Viscoelasticity Moduli of Aqueous C14EO8 Solutions as Studied by

May 20, 2013 - Viscoelasticity Moduli of Aqueous C14EO8 Solutions as Studied by Drop and Bubble ... Accounting for optical errors in microtensiometry...
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Viscoelasticity Moduli of Aqueous C14EO8 Solutions as Studied by Drop and Bubble Profile Methods V. B. Fainerman,† E. V. Aksenenko,‡ J. Krag̈ el,§ and R. Miller*,§ †

Donetsk Medical University, 83003 Donetsk, Ukraine Institute of Colloid Chemistry and Chemistry of Water, 03680 Kiev, Ukraine § Max-Planck-Institut für Kolloid-und Grenzflächenforschung, 14424 Potsdam, Germany ‡

ABSTRACT: The drop and bubble profile methods are used to study the viscoelasticity modulus of C14EO8 aqueous solutions within a wide concentration range. To determine the equilibrium concentration of the surfactant in the drop bulk, the correction is introduced for the surfactant losses caused by its adsorption on the drop surface. It is shown that with this correction the frequency dependencies of the viscoelasticity modulus measured by either of the two experimental techniques are almost the same. The theoretical model is used, which describes the surfactant dilational rheology assuming the diffusion-governed adsorption. The experimental data for C14EO8 solutions is described by the reorientation model that assumes the two states of surfactant molecules with different molar areas in the surface layer and the intrinsic compressibility of the molecules.

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INTRODUCTION Theories for adsorption layers at the solution/air and solution/ oil interfaces formed from aqueous surfactant solutions provide equations of state that express the surface tension γ as a function of the surface layer composition and the adsorption isotherms that represent the dependencies of the adsorption on the bulk concentration. Various theoretical models for equations of state and adsorption isotherms for surfactants have been proposed so far. There is a vast number of relevant publications.1−15 The oxyethylated alcohols represent a special class of nonionic surfactants. The important feature of their aqueous solutions is that the surfactant molecule can adsorb at the interface in different states with different molar areas. However, the theories developed up to 1991 do not account for this important adsorption-related behavior of oxyethylated surfactants. An approximate relation for expressing the ratio between the adsorptions of the molecules (or states) that possess different molar areas (physicochemical principle of Braun−Le Châtelier) was first derived by Joos and Serrien.5 Respective equations of state and adsorption isotherms for the adsorbed layer of reorientable molecules were obtained in ref 16 on the basis of the analysis of the chemical potentials. Various approximate and rigorous models to describe the reorientation of adsorbed molecules were proposed in ref 17. Experimental studies of C14EO8 solutions using drop and bubble profile tensiometries18 have shown that the adsorption characteristics of these solutions are best described by this reorientation model. In ref 19, the reorientation model was also generalized to consider the nonideality of the entropy of the components in a mixed surface layer. The agreement with experimental data becomes even better if the model assumes a so-called intrinsic © XXXX American Chemical Society

compressibility of the molecule in the state of minimum area at the interface.18,19 Many authors also employed the reorientation model to describe the adsorption of various oxyethylated alcohols.20−22 The data obtained by the dilational rheology of adsorption layers are especially sensitive with respect to the choice of the theoretical adsorption model. It was shown in ref 23 that by assuming a 2D compressibility and the reorientation of the surfactant molecules in the adsorption layer one can satisfactorily reproduce the equilibrium and dynamic surface tension and the surface viscoelastic behavior of these surfactant solutions. The experimental techniques based on the drop or bubble profile analysis tensiometry can be employed to obtain information about the adsorption behavior of surfactants. From a comparison of surface tension isotherms obtained by drop profile and bubble profile methods and by employing the surfactant mass balance in the drop, the adsorbed amount of surfactant can be calculated.18,19 In ref 18, the pendant drop and emerging (prolate) bubble methods were employed to study the surface tension and the amount of C14EO8 adsorbed from the solution at the interface whereas the rheological characteristics of the adsorbed layers subjected to harmonic oscillations of the surface area were measured only using the pendant drop technique. In the present study, the buoyant (oblate) bubble method was used; in addition, the rheological characteristics were measured at various frequencies of the bubble surface area Received: April 5, 2013 Revised: May 16, 2013

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Figure 1. Dependence of the equilibrium surface tension of C14EO8 solutions on initial the concentration measured in ref 18 by the pendant drop method (◊) and the emerging bubble method (□) and in this study by the buoyant bubble method (▲). Bold black curve: theoretical calculations with the parameters shown in Table 1 (rightmost column). Lines 1 and 2 illustrate the procedure used to determine the C14EO8 equilibrium concentration in the drop. Thin solid curves: theoretical calculations with the concentration-related parameters shown in Table 1 for 1.0 (red curve), 2.0 (blue curve), and 5.0 μmol/L (green curve).

where Π = γ0 − γ is the surface pressure, γ and γ0 are the surface tensions of the solution and pure solvent, respectively, R is the gas law constant, T is the temperature, ω = (ω1Γ1 + ω2Γ2)/Γ is the mean molar area, θ = ωΓ= ω1Γ1 + ω2Γ2 is the surface coverage, Γ = Γ1 + Γ2 is the total adsorption, ω1 and ω2 are the molar areas of the two orientations of the molecules adsorbed at the interface, ω2 > ω1 and ω1 = ω0(1 − εΠθ), ω0 is the molar area of the surfactant at Π = 0 or the molar area of the solvent, factor ε is the relative 2D compressibility coefficient, and a is the intermolecular interaction constant. The adsorption isotherms for states 1 and 2 are respectively

oscillations. These results, combined with the pendant drop data reported in ref 18, were used to calculate the viscoelasticity modulus for C14EO8 solutions on the basis of the relationships proposed by Joos.4 The values obtained by the two methods were found to be virtually the same if the losses of C14EO8 caused by its adsorption in the pendant drop experiments are properly taken into account. It will also be shown that the values of the rheological parameters calculated from the reorientation model, assuming the intrinsic compressibility and the nonideality of entropy, agree well with the available experimental data.

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EXPERIMENTAL SECTION

b1c =

The experiments were performed with bubble/drop profile analysis tensiometer PAT (SINTERFACE Technologies, Germany), the principle of which was described in detail elsewhere.19,24 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 (oblate) bubble formed at the bottom tip of a vertical Teflon capillary with an external diameter of 3 mm. To study the dilational viscoelasticity E, after having reached the adsorption equilibrium (aging time no less than 25 000 s for diluted solutions and 7000 s for concentrated solutions), we subjected the bubble to harmonic oscillations at frequencies of between 0.005 and 0.2 Hz with surface area oscillation amplitudes of 5−7%. The results of these resulting harmonic surface tension changes were analyzed using a Fourier transformation. Oxyethylated alcohol C14EO8 was purchased from Sigma Chemical and used without further purification. The solutions were prepared using Milli-Q water with a surface tension of 72.0 ± 0.2 mN/m at 25 °C, which was constant over a time interval of up to 105 s (about 28 h).

b2 c =

(1 − Γω)

⎛ ω ⎞ exp⎜ −2aΓω 1 ⎟ ω0 ⎠ ⎝

Γ2ω0 α

( ) (1 − Γω) ω2 ω1

ω2 / ω0

⎛ ω ⎞ exp⎜ −2aΓω 2 ⎟ ω0 ⎠ ⎝

(2)

(3)

α

( ) ω1 ω2

⎛ (ω2 − ω1) ⎞ Γ1 exp 2 a = − Γ ω ⎜ ⎟ Γ2 ω0 ⎝ ⎠ (1 − Γω)(ω2 − ω1)/ ω0

(4)

The physical picture behind this model can vary from surfactant to surfactant, such as a change in the tilt angle of the aliphatic chain in respect to the surface or a partial change in the EO chain from a less to a more hydrophobic character.19 Figure 1 illustrates the surface tension isotherms measured by the pendant drop and emerging bubble profile methods.18 The C14EO8 adsorption isotherm at the solution/air interface was calculated in ref 18 using an approximate reorientation model. The improved model that also assumes the nonideality of entropy and the intrinsic compressibility of the adsorbed layer was employed for these solutions in ref 24. The

RESULTS AND DISCUSSION The theoretical models used to describe the equilibrium surface tension and adsorption of surfactants were described in detail elsewhere.19 Therefore, only the main equations for the reorientation model applied in this study are given here. The surface equation of state reads Πω0 = ln(1 − Γω) + Γ(ω − ω0) + a(Γω)2 RT

ω1/ ω0

where c is the concentration of the surfactant, α is the power law exponent, and bi is the adsorption equilibrium constant. The ratio of adsorptions in the two possible states of the adsorbed molecules is expressed by a relationship that follows from eqs 2 and 3 when assuming b1 = b2:

■

−

Γ1ω0

(1) B

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Table 1. Adsorption Model Parameters and Diffusion Coefficient for Various C14EO8 Concentrations in the Solutions as Estimated from the Viscoelasticity Modulus Data and from the Isotherm (Rightmost Column)a concentration (μmol/L) ω10 (105 m2/mol) ω2 (106 m2/mol) α a ε (10−3 m/mN) b (105 m3/mol) D (10−9 m2/s) a

0.1 5.8/5.9 1.0 2.5 0.5 7.0/6.7 0.92/0.98 1.3/1.0

0.5 5.7 1.0 2.8 0.4/0.6 7.0 0.91/0.7 1.0

1.0 5.7 1.0 2.9 0.6 7.0 0.74/0.7 1.0

2.0 5.7 1.0 2.8 0.6/0.7 8.5 0.63/0.5 1.4/1.3

5.0 5.9 1.0 2.9/2.8 0.6 9.0 0.68/0.65 1.2/1.3

isotherm 5.7 1.0 2.8 0.3 7.0 1.0 1.0

The values obtained from the bubble/drop profile method (when different) are shown before/after the slash, respectively.

parameters of the theoretical model obtained there are listed in the rightmost column of Table 1. The only difference is the value a = 0.3 as obtained here from the best fit of the experimental data, in contrast to ref 24 where a = 0 was assumed in order to minimize the number of fitting parameters. This small discrepancy could be justified by the fact that in the present study additional experimental data for several C14EO8 concentrations using the buoyant bubble profile method were taken into account. These values, which are shown by black triangles in Figure 1, agree well with those obtained by the emerging bubble profile method in ref 18. It is seen from Figure 1 that the experimental isotherm measured by drop profile analysis tensiometry, as compared to that obtained using the bubble profile method, is essentially shifted toward higher surfactant bulk concentrations. This discrepancy is due to the adsorption of C14EO8 at the drop surface, which results in a depletion of its bulk concentration. For equal values of the equilibrium surface tension for these two isotherms (and therefore equal adsorption values in the drop and bubble profile methods, respectively), the adsorption values at these concentrations could be determined similarly to what was done in ref 18. This procedure for two concentrations is illustrated by the two straight lines in Figure 1. It was found that after the equilibrium is attained and the concentration in the drop bulk is decreased the concentrations of 3.0, 5.0, 6.0, 7.0, and 10.0 μmol/L in the drop bulk are approximately equivalent to the concentrations of 0.1, 0.5, 1.0, 2.0, and 5.0 μmol/L, respectively, in the bubble profile method. The rheological characteristics of C14EO8 solutions were measured after the adsorption equilibrium was attained. The dependencies of the viscoelasticity modulus on the surface oscillation frequency at various bulk concentrations, as obtained by the drop and bubble profile methods, are shown in Figures 2 to 3d. The experimental values obtained by the drop profile method (open squares) are reproduced from ref 18 whereas the black squares show the results obtained here by the bubble profile method. We can see that the viscoelasticity modulus values obtained by the two methods are quite close to each other if the correction for the C14EO8 adsorption on the drop surface is taken into account: the maximum deviation between the values does not exceed ±10%, while the average deviation is ±4%. This fact confirms our supposition that equal values of the equilibrium surface tension obtained from drop and bubble profile tensiometry correspond to equal values of the bulk concentrations, and the difference in geometry of these objects does not affect the results remarkably. The experimental results were theoretically processed using expressions proposed by Joos for the viscoelasticity modulus.4 For low oscillation frequencies and small oscillating bubbles or drops (in the present study, their radii were between 1.5 and 2

Figure 2. Dependence of viscoelasticity modulus on surface oscillation frequency at the C14EO8 concentration of 0.1 μmol/L, measured after equilibrium is attained in the drop (□) and by the bubble profile method (■). Red curves were calculated using the parameters listed in Table 1 (rightmost column), which were also used to calculate the isotherm in Figure 1. Black curves were calculated with the corrected model parameters listed in Table 1. Solid lines, bubble profile method; dashed lines, drop profile method.

mm), the actual geometry of the oscillating object can play a significant role. These expressions for the complex elasticity E = Er + iEi for spherical objects read4 for the adsorption from the solution bulk onto a bubble surface

{

E(Ω) = E0 1 − i

D dc (1 + nR ) ΩR dΓ

−1

}

(5)

and for the adsorption from the drop bulk onto its surface

{

E(Ω) = E0 1 − i

D dc [nR coth(nR ) − 1] ΩR dΓ

−1

}

(6)

Here, E0(c) = dΠ/d(ln Γ) is the elasticity modulus at an infinitely high oscillation frequency, n2 = iΩ/D, D is the bulk diffusion coefficient, Ω = 2πf, R is the drop/bubble radius, and f is the oscillation frequency. For large R values, eqs 5 and 6 become equivalent to the expression for a plane surface as derived by Lucassen:25−27 E(Ω) =

E0(1 + ζ + iζ ) 1 + 2ζ + 2ζ 2

(7)

with ζ=

D dc 2Ω dΓ

The viscoelasticity characteristics measured in the experiments and compared with the theoretical predictions were the viscoelasticity modulus |E| = (Er2 + Ei2)1/2 and phase angle ϕ = arctg(Ei/Er). C

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at the same equilibrium concentrations in the drop bulk were determined as follows. First, the viscoelasticity modulus was calculated from the parameters of the isotherm given in Figure 1 and obtained with the bubble profile method (the rightmost column in Table 1). The diffusion coefficients for all concentrations and methods were taken to be constant at 10−9 m2/s, as proposed in ref 24. The results obtained in this way correspond to the red curves in Figures 2 and 3a−d. Note that the solid lines correspond to the values calculated from eq 5 for bubbles whereas the dashed lines were calculated from eq 6 for drops. It is seen that in the range of very low oscillation frequencies the modulus values obtained for drops are higher than those for bubbles whereas with increasing frequency the two curves approach each other and become almost equal to those obtained from eq 7. It is seen, especially in Figure 3b−d at higher frequencies, that the values calculated in this way are slightly different from the experimental data. To improve the agreement between experiment and theory, small corrections to the values of the model parameters were introduced, depending on the C14EO8 bulk concentration in the bubble and drop profile methods. These corrected values obtained from the best fit of the experimental data are listed in Table 1. The diffusion coefficients estimated from this best fit at the various concentrations varied between 10−9 and 1.4 × 10−9 m2/s, in agreement with the values reported in refs 18 and 24. The results after this correction are also shown in Figures 2 and 3a−d as black lines, where the dotted and solid lines correspond to the drop and bubble profile methods, respectively. It is seen that these small corrections result in an essential improvement of the agreement between calculated values and experimental data. All model parameters are quite close to those calculated from the isotherm, except for parameter a. It should be noted that in the fitting procedure all model parameters except b are initially chosen whereas the b value is determined by the software from the minimization of the deviation between the calculated surface pressure values and the experimental data. The results obtained for the phase angle by the two experimental methods are also quite similar to each other. For example, with equilibrium concentration 0.5 μmol/L at frequencies 0.01 and 0.1 Hz the phase angle values measured for the bubble were 9.9 and 3.0° and those measured for the drop were 12.9 and 2.7°, respectively. The values for the bubble at these oscillation frequencies calculated using the isotherm parameters (rightmost column in Table 1) are 10.1 and 3.2°, and the calculations using the corrected (concentrationdependent) parameters yield 10.0 and 3.1°. Figure 1 illustrates the isotherms calculated for the model parameters listed in Table 1, except for those corresponding to the two smallest concentrations (0.1 and 0.5 μmol/L), for which the deviation from the bold curve based on the isotherm does not exceed ±0.3 mN/m (i.e., the corresponding curves are virtually indistinguishable from the isotherm-based curve). For more concentrated solutions (thin colored lines), the deviation does not exceed ±0.7 mN/m. We can conclude therefore that the drop and bubble profile methods are applicable to studying the rheological characteristics of surfactants that possess high surface activity. From the results obtained by these two methods, the equilibrium concentration of the solution in the drop can be assessed, which then can be used to calculate directly the adsorbed amounts18,19 and the rheological parameters.

Figure 3. The same as in Figure 2 for additional C 14 EO 8 concentrations: (a) 0.5, (b) 1.0, (c) 2.0, and (d) 5.0 μmol/L.

Note that whereas viscoelasticity E(Ω) is a dynamic interfacial quantity because it depends on the frequency of deformation Ω, it is directly linked to important equilibrium quantities of the adsorption layer. Modulus E0, for example, is a derivative of surface pressure with respect to relative adsorption, and also the relaxation parameter ζ depends inversely on the derivative of the adsorption isotherm. This demonstrates how sensitive the viscoelasticity is with respect to the chosen adsorption model. The values of the viscoelasticity modulus |E(Ω)| at various C14EO8 concentrations in the solution around the bubble and D

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(9) Douillard, R.; Lefebvre, J. Adsorption of Proteins at the Gas Liquid Interface: Models for Concentration and Pressure Isotherms. J. Colloid Interface Sci. 1990, 139, 488−499. (10) Stoyanov, S. D.; Rehage, H.; Paunov, V. N. A New Class of Interfacial Tension Isotherms for Nonionic Surfactants Based on Local Self-Consistent Mean Field Theory: Classical Isotherms Revisited. Phys. Chem. Chem. Phys. 2004, 6, 596−603. (11) Frumkin, A. N. Electrocapillary Curve of Higher Aliphatic Acids and the State Equation of the Surface Layer. Z. Phys. Chem. (Leipzig) 1925, 116, 466−484. (12) Rusanov, A. I. New Theory of Equation of State for Surface Monolayer. J. Chem. Phys. 2004, 120, 10736−10747. (13) Lucassen-Reynders, E. H. Surface Equation of State for Ionized Surfactants. J. Phys. Chem. 1966, 70, 1777−1785. (14) Borwankar, R. P.; Wasan, D. T. Equilibrium and Dynamics of Adsorption of Surfactants at Fluid-Fluid Interface. Chem. Eng. Sci. 1988, 43, 1323−1337. (15) Scheutjens, J. M. H. M.; Fleer, G. J. Statistical Theory of the Adsorption of Interacting Chain Molecules 1. Partition Function; Segment Density Distribution; and Adsorption Isotherms. J. Phys. Chem. 1979, 83, 1619−1634. (16) Fainerman, V. B.; Miller, R.; Wüstneck, R.; Makievski, A. V. Adsorption Isotherm and Surface Tension Equation for a Surfactant with Changing Partial Molar Area. 1. Ideal Surface Layer. J. Phys. Chem. 1996, 100, 7669−7675. (17) Fainerman, V. B.; Zholob, S. A.; Lucassen-Reynders, E. H.; Miller, R. Comparison of Various Models Describing the Adsorption of Surfactant Molecules Capable of Interfacial Reorientation. J. Colloid Interface Sci. 2003, 261, 180−183. (18) Fainerman, V. B.; Zholob, S. A.; Petkov, J. T.; Miller, R. C14EO8 Adsorption Characteristics Studied by Drop and Bubble Profile Tensiometry. Colloids Surf., A 2008, 323, 56−62. (19) Fainerman, V. B.; Lylyk, S. V.; Aksenenko, E. V.; Makievski, A. V.; Petkov, J. T.; Yorke, J.; Miller, R. Adsorption Layer Characteristics of Triton Surfactants 1. Surface Tension and Adsorption Isotherms. Colloids Surf., A 2009, 334, 1−7. (20) Valenzuela, M. Á .; Gárate, M. P.; Olea, A. F. Surface Activity of Alcohols Ethoxylates at the n-Heptane/Water Interface. Colloids Surf., A 2007, 307, 28−34. (21) Ritacco, H. A.; Busch, J. Dynamic Surface Tension of Polyelectrolyte/Surfactant Systems with Opposite Charges: Two States for the Surfactant at the Interface. Langmuir 2004, 20, 3648− 3656. (22) Lee, Y.-C.; Liu, H.-S.; Lin, S.-Y. Adsorption Kinetics of C10E4 at the Air−Water Interface: Consider Molecular Interaction or Reorientation. Colloids Surf., A 2003, 212, 123−134. (23) Kovalchuk, V. I.; Miller, R.; Fainerman, V. B.; Loglio, G. Dilational Rheology of Adsorbed Surfactant Layers  Role of the Intrinsic Two-Dimensional Compressibility. Adv. Colloid Interface Sci. 2005, 114−115, 303−313. (24) Fainerman, V. B.; Aksenenko, E. V.; Petkov, J. T.; Miller, R. Adsorption Layer Characteristics of Mixed Oxyethylated Surfactant Solutions. J. Phys. Chem. B 2010, 114, 4503−4508. (25) Lucassen, J.; van den Tempel, M. Dynamic Measurements of Dilational Properties of a Liquid Interface. Chem. Eng. Sci. 1972, 27, 1283−1291. (26) Lucassen, J.; Hansen, R. S. Damping of Waves on MonolayerCovered Surfaces: II. Influence of Bulk-to-Surface Diffusional Interchange on Ripple Characteristics. J. Colloid Interface Sci. 1967, 23, 319−328. (27) Lucassen-Reynders, E. H.; Cagna, A.; Lucassen, J. Gibbs Elasticity, Surface Dilational Modulus and Diffusional Relaxation in Nonionic Surfactant Monolayers. Colloids Surf., A 2001, 186, 63−72.

CONCLUSIONS The drop and bubble profile methods are used to study the dependence of the viscoelasticity modulus of aqueous C14EO8 solutions on the frequency of oscillating drops and bubbles, respectively, within a broad concentration range. To determine the equilibrium surfactant concentration in the drop bulk, a correction is introduced for the surfactant losses caused by its adsorption at the drop surface. This correction is based on the results of bubble profile analysis tensiometry experiments performed in the presence of a large solution reservoir (i.e., when the adsorption-related losses are negligibly small). It is shown that, with this correction performed, the frequency dependencies of the viscoelasticity modulus measured by either of the two experimental techniques are almost identical. For the analysis of the experimental data, a theoretical model is used that describes the surfactant dilational rheology assuming a diffusion-controlled adsorption.4,19 The complete set of experimental data for C14EO8 solutions was described by the reorientation model that assumes two states of the adsorbed surfactant molecules with different molar areas in the surface layer and an intrinsic compressibility of the molecule in the state of minimal molar area. It is shown that the results obtained via this model are in good agreement with the experimental data for the viscoelasticity modulus. In contrast, by using just the classical model without taking into consideration the limited volume of a single drop, up to a 1 order of magnitude shift on the frequency axis would result, as demonstrated in Figure 3.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was financially supported by the European Space Agency (PASTA) and the Deutsche Forschungsgemeinschaft SPP 1506 (Mi418/18-2). This work was performed in the framework of COST actions CM1101 and MP1106.

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REFERENCES

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