pubs.acs.org/Langmuir © 2009 American Chemical Society
Adsorption Layer Characteristics of Mixed SDS/CnEOm Solutions. II. Dilational Viscoelasticity V. B. Fainerman,† E. V. Aksenenko,‡ S. A. Zholob,† 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 Research and Development Port Sunlight, Quarry Road East, Bebington, CH63 3JW, United Kingdom, and ^Max-Planck-Institut f€ ur Kolloid und Grenzfl€ achenforschung, Am M€ uhlenberg 1, 14424 Potsdam, Germany Received July 10, 2009. Revised Manuscript Received August 12, 2009 Bubble profile analysis tensiometry is used to study the surface rheological behavior of mixed SDS/C12EO5 and SDS/ C14EO8 solutions. The experimental dependencies of the viscoelasticity modulus and phase angle are studied in a wide range of surfactant concentrations of the individual sodium dodecyl sulfate (SDS) and CmEOn solutions and SDS/ CnEOm mixtures at various mixing ratios. By generating harmonic oscillations of the bubble area at low oscillation amplitudes, the relaxation behavior at oscillation frequencies between 0.005 and 0.2 Hz was studied. The applied theoretical approach to describe the dilational rheology of surfactant mixtures requires the specification of the equations of state of the mixed surface layer and the adsorption isotherm of the mixture’s components. For the systems studied, the theoretical model considers different adsorption mechanisms for the different surfactants. In particular, the adsorption behavior of oxyethylated surfactants was described by the reorientation model (assumes two adsorption states of surfactant molecules with different molar areas), including an intrinsic compressibility of molecules in the state of minimal area. For the SDS component, the adsorption was assumed to be governed by the Frumkin model, which also accounts for the intrinsic compressibility. Satisfactory agreement between experimental data and theoretical calculations of the viscoelasticity modulus and the phase angle is obtained.
Introduction There is a number of experimental and theoretical studies on the dilational rheology of surfactant mixtures.1-11 The theory by Lucassen and van den Tempel developed for single surfactants12,13 was generalized in refs 6-11. In particular, for mixtures of two surfactants, analytical expressions for the complex dilational modulus were independently derived in refs 6 and 7. In refs 8 and 9, approaches based on the irreversible thermodynamics have been presented, and a general method for the derivation of equations for the surface elasticity of mixtures of an arbitrary number of surfactants and the mixed adsorption kinetics was proposed. Expressions for the complex elasticity modulus of surfactant mixtures were recently derived in ref 10, and a numerical procedure was proposed for calculating its real and imaginary parts. This procedure was illustrated for protein/nonionic surfactant mixtures. In ref , the theory was applied to mixed surfactants, and theoretical calculations were compared with experimental *Corresponding author. (1) Garrett, P. R.; Joos, P. J. Chem. Soc., Faraday Trans. 1 1975, 69, 2174. € (2) Wantke, K.-D.; Fruhner, H.; Ortegren, J. Colloid Surf., A 2003, 221, 185. (3) Noskov, B. A.; Loglio, G.; Miller, R. J. Phys. Chem. 2004, 108, 18615. (4) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1973, 42, 573. (5) Garrett, P. R.; Joos, P. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2161. (6) Jiang, Q.; Valentini, J. E.; Chiew, Y. C. J. Colloid Interface Sci. 1995, 174, 268. (7) Joos, P. Dynamic Surface Phenomena, VSP: Dordrecht, The Netherlands, 1999. (8) Noskov, B. A.; Loglio, G. Colloids Surf., A 1998, 141, 167. (9) Ivanov, I. B.; Danov, K. D.; Ananthapadmanabhan, K. P.; Lips, A. Adv. Colloid Interface Sci. 2005, 114-115, 61. (10) Aksenenko, E. V.; Kovalchuk, V. I.; Fainerman, V. B.; Miller, R. Adv. Colloid Interface Sci. 2006, 122, 57. (11) Aksenenko, E. V.; Kovalchuk, V. I.; Fainerman, V. B.; Miller, R. J. Phys. Chem. C 2007, 111, 14713. (12) Lucassen, J.; van den Tempel, M. Chem. Eng. Sci. 1972, 27, 1283. (13) Lucassen, J.; van den Tempel, M. J. Colloid Interface Sci. 1972, 41, 491.
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data of the rheological behavior obtained for mixtures of the two nonionic surfactants, C10DMPO and C14DMPO, and the two ionic surfactants, sodium dodecyl sulfate (SDS) and dodecanol, respectively. The present work complements the recent study of the adsorption behavior of mixtures of the ionic surfactant SDS with oxyethylated alcohols (C14EO8 and C12EO5)14, where the dynamic and equilibrium surface tensions of these mixed solutions at various mixing ratios were discussed. In the present paper, results on the dilational rheology of these mixtures are analyzed using theoretical models developed earlier. In particular, the set of equations, which describes the rheological behavior of the surfactant mixtures in the most general form,11 is solved simultaneously with the surface equations of state and the adsorption isotherms for mixtures of two surfactants governed by different adsorption models.14 The adsorption of SDS is governed by a Frumkin model including an intrinsic compressibility (which means that the surfactant molecules have the capability to change the tilt of the hydrophobic chain upon increased surface pressure or degree of surface coverage and leads to a decrease in the molar area). In contrast, the adsorption behavior of oxyethylated surfactants is best described by the reorientation model, which assumes two adsorption states of surfactant molecules characterized by different molar areas in the surface layer, with the additional consideration of the intrinsic compressibility of the molecule in the state of minimal area. Good qualitative and quantitative agreement of the experimental data is obtained with theoretical predictions for the viscoelasticity modulus and the phase angle, as a function of the surfactants concentration and the oscillation frequency. (14) Fainerman, V. B.; Aksenenko, E. V.; Lylyk, S. V.; Petkov, J. T.; Yorke J.; Miller, R. Langmuir 2009, in press.
Published on Web 09/15/2009
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Experimental Section The experiments were performed, similar to ref 14, with the bubble/drop profile analysis tensiometers PAT-1 and PAT-2P (SINTERFACE Technologies, Germany). The temperature of the measuring glass cell (volume, V=20 mL) was kept constant at 25 C. In this study, we used a buoyant (oblate) bubble formed in a Teflon capillary with a tip diameter of 3 mm. The SDS with a purity higher than 99% (SigmaUltra, SigmaAldrich) was used without further purification. The oxyethylated surfactants C14EO8 and C12EO5 were purchased from Sigma Chemical and also used without purification. All individual and mixed SDS/CnEOm solutions were prepared using Milli-Q water in presence 0.01 M NaCl. To study the dilational viscoelasticity E after having reached the adsorption equilibrium, the bubble was subjected to harmonic oscillations with frequencies f between 0.005 and 0.2 Hz and surface area oscillation amplitudes of 5-7%. The results of these harmonic relaxation experiments were analyzed using a Fourier transformation: 15 EðiωÞ ¼ A0
F½Δγ F½ΔA
ð1Þ
where A0 is the initial area of the bubble surface, γ is the surface tension, and ω is the angular frequency of generated oscillations.
Theory The surface dilational modulus in compression and expansion is defined as the variation in surface tension γ for a corresponding small relative variation of surface area A:11,16 E¼
dγ d ln A
mixture, ω = 2πf is the angular frequency, aij ¼ ðDΓi =Dcj Þjck6¼j are the partial derivatives, which result from the adsorption isotherm, and pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ 1 þ iω=D1 a11 þ iω=D2 a22 pffiffiffiffiffiffiffiffiffiffiffi þ ðiω= D1 D2 Þða11 a22 - a12 a21 Þ pffiffiffi pffi With i ¼ ð1 þ iÞ= 2, the expressions for the real and the imaginary parts of the viscoelasticity given in eq 3 are obtained11 Er ¼ ðPR þ QSÞ=ðP2 þ Q2 Þ, E i ¼ ½PS - QR=ðP2 þ Q2 Þ
The viscoelasticity modulus |E| and phase angle φ between the stress (dγ) and the strain (dA) are11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEj ¼ ðR2 þ S 2 Þ=ðP2 þ Q2 Þ, j ¼ arctgðEi =Er Þ ð5Þ The coefficients P, Q, R, and S contained in eqs 4 and 5 are defined as follows: pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi P ¼ 1 þ ð ω=D1 a11 þ ω=D2 a22 Þ= 2 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Q ¼ ð ω=D1 a11 þ ω=D2 a22 Þ= 2 pffiffiffiffiffiffiffiffiffiffiffi þ ðω= D1 D2 Þða11 a22 - a12 a21 Þ
ð2Þ
The measured modulus E is viscoelastic and can be presented by E = Er + iEi with an elastic part Er, accounting for the recoverable energy stored in the interface, and a viscous contribution Ei, reflecting the loss of energy due to relaxation processes. Using the results in refs 6 and 7, expressions for the complex elasticity modulus for a mixture of two surfactants (i, j = 1,2) and a harmonic perturbation of frequency f and and amplitude ΔA, assuming a diffusion-controlled adsorption mechanism of the mixture components were derived in:10,11 "rffiffiffiffiffiffi 1 DΠ iω a11 E ¼ B D ln Γ1 Γ2 D1 # rffiffiffiffiffiffi iω Γ2 iω a12 þ pffiffiffiffiffiffiffiffiffiffiffi ða11 a22 -a12 a21 Þ þ D2 Γ1 D1 D2 "rffiffiffiffiffiffi 1 DΠ iω Γ1 a21 þ B D ln Γ2 Γ1 D1 Γ 2 # rffiffiffiffiffiffi iω iω a22 þ pffiffiffiffiffiffiffiffiffiffiffi ða11 a22 -a12 a21 Þ þ D2 D1 D2
qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Pi ¼ ½ ω=Di aii þ ω=Dj aij ðΓj =Γi Þ= 2 pffiffiffiffiffiffiffiffiffiffiffi Qi ¼ Pi þ ðω= D1 D2 Þða11 a22 - a12 a21 Þ R ¼ P1 ðDΠ=D ln Γ1 ÞjΓ2 þ P2 ðDΠ=D ln Γ2 ÞjΓ1 S ¼ Q1 ðDΠ=D ln Γ1 ÞjΓ2 þ Q2 ðDΠ=D ln Γ2 ÞjΓ1 The rheological dilational characteristics of the surface layer for a single surfactant solution, assuming a diffusion controlled adsorption mechanism, were derived by Lucassen and van den Tempel12,13 as: jEj ¼ E 0 ð1 þ 2ζ þ 2ζ2 Þ -1=2 ,
ð3Þ
where Π = γ0 - γ is the surface pressure, γ0 is the surface tension of the solvent, Γj is the adsorption, cj the concentration, Dj is the diffusion coefficient of the j-th component of the (15) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Ravera, F.; Ferrari, M.; Liggieri, L. Studies in Interface Science. In Novel Methods to Study Interfacial Layers, Vol. 11; M€obius, D., Miller, R., Eds.; Elsevier: Amsterdam, The Netherlands, 2001; pp. 439. (16) Interfacial Rheology. In Progress in Colloid and Interface Science, Vol. 1; Miller R., Liggieri, L., Eds.; Brill Publishing: Leiden, The Netherlands, 2009.
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ð4Þ
with E 0 ¼ dΠ=d ln Γ,
ζ ¼
j ¼ arctg½ζ=ð1 þ ζÞ
ð6Þ
D dc 2 2 dΓ
ð7Þ
1=2 ωD , ω
ωD ¼
The expressions for the surface viscoelastic properties, given by eq 6 for single surfactant layers and eqs 3-5 for mixtures, involve two and six derivatives, respectively. To determine these values, certain models of the surface adsorption layer have to be assumed, which obey the equations of state and adsorption isotherms. For mixtures of two surfactants governed by different adsorption models, the following equations were derived:14 DOI: 10.1021/la9024926
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The equation of state of the surface layer:
-
Πω0 ¼ lnð1 - θR - θF Þ þ θR ð1 - ωR0 =ωR Þ þ aR θ2R RT þ aF θ2F þ 2aRF θR θF
ð8Þ
with
ω0 ¼
ωR0 θR þ ωF0 θF θR þ θF
where R is the gas constant, T is the absolute temperature, θi = ωiΓi is the surface coverage by surfactant molecules of component i (the subscript i=R stands for the reorientation model and i=F for the Frumkin model), Γi is the adsorption, ωi is molar area, ωi0 is the molar area at zero surface pressure, ai and aRF are the interaction constants. The molar areas of the reorientable surfactants adsorbed in states 1 and 2 are denoted by ω1 and ω2, hence the coverage of the surface layer is given by θR =ωRΓR =ω1Γ1 + ω2Γ2. The molar areas of the two surfactants (ωF for the Frumkin surfactant and ω1 for the reorientable surfactant in the state with minimal area) can be approximated by a linear dependence on surface pressure Π and total surface coverage θ=θR + θF:14 ω1 = ωR0 (1- εRΠθ) and ωF = ωF0(1 - εFΠθ). Here εR and εF are the respective compressibilities for the two surfactants. The adsorption isotherms for the states 1 and 2 of the reorientable surfactant (in our case for CnEOm) are given by b R cR ¼
Γ1 ωR0 ð1 - θÞω1 =ωR0
ω1 exp ð2aR θR Þ - 2aRF θF ωR0
Γ2 ωR0
ð9Þ
ω2 b R cR ¼ exp ð2aR θR Þ - 2aRF θF RR ω2 =ω R0 ω R0 ðω2 =ω1 Þ ð1 - θÞ
ð10Þ where bR is the adsorption equilibrium constant, cR is the surfactant bulk concentration, and RR is the constant which stands for an increase in the adsorption constant for state 2 with maximum surface area. The adsorption isotherm for the Frumkin component (in our case for SDS) is given by b F cF ¼
θF exp½ -2aF θF - 2aRF θR ð1 - θÞ
ð11Þ
where bF is the adsorption equilibrium constant and cF is the surfactant bulk concentration. Using the model parameters of the mixture compounds, and the relevant concentrations of the components cR and cF, the values of Π, ΓR, and ΓF can be determined from a simultaneous solution of eqs 8-11, following the procedure described in detail in ref 11.
Results and Discussion The dilational viscoelasticity of SDS/CnEOm mixed adsorption layers in the presence of 0.01 M NaCl was measured for the following mixing ratios: 3:1, 10:1, and 30:1 for SDS/C12EO5 and 5:1, 30:1, and 200:1 for SDS/C14EO8 mixtures. The dynamic and equilibrium surface tensions for the same mixtures were reported in ref 14. Figure 1a and b shows the experimental dependencies of the viscoelasticity modulus |E| and phase angle φ, respectively, on 1798 DOI: 10.1021/la9024926
Figure 1. Experimental dependence of the viscoelasticity modulus (a) and phase angle (b) on the oscillation frequency for individual C12EO5 (9) and SDS (b) solutions and SDS/C12EO5 mixtures at molar mixing ratios of 3:1 (]), 10:1 (4) and 30:1 (0), and a total solution concentration of 50 μmol/L.
the oscillation frequency, as measured by the oscillating bubble profile method for the individual solutions of SDS and C12EO5, and the mixtures of SDS/C12EO5 (mixing ratios 3:1, 10:1, and 30:1). In these experiments, the total surfactant concentration for all studied solutions was 50 μmol/L. Note that, according to the surface tension isotherms reported in ref 14 (see Figure 4 therein), this concentration is below the critical micelle concentration for both the individual solutions of SDS and C12EO5 and the mixtures. It is seen from Figure 1 that the viscoelasticity modulus is minimum for the individual solutions, while its maximum value is observed for the mixture at the mixing ratio of 30:1. The phase angle φ values for this concentration ratio are minimal for the individual SDS solution and very high (ca. 45) for the individual C12EO5 solutions. These high values of φ should be ascribed to the fact that the C12EO5 concentration is quite close to the respective CMC (ca. 60 μmol/L, see ref 14). The phase angles for the mixtures are in between those for the solutions of individual substances and exhibit a monotonous decrease with increasing ratios of SDS/C12EO5. The experimental dependencies will be compared now with the calculated values at two fixed frequency values 0.01 and 0.1 Hz as functions of the total surfactant concentration. Figures 2a and 2b illustrate the experimental dependencies of the viscoelasticity modulus and phase angle, respectively, on the concentration of SDS, C12EO5 and SDS/C12EO5 mixtures at a frequency of Langmuir 2010, 26(3), 1796–1801
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Figure 3. The same as in Figures 2 for the oscillation frequency Figure 2. Dependence of the viscoelasticity modulus (a) and
of 0.1 Hz.
phase angle (b) on concentration at the oscillation frequency of 0.01 Hz for individual C12EO5 (red points, red curve) and SDS (blue points, blue curve) solutions and SDS/C12EO5 mixtures with molar ratio of the components 3:1 (], solid curve), 10:1 (4, dotted curve), and 30:1 (0, dashed curve). Points are experimental data and lines are calculated dependencies.
the optimum diffusion coefficient value was essentially lower: 10-12 m2/s. This fact presumably indicates that for the individual SDS solutions an adsorption barrier exists, which becomes essentially lower if the nonionic surfactant is added to the solution. Also, similarly to the kinetics of adsorption from both the individual and mixed SDS solutions onto a fresh surface,14 the apparent SDS diffusion coefficient is much lower than its real value; this discrepancy between the fitted values and the experimental data is due to the fact that the theoretical model does not account for the presence of surface active contaminations in the solution. For the SDS solutions, this impurity is the dodecanol, which is always formed in aqueous SDS solutions due to its hydrolysis.17 It was shown there, that if the presence of dodecanol is accounted for, i.e., the individual SDS solution is considered as a SDS/dodecanol mixture, the apparent SDS diffusion coefficient increase by about 1 order of magnitude. It is seen from Figures 2 and 3 that quite good qualitative correspondence exists between the calculated and measured values of viscoelasticity characteristics; note, for example, the locations of the extrema in the |E|(c) dependencies. Also wave-like dependencies of the viscoelasticity modulus (both experimental and calculated ones) are clearly seen, especially at a frequency of 0.1 Hz (Figure 3a) for the individual C12EO5 solutions and its mixtures with SDS. The quantitative agreement between the experimental data and the theoretical predictions is also quite good. Figures 4a and b illustrate the experimental and calculated dependencies of the viscoelastic modulus and the phase angle,
0.01 Hz, while corresponding dependencies for the frequency 0.1 Hz are shown in Figures 3a and 3b. For all studied solutions the viscoelasticity modulus exhibits maximum, while the phase angle increases monotonously with the surfactant concentration. These experimental dependencies were compared with the values calculated using the theoretical model defined above by eqs 5-11. The calculations were performed with the following values of the model parameters for the individual surfactants: SDS: ωF0 = 3.0 105 m2/mol, aF = 0, εF = 0.008 m2/mN, and bF = 25.0 m3/mol; C12EO5: ωR0 = 4.2 105 m2/mol, ω2 = 1.0 106 m2/mol, aR =0.2, RR =2.5, εR =0.006 m/mN, and bR =5.1 103 m3/mol. The only additional parameter for mixtures is the intermolecular interaction coefficient aRF = 1.3. Note, these parameters of the theoretical models are exactly the same as those given in ref 14. From these values, the surface tension isotherms for the individual surfactants and the SDS/C12EO5 mixtures and also the kinetic dependencies, corresponding to the adsorption of these mixtures from the solutions, were calculated. The values of the diffusion coefficients needed for the calculations of the rheological parameters of the SDS/C12EO5 mixtures were also close to those reported in ref 14: (4-5) 10-10 m2/s for C12EO5 and (3-10) 10-11 m2/s for SDS. Note that for the individual solutions of C12EO5, the diffusion coefficient was assumed to be the same as that for the mixture, while for the individual SDS solution Langmuir 2010, 26(3), 1796–1801
(17) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Petkov, J. T.; Yorke, J.; Miller, R. Colloids Surf., A 2009; doi: 10.1016/j.colsurfa.2009.02.022.
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Figure 4. Dependence of the viscoelasticity modulus (a) and phase angle (b) on concentration at the oscillation frequency of 0.01 Hz for individual C14EO8 (red points, red curve) and SDS (blue points, blue curve) solutions and SDS/C14EO8 mixtures with molar ratio of the components 5:1 (], solid curve), 30:1 (4, dotted curve), and 200:1 (0, dashed curve). Points are experimental data and lines are calculated dependencies.
respectively, on the concentration of individual SDS and C14EO8 solutions and SDS/C14EO8 mixtures at the frequency 0.01 Hz, while similar dependencies at the frequency of 0.1 Hz are shown in Figures 5a and b. One can note that the behavior of the viscoelastic characteristics for SDS/C14EO8 mixtures is similar to that for the SDS/C12EO5 mixtures: for all studied solutions, the concentration dependence of |E|(c) exhibits a maximum or is wavelike, while the phase angle increases monotonously with increasing the total surfactant concentration. The model parameters used in the calculations for C14EO8 were identical to those obtained in ref 14: ωR0 = 5.7 105 m2/mol, ω2 = 1.0 106 m2/mol, aR =0.0, RR =2.8, εR =0.007 m/mN, and bR =1.0105 m3/mol, while the intermolecular interaction coefficient for SDS/C14EO8 mixtures was aRF = 1.4. The diffusion coefficients for the SDS/ C14EO8 mixtures were assumed to be the same as that of the SDS/ C12EO5 mixtures. In contrast to the SDS/C12EO5 mixtures and C12EO5 individual solutions, the SDS/C14EO8 mixtures and C14EO8 individual solutions exhibit a higher (approximately by a factor of 2) viscoelastic modulus, and also the limiting value of the phase angle (at high concentrations) is lower. These differences should be ascribed to a higher surface activity of C14EO8 as compared to that of C12EO5. Therefore, the values of dΓ/dc for individual C14EO8 solutions are high (cf. eq 7), and the resulting values of |E| become closer to the limiting elasticity modulus E0 = dΠ/d lnΓ, while the phase angle becomes lower. Also, the partial derivatives aRR ¼ ðDΓR =DcR ÞjcF for C14EO8 in its mixtures with SDS are 1800 DOI: 10.1021/la9024926
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Figure 5. The same as in Figures 4 for the oscillation frequency 0.1 Hz.
higher than those of C12EO5 (see the concentration dependencies of the adsorption of the mixture components shown in Figures 6 and 7 in ref 14). According to eq 5, this results in an increase of the viscoelastic modulus and a decrease of phase angle for SDS/ C14EO8 mixtures as compared to that of SDS/C12EO5 mixtures. One can see from the data shown in Figures 4 and 5 that the agreement between the theory and experimental data is quite good. Moreover, the calculated curves for the mixtures reproduce very well the particularities of the experimental dependencies, for example, the presence of two maxima in the dependence |E|(c) in Figure 5a for the 200:1 ratio of SDS/C14EO8 and the complicated shape of the concentration dependence of the phase angle for the same mixture in Figure 5b. This agreement between theory and experiment is caused mainly by two factors in the proposed model: the intrinsic compressibility of each surfactant in the surface layer and the reorientation of C12EO5 and C14EO8 molecules. This is illustrated by Figure 6 in which the calculated dependencies of the viscoelasticity modulus for SDS/C14EO8 mixtures (presented also in Figure 5a) are shown by black lines. The red lines in Figure 6 correspond to theoretical curves for the same mixtures calculated with zero compressibility (εF =εR =0). Moreover, for the C14EO8 molecules, we did not assume the reorientation model, but instead the Frumkin model defined by eq 11 was used with the following parameters: ωF0 =4.0 105 m2/mol, aF =-1.5, εF =0, and bF = 3 105 m3/mol. The calculations of the rheological properties of SDS/C14EO8 mixtures, assuming that both components are incompressible and obey the Frumkin model, were based on the procedure described in ref 11. It is clearly seen in Figure 6 that the neglecting of intrinsic compressibility and reorientation makes it impossible to reproduce the experimental viscoelasticity modulus Langmuir 2010, 26(3), 1796–1801
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Figure 6. The calculated dependencies of the viscoelasticity modulus on concentration at the oscillation frequency 0.1 Hz for SDS/ C14EO8 mixtures with molar ratio of the components 5:1 (solid curve), 30:1 (dotted curve) and 200:1 (dashed curve). Black curves are calculations using the model defined by eqs 8-11 and red curves are calculations neglecting the internal compressibility and reorientation.
by the theory: the calculated maximum of |E| is by a factor of 2-2.5 higher than the experimental data and the values calculated using the model defined by eqs 8-11. Also this approximation predicts the location of the maxima on the theoretical curves in the |E|(c) dependence apart from that obtained experimentally. On the other hand, if one neglects the reorientation of the oxyethylated component in the mixture, the values for |E| in the lowconcentration range become too small (see Figure 6), and the concentration range in which the maximum of |E| is observed becomes too narrow; and also the wave-like shape of the dependence |E|(c) vanishes. In Figure 7 the same dependencies as in Figure 6 are presented but calculated using the theoretical model (eqs 5-11) for an extremely high oscillations frequency of 105 Hz, when |E| ≈ E0 and φ ≈ 0. At concentrations close to the CMC the viscoelastic modulus calculated for the approximation without internal compressibility becomes as high as 400-600 mN/m, while from the model with internal compressibility one obtains values below 100 mN/m. These results are in a good agreement with data obtained earlier for the individual surfactant solutions.18-20 It was shown in ref 21 that the dependence |E|(c) for solutions of Triton X-100 (with average number of oxyethylene groups equal to 10) and Tritons with a higher degree of oxyethylation (X-165 and X-405) clearly exhibits two maxima. It was concluded that the first maximum is caused by the transition of the expanded state of the adsorbed Triton molecules in the surface layer (ethylene oxide groups are adsorbed at the interface) to a more (18) Fainerman, V. B.; Kovalchuk, V. I.; Aksenenko, E. V.; Michel, M.; Leser, M. E.; Miller, R. J. Phys. Chem. B 2004, 108, 13700. (19) Kovalchuk, V. I.; Miller, R.; Fainerman, V. B.; Loglio, G. Adv. Colloid Interface Sci. 2005, 114-115, 303–313. (20) Fainerman, V. B.; Zholob, S. A.; Petkov, J. T.; Miller, R. Colloids Surf., A 2008, 323, 56. (21) 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, 16.
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Figure 7. The same as in Figure 6 for the limiting oscillation frequency.
compact state (increase in surface pressure leads to a desorption of the ethylene oxide groups from the interface). The existence of the second maximum in concentrated solutions (at sufficiently low oscillation frequencies) is caused by a diffusional exchange between the solution bulk and surface layers (the derivative dc/dΓ is increased with increasing concentration) and the intrinsic compressibility of Triton molecules at high surface pressure. A similar mechanism (while not so pronounced because, in this case, the oxyethylation degree is relatively small) governs the adsorption of the oxyethylated alcohols studied here. Another evidence in favor of the theoretical model given by eqs 8-11 is the fact that the data obtained in different independent experiments (dilational rheology and dynamic and equilibrium adsorption) can be described by a single set of model parameters. It should be noted also that the values of the adsorption of oxyethylated surfactants assumed in this study agree well with the experimental data determined by neutron reflection.22
Conclusions In this work we discussed the rheological behavior of mixtures of SDS with the oxyethylated alcohols C12EO5 and C14EO8. The experimental dependencies of |E|(c) and φ(c) in a wide range of surfactant concentrations are obtained by using the buoyant (oblate) bubble profile method involving harmonic oscillations of the bubble area at low oscillation frequencies (0.005-0.2 Hz). For the analysis of the experimental data, the theoretical approach proposed earlier11 was employed, which describes the dilational rheology of the mixtures of two surfactants with a diffusional adsorption mechanism. This approach requires the refinement of the equations of state of the mixed surface layer and the adsorption isotherm. For the systems studied, the theoretical model assumes different adsorption mechanisms for different surfactants. A satisfactory agreement between the experimental data and the calculations of the viscoelasticity modulus and the phase angle of the mixed adsorption layers was achieved. (22) Lu, J. R.; Thomas, R. K.; Penfold, J. Adv. Colloid Interface Sci. 2000, 84, 143.
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