their mixtures for various mixing ratios. The experimental values for the individual C14EO8 solutions obtained by bubble Langmuir 2010, 26(1), 284–292
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Article Table 1. Model Parameters for the Studied Surfactants (i stands for R or F)
surfactant
ωi0 (105 m2/mol)
SDS C12EO5 C14EO8
3.0 4.2 5.7
ω2 (106 m2/mol)
ai
bi (103 m3/mol)
Ri
εi (m/mN)
1.0 1.0
0 0.2 0
0.025 5.1 100.0
2.5 2.8
0.008 0.006 0.007
Figure 5. Equilibrium surface tension isotherms for individual solutions of C14EO8 (() and SDS (9), and SDS/C14EO8 mixtures at molar ratios of 5:1 (4), 30:1 ()) and 200:1 (O). Theoretical curves were calculated with the proposed model using the parameters listed in the text.
Figure 6. Theoretical dependencies of equilibrium adsorption of SDS (dotted lines) and C12EO5 (solid lines) on total surfactant concentration for the SDS/C12EO5 mixtures at various mixing ratios (as shown in the figure).
profile analysis tensiometry are reproduced from refs 33 and 45. The same theoretical model used for C12EO5 yields the parameter values for C14EO8 shown in Table 1. Mixed solutions were described by the model defined by eqs 1-8 with the SDS and C12EO5 parameters listed above; the intermolecular interaction coefficient for SDS/C14EO8 mixtures was determined by fitting to be aRF = 1.4. The calculated equilibrium adsorption values of the components in the SDS/C12EO5 and SDS/C14EO8 mixtures are shown in Figures 6 and 7, respectively. It is seen that the adsorption of C12EO5 and C14EO8 in the mixtures exhibits a monotonous increase with increasing total surfactant concentration. At the same time, it becomes higher with the increase of the fraction of nonionic surfactant. It is clearly seen that the (45) Fainerman, V. B.; Petkov, J. T.; Miller, R. Langmuir 2008, 24, 6447.
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SDS adsorption exhibits a maximum value, and this value becomes higher with the increase of the SDS fraction in the mixture, while the location of the maximum shifts toward higher concentrations. This extreme behavior of the SDS adsorption in its mixtures with oxyethylated alcohols indicates the competitive character of adsorption in mixed surface layers, if the fraction of ionic surfactants is close to saturation. Comparing the dependencies in Figures 6 and 7 with the isotherms presented in Figures 4 and 5, respectively, one can see that the maximum SDS adsorption (for various molar ratios of the components) corresponds to a surface tension value of approximately 50 mN/m. This surface tension corresponds to a surface layer coverage of approximately 99% for all studied mixtures. Note that, using the parameters given in Table 1, we can also calculate the adsorption values Γ(c) of the single compounds. If we do so for C12EO5, for example, and DOI: 10.1021/la902072v
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Figure 7. The same as in Figure 6 for the SDS/C14EO8 mixtures.
Figure 8. Theoretical dependencies of the adsorption values of SDS (dotted lines) and CmEOn (solid lines) for SDS/CmEOn mixtures at a mixing ratio of 30:1 and total surfactant concentration of 50 μmol/L; red line = SDS/C12EO5 mixture; blue lines = SDS/C14EO8 mixture.
compare the results with the data directly measured by neutron reflection,14 we find excellent agreement. Turning again to Figures 1-3, we discuss now the calculated dynamic surface tensions for the SDS/C12EO5 and SDS/C14EO8 mixtures, respectively. These theoretical dependencies were calculated by a procedure based on the numerical solution of Fick’s equations for two-component mixtures (see section 3.2.). In this modeling, only the diffusion coefficients DF and DR for SDS and CnEOm are adjustable parameters. For the curves shown in Figures 1-3, and for similar dependencies corresponding to other mixtures, the optimum value of the diffusion coefficient DR was found in the range between 3 10-10 and 4 10-10 m2/s. These values are physically realistic, which supports the diffusion mechanism of adsorption of oxyethylated alcohols both in individual solutions and in mixtures with SDS. On the contrary, the behavior of SDS in its mixture with CnEOm is rather different. For the minimum ratios (3:1 and 5:1 for SDS/C12EO5 and SDS/ C14EO8 mixtures, respectively), the apparent diffusion coefficient is DF ∼ 10-11 m2/s. This value is more than 1 order of magnitude smaller than the physically expected one, which could be attributed to the presence of an adsorption barrier, presumably of electrostatic nature. With the increase of the SDS fraction in the mixtures with oxyethylated alcohols, the apparent diffusion coefficient of SDS becomes even smaller (up to 10-12 m2/s), 290 DOI: 10.1021/la902072v
which would correspond to an increase of the adsorption barrier. These results agree with those reported in ref 41, where the dynamic surface tension and dilation rheology of SDS solutions were studied in a wide concentration range. It was shown that, with the increase in SDS concentrations, to obtain agreement with the experimental data, the values of apparent diffusion coefficient should be decreased from 10-11 to 10-12 m2/s. Besides an adsorption barrier, there is another possible explanation for the low diffusion coefficient determined for SDS: because of the ongoing hydrolysis in aqueous solution, the presence of dodecanol cannot be excluded. It was demonstrated in ref 41 that the consideration of small amounts of dodecanol in the analysis of the adsorption kinetics of SDS solutions, i.e., by handling the solution as a mixture of two surfactants, leads to increased diffusion coefficients for SDS by orders of magnitude and, at the same time, to realistic values for the dodecanol (2 10-10 m2/s). Figure 8 shows the theoretical curves of the dynamic adsorption Γ(t) of SDS and CnEOm in the SDS/C12EO5 and SDS/ C14EO8 mixtures. These curves were calculated for one concentration of the mixture (50 μmol/L) and a molar ratio of 30:1. It is seen that, in spite of the fact that the value of the SDS apparent diffusion coefficient (values of 5 10-12 to 10-11 m2/s were used for the theoretical calculations in Figures 1-3) was lower than the value for CnEOm (310-10 m2/s), the dynamic adsorption of SDS Langmuir 2010, 26(1), 284–292
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in the short time range is noticeably higher than that of CnEOm. This essential difference between the dynamic adsorption of SDS and its equilibrium adsorption (see Figures 6 and 7) can be explained in the following way: In the conditions corresponding to Figure 8, the SDS concentration is by a factor of 30 higher than that of CnEOm, while the effective diffusion coefficient of SDS is by a factor of 30-60 lower than the value for CnEOm. The dynamic adsorption of the mixture components in the short time range obeys the approximate expression46 Γi = 2ci(Dit/π)1/2. Therefore, the dynamic adsorption of SDS at low adsorption times could be several times higher than the adsorption of CnEOm. In the time range of 300-500 s, the dynamic adsorption of SDS exhibits a maximum, followed at higher times by a significant decrease. Similarly to its equilibrium behavior, this fact is due to the competitive adsorption by CnEOm. On the contrary, the dynamic adsorption of much more surface active substances C12EO5 and C14EO8 increases with time, similarly to its equilibrium behavior.
5. Conclusions In the presented work, which represents the first part of a comprehensive investigation of the adsorption and rheological behavior of mixed surface layers, bubble profile analysis tensiometry is used to study the dynamic and equilibrium surface tensions of mixed SDS/CnEOm solutions. For the analysis of the experimental data, a new theoretical model was employed, which assumes different adsorption mechanisms for each surfactant. In particular, the adsorption behavior of the nonionic oxyethylated surfactants was described by the reorientation model, which assumes two states of surfactant molecules with different molar areas in the surface layer, and an intrinsic compressibility of the molecule in the state with minimum area. For the anionic SDS, we assumed the adsorption is governed by a Frumkin-type model, which also accounts for the intrinsic compressibility. For the theoretical analysis of the dynamic surface tensions, the developed model was based on the numerical solution of Fick’s equations for the actual geometry of the experimental system, i.e., diffusion to the surface of a spherical bubble. The theoretical models allow an accurate and consistent description of the experimental results for the equilibrium surface tensions of all studied SDS/CnEOm solutions. For dynamic surface tension, the theory can be fitted satisfactorily to the experimental data.
The initial conditions (eq 10) transform into U i ðF, 0Þ ¼ F
where an homogeneous distribution of the surfactants throughout the solution bulk is assumed at t=0. The boundary conditions at the external surface (eq 11) now become
DU i U i DF F
Appendix: Numerical Procedure
ð0Þ
C i ¼ ci =ci , U i ¼ FC i ,
F ¼ r=RI ,
τ ¼ tDR =k 2 ,
ð0Þ Γi =Γi ,
ð0Þ Γð0Þ r =cR
Gi ¼
k ¼
ðA1Þ
where the superscript (0) refers to values at t = 0. Then the bubble/ solution interface is located at F = 1, and Fick’s eqs 9 become DU i D2 U i ¼ q2i Dτ DF2 where qR ¼ k=RI ,
at 1 < F < FE RE =RI pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qF ¼ qR DF =DR
(46) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 543.
Langmuir 2010, 26(1), 284–292
ðA2Þ
ðA3Þ
F ¼FE-
¼0
ðA5Þ
and the boundary conditions at the bubble/solution interface, eq 12, read " # dΓi D Ui ¼ pi DF F dτ
ðA6Þ F ¼1 -
with ð0Þ ð0Þ
pR ¼ qR ,
pF ¼ qR
ΓR c F D F ð0Þ ð0Þ c Γ DR R
ðA7Þ
F
Note that, while the initial conditions 9-12 are symmetric with respect to the interchange of subscripts i (i = R or F), it is necessary to introduce a single time scale (and hence a unique dimensionless temporal variable τ) based on one of the two diffusion coefficients involved. This leads to a certain asymmetry in the governing equations, as is seen from eqs A3 and A7. Here the diffusion coefficient of the reorientable surfactant is chosen to be the “basic” one. To perform the numerical calculations, a finite difference scheme was defined, rather similar to that employed in ref 40. As the maximum gradients of surfactant concentrations (and hence Ui) exist in the vicinity of the interface, a scheme with nonequidistant spatial nodes was chosen. The positions of the nodes Fn are defined on the basis of the value of δ, which is the distance between the interface and the node No. 1 nearest to the interface. The spatial grid is then defined as Fn ¼ 1 þ δ
Following the lines given in ref 40, we introduced the dimensionless quantities
ðA4Þ
βn -1 , β -1
n ¼ 1, :::, N
ðA8Þ
where the value β is chosen such that it satisfies the equation βN -1 1 ¼ ðFE -1Þ β -1 δ
ðA9Þ
Here N is the number of nodes. The actual discretization procedure was performed as follows: First, the values of δ and an “approximate” β were chosen. Then the “approximate” number of nodes N0 was calculated using eqs A8. This value is generally a noninteger. Therefore N was taken to be the smallest integer larger than or equal to N0 , and β was calculated as the solution of eq A9. As it is necessary to simulate the temporal evolution of the system over a large time range, the time discretization interval is also chosen to be variable: τj ¼
Rj -1 -1 ε R -1 DOI: 10.1021/la902072v
ðA10Þ 291
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with the jth temporal step ( j > 1) Δτj ¼ τj þ1 -τj ¼ R 3 ðτj -τj -1 Þ
ðA11Þ
To solve the set of differential eqs A2, the CrankNicolson scheme as described in ref 47 is implemented,
modified to account for nonequidistant nodes. With the derivatives calculated using the Lagrange interpolation formula for a function defined in three adjacent nodes, the Crank-Nicolson scheme at the jth temporal step yields two tridiagonal sets of equations in the nodes n = 1, ..., N - 1:
ð1, jÞ ð1, jÞ ð2, jÞ ð1, jÞ ð1, j -1Þ ð2, j -1Þ ðjÞ ðj -1Þ Ai U i þ Ui ¼ Bi U i -U i -βðui þ ui Þ n ¼1: ðn -1 , jÞ ðn , jÞ ðn , jÞ ðn þ1 , jÞ ðn -1 , j -1Þ ðn , jÞ ðn , j -1Þ ðn þ1, j -1Þ 2eneN -2 : βU i þ Ai U i þ Ui ¼ -βU i þ Bi U i -U i ðN -2 , jÞ ðN -1 , jÞ ðN -1 , jÞ ðN -2 , j -1Þ ðN -1 , jÞ ðN -1, j -1Þ n ¼ N -1 : βU i þ ðAi -λÞU i ¼ -βU i þ ðBi -λÞU i Here U(n,j) is the value of Ui in the nth spatial node at the jth i temporal step, ! 1 δ2 2n -1 ðn, jÞ , Ai ¼ -ð1 þ βÞ 1 þ 2 β qi Δτj ðn, jÞ Bi
1 δ2 2n -1 ¼ ð1 þ βÞ 1 - 2 β qi Δτj
! ðA13Þ
+ and u(j) i = Ui (F = 1 , τ = τj) is the subsurface value of Ui at the jth temporal step. The last equation of A12 accounts for the boundary condition A5, which implies that the concentrations of the ith surfactant at the (N - 1)th and Nth spatial nodes are equal to each other, i.e.,
ðN -1, jÞ
Ui
ðN -1, jÞ
=FN -1 ¼ Ui
=FN
ðA14Þ
from which the value of λ is calculated: λ ¼ FE =ðFE -δβN -1 Þ
ðA15Þ
To couple the two sets of eqs A12, the boundary conditions of eq A6 are applied, which are discretized according to the threepoint Lagrange interpolation formula. This yields a set of two (47) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes; Cambridge University Press: Cambridge, U.K., 2007.
292 DOI: 10.1021/la902072v
ðA12Þ
equations: ðj -1Þ
δ=pi
Gi ðuR
ðj -1Þ
, uF
ðjÞ
ðjÞ
Þ -Gi ðuR , uF Þ Δτj
ðjÞ ð1, j -1Þ ð1, jÞ þ ui 1 þ β 1 Ui þ Ui β 1 -δ 2 2 ð2, j -1Þ ð2, jÞ 1 1 Ui þ Ui ¼0 ðA16Þ þ β 3 ð1 þ βÞ 1 -δð1 þ βÞ 2 ðj -1Þ
þ
2 þ β ui 1 þβ
where the values of the dimensionless adsorptions Gi for the and u(j) dimensionless concentration-related variables u(j-1) i i should be calculated via the equation of state and adsorption isotherms. Now the calculation procedure is quite straightforward. At and U(n,j-1) are known each jth temporal step, the values of u(n,j) i i from the previous step. Then, each of the two sets of linear inhomogeneous eqs A12 can be solved for u(j) i , which yields the (1 e n e N - 1). This solution procedure was desvalues U(n,j) i cribed earlier in ref 40. Therefore, the set of two eqs A16 becomes dependent on the two variables u(j) i (i = R and F). This set of two equations can be solved using any suitable standard procedure. To control the consistency of the computation process, at the end of each temporal step, the mass balance condition, which requires the conservation of matter in the system, Z RE ci ðr, tÞr2 dr ¼ const ðA17Þ M i ¼ 4π½rI Γi ðtÞþ RI
was verified. In all studied cases, the deviation from this mass balance did not exceed 1%.
Langmuir 2010, 26(1), 284–292
