Surface Tension of an Anionic Surfactant: Equilibrium, Dynamics, and

Jun 9, 2001 - The equilibrium and dynamic surface tension of sodium bis(2-ethylhexyl) sulfosuccinate (Aerosol-OT) are studied as a function of concent...
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Langmuir 2001, 17, 4287-4296

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Surface Tension of an Anionic Surfactant: Equilibrium, Dynamics, and Analysis for Aerosol-OT Sammy S. Datwani† and Kathleen J. Stebe*,†,‡ Departments of Chemical Engineering and of Materials Science and Engineering, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2689 Received November 30, 2000. In Final Form: April 10, 2001 The equilibrium and dynamic surface tension of sodium bis(2-ethylhexyl) sulfosuccinate (Aerosol-OT) are studied as a function of concentration and ionic strength controlled by the addition of either the monovalent salt sodium chloride or the divalent salt calcium chloride. These data are compared to a surfactant mass-transfer model with a quasi-equilibrium treatment of the electrostatics. The Davies adsorption isotherm and surface equation of state relate the bulk concentration, surface concentration, and surface tension. At equilibrium, the surface concentration increases with the ionic strength of the electrolyte, so the surface tension reduces more strongly. The data at all ionic strengths are well described by the Davies model. Because the characteristic diffusion time scale increases as the square of the surface concentration, an increasing equilibration time with ionic strength might be anticipated for this molecule. However, the time required for the surface tension relaxation observed in experiment is fairly insensitive to changes in the ionic strength over the range of surfactant concentrations studied for both monovalent and divalent electrolytes at fixed surfactant bulk concentration. When these data are compared to a full integration of the surfactant transport equations, they are found to agree with a diffusion-controlled mass-transfer mechanism. The key issue behind the apparently contradictory behavior of increased adsorption resulting in lower equilibrium surface tensions, while diffusion time scales remain essentially unchanged, is the high surface activity of Aerosol-OT. Even at the most dilute concentrations studied, Aerosol-OT adsorbs close to its maximum packing limit. The surface concentration increases weakly near this value with ionic strength. Therefore, the diffusion time scale also changes weakly. Concomitantly, the equilibrium surface tension changes strongly because it is highly sensitive to small changes in surface concentration near this limit.

1. Introduction In this paper, the surface tension evolution caused by the adsorption of the ionic surfactant sodium bis(2ethylhexyl) sulfosuccinate (Aerosol-OT) is studied using the pendant bubble technique. These equilibrium data are compared to the Davies adsorption isotherm and equation of state which relate the surface concentration, bulk concentration, and surface tension. The dynamic data are compared to a diffusion-controlled surfactant masstransfer argument with a quasi-equilibrium model for electrostatics in which the surface charge density evolves as surfactant adsorbs to the interface. There have been prior studies in which the dynamic surface tension for ionic surfactants has been compared to models which account for the effects of electrostatics in the surfactant mass transfer to the interface outside of asymptotic analyses restricted to short or long times. For example, both Borwankar and Wasan1 and Chang and Franses2 compared models for surfactant diffusion and adsorption-desorption kinetics to the experimental data of Fainerman3 obtained by the maximum bubble pressure method for sodium dodecyl sulfate (SDS). Borwankar and Wasan used a quasi-equilibrium model for the electrostatics with a Gouy-Chapman solution to relate the surface charge density to the surface potential. The Davies model was used to relate the surface tension to the instantaneous surface concentration. Chang and Franses * To whom correspondence should be addressed. E-mail: [email protected]. Phone: (410) 516-7769. Fax: (410) 516-5510. † Department of Chemical Engineering. ‡ Department of Materials Science and Engineering. (1) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1986, 41, 199. (2) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (3) Fainerman, V. B. Colloid J. USSR 1978, 40, 769.

used a linear argument for electrostatics, assuming that a repulsive barrier develops which is proportional to the surface concentration. In their model, this barrier weights the rates of both adsorption and desorption, so the effects of electrostatics do not alter the equilibrium state. These latter authors also compared their model to data for Aerosol-OT obtained by Hua and Rosen4 by the maximum bubble pressure method. Both studies conclude that the surfactants exhibit mixed kinetic-diffusion-controlled mass transfer to the interface. Fainerman considered the adsorption of sodium alkyl sulfates5 and SDS3 by the maximum bubble pressure method to be purely kinetically controlled. While the maximum bubble pressure technique allows surface tension data to be obtained at early times after surface creation, there is a leading order dilatation of the interface throughout the measurement that can dilute the interface and create convective fluxes which are not accounted for in the mass-transfer analysis. In another paper from the Franses group (Pinazo et al.6), their kinetic model is compared to data for an aminobetaine surfactant obtained from pendant bubbles under pulsating and constant-area conditions; the authors conclude that the surfactant transport exhibits mixed kinetic-diffusion control. In Bonfillon et al.,7 dynamic surface tension data of SDS solutions obtained by pendant drop data at aqueous-dodecane interfaces are compared to a model for diffusion and adsorption which accounts for the time evolving surface charge density and assumes that the fluid surrounding the bubble is at rest, an (4) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1988, 124, 652. (5) Fainerman, V. B. Colloids Surf. 1991, 57, 249. (6) Pinazo, A.; Chang, C. H.; Franses, E. I. Colloid Polym. Sci. 1994, 272, 447. (7) Bonfillon, A.; Sicoli, F.; Langevin, D. J. Colloid Interface Sci. 1994, 168, 497.

10.1021/la001676a CCC: $20.00 © 2001 American Chemical Society Published on Web 06/09/2001

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Figure 1. Chemical structure of the anionic surfactant AerosolOT.

assumption which is closely approximated in the pendant bubble and pendant drop techniques. In this paper, the Davies model8 is adopted to describe the equilibrium behavior of this system, with a GouyChapman treatment for the electrostatics. This model predicts a variation in equilibrium adsorption of an ionic surfactant as a function of ionic strength. As the ionic strength increases, the amount of ionic species adsorbed also increases as electrostatic repulsion is screened. The manner in which these changes in equilibrium adsorption alter the dynamic surface tension is studied using the quasi-equilibrium approach delineated by MacLeod and Radke,9 who compared the solution to the Nernst-Planck diffusion migration equation to the quasi-equilibrium model, and noted that the two solutions superpose for times accessible by the pendant bubble method. This model is compared to data taken by the pendant bubble technique for Aerosol-OT over times spanning hundredths of seconds to hours. Other models that have been adopted to study the surface tension of ionic solutions are reviewed elsewhere.10 Aerosol-OT is known to hydrolyze to form the product 2-ethyl-1-hexanol; since this product is less surface active than its parent molecule, and is present at very low concentrations, it plays a negligible role in determining the dynamic or equilibrium properties of this system. This is confirmed in control studies in which 2-ethyl-1-hexanol is deliberately added in excess. The equilibrium isotherm for the surface tension as a function of bulk concentration is obtained for two salts, sodium chloride and calcium chloride. Both are described well by the Davies equation; the inverse maximum packing fitted to the data agrees very closely with that reported in neutron reflectivity measurements.11,12 The characteristic time scale for diffusion-controlled adsorption is given by

τD ) h2/D; h ) Γeq/C1∞

(1)

where D is the surfactant diffusivity. In this expression, h is the adsorption depth, the characteristic length scale depleted by surfactant adsorption. This depth is given by the ratio of the surface concentration of surfactant Γeq over the bulk concentration of surfactant C1∞. According to this ratio, if Γeq increases at a given bulk concentration, τD also increases. The Davies equation predicts greater adsorption the higher the ionic strength. Therefore, according to eq 1, the time required for the surface tension to equilibrate should also increase with ionic strength. In (8) Davies, J. T. Proc. R. Soc. London, A 1958, 245, 417. Davies, J. T. Proc. R. Soc. London, A 1958, 245, 429. (9) MacLeod, C. A.; Radke, C. J. Langmuir 1994, 10, 3555. (10) . Datwani, S. S.; Stebe, K. J. J. Colloid Interface Sci. 1999, 219, 282. (11) Li, Z. X.; Lu, J. R.; Thomas, R. K. Langmuir 1997, 13, 3681. (12) Li, Z. X.; Lu, J. R.; Thomas, R. K.; Penfold, J. J. Phys. Chem. B 1997, 101, 1615.

Figure 2. The pendant bubble apparatus consists of a quartz cell (A) filled with surfactant and electrolyte solution aligned perpendicularly to a collimated beam of visible light. A bubble is formed at the tip of an inverted needle immersed in the solution. The light casts a silhouette of the bubble onto a CCD camera (B). Digital images of the bubble silhouette are stored on disk (C).

Figure 3. Dynamic surface tension of a 2-ethyl-1-hexanol/ Aerosol-OT mixture of 4 wt %. The Aerosol-OT concentration was C1∞ ) 5 × 10-8 mol/cm3, and the 2-ethyl-1-hexanol concentration was 1 × 10-8 mol/cm3 at 0.05M NaCl (filled triangles). These data nearly superpose with the dynamic surface tension of Aerosol-OT solution at C1∞ ) 5 × 10-8 mol/ cm3 prepared from AOT as received from Sigma without any deliberately added alcohol (filled circles).

this work, it is shown that the time required for AerosolOT to equilibrate at a given bulk concentration is nearly constant as ionic strength is varied, in apparent contradiction to this argument. However, upon detailed analysis, the mass transfer of Aerosol-OT is found in this work to be diffusion-controlled. The key issue is the very high surface activity of this molecule. The structure of AerosolOT is presented in Figure 1; this molecule contains 20 carbon atoms and has a tendency to adsorb to the interface that is far stronger than the smaller ionic species such as SDS. Even at the lowest concentrations studied, AerosolOT is adsorbed at surface concentrations close to its maximum packing. Small changes about this limit occur as the ionic strength is varied. These changes strongly alter the equilibrium surface tension, but yield only small changes in the equilibration time scale. Using a Davies model and a quasi-equilibrium approach to the surfactant transport generalized for nonequivalent systems,10 these results are confirmed for monovalent and divalent added salts.

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Figure 4. The dynamic surface tensions of mixtures of AerosolOT and EDTA were compared to that in the absence of EDTA to assess the role of trace Ca2+ in our experiments. For concentrations of 5 × 10-9 mol/cm3 Aerosol-OT and (a, top) 0.05 M NaCl + 0.001 M EDTA (filled triangles) and 0.05 M NaCl without EDTA (filled circles) and (b, bottom) 0.5 M NaCl + 0.001 M EDTA (open triangles) and 0.5 M NaCl without EDTA (open squares).

Figure 5. Equilibrium surface tensions as a function of the bulk surfactant concentration for three different ionic strengths of NaCl (500 mM (open circles), 50 mM (filled circles), and 10 mM (plus signs) are compared to those predicted by the Davies isotherm (500 mM (solid curve), 50 mM (dashed curve), and 10 mM (dotted curve) for Γ∞ ) 2.15 × 10-10 mol/cm2 and β/R ) 5.4 × 1011 cm3/mol.

2. Experimental Section 2.1. Apparatus. The pendant bubble apparatus is shown in Figure 2. In this method, a quartz cell is filled with surfactant and electrolyte solution. This cell is aligned perpendicularly to

Figure 6. The dynamic surface tensions of Aerosol-OT as a function of bulk surfactant concentration are presented for fixed concentrations of NaCl: (a, top) 500 mM, (b, middle) 50 mM, (c, bottom) 10 mM. For (a) and (b) the fastest to slowest relaxations correspond to 2 × 10-7, 1 × 10-7, 5 × 10-8, 2 × 10-8, 1 × 10-8, 5 × 10-9, 2 × 10-9, 1 × 10-9, 5 × 10-10, and 2 × 10-10 mol/cm3, respectively. For (c) the fastest to slowest relaxations correspond to 5 × 10-7, 2 × 10-7, 1 × 10-7, 5 × 10-8, 2 × 10-8, 1 × 10-8, 5 × 10-9, 2 × 10-9, 1 × 10-9, and 5 × 10-10 mol/cm3, respectively. a collimated beam of visible light. A bubble is rapidly formed at the tip of a Teflon-coated stainless steel inverted needle which is immersed in the surfactant/electrolyte solution. The collimated beam of light casts a silhouette of the bubble onto a CCD camera; using the camera and a frame grabber, digital images of the bubble silhouette are obtained. The locus of the bubble shape is determined using an edge detection routine and matched to a numerical solution of the Young-Laplace equation to determine the surface tension. For dynamic surface tension experiments, a bubble is formed rapidly and the CCD camera is set to record

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Figure 7. The surface tension evolution for diffusion-controlled mass transfer is compared to experiment for (a, top left) C1∞ ) 1.0 × 10-9 mol/cm3, 0.5 M NaCl, (b, top right) C1∞ ) 1.0 × 10-9 mol/cm3, 0.05 M NaCl, (c, bottom left) C1∞ ) 5.0 × 10-9 mol/cm3, 0.5 M NaCl, and (d, bottom right) C1∞ ) 5.0 × 10-9 mol/cm3, 0.05 M NaCl. The diffusivity D ) 6.5 × 10-6 cm2/s is fixed for all of the curves. images at given time intervals. The surface tension corresponding to each image is determined. The long-time asymptotes of the surface tension profiles give the equilibrium surface tension. 2.2. Materials. Sodium bis(2-ethylhexyl) sulfosuccinate (Aerosol-OT), 99% purity, was obtained from Sigma Chemical Co. (St. Louis, MO). For control experiments, 2-ethyl-1-hexanol was obtained from Aldrich with a reported purity of >99%. Sodium chloride and calcium chloride were also obtained from Sigma, who report a minimum of 99% and 99.6% purity, respectively. Deionized water was obtained from a Millipore Milli-Q 50 purification system (Bedford, MA) which had a resistivity of not less than 18.0 MΩ cm. This water was used for all rinsing and cleaning procedures and to prepare all surfactant and electrolyte solutions. In all surface tension experiments, the chemicals were used as received. 2.3. Analysis for Impurities. To evaluate the purity of the Aerosol-OT, a series of experiments were performed. The AerosolOT was further purified by two standard procedures: ion exchange, and recrystallization with methanol and deionized water. The amount of the dominant trace impurity, 2-ethyl-1hexanol (formed by hydrolysis of either one or both of the two hydrocarbon tails), was determined for the three different samples using gas chromatography (GC). The three samples were each dissolved in ACS reagent grade acetonitrile (0.1-0.2 mg/mL) and injected into the gas chromatograph with splitless injection and FID, with an inlet temperature set at 250 °C. The standard concentration was determined to be 0.1 µg/mL. For the AerosolOT as obtained from Sigma and for the surfactant that was methanol/water purified, the GC results indicate that the samples were 0.4 wt % (2-ethyl-1-hexanol/Aerosol-OT). For the surfactant that was purified via ion exchange, the GC results indicate that the sample was 0.6 wt % (2-ethyl-1-hexanol/Aerosol-OT). 2.4. Control Experiments. Because 2-ethyl-1-hexanol is far less surface active than Aerosol-OT, this impurity weakly affects

the surface tension. To verify that the impurity did not appreciably alter our experimental results, a series of pendant bubble experiments were performed using a mixture of 2-ethyl-1-hexanol/ Aerosol-OT fixed at 4 wt %, an order of magnitude more concentrated than the impurity in our sample. Specifically, the Aerosol-OT concentration was C1∞ ) 5 × 10-8 mol/cm3 and the 2-ethyl-1-hexanol concentration was 1 × 10-8 mol/cm3 at 0.05 M NaCl. The repeated dynamic surface tension traces are shown in Figure 3 in comparison to an Aerosol-OT solution at C1∞ ) 5 × 10-8 mol/cm3 without any deliberately added alcohol. The traces nearly superpose; the equilibrium surface tensions are the same to within the error associated with the experimental technique ((0.1 mN/m). It has been suggested that the behavior of Aerosol-OT or related compounds (i.e., sodium dihexyl sulfosuccinate) at aqueous-air interfaces is influenced by the presence of trace Ca2+ ions in solution. In a comparison of neutron reflectivity results obtained from a specially synthesized sample of deuterated Aerosol-OT, and surface tension studies of commercially acquired AerosolOT by Li et al.,11 the equilibrium surface tension was observed to increase after purification in an ion-exchange resin followed by the addition of EDTA, an agent which sequesters Ca2+. Furthermore, the surface concentration of surfactant solution as a function of bulk concentration inferred from the Gibbs adsorption equation increased and approached the neutron reflectivity result. In studies of sodium dihexyl sulfosuccinate by Eastoe et al.,13 reductions in equilibrium surface tension with the addition of EDTA were reported. To confirm that trace Ca2+ did not play a role in our experiments, all of which were performed in the presence of added salt, 0.001 M EDTA was added to surfactant solutions of both (13) Eastoe, J.; Nave, S.; Downer, A.; Paul, A.; Rankin, A.; Tribe, K.; Penfold, J. Langmuir 2000, 16, 4511.

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Figure 8. Dynamic surface tensions at a fixed surfactant concentration and three ionic strengths are plotted on the same graph: (a, top left) C1∞ ) 1.0 × 10-9 mol/cm3; (b, top right) C1∞ ) 5.0 × 10-9 mol/cm3; (c, bottom left) C1∞ ) 5.0 × 10-8 mol/cm3; (d, bottom right) 1.0 × 10-7 mol/cm3. Plus signs represent 10 mM NaCl, the filled symbols represent 50 mM NaCl, and the open symbols represent 500 mM NaCl. 0.05 and 0.5 M NaCl for an Aerosol-OT concentration of 5 × 10-9 mol/cm3. The dynamic surface tension data are reported in Figure 4. For low ionic strength, there is a weak effect on the surface tension caused by the addition of EDTA (which may be caused by the increase in ionic strength owing to the presence of this ionic species). For high ionic strength, the curves nearly superpose.

3. Results and Discussion 3.1. The Equivalent System: Aerosol-OT and Sodium Chloride. The equilibrium surface tensions as a function of the bulk surfactant concentration for three different ionic strengths of sodium chloride (500, 50, and 10 mM) are reported in Figure 5. These data are compared to the Davies equation:

βC1∞/R Γeq ) Γ∞ exp(z1FΨs/RT) + βC1∞/R

(2)

where z1 is the valence of the surfactant, F is Faraday’s constant, and Γ∞ and β/R are the maximum packing and the tendency to adsorb at the interface, respectively. These latter two parameters are fit to the data. The surface potential Ψs is related to the surface charge density σ by the Gouy-Chapman relationship for the equivalent species: 4

Ci∞)1/2 sinh(zFΨs/2RT) ∑ i)1

σ ) z1FΓeq ) 2(RT

(3)

where C2∞ is the bulk concentration of the surfactant counterion and C3∞ and C4∞ are the bulk concentrations

of the fully disassociated ions from the added salt (where C3∞ corresponds to the co-ion and C4∞ to the counterion). The surface equation of state for the equilibrium surface tension γeq is

(

γeq ) γ0 + RTΓ∞ ln 1 -

)

Γeq Γ∞

[

(

)]

z1FΨs 4RT (2RT(C1∞ + C3∞))1/2 1 - cosh z1F 2RT

(4)

In these expressions,  is the permittivity of water, set to 80 in the simulations. The two fitted parameters are obtained by minimizing the error between these data and the Davies equation. The resulting maximum packing parameter Γ∞ is 2.15 × 10-10mol/cm2, and the resulting value for β/R is 5.40 × 1011 cm3/mol. The area per molecule at maximum surface coverage, estimated from Γ∞-1, corresponds to 77.2 Å2/molecule, which is in agreement with an independent study by Li et al.,11,12 who report a value of 78.0 Å2/molecule using neutron reflectivity on a purified sample of Aerosol-OT. The description of electrostatics in the Davies model requires no additional fitted parameters, only known electrochemical material parameters. Note that the dependence of the surface tension on ionic strength predicted by this model agrees well with the experimental data at the three ionic strengths reported; that is, the Gouy-Chapman model allows the changes in surface tension with ionic strength to be captured.

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Figure 9. Rescaled dynamic surface tensions θ at a fixed surfactant concentration and three ionic strengths are plotted for the same conditions as in Figure 8.

The coupling among the surface tension, the surface concentration, and the ionic strength of the solution depends on the dimensionless group

are modeled for a spherical bubble. The diffusioncontrolled evolution of the surface concentration on a spherical surface that is initially free of adsorbed surfactant is given by

4

Ci∞) ∑ i)1

κhion ) κ(Γeq/

(5)

Γ(t) )

2D1/2 [C1∞t1/2 π1/2

∫0t

1/2

C1s(t - τ) dτ1/2] + D [C t a 1∞

where the inverse Debye length κ is given by

∫0tC1s dτ]

(7)

4

Ci∞/RT)1/2 ∑ i)i

κ ) (F2

(6)

The ratio in eq 5 decreases with ionic strength. Therefore, the smaller the κhion, the weaker the coupling between the electrostatics and the surfactant behavior. The importance of this ratio is discussed in detail by Datwani and Stebe.10 All results are discussed in dimensional form in this paper for conciseness. In Figure 6 the families of dynamic surface tension curves for Aerosol-OT as a function of bulk surfactant concentration are presented. There are three families of curves, corresponding to 500, 50, and 10 mM sodium chloride, respectively. Different symbols on the nearly superposing curves are runs performed with different solutions, indicating the reproducibility of the experiment. As expected, as C1∞ increases, the time for equilibration decreases. To better understand the controlling masstransfer mechanisms, these data are compared to a model for the diffusion-controlled evolution of the surface tension. Since the pendant bubble shape deviates weakly from a spherical geometry, surfactant diffusion and adsorption

This equation relates the instantaneous adsorption Γ to the instantaneous sublayer concentration C1s. The surface concentration is assumed to evolve in instantaneous equilibrium with the sublayer concentration; thus, the relationships which close these equations are given by eqs 2-4, with the equilibrium values for the surface concentration, bulk concentration, surface potential, and surface tension replaced by the instantaneous values of Γ(t), C1s(t), Ψs(t), and γ(t), respectively. The predicted surface tension evolution is compared to experiment in Figure 7a,b for a fixed bulk surfactant concentration assuming diffusion-controlled adsorption as a function of ionic strength for a monovalent salt, NaCl. The best-fit diffusivity D was determined to be 6.5 × 10-6 cm2/s. Using the same diffusion coefficient for the remaining bulk concentrations of Aerosol-OT, there is adequate agreement between the predicted profiles and the experiment, as shown in Figure 7c,d. The largest discrepancies between the predicted and experimental surface tensions occur during the induction time, which is evident in the experimental traces. Induction times have been established to result from intermolecular attrac-

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tions14-16 which occur between the hydrocarbon moieties. Since only electrostatic interactions have been included in our analysis, the induction time does not appear in the model curves. When the dynamic surface tension data for the three ionic strengths at a fixed bulk concentration are plotted on the same graph as in Figure 8, it is apparent that the time scale for surface tension relaxation does not vary with ionic strength. This is so for bulk concentrations spanning from 1.0 × 10-9 to 1.0 × 10-7 mol/cm3. A similar insensitivity in equilibration times for Aerosol-OT was reported by Hua and Rosen4 in their maximum bubble pressure experiments. To focus only on the time scale for surface tension relaxation, rather than the change in equilibrium with ionic strength, the data in Figure 8 are rescaled in Figure 9 as

θ(t) )

γ(t) - γeq γ0 - γeq

(8)

In Figure 9, note that time is reported in seconds; that is, it is not rescaled. These curves all equilibrate at nearly the same time regardless of ionic strength. 3.2. A Characteristic Time Scale for DiffusionControlled Adsorption. There is a characteristic time scale for diffusion-controlled adsorption which can be derived from a dimensional analysis of eq 7. Its physical significance is discussed here. Surfactant immediately adjacent to the interface adsorbs, diluting the region adjacent to the interface and setting up a diffusive flux from the bulk toward the interface. The greater the depth depleted by surfactant adsorption, the longer the time required for diffusion to replenish the interface with surfactant. The characteristic depleted depth is termed the adsorption depth h; this length scale is determined by the ability of the surfactant to adsorb at the interface. Performing a mass balance on an area element, dA of the fluid interface, the adsorbed mass on an area element dA at equilibrium is given by Γeq dA. The total mass of surfactant in the volume spanning the depth h beneath this area element is given by C1∞h dA. Equating the mass of surfactant in the area element and the mass of surfactant in the volume element and solving, h is found to be given by the ratio in eq 1. This length scale determines the diffusion time scale, τD, also defined in eq 1. To calculate h and τD, the isotherm parameters Γ∞ and β/R and the surfactant diffusivity must be known. Adopting the values found for Aerosol-OT, the predicted Γeq, h, and τD are reported in Figure 10. In Figure 10a, Γeq increases weakly with C1∞ for the three ionic strengths studied in the experiment. Since Aerosol-OT is so strongly surface active, the surface concentration is comparable to its maximum packing even at the most dilute concentrations in the range studied. Because of this pronounced adsorption, Γeq is only weakly dependent on ionic strength. This dependence is weaker the greater the bulk concentration. For example, the Davies model predicts an increase in Γeq as ionic strength increases from 10 to 500 mM sodium chloride of approximately 15% at a C1∞ of 1.0 × 10-8 mol/ cm3, but only 3% for a C1∞ of 1.0 × 10-7 mol/cm3. In Figure 10b, h versus C1∞ is reported; as the ionic strength increases at fixed C1∞, h increases weakly as Γeq. However, since Γeq approaches Γ∞, h asymptotes to Γ∞/C1∞ (14) Lin, S.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (15) Ferri, J.; Stebe, K. J. J. Colloid Interface Sci. 1999, 209, 1. (16) Pollard, M. L.; Pan, R.; Steiner, C.; Maldarelli, C. Langmuir 1998, 14, 7222.

Figure 10. The Davies isotherm is used to predict (a, top) the equilibrium adsorption Γeq versus C1∞, (b, middle) the adsorption depth h versus C1∞, and (c, bottom) the diffusion time scale equilibrium τD versus C1∞ for NaCl concentrations of 500 mM (solid curve), 50 mM (dashed curve), and 10 mM (dotted curve).

with increasing concentration. In this range, h becomes insensitive to ionic strength; for bulk surfactant concentrations exceeding roughly 1.0 × 10-8 mol/cm3, the graphs of h versus C1∞ for the three ionic strengths converge. In Figure 10c, τD versus C1∞ is shown; because τD goes as h2, τD converges for the three ionic strengths at lower concentrations than did the curves for h. The predicted equilibration times superpose to within seconds for concentrations exceeding C1∞ ) 1.0 × 10-8 mol/cm3. Thus, the observed insensitivity of the equilibration times for Aerosol-OT with changing ionic strength is consistent with a diffusion-controlled mechanism, and can be explained

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Figure 11. The predicted equilibrium surface tension as a function of the surface concentration is plotted for NaCl concentrations of 500, 50, and 10 mM. The symbols indicate the changes in equilibrium tension with ionic strength for a fixed Aerosol-OT concentration of C1∞ ) 1.0 × 10-8 mol/cm3. Details are given in the text.

Figure 12. Equilibrium surface tension as a function of the bulk surfactant concentration for two ionic strengths of CaCl2 (16.7 and 3.33 mM). The solid curve indicates the best-fit Davies isotherm with β/R ) 1.10 × 1012 cm3/mol for 16.7 mM, and the dashed curve represents 3.33 mM CaCl2. The best-fit Γ∞ remains unchanged from the monovalent salt case.

in terms of the characteristic time scale for diffusioncontrolled adsorption. This insensitivity is a direct result of the high surface activity of this molecule, which causes it to adsorb close to its maximum packing even when electrostatic repulsion is appreciable. (This near insensitivity of the surface concentration is also apparent in the neutron reflectivity data for the molecule, obtained by Li et al.11 for concentration ranges from 1 × 10-8 to 5 × 10-6 mol/cm3. Over this concentration range, the surface concentration changes weakly, from 1.38 × 10-10 to 2.15 × 10-10 mol/cm2.) While the time required for surface tension to equilibrate changes negligibly with ionic strength, the equilibrium surface tension realized changes strongly. This is because γeq changes steeply for small changes in Γeq near Γ∞. This is shown for the Davies equation of state in Figure 11 for the constants which describe the Aerosol-OT data. A steep drop in the surface tension occurs as Γeq approaches Γ∞. For example, for the case where C1∞ ) 1.0 × 10-8 mol/cm3, the equilibrium surface concentration and resulting surface tension predicted for Aerosol-OT are indicated by the filled circles in the figure. The surface concentration and surface tension change from Γeq ) 0.86Γ∞ and γeq ) 54 mN/m at 10 mM NaCl, to Γeq ) 0.95Γ∞ and γeq ) 47

Figure 13. A family of dynamic surface tension curves of Aerosol-OT for various bulk surfactant concentrations are presented for CaCl2 concentrations of (a, top) 16.7 mM and (b, bottom) 3.33 mM. For (a) the fastest to slowest relaxations correspond to 2 × 10-7, 1 × 10-7, 5 × 10-8, 2 × 10-8, 1 × 10-8, 5 × 10-9, 1 × 10-9, 5 × 10-10, 2 × 10-10, and 1 × 10-10 mol/cm3, respectively. For (b) the fastest to slowest relaxations correspond to 5 × 10-7, 2 × 10-7, 1 × 10-7, 5 × 10-8, 2 × 10-8, 1 × 10-8, 5 × 10-9, 2 × 10-9, 1 × 10-9, 5 × 10-10, 2 × 10-10, and 1 × 10-10 mol/cm3, respectively.

mN/m at 50 mM NaCl, and finally to Γeq ) 0.99Γ∞ and γeq ) 37 mN/m at 500 mM NaCl, respectively. 3.3. The Nonequivalent System: Aerosol-OT and Calcium Chloride. Equilibrium surface tension data as a function of the bulk surfactant concentration for two ionic strengths of calcium chloride (3.33 and 16.7 mM) are presented in Figure 12. As in the monovalent case, when the ionic strength is increased, the electrostatic repulsion is screened; more surfactant adsorbs, thus reducing the surface tension. These data are compared to the Davies adsorption isotherm (2), with the surface tension and the Gouy-Chapman model extended to treat nonequivalent species. The equilibrium surface tension is given by

∫0Ψ ∇Ψ dΨ]

(9)

Ci∞[exp(-ziFΨs/RT) - 1]}1/2 ∑ i)1

(10)

γeq ) γ0 + RTΓ∞[ln(1 - Γeq/Γ∞) - 

s

where 4

σ ) -2(RT)1/2{

The sign of the right-hand side of eq 10 is chosen so that the surface charge density repels like charged species. In

Surface Tension of an Anionic Surfactant

Figure 14. Dynamic surface tension traces are presented for a fixed ionic strength for different Aerosol-OT concentrations: (a, top) C1∞ ) 1.0 × 10-9 mol/cm3, CaCl2 concentration of 16.7 mM (filled triangles); (b, bottom) C1∞ ) 5.0 × 10-9 mol/cm3, CaCl2 concentration of 16.7 mM (filled triangles). The solid lines indicate the predicted surface tension evolution for diffusion control.

eq 9, the electrostatic contribution to the surface tension is determined numerically. This model agrees well with the equilibrium experimental data providing β/R is refit to this data set; for these data β/R ) 1.10 × 1012 cm3/mol. The best-fit Γ∞ remains unchanged from the monovalent salt case. As in the monovalent salt case, the Gouy-Chapman model for electrostatics accurately captures the change in equilibrium surface tension with ionic strength for a given bulk surfactant concentration. A family of dynamic surface tension traces for AerosolOT are shown in Figure 13 for two concentrations of CaCl2. As in the monovalent salt case, the equilibration time scales decrease as the bulk concentration of surfactant increases. These data are compared favorably to the surface tension evolution predicted for diffusion-controlled mass transfer, as shown in Figure 14. Therefore, even in the presence of divalent salts, diffusion is the controlling mechanism for Aerosol-OT adsorption onto freshly formed aqueous-air interfaces. (In this analysis, eqs 2, 7, 9, and 10 are solved simultaneously for the equilibrium values for the surface concentration, bulk concentration, surface potential, and surface tension replaced by the instantaneous values of Γ(t), C1s(t), Ψs(t), and γ(t), respectively.) The equilibration time scale is insensitive to changes in ionic strength, as shown in Figure 15. The integration of the full surfactant transport equations predicts a near superposition of time scales. Even in the presence of a

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Figure 15. Dynamic surface tensions at a fixed surfactant concentration and two ionic strengths are plotted on the same graph: (a, top) C1∞ ) 5.0 × 10-9 mol/cm3; (b) C1∞ ) 5.0 × 10-8 mol/cm3. Filled triangles represent 3.3 mM CaCl2, and open triangles represent 16.7 mM CaCl2.

divalent salt, the high surface activity of this molecule dictates that the surface concentration is near its maximum packing even at dilute concentrations and low ionic strengths. An analysis of the adsorption isotherm and τD for the divalent salt case yields results similar to those of the monovalent salt case discussed above. These results are not repeated here for conciseness. 3.4. Comparison to Predictions in Prior Work. The insensitivity of the equilibration time scale for AerosolOT to changes in ionic strength is not universal behavior for all ionic surfactants undergoing diffusion-controlled adsorption to the interface. For example, the predicted surface tension evolution for the ionic surfactant SDS has been studied. Assuming diffusion-controlled adsorption for this molecule, the equilibration time is predicted to increase as the ionic strength increases, owing to screening of electrostatic repulsion and a pronounced increase in the surface concentration.9,10 The differences in the dependence of the equilibration time on ionic strength can be attributed to the differing surface activities of these molecules. The tendency to partition to the interface for SDS, β/R, was found by Fainerman3 to be 4.1 × 108 cm3/mol. The concentration range over which SDS just reduces the surface tension up to the cmc spans from 1 × 10-8 to roughly 1 × 10-6 mol/ cm3. Consider the Davies adsorption isotherm, recast in terms of the adsorption number k:

Γeq k ) Γ∞ exp(z1FΨs/RT) + k

(11)

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where k is given by

k ) βC1∞/R

(12)

For SDS, the adsorption number spans the values 4.1 < k < 500 over this concentration range. For k small enough, changes in electrostatic repulsion in eq 11 yield pronounced changes in Γeq and, hence, strong changes in τD. For k . 1, electrostatics become less important. In contrast, for Aerosol-OT, the concentration range of interest spanned from 1 × 10-9 to roughly 1 × 10-7 mol/ cm3, for which the adsorption number spans the values 5.4 × 102 < k < 5.4 × 104. Since k . 1 for this entire range, changes in electrostatic repulsion in eq 11 change Γeq and hence τD weakly. 4. Conclusions The equilibrium and dynamic surface tension of AerosolOT were obtained in the presence of NaCl and CaCl2. As ionic strength increased, the surface tension decreased in a manner which agreed well with the predictions of the Davies isotherm and a Gouy-Chapman model extended to treat nonequivalent species. The surface tension evolution agrees well with diffusion-controlled adsorption. Because of the large number of carbons in this molecule, Aerosol-OT is highly surface active at all ionic strengths, establishing a surface concentration close to its maximum packing even at dilute bulk concentrations. At a given bulk concentration, the amount of surfactant adsorbed increases only weakly with ionic strength. Since the diffusion time scale goes as the square of the surface concentration over the bulk concentration, the time scale required for the surface tension to equilibrate is nearly insensitive to changes in ionic strength. This behavior is directly attributable to the strong surface activity of this molecule. Nomenclature Ci∞

concentration of component i, in the bulk phase

Ci C1s D F h hion i k R t T zi R β Γ Γ∞ Γeq γ γel γeq γ0  θ κ σ τD Ψ

concentration of component i sublayer concentration of component 1 surfactant bulk diffusion coefficient Faraday’s constant adsorption depth ion depletion depth component i adsorption number gas constant time temperature valence of species i kinetic constant for adsorption kinetic constant for desorption surface concentration maximum surface concentration equilibrium surface concentration surface tension electrical contribution to the surface tension equilibrium surface tension initial (clean) surface tension dielectric permittivity dimensionless surface tension reduction inverse Debye length surface charge density diffusion time scale electrostatic potential

Acknowledgment. We acknowledge the contributions of Dr. Greg Dado and Air Products and Chemicals, Inc., for purifying the commercially obtained Aerosol-OT, Dr. Yingru Zhang and American Cyanamid Corp. for the GC measurements, and Dr. James Ferri (Lafayette College) for his work in obtaining some of the pendant bubble data. LA001676A