J. Phys. Chem. C 2007, 111, 14713-14719
14713
Surface Dilational Rheology of Mixed Surfactants Layers at Liquid Interfaces E.V. Aksenenko,† V.I. Kovalchuk,‡ V.B. Fainerman,§ and R. Miller*,| Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky AVenue, 03680 KyiV (KieV), Ukraine, Institute of Biocolloid Chemistry, 42 Vernadsky AVenue, 03680 KyiV (KieV), Ukraine, Medical Physicochemical Centre, Donetsk Medical UniVersity, 16 Ilych AVenue, 83003 Donetsk, Ukraine, and Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, Am Mu¨hlenberg 1, 14424 Potsdam, Germany ReceiVed: May 21, 2007
Experimental data on surface tension and dilational rheology for mixtures of two non-ionic surfactants (decyl and tetradecyl dimethyl phosphine oxide) are analyzed on the basis of a recently derived model. The chosen surfactant mixtures are suitable for studies of the dilational rheology at low frequencies, as provided by slow area oscillations in drop and bubble profile analysis tensiometry. The results show that the compressibility of adsorbed layers given by the parameter has negligible effect on the equilibrium adsorption behavior but have significant influence on the frequency dependence of the viscoelasticity modulus and phase angle. The model allows also satisfactory analysis of literature data obtained for mixed anionic (sodium dodecyl sulfate) and non-ionic (dodecanol) surfactant systems.
1. Introduction Under practical conditions, surfactants are typically mixtures of components of different surface activity. In modern technologies, surfactant mixtures are used to reach special effects that cannot be achieved by single compounds. This fact makes a deeper understanding of the adsorption process from solutions of surfactant mixtures necessary. The adsorption behavior of mixed surfactant solutions at liquid interfaces was considered in several papers, most of them published only very recently.1-12 The theoretical analysis of the dilational rheology of surfactant mixtures was presented in refs 13-18, in which the theory by Lucassen and van den Tempel for single surfactants19,20 was generalized. In particular, for the mixture of two surfactants, analytical expressions for the complex dilational modulus were derived,15,16 irreversible thermodynamic approaches have been applied, and a general method for the derivation of equations for the surface elasticity of mixtures of an arbitrary number of surfactants and mixed adsorption kinetics was proposed.17,18 It was shown by Joos16 that, by introducing partial elasticities, it becomes possible to express the limiting high-frequency elasticity of surfactant mixtures in terms of elasticities of the individual components. At the same time, experimental studies of surface dilational rheology of surfactant mixtures are very scarce.21-23 In a recent review, rigorous expressions for the complex elasticity modulus of surfactant mixtures were derived,24 and a numerical procedure was proposed for calculating its real and imaginary parts. This procedure was illustrated for protein/ nonionic surfactant mixtures. In the present paper the theory for mixed surfactants developed earlier24 is generalized, and theoretical calculations are compared with experimental data obtained for mixtures of the two nonionic surfactants C10-dimethyl phosphine oxide * To whom correspondence should be addressed. E-mail: miller@ mpikg.mpg.de. † Institute of Colloid Chemistry and Chemistry of Water. ‡ Institute of Biocolloid Chemistry. § Donetsk Medical University. | Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung.
(DMPO)and C14DMPO. We also present a theoretical analysis of the rheological behavior of sodium dodecyl sulfate (SDS)/ dodecanol mixtures studied in ref 22. The theoretical dependencies of the elasticity modulus on frequency agree qualitatively well with the experimental finding obtained by the oscillating bubble method. 2. Theory Surface Elasticity of Mixed Adsorption Layers. The surface dilational modulus is defined as the increase in surface tension for a small relative increase of surface area (eq 1),
E)
dγ d(ln A)
(1)
where γ is the surface tension, and A is the area of the surface. The measured modulus (E) is viscoelastic, with an elastic part accounting for the recoverable energy stored in the interface and a viscous contribution reflecting the loss of energy through relaxation processes. Expressions for the complex elasticity modulus for a mixture of two surfactants (i, j ) 1, 2) and a harmonic perturbation of frequency f and an amplitude ∆A were derived in eq 2,15,16,24
E)
( ) [x
1 ∂Π B ∂ ln Γ1 iω
Γ2
iω a + D1 11
x
iω Γ2 a + D2 12Γ1
] ( ) [x
(a11a22 - a12a21) +
xD1D2
x
iω a + D2 22
1 ∂Π B ∂ ln Γ2 iω
xD1D2
Γ1
iω Γ1 a + D1 21Γ2
]
(a11a22 - a12a21) (2)
where Π ) γ0 - γ is the surface pressure, γ0 is surface tension of the solvent, Γj is the adsorption, cj the concentration and Dj the diffusion coefficient of the jth component of the mixture, T is the temperature, ω ) 2πf is the angular frequency, aij ) (∂Γi/ ∂cj)|ck*j are the partial derivatives which should be determined
10.1021/jp073904g CCC: $37.00 © 2007 American Chemical Society Published on Web 09/19/2007
14714 J. Phys. Chem. C, Vol. 111, No. 40, 2007
Aksenenko et al.
from the adsorption isotherm, and B is defined by the following equation.
B ) 1 + xiω/D1a11 + xiω/D2a22 + (iω/xD1D2) × (a11a22 - a12a21) The viscoelastic modulus can be expressed as a complex number, E ) Er + iEi, where the real part Er corresponds to the storage modulus equal to the dilational elasticity, and the imaginary part Ei corresponds to the loss modulus given by the product of the dilational viscosity ηd and the imposed angular frequency (ω) of the area variations. With xi ) (1 + i)/x2, we finally obtain the expressions for the real and imaginary parts of the viscoelasticity given in eq 2 as eq 3:24
Er ) (PR + QS)/(P2 + Q2) Ei ) [PS - QR]/(P2 + Q2)
(3)
and the expressions for the viscoelasticity modulus |E| and phase angle (φ) between stress (dγ) and strain (dA) is given by eq 4.
|E| ) x(R2 + S2)/(P2 + Q2)
φ ) arctg (Ei/Er)
(4)
transforms into eq 5. In fact, for this case, the equation of state (expressed as the functional dependence of Π on Γ, with Γ ) Γ1 + Γ2) and the adsorption isotherm (expressed as the functional dependence of c on Γ, with c ) c1 + c2) for the mixture of two surfactants, 1 and 2, are shown by eqs 8 and 9,
Π ) Π(Γ)
(8)
c ) c(Γ)
(9)
and the functional dependencies of Π on Γ and c on Γ are exactly the same as those for the case of a single surfactant. Differentiation of eqs 8 and 9 yields eqs 10 and 11.
( )
Ei0 ) Γi
[
]
∂(c1 + c2) ∂ci
cj
∂Π ∂Γi
dΠ dΠ ) Γi ) Γi |Γ)Γ1+Γ2 (10) dΓ d(Γ + Γ ) Γj 1 2
)1)
[( ) ( ) ]
∂Γ1 dc d(Γ1 + Γ2) ∂ci
Pi ) [xω/Diaii + xω/Djaij(Γj/Γi)]/x2
R ) P1(∂Π/∂ ln Γ1)|Γ2 + P2(∂Π/∂ ln Γ2)|Γ1 S ) Q1(∂Π/∂ ln Γ1)|Γ2 + Q2(∂Π/∂ ln Γ2)|Γ1 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 Tempel19,20 as eq 5,
( x )( dΓ dc
xiωD ) ) E 11++2ζζ ++ 2ζiζ
dΓ iω / 1+ D dc
0
2
(5)
or
|E| ) E0(1 + 2ζ + 2ζ2)-1/2
φ ) arctg [ζ/(1 + ζ)] (6)
with the following definitions (eqs 7).
E0 ) dΠ/d ln Γ ζ)
( )
ωD )
ωD ω
1/2
D dc 2 2 dΓ
( )
dc | (11) dΓ Γ)Γ1+Γ2
dΓ dΓ | ) | (12) dc Γ)Γ1+Γ2 dc c)c1+c2
From eq 12 we get a22 ) a11 + a21 - a12, and therefore we obtain eq 13.
a11a22 - a21a12 ) (a11 + a21) (a11 - a12)
(13)
Introducing this expression into eq 2, one finally obtains eq 14,
Qi ) Pi + (ω/xD1D2) × (a11a22 - a12a21)
E ) E0
)
cj
It is seen from eq 11 with i, j ) 1, 2 that the following is obtained.
a11 + a21 ) a12 + a22 )
Q ) (xω/D1a11 + xω/D2a22)/x2 + (ω/xD1D2) × (a11a22 - a12a21)
cj
∂Γ2 ∂ci
(aii + aji) ×
The coefficients P, Q, R, and S have the following form.
P ) 1 + (xω/D1a11 + xω/D2a22)/x2
+
(7)
It can be shown that, if the properties (parameters of the equation of state, adsorption isotherm, and diffusion coefficients) of the two surfactants in the mixture are the same, then eq 2
x x
iω (a + a21) D 11 dΠ ) E ) (Γ1 + Γ2) |Γ)Γ1+Γ2 dΓ iω 1+ (a + a21) D 11 iω dΓ | D dc Γ)Γ1+Γ2 E0(Γ) (14) iω dΓ | 1+ D dc Γ)Γ1+Γ2
x x
where E0(Γ) ) Γ dΠ/dΓ|Γ)Γ1+Γ2. Therefore, for Γ ) Γ1 + Γ2 (and also for c1 + c2 ) c) eq 14 becomes equal to eq 5, which expresses the viscoelasticity for a single surfactant solution. The expressions for the surface viscoelastic properties, given by eq 5 for the single surfactant layers and eqs 2 and 4 for mixtures, involve two and six derivatives, respectively. To determine these values, certain models of the surface adsorption layer, which obey the equation of state and adsorption isotherm, have to be assumed. In what follows, we consider two types of systems: (i) the adsorption of an individual surfactant, and (ii) the competitive adsorption from a mixed surfactant solution. Individual Surfactant Solutions. As was shown in a number of experimental studies with different surfactant systems, the limiting elasticity (E0) obtained from rheological studies levels off or even passes through a maximum with increasing concentration. A new interpretation of these dependencies was recently proposed, which explains this effect on the basis of a finite compressibility of adsorbed surfactant molecules.25-27 The
Mixed Surfactants Layers at Liquid Interfaces
J. Phys. Chem. C, Vol. 111, No. 40, 2007 14715
rigorous theoretical model given in ref 26 can be simplified by neglecting the contribution of the nonideality of entropy, giving the equation of state and adsorption isotherm of the Frumkin model (eqs 15 and 16),
bc ) Π)-
θ exp(-2Rθ) (1 - θ)
(15)
RT [ln(1 - θ) + Rθ2] ω0
(16)
where θ ) ω × Γ is the surface coverage by surfactant molecules, Γ is the adsorption of the surfactant, b is the adsorption equilibrium constant, and R is an interaction constant. As recently shown, the molar area of a surfactant (ω) can be approximated by a linear dependence on surface pressure Π (eq 17),26,27
ω ) ω0(1 - Πθ)
(17)
where ω0 is the molar area at zero surface pressure, and is the two-dimensional (2D) 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 molecules upon surface layer compression, accompanied by an increase in the thickness of the surface layer.27 In contrast to the expression for ω used in the theory described in refs 25-27, eq 17 involves the additional factor θ, which roughly accounts for the fact that an unsaturated adsorption layer (formed by individual or for the mixtures by two or more surfactants) is different from a saturated monolayer. For a saturated monolayer of an individual surfactant or for the mixture of two or more surfactants the parameter θ (or the total coverage) becomes unity, and eq 17 turns into the expression used in ref 27 for Grazing Incidence X-ray Diffraction experiments of condensed insoluble monolayers.28 Accordingly, one can express θ in eqs 15 and 16 as follows (eq 18).
θ ) Γω ) Γω0(1 - Πθ)
(18)
Then, for any set of model parameters R, ω0, and b, one can solve eq 15 for any c, from which the solution θ ) θ(c) and hence the values for Π and Γ can be calculated via eqs 16 and 18. Mixture of Two Surfactants. If we can apply the approximation ω10 = ω20 for surfactants 1 and 2, the following generalized Frumkin equation of state for a nonionic surfactants mixture (neglecting the contribution of the nonideality of entropy) results;11,29
-
Πω/0 RT
) ln(1 - θ1 - θ2) +
R1θ21
+
R2θ22
+ 2R12θ1θ2 (19)
with
ω/0 )
ω10θ1 + ω20θ2 θ1 + θ2
(20)
where θi ) ωi × Γi is the surface coverage by surfactant molecules of component i, Γi is the adsorption, and Ri the interaction constant. The molar area of a surfactant (ωi) can be approximated by a linear dependence on surface pressure Π and surface coverage, and therefore we get the following relations (eqs 21 and 22),
ω1 ) ω10(1 - 1Πθ)
ω2 ) ω20(1 - 2Πθ)
(22)
θ1 ) Γ1ω1 ) Γ1ω10[1 - 1Πθ] θ2 ) Γ2ω2 ) Γ2ω20[1 - 2Πθ] (21) where θ ) θ1 + θ2. The adsorption isotherms turn into the generalized Frumkin equation,28 that is, for surfactant 1 we have eq 23,
b1c1 )
θ1 (1 - θ1 - θ2)
exp[-2R1θ1 - 2R12θ2]
(23)
and for surfactant 2 we have eq 24.
b2c2 )
θ2 (1 - θ1 - θ2)
exp[-2R2θ2 - 2R12θ1]
(24)
Here bi are the adsorption equilibrium constants, and ci are surfactant bulk concentrations. The procedure used for the numerical calculations is quite straightforward; given the known values of T, ω10, ω20, R1, R2, R12, 1, 2, b1, and b2, a computation procedure was developed that, for any given values of surfactant concentrations c1 and c2, determines θ1 and θ2 from eqs 23 and 24. These θ1 and θ2 values are then used to calculate the values of individual adsorption via eqs 21 and, finally, the surface pressure value is calculated via eqs 19 and 20. Numerical Calculation of Partial Derivatives. If the system considered involves single surfactants only, then the numerical calculation of the derivatives involved in eqs 5-7 is straightforward. Provided that a suitable procedure exists that enables one to calculate the surface pressure and adsorption at any value of the surfactant concentration, the relevant derivatives can be calculated from finite differences. For a system that involves two (or more) surfactants, the procedure is more involved. Assume that for each pair of concentrations {c1, c2} it is possible to calculate the values of surface pressure Π ) Π(c1, c2) and adsorption of the components Γ1 ) Γ1(c1, c2) and Γ2 ) Γ2(c1, c2). For sufficiently small deviations of the concentrations δc1 and δc2 in the vicinity of the point {c1, c2} (i.e., in the points {0} ) {c1, c2}, {1} ) {c1 + δc1, c2} and {2} ) {c1, c2 + δc2}), one can calculate the values given by eqs 25 and 26.
Π{0} ) Π(c1, c2)
Π{1} ) Π(c1 + δc1, c2) Π{2} ) Π(c1, c2 + δc2) (25)
Γi{0} ) Γi(c1, c2)
Γi{1} ) Γi(c1 + δc1, c2) Γi{2} ) Γi(c1, c2 + δc2)
(i ) 1,2) (26)
Then, the partial derivatives of adsorptions with respect to concentrations aij ) (∂Γi/∂cj)|ck*j given in eq 2 can be approximately calculated from the finite differences (eqs 27):
a11 ) (∂Γ1/∂c1)|c2 ) (Γ1{1} - Γ1{0})/δc1 a12 ) (∂Γ1/∂c2)|c1 ) (Γ1{2} - Γ1{0})/δc2 a21 ) (∂Γ2/∂c1)|c2 ) (Γ2{1} - Γ2{0})/δc1 a22 ) (∂Γ2/∂c2)|c1 ) (Γ2{2} - Γ2{0})/δc2 (27) To calculate the partial derivatives (∂Π/∂lnΓi)|Γj*i ≡ (∂Π/ ∂lnΓi)|lnΓj*i, one should perform the change of variables, noting that the surface pressure Π ) Π(c1, c2) can be considered as a
14716 J. Phys. Chem. C, Vol. 111, No. 40, 2007
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function of two other variables, namely ln(Γ1) and ln(Γ2), which could be thought of as independent, instead of the concentrations. Therefore, the small deviations of Π that correspond to small deviations of ln(Γ1) and ln(Γ2) (in turn caused by small deviations of c1 and c2) can be expressed as shown in eq 28:
(δΠ){k} ) (∂Π/∂ ln Γ1)|ln Γ2(δ ln Γ1){k} + (∂Π/∂ ln Γ2)|ln Γ1(δ ln Γ2){k}
(k ) 1, 2) (28)
where the superscript {k} denotes the points {1} and {2} as defined above, (δΠ){k} ) Π{k} - Π{0}, and (δ ln Γi){k} ) ln Γi{k} - ln Γi{0}. Finally, one can calculate the partial derivatives (∂Π/∂ ln Γi)|Γj*i ≡ (∂Π/∂ ln Γi)|lnΓj*i from the set of eqs 28 with the known coefficients determined from eqs 25 and 26, provided that a suitable procedure is available for the numerical calculation of the surface pressure and adsorption values. 3. Experimental The experiments were performed by bubble/drop profile analysis tensiometry (PAT-1, SINTERFACE Technologies, Germany).30 The temperature of the measuring glass cell (volume V ) 20 mL) was kept at 22 °C. In this study we applied the emerging bubble technique to avoid loss of surfactant because of adsorption at low bulk concentrations. The alkyl dimethyl phosphine oxides with alkyl chain lengths of 10 and 14 (C10DMPO and C14DMPO) were synthesized and purified as described earlier31 with a special grade of purity that allowed us to use them in interfacial studies. All solutions were prepared with MilliQ water. To study the dilational elasticity, the bubble, after having reached the adsorption equilibrium, was subjected to harmonic oscillations with an amplitude of ∆A/A ) (7-8% and frequencies in the range between 0.005 and 0.2 Hz. The results of the experiments with harmonic surface area oscillations were analyzed using the Fourier transformation shown in eq 29:32,33
F[∆γ] E(iω) ) A0 F[∆A]
(29)
where A0 is the initial area of the drop surface. The experimental raw data were first filtered to exclude scattered points, and also any drift of the surface tension with time was eliminated, which could be caused by a yet slower, further surface layer equilibration during the harmonic oscillations. 4. Results and Discussion The equilibrium surface tension isotherms of C10DMPO and C14DMPO solutions are shown in Figure 1. The surface tensions were measured with the ring method in refs 34 and 35. In ref 36, the data for the limiting (high-frequency) elasticity E0 for individual solutions of C10DMPO and C14DMPO were presented. These data were also used to determine the parameters involved in the theoretical model given by eqs 15-18. The model parameters obtained in this way are as follows; C10DMPO: ω0 ) 2.66 × 105 m2/mol, R ) 0, b ) 2.12 × 104 l/mol, and ) 0.012 m/mN; and C14DMPO: ω0 ) 2.52 × 105 m2/mol, R ) 0.6, b ) 1.59 × 106 l/mol, and ) 0.01 m/mN. Note that, for ) 0, eqs 15-18 become the ordinary equation of state and adsorption isotherm for the Frumkin model. Note that the introduction of the parameter does not affect the shape of theoretical isotherms presented in Figure 1, because it effects only the adsorption value Γ and, therefore, the limiting elasticity E0; see eq 18.
Figure 1. Equilibrium surface tensions of individual C10DMPO and C14DMPO solutions as a function of bulk concentration (data from refs 34 and 35) and of C10DMPO/C14DMPO mixtures as a function of C10DMPO concentration at fixed C14DMPO concentration; the labels correspond to the C14DMPO concentration in µM; theoretical isotherms were calculated from eqs 15-24 using the parameters given in the text.
Figure 2. Limiting (high-frequency) elasticity E0 ) dΠ/d ln Γ as a function of surface pressure Π for C10DMPO and C14DMPO solutions, calculated from eqs 14 and 15-18; the dotted line refers to ) 0; 0 and ] are experimental data from ref 36.
Figure 1 also shows several dependencies of equilibrium surface tensions for C10DMPO/C14DMPO mixtures plotted over the C10DMPO concentration at fixed C14DMPO concentration (the given labels are the corresponding C14DMPO concentrations in µM). The theoretical curves for the mixtures were calculated from eqs 19-24 using the parameters for the individual solutions as given above and using R12 ) 0.3 as the average value for the two surfactants studied here. It is seen that the theoretical curves describe the experimental surface tension isotherms for the C10DMPO/C14DMPO mixtures very well. The same set of model parameters for the equations of state and adsorption isotherm describes the limiting (high-frequency) elasticity E0 for C10DMPO and C14DMPO satisfactorily well, as shown in Figure 2. The experimental E0 values taken from ref 36 are compared with those calculated from eqs 15-18. Note that for ) 0 we obtain the Frumkin model and a good description of the surface tension data of Figure 1, whereas the resulting high-frequency elasticities increase much stronger with surface pressure than those observed experimentally. The theoretical curve for C14DMPO calculated for ) 0 (Fr) is also shown in Figure 2. Figures 3 and 4 illustrate the dependencies of viscoelasticity modulus and phase angle on the frequency (f) calculated using eqs 4 for the mixtures C10DMPO/C14DMPO in a wide concen-
Mixed Surfactants Layers at Liquid Interfaces
Figure 3. (a) Viscoelastic modulus |E| as a function of frequency (f) for mixed C14DMPO/C10DMPO solutions at a fixed C14DMPO concentration of 2 µM, calculated from eq 4 (solid lines and labels are the C14DMPO/C10DMPO concentrations expressed in µM); dotted line calculated from eq 6 for a pure 2 µM C14DMPO solution; thin lines are results for pure C10DMPO solutions at the same concentration as in the corresponding mixtures. (b) The same as in Figure 3a for 5 µM C14DMPO solutions.
trations range and for a realistic surfactant diffusion coefficient of D ) 4 × 10-10 m2/s. In these figures the dependencies of the viscoelasticity modulus |E| and phase angle (φ) on frequency for individual solutions at the same concentrations as calculated from eqs 6 by Lucassen and van den Tempel are also shown for comparison. The values of |E| and φ for the individual C10DMPO and C14DMPO solutions calculated from the theory agree satisfactorily with the experimental values obtained from oscillating bubble experiments.36 In Figures 3 and 4 we can see a monotonous increase of the viscoelasticity modulus with frequency. At the same time, for individual solutions the phase angle decreases with increasing frequency, whereas for mixtures this dependence can exhibit an extremum. Whereas for low C10DMPO concentrations a monotonous decrease in φ is seen, with increasing C10DMPO concentration, an increasing maximum in the φ(f) curves is observed. This increasing maximum is attributable to the increasing coverage of the surface by C10DMPO, because a viscous behavior is observed for pure C10DMPO solutions in this frequency range. In Figure 5 the experimental dependencies of the viscoelasticity modulus on frequency for various C10DMPO and C14DMPO concentrations in the mixture are compared with the theoretical values calculated from eq 4. We observe an increase in the viscoelasticity modulus with increasing C14DMPO concentration, whereas an increasing C10DMPO concentration leads to a decrease of |E|. The theoretical dependencies shown in Figure 5 agree well with the experimental data. Also, the phase angles (φ) agree well with the observed data; the
J. Phys. Chem. C, Vol. 111, No. 40, 2007 14717
Figure 4. (a) Phase angle (φ) as a function of frequency (f) for mixed C14DMPO/C10DMPO solutions at a fixed C14DMPO concentration of 2 µM, calculated from eq 4 (solid lines and the labels are the C14DMPO/C10DMPO concentrations expressed in µM); dotted line are calculated from eq 6 for pure 2 µM C14DMPO solution, thin lines are the results for pure C10DMPO solutions at the same concentration as in the corresponding mixtures. (b) The same as in Figure 4a for a 5 µM C14DMPO solution.
Figure 5. Viscoelastic modulus |E| as a function of frequency (f) for mixed C14DMPO/C10DMPO solutions at different concentrations; labels refer to the C14DMPO/C10DMPO concentrations in the mixture in µM; symbols are experimental points.
experimental values vary in the range between 20 and 40 degrees, whereas the theoretical predictions are 10 to 30°. Using the model given by eq 4, the rheological behavior of (SDS)/dodecanol mixtures experimentally studied in ref 22 was also analyzed. The experimental surface tension data for the single surfactant solutions were presented in refs 22 and 29 for SDS and in ref 37 for dodecanol. Also in ref 22, the data for the limiting (high-frequency) elasticity for the individual solu-
14718 J. Phys. Chem. C, Vol. 111, No. 40, 2007
Figure 6. Equilibrium surface tension (γ) and limiting elasticity E0 of SDS and dodecanol solutions as a function of concentration; data are from refs 22, 29, and 37; filled points, surface tension; open points, limiting elasticity; theoretical curves calculated from eqs 15-18 using the parameters given in the text.
Figure 7. Dilational elasticity modulus |E| as a function of oscillation frequency at various dodecanol concentrations; labels are the concentrations expressed in mM; experimental data are from ref 22; theoretical curves calculated from eq 6 using the parameters given in the text.
tions of SDS and dodecanol were presented. The parameters of the theoretical model given by eqs 15-18 for SDS and dodecanol are as follows; SDS: ω0 ) 1.7 × 105 m2/mol (per one ion), R ) 1.4, b ) 3.0 × 102 l/mol, and ) 0.013 m/mN; Dodecanol: ω0 ) 1.2 × 105 m2/mol, R ) 1.55, b ) 1.12 × 105 l/mol, and ) 0.004 m/mN; From Figure 6 we see that, with these parameters, eqs 1518 reproduce the surface tension isotherms and the experimental dependencies of limiting elasticity E0 for both surfactants quite well. Figure 7 illustrate the dependencies of the viscoelasticity modulus |E| on frequency for dodecanol solutions, as calculated from eq 6 for D ) 5 × 10-10 m2/s. The calculated dependencies shown in Figure 7 are almost identical to the experimental data,22 and the calculated phase angles are also similar to the reported data. For f > 10 Hz, the experimental and theoretical values were zero. In addition, the viscoelasticity modulus on frequency |E|(f) for the SDS solutions calculated from eq 6 and for D ) 4 × 10-10 m2/s agree quite well with the experimental data reported in ref 22. In the 0.001-1000 Hz frequency range the theory predicts a monotonous increase of |E| and a monotonous decrease of φ. Even at a frequency of 1000 Hz the phase angle φ is not yet zero. The theoretical predictions for the frequency dependence of the phase angle φ(f) disagree with the experimental data given in ref 22; the phase angle increases with
Aksenenko et al.
Figure 8. Dilational elasticity modulus |E| as a function of oscillation frequency at various concentrations in the SDS/dodecanol mixtures; theoretical curves calculated from eq 4 using parameters given in the text; experimental data are from ref 22; labels refer to the concentration ratio of SDS/dodecanol: (1) 3 mM + 0.0003 mM; (2, ]) 3 mM + 0.0015 mM; (3, [) 5 mM + 0.003 mM; (4, 0) 3 mM + 0.012 mM; (5, 9) - 5 mM + 0.02 mM.
Figure 9. The dependencies of phase angle (φ) on the oscillation frequency at various concentrations in the SDS/dodecanol mixtures; theoretical curves were calculated from eq 4 with the parameters given in the text; experimental data are from ref 22; labels correspond to the concentration ratio of SDS/dodecanol: (1) 3 mM + 0.0003 mM; (2, ]) 3 mM + 0.0015 mM; (3, [) 5 mM + 0.003 mM; (4, 0) 3 mM + 0.012 mM; (5, 9) 5 mM + 0.02 mM.
frequency in the range 1-500 Hz. This experimental behavior could possibly be ascribed to the presence of trace quantities of dodecanol (at a level of 0.01% mol) in the SDS solution. Figures 8 and 9 illustrate the theoretical dependencies of the viscoelasticity modulus and phase angle on frequency for SDS/ dodecanol mixtures as calculated from eqs 4 and 19-24 using the parameters of the individual solutions listed above. The R12 value was taken to be 1.5 (i.e., the average between the R-values for dodecanol and SDS). We see that the modulus increases with frequency (see Figure 8), whereas the phase angle first attains a minimum and then a maximum (see Figure 9). All theoretical curves are in a qualitative agreement with the experimental data reported in ref 22. A quantitative agreement cannot be reached for the two surfactants dodecanol and SDS; the set of eqs 19-24 are not optimum because they disregard both the average activity of SDS ions and a cluster formation in the dodecanol surface layer. Figures 8 and 9 show the theoretical curves (1) for a 3 mM SDS solution in the presence of 0.0003 M dodecanol (i.e., 0.01 mol %). From Figure 9 we see that even this small admixture of dodecanol leads to a temporary increase of the phase angle
Mixed Surfactants Layers at Liquid Interfaces in a certain frequency range, which is in qualitative agreement with the data reported in ref 22 for pure SDS solutions. Hence, it is quite possible that the anomalous character of the phase angle dependence on frequency for pure SDS is caused by the presence of small traces of dodecanol. It can be shown by calculations that, only if the dodecanol concentration in the solution is below 0.001 mol %, any presence of dodecanol does not affect the rheological properties of SDS, which is hard to establish experimentally. 5. Conclusions Expressions for the complex elasticity modulus of mixtures of two surfactants are presented. These equations transform into the equations derived by Lucassen and van den Tempel19,20 for the single surfactant solution if the adsorption characteristics of the two surfactants are assumed to be identical. A numerical procedure is proposed for calculating the real and imaginary parts of the elasticity modulus and are illustrated for some surfactants mixtures. The adsorption layer model used here assumes the internal compressibility of the surfactant adsorption layer.25-27 The developed procedure for a numerical solution of the set of equations that govern the adsorption behavior from mixed surfactant solution involves only the parameters of the individual solutions of the components as determined from surface tension, adsorption, and dilational rheology data of the individual solutions. The presented experimental data were obtained from slow oscillating drop studies using profile analysis tensiometry of mixed adsorption layers containing C10DMPO and C14DMPO. These data are in good agreement with the predictions of the model. Also, a theoretical analysis of the rheological behavior of a SDS/dodecanol mixture is presented, whereas the respective data were taken from literature.22 The calculated dependencies of the elasticity modulus on frequency for individual SDS and dodecanol solutions agree qualitatively with the experiments obtained from the oscillating bubble method in the frequency range between 1 and 500 Hz. To perform the calculations according to the theoretical approach described in Section 2, software was developed that implements the corresponding models. In particular, using these programs, one can calculate the thermodynamic and rheological characteristics of individual surfactants and surfactant mixtures, and some of these programs even provide a visual comparison between experimental data and calculated curves. The program packages, partly described earlier in ref 38, which include software, manuals, and examples of experimental data, are available via the Internet.39 Acknowledgment. The work was financially supported by projects of the European Space Agency (FASES MAP AO-99052) and the DFG (Mi418/15-1). The authors want to thank Emmie Lucassen-Reynders and Jaap Lucassen for very helpful discussions. References and Notes (1) Lucassen-Reynders, E. H. In: Anionic Surfactants - Physical Chemistry of Surfactant Action; Marcel Dekker Inc.: New York - Basel, 1981; p 1.
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