Possible Existence of Convective Currents in Surfactant Bulk Solution

Jan 7, 2009 - In this Article, we analyze the experimental pendant-bubble DST data of the alkyl ... Dynamic surface tension (DST) measurements of surf...
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Langmuir 2009, 25, 1434-1444

Possible Existence of Convective Currents in Surfactant Bulk Solution in Experimental Pendant-Bubble Dynamic Surface Tension Measurements Srinivas Nageswaran Moorkanikkara and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed August 6, 2008. ReVised Manuscript ReceiVed October 16, 2008 Traditionally, surfactant bulk solutions in which dynamic surface tension (DST) measurements are conducted using the pendant-bubble apparatus are assumed to be quiescent. Consequently, the transport of surfactant molecules in the bulk solution is often modeled as being purely diffusive when analyzing the experimental pendant-bubble DST data. In this Article, we analyze the experimental pendant-bubble DST data of the alkyl poly (ethylene oxide) nonionic surfactants, C12E4 and C12E6, and demonstrate that both surfactants exhibit “superdiffusive” adsorption kinetics behavior with characteristics that challenge the traditional assumption of a quiescent surfactant bulk solution. In other words, the observed superdiffusive adsorption behavior points to the possible existence of convection currents in the surfactant bulk solution. The analysis presented here involves the following steps: (1) constructing an adsorption kinetics model that corresponds to the fastest rate at which surfactant molecules adsorb onto the actual pendant-bubble surface from a quiescent solution, (2) predicting the DST behaviors of C12E4 and C12E6 at several surfactant bulk solution concentrations using the model constructed in step 1, and (3) comparing the predicted DST profiles with the experimental DST profiles. This comparison reveals systematic deviations for both C12E4 and C12E6 with the following characteristics: (a) the experimental DST profiles exhibit adsorption kinetics behavior, which is faster than the predicted fastest rate of surfactant adsorption from a quiescent surfactant bulk solution at time scales greater than 100 s, and (b) the experimental DST profiles and the predicted DST behaviors approach the same equilibrium surface tension values. Characteristic (b) indicates that the cause of the observed systematic deviations may be associated with the adsorption kinetics mechanism adopted in the model used rather than with the equilibrium behavior. Characteristic (a) indicates that the actual surfactant bulk solution in which the DST measurement was conducted, most likely, cannot be considered to be quiescent at time scales greater than 100 s. Accordingly, the observed superdiffusive adsorption behavior is interpreted as resulting from convection currents present in a nonquiescent surfactant bulk solution. Convection currents accelerate the surfactant adsorption process by increasing the rate of surfactant transport in the bulk solution. The systematic nature of the deviations observed between the predicted DST profiles and the experimental DST behavior for C12E4 and C12E6 suggests that the nonquiescent nature of the surfactant bulk solution may be intrinsic to the experimental pendant-bubble DST measurement approach. To validate this possibility, we identified generic features in the experimental DST data when DST measurements are conducted in a nonquiescent surfactant bulk solution, and the DST measurements are analyzed assuming that the surfactant bulk solution is quiescent. An examination of the DST literature reveals that these identified generic features are quite general and are observed in the experimental DST data of several other surfactants (decanol, nonanol, C10E8, C14E8, C12E8, and C10E4) measured using the pendantbubble apparatus.

1. Introduction Dynamic surface tension (DST) measurements of surfactant solutions using the pendant-bubble apparatus are often carried out to study the adsorption kinetics behavior of surfactants.1 In a typical pendant-bubble experiment, an air bubble is created at the tip of an inverted needle that is immersed in a quartz cell filled with the surfactant solution of interest. As surfactant molecules adsorb from the bulk solution onto the freshly formed bubble surface, the surface tension of the bubble decreases as a function of time. Digital images of the bubble profile are taken at regular time intervals, and the instantaneous surface tension is calculated by numerically matching the solution of the Young-Laplace equation to the measured bubble profile. A detailed description of the manner in which DST measurements are conducted using the pendant-bubble apparatus can be found in ref 2. Experimental pendant-bubble DST data have been used to investigate the adsorption kinetics behavior of several alkyl * To whom correspondence should be addressed. Tel.: (617) 253-4594. Fax: (617) 252-1651. E-mail: [email protected]. (1) Eastoe, J.; Dalton, J. S. AdV. Colloid Interface Sci. 2000, 85, 103–144.

poly(ethylene) oxide, CiEj, nonionic surfactants,2-11 alcohols,12-16 trisiloxane surfactants,17 and gemini surfactants.18 Specifically, the experimental pendant-bubble DST data is used (2) Lin, S.-Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785–1795. (3) Lee, Y.-C.; Liu, H.-S.; Lin, S.-Y. Colloids Surf., A 2003, 212, 123–134. (4) Lee, Y.-C.; Stebe, K. J.; Liu, H.-S.; Lin, S.-Y. Colloids Surf., A 2003, 220, 139–150. (5) Chang, H.-C.; Hsu, C.-T.; Lin, S.-Y. Langmuir 1998, 14, 2476–2484. (6) Pan, R.; Green, J.; Maldarelli, C. J. Colloid Interface Sci. 1998, 205, 213– 230. (7) Hsu, C.-T.; Shao, M.-J.; Lin, S.-Y. Langmuir 2000, 16, 3187–3194. (8) Lin, S.-Y.; Tsay, R.-Y.; Lin, L.-W.; Chen, S.-I. Langmuir 1996, 12, 6530– 6536. (9) Lin, S.-Y.; Lee, Y.-C.; Shao, M.-J.; Hsu, C.-T. J. Colloid Interface Sci. 2001, 244, 372–376. (10) Lee, Y.-C.; Lin, S.-Y.; Shao, M.-J. J. Chin. Inst. Chem. Eng. 2002, 33, 631–643. (11) Subramanyam, R.; Maldarelli, C. J. Colloid Interface Sci. 2002, 253, 377–392. (12) Lee, Y.-C.; Liou, Y.-B.; Miller, R.; Liu, H.-S.; Lin, S.-Y. Langmuir 2002, 18, 2686–2692. (13) Lin, S.-Y.; Lu, T.-L.; Hwang, W.-B. Langmuir 1995, 11, 555–562. (14) Lin, S.-Y.; Hwang, W.-B.; Lu, T.-L. Colloids Surf., A 1996, 114, 143– 153. (15) Lin, S.-Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055–1066. (16) Tsay, R.-Y.; Wu, T.-F.; Lin, S.-Y. J. Phys. Chem. B 2004, 108, 18623– 18629.

10.1021/la802555p CCC: $40.75  2009 American Chemical Society Published on Web 01/07/2009

Pendant-Bubble DST Measurements

in conjunction with a known model of the equilibrium adsorption behavior of the surfactant (specifically, the equilibrium surface tension vs bulk solution concentration - ESTC behavior) to understand the underlying adsorption kinetics mechanism and to determine the values of the associated kinetics parameters. Note that one of the key assumptions made in the traditional analysis of experimental pendant-bubble DST data is that the surfactant bulk solution is quiescent. Accordingly, the transport of surfactant molecules in the bulk solution is assumed to occur purely via diffusion.2-18 Specifically, the experimental DST behavior is rationalized in the context of either (1) a diffusioncontrolled adsorption model, which assumes that only diffusion of surfactant molecules in the bulk solution controls the overall rate of surfactant adsorption, or (2) a mixed diffusion-barrier controlled adsorption model,19 which assumes that both diffusion of surfactant molecules in the bulk solution and adsorption of surfactant molecules from the subsurface20 onto the surface control the overall rate of surfactant adsorption. In this Article, we present an analysis of the experimental pendant-bubble DST data of C12E4 and C12E6, originally published in refs 7 and 10, respectively, which reveals “superdiffusive” kinetics adsorption behavior of these two nonionic surfactants and challenges the traditional assumption of a quiescent surfactant bulk solution in the pendant-bubble experimental setup. Specifically, the analysis presented here involves the following steps: (1) constructing an adsorption kinetics model to represent the fastest rate at which surfactant molecules adsorb onto the actual pendant-bubble surface from a quiescent solution; (2) predicting the DST behavior of C12E4 and C12E6 at several surfactant bulk solution concentration (Cb) values using the model constructed in step 1 (since the model predicts the fastest rate of surfactant adsorption from a quiescent solution, the DST predicted by the model corresponds to the lowest DST attainable using the actual pendant-bubble experimental DST data at every instant of time if surfactant adsorption occurs from a quiescent surfactant bulk solution); and (3) comparing the predicted DST profiles with the corresponding experimental pendant-bubble DST data measured at the Cb values considered in step 2. For both C12E4 and C12E6, a comparison of the DST profiles predicted following steps 1 and 2 above with the experimental pendant-bubble DST data reveals systematic deviations, where the experimental DST values decrease faster than the predicted DST Values at time scales greater than 100 s, but approach the same equilibrium surface tension Value. Approaching the same equilibrium surface tension value indicates that the cause of the observed systematic deviations is most probably not due to the surfactant equilibrium adsorption behavior (ESTC behavior), but instead is most probably associated with the surfactant kinetics mechanism, assumed in the adsorption kinetics model constructed in step 1. In other words, the observed deviations indicate a faster adsorption kinetics mechanism in the experimental pendantbubble DST data relative to the predicted fastest rate of surfactant adsorption from a quiescent solution. This observation suggests that the actual surfactant bulk solutions in which experimental DST measurements were conducted may not be considered to be quiescent at time scales greater than 100 s. More specifically, the observed superdiffusive behavior may result from convective currents present in a nonquiescent surfactant bulk solution. The (17) Kumar, N.; Couzis, A.; Maldarelli, C. J. Colloid Interface Sci. 2003, 267, 272–285. (18) Ferri, J. K.; Stebe, K. J. Colloids Surf., A 1999, 156, 567–577. (19) We shall refer hereafter to the diffusion-controlled adsorption model as the diffusion-controlled model, and to the mixed diffusion-barrier controlled adsorption model as the mixed-controlled model. (20) The sub-surface is the zone of a few angstroms thickness adjacent to the surface.

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systematic nature of the deviations observed between the predicted DST profiles and the experimental DST behavior for C12E4 and C12E6 suggests that the nonquiescent nature of the surfactant bulk solution may, in fact, be intrinsic to the experimental pendantbubble DST measurement approach. To validate this possibility, we identified generic features when DST measurements are conducted in a nonquiescent surfactant bulk solution, and the DST measurements are analyzed assuming that the surfactant bulk solution is quiescent. An examination of the DST literature reveals that the identified generic features are quite general and are observed in the experimental DST data of several other surfactants (decanol, nonanol, C10E8, C14E8, C12E8, and C10E4) measured using the pendant-bubble apparatus. The remainder of this Article is organized as follows. In section 2, we construct an adsorption kinetics model that corresponds to the fastest rate of surfactant adsorption onto the pendantbubble surface from a quiescent solution. In section 3, we (i) predict the DST profiles of C12E4 and C12E6 at several Cb values using the model constructed in section 2, and (ii) compare the predicted DST profiles with the experimental pendant-bubble DST data of these two nonionic surfactants reported in refs 7 and 10. In section 4, we analyze the observed systematic deviations between the predicted DST profiles and the experimentally measured DST behavior and demonstrate that the surfactant bulk solutions used in the pendant-bubble measurements, most likely, are not quiescent. In section 5, we discuss the implications of having a nonquiescent surfactant bulk solution in conducting experimental DST measurements using the pendant-bubble apparatus, and we compile previously reported DST data of several surfactants (decanol, nonanol, C10E8, C14E8, C12E8, and C10E4) where signatures of a nonquiescent surfactant bulk solution are observed in the experimental pendant-bubble DST data. Due to space limitations, in section 5, we only discuss the DST data of C10E8, and we compile the reported DST data of decanol, nonanol, C14E8, C12E8, and C10E4 in the Supporting Information. Finally, in section 6, we summarize the main results of this Article. In Appendix A, we summarize the modeling equations used to describe diffusion-controlled adsorption onto a spherical surface. In Appendix B, we describe the steps followed to identify the equilibrium surface tension versus bulk solution concentration (ESTC) models for C12E4 and C12E6.

2. Fastest Adsorption onto the Pendant-Bubble Surface In this section, we construct an adsorption kinetics model that corresponds to the fastest rate of surfactant adsorption onto the actual pendant-bubble surface from a quiescent solution. Specifically, in section 2.1, we specify the system modeled, in section 2.2, we describe the adsorption kinetics mechanism, and in section 2.3, we discuss the implementation of the model. 2.1. System Specification. Figure 1a shows the typical shape of a pendant bubble.21 Note that the pendant-bubble shape deviates from the energy-minimizing spherical shape due to the action of the buoyant force that elongates the bubble at the tip of the needle. To model the adsorption kinetics onto the pendant-bubble surface, it is traditionally assumed that2-18 (i) the pendant-bubble surface can be approximated to be a spherical surface with a radius equal to the radius of curvature of the pendant bubble at the apex (r0), as shown by the dashed line in Figure 1a, and (ii) the effect of the needle on surfactant adsorption kinetics is negligible. Accordingly, surfactant adsorption onto the actual pendant-bubble surface at the tip of the inverted needle is typically (21) Lin, S.-Y.; Wang, W.-J.; Lin, L.-W.; Chen, L.-J. Colloids Surf., A 1996, 114, 31–39.

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Figure 1. Spherical approximation of a pendant-bubble surface: (a) actual snapshot of a pendant-bubble profile with the radius of curvature at the apex indicated by the dashed line, and (b) spherical approximation of the pendant bubble in (a), where the sphere radius is equal to the radius of curvature of the pendant bubble at the apex.

modeled as adsorption onto a stationary spherical bubble submerged in an infinite solution (see Figure 1b). Recently, a rigorous numerical analysis was carried out to test the validity of assumptions (i) and (ii) above.22 The authors concluded that adsorption onto a spherical bubble surface in an infinite solution leads to faster adsorption relative to that onto the actual pendant-bubble surface, whenever r0 is larger than the radius of the needle (rn).22 The conclusion above can be rationalized as follows: When the radius of the pendant bubble is large relative to that of the needle tip, elongation of the bubble due to buoyancy is significant, and the radius of curvature of the bubble surface attains its minimum value at the apex (see Figure 1a). In this case, by modeling the actual pendant-bubble surface as a spherical surface, one assumes a constant value of the radius of curvature (r0) at all sites on the pendant-bubble surface. For a spherical bubble, it is known that the rate of surfactant adsorption increases as the radius of curvature decreases.23 Consequently, by assuming a constant low value of r0 at all sites on the pendant-bubble surface, the predicted rate of surfactant adsorption yields a higher estimate of the rate of surfactant adsorption onto the actual pendantbubble surface. In the pendant-bubble experiments conducted for C12E4 and C12E6 in refs 7 and 10, the radius of the needle used was rn ) 0.0535 cm, and the minimum radius of curvature of the bubble at the apex r0 was 0.10 cm. Because r0 is greater than rn in this case, according to the results presented in ref 22, surfactant adsorption onto a spherical surface of radius r0 in an infinite solution should be faster relative to that onto the actual pendantbubble surface. With the above in mind, we consider surfactant adsorption onto a stationary spherical bubble surface in an infinite solution as our system when analyzing the pendant-bubble experiments conducted for C12E4 and C12E6 in refs 7 and 10. 2.2. Adsorption Kinetics Mechanism. Traditionally, the surfactant adsorption process is viewed as consisting of the following three steps.1,24-31 Step 1 involves transport of the surfactant molecules from the bulk solution to the subsurface. Typically, the surfactant bulk solution is assumed to be quiescent, and, consequently, surfactant transport in the bulk solution is considered to be purely diffusive. Step 2 involves adsorption of the surfactant molecules from the subsurface onto the surface. (22) Yang, M.-W.; Wei, H.-H.; Lin, S.-Y. Langmuir 2007, 23, 12606–12616. (23) Ferri, J. K.; Lin, S. Y.; Stebe, K. J. J. Colloid Interface Sci. 2001, 241, 154–168.

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Step 3 involves possible reorganization of the surfactant molecules at the surface. In the context of the above mechanistic description, the classical diffusion-controlled model assumes that step 1 controls the overall rate of surfactant adsorption, and that steps 2 and 3 are instantaneous.1 Note that the diffusion-controlled model corresponds to the fastest rate of surfactant adsorption from a quiescent fluid for C12E4 and C12E6. This follows because, if step 2 were not instantaneous, it would effectively reduce the rate of surfactant adsorption onto the surface, which in turn would result in a rate of DST reduction that is slower than that predicted by the diffusion-controlled model. The assumption that step 3 is instantaneous is likely valid for C12E4 and C12E6, because in the molecular dynamics simulations conducted to study the adsorption properties of C12E6 and the structurally similar surfactant C12E5, equilibrium behavior was attained within about 4 ns.32-35 On the other hand, typical time scales associated with the adsorption kinetics of the C12E4 and C12E6 nonionic surfactants considered here vary from seconds to hours (see Figures 2a-d and 3a-d, and also section 3.5), which are several orders of magnitude greater than the time scale associated with molecular reorganization at the surface.32-35 In view of the above, it is reasonable that the diffusioncontrolled adsorption of surfactant molecules onto a stationary spherical bubble in an infinite solution corresponds to the fastest rate of surfactant adsorption onto an actual pendant-bubble in a quiescent solution. 2.3. Model Implementation. Detailed descriptions of the implementation of the diffusion-controlled adsorption model to describe the adsorption kinetics onto a stationary spherical bubble in a quiescent solution, including prediction of the dynamic surface tension (DST) at a given surfactant bulk solution concentration (Cb), are presented in Appendix A. Briefly, predicting the DST at a given Cb using the diffusion-controlled model involves the following four specifications: (1) the surfactant equilibrium adsorption behavior, specifically, the equilibrium surface tension versus bulk solution concentration (ESTC) model; (2) the diffusion coefficient of the surfactant molecule, D; (3) the radius of the spherical bubble, r0; and (4) the initial surfactant surface concentration, Γ0.

3. Predictions of the Diffusion-Controlled Model In this section, we identify the four input specifications (see section 2.3) of the diffusion-controlled model for C12E4 and C12E6 (sections 3.1-3.4) and predict the DST behavior corresponding to the fastest rate of surfactant adsorption onto the actual pendant bubble in a quiescent solution at several Cb values (section 3.5). 3.1. Input Equilibrium Surface Tension versus Bulk Solution Concentration (ESTC) Model. Note that since the subsurface and the surface reach equilibrium instantaneously (24) Noskov, B. A. AdV. Colloid Interface Sci. 1996, 69, 63–129. (25) Ravera, F.; Ferrari, M.; Liggieri, L. AdV. Colloid Interface Sci. 2000, 88, 129–177. (26) Joos, P. Dynamic Surface Phenomena; VSP: The Netherlands, 1999. (27) Defay, R.; Petre, G. Surface and Colloid Science; Wiley: New York, 1971; Vol. 3, pp 27-81. (28) Chang, C.-H.; Franses, E. I. Colloids Surf., A 1995, 100, 1–45. (29) Miller, R.; Joos, P.; Fainerman, V. B. AdV. Colloid Interface Sci. 1994, 49, 249–302. (30) Pillai, V., Shah, D. O., Eds. Dynamic Properties of Interfaces and Association Structures; AOCS Press: Champaign, IL, 1996. (31) Ravera, F.; Liggieri, L.; Miller, R. Colloids Surf., A 2000, 175, 51–60. (32) Kuhn, H.; Rehage, H. Colloid Polym. Sci. 2000, 278, 114–118. (33) Kuhn, H.; Rehage, H. Phys. Chem. Chem. Phys. 2000, 2, 1023–1028. (34) Kuhn, H.; Rehage, H. J. Phys. Chem. B 1999, 103, 8493–8501. (35) Chanda, J.; Bandyopadhyay, S. J. Chem. Theory Comput. 2005, 1, 963– 971.

Pendant-Bubble DST Measurements

Figure 2. Comparison of (a) the fitted polynomial EOS model with the surface-expansion measurements for C12E4, and (b) the fitted ESTC model with the equilibrium surface tension measurements for C12E4. The solid lines correspond to the fitted EOS model (Figure 2a) and ESTC model (Figure 2b) obtained following the approach described in Appendix B; the dashed lines correspond to the fitted Frumkin model (EOS - Figure 2a and ESTC - Figure 2b) reported in ref 7. The “b” in (a) correspond to the experimental surface-expansion measurements, and the “b” in (b) correspond to the experimental equilibrium surface tension measurements.

when the adsorption kinetics is diffusion-controlled, the diffusioncontrolled model uses the ESTC model to relate the instantaneous surfactant subsurface concentration, Cs(t), to the instantaneous surface tension (γ(t)) (see Appendix A). Moreover, the input ESTC model is used along with the Gibbs adsorption equation (eq A.10) to determine (a) the equilibrium adsorption isotherm (eq A.6), and (b) the equation of state (EOS) (eq A.8), used in the diffusion-controlled model. Traditionally, the input ESTC model is identified by (i) choosing an empirical parametrized ESTC model, and (ii) fitting the parameters using experimental equilibrium surface tension measurements. In refs 6, 9, 36-38 the authors used surfaceexpansion measurements, in addition to the equilibrium surface tension measurements, to further validate the accuracy of the fitted ESTC model. With the above in mind, we use both the equilibrium surface tension measurements and the surface-expansion measurements reported in refs 7 and 10 to identify the input ESTC models for C12E4 and C12E6. The steps followed to identify the ESTC models for C12E4 and C12E6 are described in Appendix B. Briefly, these steps include (1) using the surface-expansion measurements to fit a suitable polynomial EOS model, and (2) using the EOS model identified in step 1, along with the equilibrium surface tension measurements, to determine the corresponding ESTC model. (36) Hsu, C.-T.; Chang, C.-H.; Lin, S.-Y. Langmuir 1997, 13, 6204–6210. (37) Lee, Y.-C.; Lin, S.-Y.; Liu, H.-S. Langmuir 2001, 17, 6196–6202. (38) Lin, S.-Y.; Lee, Y-C.; Yang, M.-W.; Liu, H.-S. Langmuir 2003, 19, 3164– 3171. (39) Moorkanikkara, S. N.; Blankschtein, D. J. Colloid Interface Sci. 2006, 302, 1–19.

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Figure 3. Comparison of (a) the fitted polynomial EOS model with the surface-expansion measurements for C12E6, and (b) the fitted ESTC model with the equilibrium surface tension measurements for C12E6. The solid lines correspond to the fitted EOS model (Figure 3a) and ESTC model (Figure 3b) obtained following the approach described in Appendix B; the dashed lines correspond to the fitted Frumkin model (EOS - Figure 3a and ESTC - Figure 3b) reported in ref 10. The “b” in (a) correspond to the experimental surface-expansion measurements, and the “b” in (b) correspond to the experimental equilibrium surface tension measurements.

The EOS models and the ESTC models identified for C12E4 and C12E6 following steps 1 and 2 above are shown in Figures 2 and 3, respectively. Figures 2a and 3a compare the fitted EOS models for C12E4 and C12E6, respectively, with the corresponding surface-expansion measurements reported in refs 7 and 10, respectively. Figures 2b and 3b compare the fitted ESTC models identified for C12E4 and C12E6, respectively, with the corresponding equilibrium surface tension measurements reported in refs 7 and 10, respectively. In Figures 2 and 3, γe refers to the equilibrium surface tension, Ceb refers to the equilibrium surfactant bulk solution concentration, Γe refers to the equilibrium surfactant surface concentration, and Γref refers to the chosen reference value of the surfactant surface concentration. In Figures 2 and 3, (i) the solid lines correspond to the fitted EOS models (Figures 2a and 3a) and fitted ESTC models (Figures 2b and 3b), and (ii) the “b” correspond to the experimental surface-expansion measurements (Figures 2a and 3a) and experimental equilibrium surface tension measurements (Figures 2b and 3b). For comparison, in Figures 2 and 3, we show the predictions of the Frumkin model (EOS - Figures 2a and 3a, and ESTC Figures 2b and 3b), originally identified by fitting only the equilibrium surface tension measurements, and used to analyze the experimental DST data of C12E4 and C12E6 in refs 7 and 10, respectively. Specifically, in Figures 2a and 3a, the dashed lines correspond to the predictions of the Frumkin EOS models of C12E4 and C12E6, respectively. In Figures 2b and 3b, the dashed lines correspond to the best-fit Frumkin ESTC model of C12E4 and C12E6, respectively. Figures 2 and 3 clearly show that the agreement between the EOS and ESTC models identified using the procedure described in Appendix B and the experimental data is better than that of the Frumkin EOS and ESTC models.

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Figure 4. Comparison of the predicted dynamic surface tension, γ(t), as a function of time, t, using the diffusion-controlled adsorption model with the experimentally observed pendant-bubble DST behavior for C12E4 at (a) Cb ) 0.4 × 10- 8 mol/cm3, (b) Cb ) 0.6 × 10- 8 mol/cm3, (c) Cb ) 1.0 × 10- 8 mol/cm3, and (d) Cb ) 2.0 × 10- 8 mol/cm3. In these figures, the solid, dashed, and dotted lines correspond to the nominal, lower, and upper bound values of D, respectively. The “b” correspond to the experimental pendant-bubble DST data reported in ref 7.

3.2. Input Diffusion Coefficient Value, D Note that the surfactant diffusion coefficient, D, affects the rate of diffusive transport of surfactant molecules in the bulk solution and is the sole “kinetics” parameter in the diffusion-controlled adsorption model (see eq A.1). The D values of C12E4 and C12E6 have not been measured experimentally. Therefore, we estimate their D values using correlations as well as using the D values measured experimentally for structurally similar solutes. In ref 39, the Wilke-Chang correlation40 was used to estimate the D values of C12E4 and C12E6 to be 3.8 × 10- 6 and 3.4 × 10- 6 cm2/s, respectively. In ref 41, the diffusion coefficient of the structurally similar nonionic surfactant C12E5 was measured using NMR to be 3.9 × 10- 6 cm2/s. Moreoever, in ref 39, experimental shorttime DST data of C12E4 and C12E6 were analyzed, and the D values of these surfactants were determined to be (3.9 ( 0.6) × 10- 6 and (3.8 ( 0.6) × 10- 6 cm2/s, respectively, which is in close agreement with the estimates of the Wilke-Chang correlation and consistent with the measured D value of C12E5. Keeping the above findings in mind, we assume that the D values of the two nonionic surfactants considered here lie within the following ranges: D ) (3.9 ( 0.6) × 10- 6 cm2/s for C12E4, and D ) (3.8 ( 0.6) × 10- 6 cm2/s for C12E6. Accordingly, diffusion-controlled model predictions were made separately for the nominal, upper, and lower bound D values of C12E4 and C12E6 (see section 3.5). 3.3. Input Radius of the Spherical Bubble, r0. The radius of the spherical bubble, r0, is a geometrical parameter and affects the overall rate of surfactant adsorption (see eq A.5). In the pendant-bubble experiments conducted for C12E4,7 the initial value of r0 was varied between 0.10 and 0.15 cm as Cb decreased42 from 2 × 10- 8 to 0.4 × 10- 8 mol/cm3. Similarly, in the pendantbubble experiments conducted for C12E6,10 the initial value of (40) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264–270. (41) Schonhoff, M.; Soderman, O. J. Phys. Chem. B 1997, 101, 8237–8242.

r0 was varied between 0.10 and 0.15 cm as Cb decreased from 2 × 10- 8 to 0.2 × 10- 8 mol/cm3. A value of r0 ) 0.10 cm, corresponding to the lower limit of the experimental r0 values, was used for the DST predictions for both C12E4 and C12E6 at all Cb values. Note that the rate of surfactant adsorption increases as the value of r0 decreases2,23 (see also eq A.5). Therefore, choosing the lower limit value of r0 ) 0.10 cm corresponds to the case of faster adsorption. 3.4. Initial Surfactant Surface Concentration, Γ0. The initial surfactant surface concentration, Γ0, defines the condition of the surface at the beginning of the adsorption kinetics process (see eq A.4). The “b” in Figures 4 and 5 represent the experimental DST data of C12E4 and C12E6 at several Cb values (see section 3.5 for a detailed discussion of Figures 4 and 5). Figures 4 and 5 show that the experimental DST data do not approach the expected pure water/air surface tension value of 72.0 mN/m at 298 K as t f 0. This behavior of the experimental DST data may be due to the presence of trace quantities of adsorbed surfactant molecules at the beginning of the adsorption process.43 Accordingly, to set the proper initial conditions for the predictions that best represent the experimental initial conditions, the initial surfactant surface concentration values, Γ0, were chosen such that the resulting predicted initial surface tensions agreed with the experimental DST data as t f 0. 3.5. Predicted Dynamic Surface Tension. For the two nonionic surfactants considered, C12E4 and C12E6, DST profiles were predicted at several Cb values using the four inputs described in sections 3.1-3.4. Specifically, Figure 4a-d shows the predicted DST profiles for C12E4 at (a) Cb ) 0.4 × 10- 8 mol/cm3, (b) Cb ) 0.6 × 10- 8 mol/cm3, (c) Cb ) 1.0 × 10- 8 mol/cm3, and (d) (42) Note that the DST measurements at the higher Cb values are conducted at lower initial r0 values to ensure that the bubbles remain stable as the DST values decrease. (43) Schlarmann, J. S.; Stubenrauch, C.; Miller, R. Tenside, Surfactants, Deterg. 2005, 42, 307–312.

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Figure 5. Comparison of the predicted dynamic surface tension, γ(t), as a function of time, t, using the diffusion-controlled adsorption model with the experimentally observed pendant-bubble DST behavior for C12E6 at (a) Cb ) 0.2 × 10- 8 mol/cm3, (b) Cb ) 0.6 × 10- 8 mol/cm3, (c) Cb ) 1.3 × 10- 8 mol/cm3, and (d) Cb ) 2.0 × 10- 8 mol/cm3. In these figures, the solid, dashed, and dotted lines correspond to the nominal, lower, and upper bound values of D, respectively. The “b” correspond to the experimental pendant-bubble DST data reported in ref 10.

Cb ) 2.0 × 10- 8 mol/cm3. Similarly, Figure 5a-d shows the predicted DST profiles for C12E6 at (a) Cb ) 0.2 × 10- 8 mol/cm3, (b) Cb ) 0.6 × 10- 8 mol/cm3, (c) Cb ) 1.3 × 10- 8 mol/cm3, and (d) Cb ) 2.0 × 10- 8 mol/cm3. The experimental DST data measured using the pendant-bubble apparatus for C12E4 and C12E67,10 at the corresponding Cb values are also shown in Figures 4a-d and 5a-d. In Figures 4a-d and 5a-d, the solid lines correspond to the predicted DST profiles at the nominal values of D, the dashed lines correspond to the predicted DST profiles at the lower bound values of D, the dotted lines correspond to the predicted DST profiles at the upper bound values of D, and the “b” correspond to the experimental DST data reported in refs 7 and 10. Note that in Figures 4d and 5d, DST predictions are reported only for γ > 40 mN/m and γ > 43 mN/m, respectively, because reliable EOS data were available only over this range of γ values (see Figures 2 and 3). As can be seen from Figures 4a-d and 5a-d, the predicted DST profiles match the experimentally observed DST behavior at higher Cb values, while systematic deViations exist at lower Cb values, for both C12E4 and C12E6. In section 4 below, we analyze the observed systematic deviations between the predicted DST profiles and the experimentally observed DST behavior of C12E4 and C12E6.

4. Analysis On the basis of Figures 4a-d and 5a-d, the following four observations can be made on the observed systematic deviations between the DST profiles predicted using the diffusion-controlled model and the experimental DST behavior of C12E4 and C12E6. Observation 1. Figures 4a-d and 5a-d clearly show that, for both C12E4 and C12E6, the agreement between the predicted DST profiles and the experimental DST data becomes progressively better as Cb increases. Observation 2. Although the experimental DST behavior and the predicted DST profiles deviate at low Cb values, both the

predicted DST profiles and the experimental DST behavior approach the same equilibrium surface tension values. Observation 3. A closer investigation of the deviations at lower Cb values in Figures 4a-c and 5a,b reveals that the deviations occur consistently at time scales greater than 100 s. Observation 4. For those deviations that occur beyond a time scale of 100 s, the experimental DST data are consistently lower than the predicted DST profiles (see Figures 4a-c and 5a,b). Observations 1 and 2 indicate that the cause of the observed systematic deviations may not be associated with the equilibrium surfactant adsorption models used for C12E4 and C12E6, that is, the ESTC models in Figures 2b and 3b. Indeed, if the ESTC models were incorrect, then this would affect the DST predictions at all the Cb values studied, and especially at the higher Cb values where the ESTC models are used over wider ranges of surface tension values. This conclusion is at odds with observation 1, which states that the agreement between the predicted DST profiles and the experimental DST behavior becomes progressively better as Cb increases. Indeed, the excellent agreement between the predicted DST profiles and the experimental DST data at higher Cb values (Figure 4d for C12E4, and Figure 5c,d for C12E6) validates the accuracy of the ESTC models used in the predictions made. The accuracy of the ESTC models used is further validated by observation 2 above, which states that the predicted DST profiles and the experimental DST behavior approach identical equilibrium surface tension values, for both C12E4 and C12E6 and for all of the Cb values studied. In other words, observations 1 and 2 indicate that the cause of the systematic deviations between the predicted DST profiles and the experimental DST behavior may be due to a difference in the kinetics of surfactant adsorption onto the actual pendantbubble surface and the kinetics (diffusion-controlled) assumed for the predictions made. Observation 3 indicates that the underlying cause for the observed deviations between the predicted DST profiles and the

1440 Langmuir, Vol. 25, No. 3, 2009

experimental DST behavior operates at time scales greater than 100 s. Taken together, observations 1, 2, and 3 indicate that the kinetics of surfactant adsorption onto the actual pendant-bubble surface may not be diffusion-controlled at time scales greater than 100 s. Observation 4 indicates that the actual kinetics of surfactant adsorption is faster than the predicted fastest rate of surfactant adsorption from a quiescent solution at time scales greater than 100 s. This observation is at odds with the original assumption that the surfactant bulk solution is quiescent throughout the surfactant adsorption process. More specifically, this finding strongly suggests that the actual surfactant bulk solution in which the pendant-bubble DST measurements were conducted cannot be considered to be quiescent at time scales greater than 100 s, for both C12E4 and C12E6. In other words, observation 4 suggests the possible existence of convective currents operating at time scales greater than 100 s in the surfactant bulk solution in which pendant-bubble experiments were conducted for C12E4 and C12E6. Note that the existence of convective currents operating at time scales greater than 100 s (i) does not affect the surfactant equilibrium behavior, and instead affects only the kinetics of surfactant adsorption (consistent with observations 1 and 2), (ii) would affect the kinetics at time scales greater than 100 s (consistent with observation 3), and (iii) would lead to an increased rate of surfactant adsorption relative to adsorption from a quiescent solution at time scales greater than 100 s (consistent with observation 4). Accordingly, we conclude that the systematic deviations observed in Figures 4a-c and 5a,b may reflect the existence of convective currents in the surfactant bulk solution operating at time scales greater than 100 s.

5. Discussion The systematic nature of the observed deviations between the predicted DST profiles and the experimental DST behavior for C12E4 (see Figure 4a-c) and C12E6 (see Figure 5a,b) suggests that the nonquiescent nature of the surfactant bulk solution may be intrinsic to the experimental pendant-bubble DST measurement approach. Note that the traditional analysis of experimental pendant-bubble DST data involves the following:2-18 (1) assuming that the surfactant bulk solution is quiescent, and consequently assuming that the transport of surfactant molecules in the bulk solution is purely diffusive; (2) assuming the applicability of the diffusion-controlled model to describe the surfactant adsorption kinetics; and (3) numerically identifying the best-fit (bf) value for the diffusion coefficient of the surfactant molecule, Dbf, using experimental DST data measured at different Cb values. The key difference between the traditional analysis of experimental pendant-bubble DST data and the analysis presented in sections 3 and 4 above is that the traditional analysis fits the diffusion coefficient of the surfactant molecule using the actual experimental DST data, while the analysis presented here fully predicts the DST behavior. Accordingly, observations 1-4 made in section 4 are not directly applicable when interpreting published experimental DST data analyzed using the traditional analysis. With the above in mind, along with the suggestion put forward in section 4 that convective currents operating at time scales greater than 100 s may exist in the surfactant bulk solutions in which pendant-bubble DST experiments were conducted for C12E4 and C12E6, in this section, we identify distinct signatures in the traditional analysis of experimental DST data in a scenario where the experimental DST measurements used in the analysis are actually conducted in a nonquiescent surfactant bulk solution in which convective currents operate at time scales greater than

Moorkanikkara and Blankschtein

100 s. A review of previously published DST literature reveals that the identified signatures are quite general and are, in fact, observed in the analysis of the experimental pendant-bubble DST data of several surfactants. Consider a case where a pendant-bubble DST measurement is conducted in a surfactant bulk solution where convective currents are present and operate at a time scale greater than 100 s. Specifically, at time scales lower than 100 s, the effect of convective currents on the surfactant adsorption kinetics is negligible, while at time scales greater than 100 s, the effect of convective currents becomes increasingly significant. Next, assume (incorrectly) that the surfactant bulk solution is quiescent at all times, and that surfactant transport in the bulk solution occurs solely Via diffusion. Furthermore, assume the applicability of the diffusion-controlled model to describe the adsorption kinetics of the surfactant and fit for the diffusion coefficient value using the experimental DST data. In this scenario, it is reasonable that the following three features should be observed. Feature 1. Whenever a significant percentage reduction in the dynamic surface tension44 takes place at a time scale that is smaller than 100 s, there is relatively good agreement between the experimental DST data and the diffusion-controlled model fit. Feature 2. Whenever a significant percentage reduction in the dynamic surface tension takes place at a time scale that is greater than 100 s, there is relatively poor agreement between the experimental DST data and the diffusion-controlled model fit. Feature 3. In those cases where there is relatively poor agreement between the experimental DST data and the diffusioncontrolled model fit, the experimental DST data reduce faster than the diffusion-controlled model fit at later times. Indeed, the diffusion-controlled model fit is forced to account for both diffusion at earlier time scales and for diffusion-convection at later time scales, and therefore reflects an intermediate adsorption kinetics behavior. It is then reasonable to anticipate that this intermediate adsorption kinetics behavior cannot properly account for the kinetics corresponding to diffusion-convection occurring at the later times. Interestingly, an examination of reported traditional analyses of experimental pendant-bubble DST data of several surfactants (decanol, nonanol, C10E8, C14E8, C12E8, and C10E4), which make use of the diffusion-controlled model fit, reveals the existence of features 1-3 listed above. Due to of space limitations, in this section, we only present a comparison between the diffusioncontrolled model fits and the experimental pendant-bubble DST data for the nonionic surfactant C10E8 (see Figure 6a-d). However, in the Supporting Information, we provide additional evidence for other surfactants (decanol, nonanol, C14E8, C12E8, and C10E4), where similar features can be observed in the reported experimental pendant-bubble DST data. Figure 6a-d shows the experimental pendant-bubble DST data of C10E8 at (a) Cb ) 1 × 10- 9 mol/cm3, (b) Cb ) 4 × 10- 9 mol/cm3, (c) Cb ) 6 × 10- 9 mol/cm3, and (d) Cb ) 10 × 10- 9 mol/cm3, and the corresponding diffusion-controlled model fits reported in ref 5. In Figure 6a-d, the symbols correspond to the experimental pendant-bubble DST data, and the lines correspond to the reported diffusion-controlled model fits. Figure 6a-d clearly shows that the agreement between the diffusion(44) “Percentage reduction in dynamic surface tension” is defined as the actual reduction in dynamic surface tension (γw/a - γ(t)) normalized by the total reduction in surface tension (γw/a - γe), where γw/a is the pure water/air surface tension, γ(t) is the instantaneous surface tension at time t, and γe is the equilibrium surface tension. (45) Binks, B. P.; Fletcher, P. D. I.; Paunov, V. N.; Segal, D. Langmuir 2000, 16, 8926–8931.

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Figure 6. Comparison of the experimental pendant-bubble DST data with the diffusion-controlled model fit of C10E8 (adapted from Figure 2 of ref 5) at (a) Cb ) 1 × 10- 9 mol/cm3, (b) Cb ) 4 × 10- 9 mol/cm3, (c) Cb ) 6 × 10- 9 mol/cm3, and (d) Cb ) 10 × 10- 9 mol/cm3. The symbols correspond to the experimental pendant-bubble DST data, and the solid lines correspond to the diffusion-controlled model fits.

controlled model fits and the experimental DST data becomes progressiVely better as a significant percentage reduction in surface tension occurs at time scales, which are less than 100 s (features 1 and 2 above). Specifically, in Figure 6d, γe ≈ 59.5 mN/m and γ(t ) 100 s) ≈ 61 mN/m. Using γw/a ) 72 mN/m, the percentage reduction in dynamic surface tension44 at t ) 100 s is (72 - 61)/(72 - 59.5) × 100 ≈ 88%, indicating that a significant 88% of total surface tension reduction has taken place at time scales that are smaller than 100 s. Note that Figure 6d shows good agreement between the experimental DST data and the diffusion-controlled model fit, consistent with feature 1 above. Furthermore, in Figure 6a, γe ≈ 65.5 mN/m and γ (t ) 100 s) ≈ 71 mN/m. Using γw/a ) 72 mN/m, the percentage reduction in dynamic surface tension at t ) 100 s is (72 - 71)/ (72 - 65.5) × 100 ≈ 15%, indicating that a significant 85% of total surface tension reduction has taken place at time scales that are greater than 100 s. Note that Figure 6a shows relatively poor agreement between the experimental DST data and the diffusioncontrolled model fit, consistent with feature 2 above. In addition, Figure 6a-c shows that the diffusion-controlled model fits approach the corresponding equilibrium surface tension values at rates that are slower than those corresponding to the experimental DST data, consistent with feature 3 above. Accordingly, we conclude that the three features observed in Figure 6a-d are consistent with the possible existence of convective currents in the C10E8 bulk solutions, which operate at time scales greater than 100 s. Additional evidence supporting the possible existence of convective currents at time scales greater than 100 s may be obtained by studying the best-fit diffusion-coefficient (Dbf) values corresponding to Figure 6a-d. Specifically, the Dbf value was reported to increase as the concentration Cb decreases.5 Specifically, the Dbf value corresponding to Cb ) 1 × 10- 9 mol/cm3

(Figure 6a) was reported to be (6.5 ( 0.5) × 10- 6 cm2/s, while the Dbf value corresponding to Cb ) 10 × 10- 9 mol/cm3 (Figure 6d) was reported to be (4.5 ( 0.4) × 10- 6 cm2/s.5 This observation is consistent with the possible existence of convective currents operating at time scales greater than 100 s because for Cb ) 1 × 10- 9 mol/cm3, a significant percentage reduction in dynamic surface tension takes place at time scales greater than 100 s (see Figure 6a). Consequently, the best-fit Dbf value needs to account for both diffusion and convection, resulting in a higher value of Dbf. On the other hand, for Cb ) 10 × 10- 9 mol/cm3, a significant percentage reduction in dynamic surface tension takes place at time scales smaller than 100 s (see Figure 6d). Consequently, the best-fit Dbf value needs to account solely for the diffusive transport of the surfactant molecules, resulting in a lower value of Dbf. The actual diffusion coefficient value of C10E8 has not been measured experimentally to allow a direct comparison. Interestingly, however, the reported Dbf value of (4.5 ( 0.4) × 10- 6 cm2/s, corresponding to Cb ) 10 × 10- 9 mol/cm3, appears to be consistent with the measured diffusion coefficient value of 3.9 × 10- 6 cm2/s of the structurally similar C12E5 surfactant.41 The existence of features 1-3 in the analysis of the experimental DST data of C10E8, as well as the deduced higher best-fit diffusion-coefficient value at lower Cb values, suggests that convective currents, most likely, were present in the surfactant bulk solutions where pendant-bubble DST measurements were conducted for C10E8. Although the experimental pendant-bubble DST data of C10E8 (see Figure 6a-d) and of the additional surfactants (decanol, nonanol, C14E8, C12E8, and C10E4) compiled in the Supporting Information do display features 1-3 above suggesting the existence of convective currents, these features have been consistently overlooked, and the experimental DST data have been analyzed invariably assuming that the surfactant bulk

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solution is quiescent and that surfactant transport in the bulk solution is purely diffusive.3-5,7,8,10,13,15 We should point out that a few authors have acknowledged the existence of anomalous superdiffusive behavior in experimental pendant-bubble DST measurements. For example, in ref 45, dynamic ellipsometry measurements were conducted, in addition to pendant-bubble DST measurements, to directly measure the dynamic surface concentration, Γ(t), of C12E5. The authors in ref 45 observed that at lower Cb values, Γ(t) measured using dynamic ellipsometry was higher than the Γ(t) predicted using the diffusion-controlled model. In effect, they observed a larger number of surfactant molecules at the surface relative to that predicted by the diffusion-controlled model. The authors commented that the observed high dynamic surfactant surface concentration may be due to the existence of convective currents in their system. Interestingly, the deviations between the predicted and the experimental Γ(t) values reported in ref 45 began at t ≈ 5 min ()300 s), a time that is consistent with the time scales at which systematic deviations are observed in the DST analysis presented here (see Figures 4a-c, 5a,b, 6a,b, and the Supporting Information). In addition, in ref 46, pendant-bubble DST measurements of an aqueous solution of the protein bovine serum albumin were carried out, and the authors accounted for the observed dependence of the DST lag time on protein concentration using a model that includes the effect of possible convection in the protein bulk solution. The authors in ref 46 assumed that the aqueous solvent exhibits convective currents that affect the transport of the dissolved protein. Specifically, the authors modeled convective currents in the protein bulk solution as operating at length scales of δ and higher, and diffusion phenomena operating at length scales lower than δ.46 In other words, the authors modeled the protein adsorption process as being diffusive over a layer of thickness δ adjacent to the bubble surface and assumed that convection currents equalize the surfactant concentration beyond a distance δ from the bubble surface. Subsequently, the value of δ was fitted using the measured DST lag time, and a best-fit value of 0.2 mm was selected. Furthermore, the authors rationalized the fitted value of δ in terms of the thermal properties of the aqueous solvent. Note that modeling the effect of convection in surfactant bulk solution as diffusion over a finite effective length scale has been reviewed in ref 28. If the model proposed in ref 46 for convective currents in the bulk solution was adopted for the surfactant solutions considered here, then, it is reasonable that the surfactant transport should be purely diffusive until the diffusion front reaches a thickness of δ that is characteristic of the aqueous solvent. In other words, surfactant transport in the bulk solution may be considered to be diffusive until a time of δ2/D, beyond which the convective currents would begin to affect the adsorption kinetics. With this in mind, for the nonionic surfactants C12E4 and C12E6 considered in this Article, this time scale can be estimated using their respective nominal D values of 3.9 × 10- 6 and 3.8 × 10- 6 cm2/s, respectively, and a δ value of 0.2 mm to be 102 and 105 s, respectively. These time scales are in excellent agreement with the time scale of 100 s observed in Figures 4a-c, 5a,b, 6a,b, and the Supporting Information, for the onset of superdiffusive adsorption kinetics behavior.

6. Conclusions Assuming that the surfactant bulk solution is quiescent is one of the traditional assumptions made when analyzing the experimental pendant-bubble DST data of surfactant solutions. (46) Ybert, C.; di Meglio, J. M. Langmuir 1998, 14, 471–475.

Moorkanikkara and Blankschtein

In this Article, we presented an analysis of the experimental pendant-bubble DST data of two alkyl poly (ethylene oxide) nonionic surfactants, C12E4 and C12E6, which reveals that the traditional assumption of a quiescent solution may not be valid. Specifically, the analysis involved (i) constructing an adsorption kinetics model that predicts the fastest rate of surfactant adsorption onto the actual pendant-bubble surface from a quiescent solution, (ii) predicting the DST profiles using the constructed adsorption kinetics model at several initial surfactant bulk solution concentrations (Cb’s) for the two nonionic surfactants considered, and (iii) comparing the predicted DST profiles with the experimental DST data measured using the pendant-bubble apparatus. The comparison revealed systematic deviations for both C12E4 and C12E6 with the following characteristics: (a) the experimental DST behavior decreased at a rate that is faster than the predicted fastest rate of surfactant adsorption from a quiescent solution at time scales greater than 100 s, and (b) both the experimental DST data and the predicted DST profiles approached the same equilibrium surface tension value for all of the Cb values studied. These characteristics indicate that the kinetics in the actual experimental setup was faster than the fastest adsorption kinetics from a quiescent solution at time scales greater than 100 s. On the basis of this analysis, we proposed that the observed systematic deviations are consistent with the existence of convective currents in the surfactant bulk solution, which operate at time scales greater than 100 s. This important finding leads to the conclusion that surfactant transport in the surfactant bulk solution cannot be assumed to be purely diffusive over the entire relaxation time scales probed, contrary to what is typically assumed when analyzing experimental pendant-bubble DST data. The systematic nature of the deviations observed between the predicted DST profiles and the experimental DST behavior suggests that the nonquiescent nature of the surfactant bulk solution may be intrinsic to the experimental pendant-bubble DST measurement approach. With this in mind, we identified generic signatures when one assumes (incorrectly) that the surfactant bulk solution is quiescent in analyzing experimental pendant-bubble DST data conducted in a nonquiescent surfactant bulk solution. An examination of the pendant-bubble DST literature revealed that the identified signatures are observed in the experimental pendant-bubble DST data of several surfactants (decanol, nonanol, C10E8, C14E8, C12E8, and C10E4), but have been consistently overlooked. While the results reported in this Article support the existence of convective currents in surfactant bulk solutions when measuring DST using the pendant-bubble apparatus, clearly, additional experimental analysis of the pendant-bubble technique is required to understand its source and effect in detail. In conclusion, since a central aim of the pendant-bubble method is to investigate the adsorption kinetics behavior of surfactant molecules onto liquid/ liquid interfaces, it is essential to elucidate and characterize the underlying transport mechanism to carry out a reliable analysis of the experimental DST data measured using this method. Acknowledgment. We are grateful to Professor Shi-Yow Lin for kindly sharing with us his actual experimental DST data measured using the pendant-bubble apparatus for C12E4 and C12E6, which was originally published in refs 7 and 10, respectively. We are also grateful to Professor Charles Maldarelli for helpful and insightful discussions on the pendant-bubble apparatus, in particular, and on dynamic surface tension, in general. Supporting Information Available: Compilation of published experimental pendant-bubble DST data of several surfactants where consistent deviations can be observed between the experimental pendant-

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Langmuir, Vol. 25, No. 3, 2009 1443

bubble dynamic surface tension data and the diffusion-controlled model fits. This material is available free of charge via the Internet at http:// pubs.acs.org.

Appendix A: Diffusion-Controlled Model This Appendix summarizes the model equations for diffusioncontrolled adsorption of surfactant molecules onto a stationary spherical bubble, and the prediction of the dynamic surface tension (DST) at a given surfactant bulk solution concentration, Cb. A detailed derivation of the model equations can be found in refs 2, 6, 15, 23. Below, we summarize the key governing equations and the associated boundary and initial conditions. The surfactant diffusive transport along the radial direction, r, towards the spherical surface is given by:

∂C D ∂ 2 ∂C r , r g r0, t g 0 ) ∂t r2 ∂r ∂r

( )

(A.1)

where C is the surfactant concentration, and r0 is the radius of the pendant bubble. The rate of surfactant adsorption at the surface is equal to the net flux of surfactant molecules leaving the sub-surface towards the surface, that is,

|

dΓ ∂C ,tg0 )D dt ∂r r)r0

(A.2)

where Γ is the surfactant surface concentration. With the far field (r f ∞) surfactant concentration being uniform at a value Cb, the boundary condition for eq A.1 is given by:

C(t, r f ∞) ) Cb, t g 0

(A.3)

For a surfactant surface concentration, Γ0, and a homogeneous bulk solution at the beginning of the adsorption process, the initial conditions associated with the governing equations (eqs A.1 and A.2) can be expressed as follows:

Γ(t ) 0) ) Γ0 and C(t ) 0, r) ) Cb, r g r0

(A.4)

Equations A.1-A.4 can be solved using the method of Laplace transforms to yield:

Γ(t) ) Γ0 +

D C tr0 b

[

∫0t Cs(t) dt] + 2 Dπ [Cb√t -

∫0√t Cs(t - η) d√η](A.5)

where Cs(t) is the instantaneous surfactant sub-surface concentration (C(t, r ) r0)) and is related to the instantaneous surfactant surface concentration, Γ(t), through the equilibrium adsorption isotherm (EAI):

Γe ) g(Ceb)

(A.6)

where Γe is the equilibrium surfactant surface concentration, and Cbe is the equilibrium surfactant bulk solution concentration. Specifically, since the diffusion-controlled model assumes that the sub-surface and the surface reach equilibrium instantaneously, Cs(t) is related to Γ(t) through the EAI in eq A.6, that is,

Γ(t) ) g(Cs(t))

(A.7)

Solution of the diffusion-controlled model involves solving eqs A.5 and A.7 simultaneously to predict Γ(t). Once Γ(t) is known, the following equilibrium equation of state (EOS) is used to predict the DST, γ(t):

γw⁄a - γe ) f(Γe)

(A.8)

where γw/a is the pure water/air surface tension, γe is the equilibrium surface tension, and the function f(Γe) is the EOS. Specifically, since the diffusion-controlled model assumes that any reorientation of the adsorbed surfactant molecules at the surface occurs instantaneously, Γ(t) is also related to γ(t) through the equilibrium EOS in eq A.8, that is,

γw⁄a - γ(t) ) f(Γ(t))

(A.9)

Note that the EAI, eq A.6, is related to the equilibrium EOS, eq A.8, through the Gibbs adsorption equation:47

Γe ) -

1 dγe RT d ln Ce

(A.10)

b

Typically, the equilibrium adsorption behavior of the surfactant is obtained by measuring the equilibrium surface tension, γe, as a function of the equilibrium surfactant bulk solution concentration, Cbe (ESTC). One then uses the deduced ESTC relation in conjunction with the Gibbs adsorption equation, eq A.10, to determine the EOS of the surfactant. The EAI, eq A.6, is then determined by eliminating γe between the specified ESTC behavior and the EOS determined through the application of eq A.10. Accordingly, the diffusion-controlled model of surfactant adsorption onto a stationary spherical bubble involves the following four specifications: (1) the surfactant ESTC model (used to determine the EAI in eq A.6 and the EOS in eq A.8); (2) the diffusion coefficient of the surfactant molecule, D (see eq A.5); (3) the radius of the spherical bubble, r0 (see eq A.5); and (4) the initial surfactant surface concentration, Γ0 (see eq A.5). With the four specifications listed above, the diffusioncontrolled model can predict the DST behavior at a given initial surfactant bulk solution concentration, Cb.

Appendix B: Equilibrium Surface Tension versus Bulk Solution Concentration (ESTC) Model for C12E4 and C12E6 This Appendix describes the steps followed to identify the ESTC models for C12E4 and C12E6 that fit both the equilibrium surface tension measurements as well as the surface-expansion measurements. Recall that in the diffusion-controlled model (see Appendix A), the input ESTC model is used to determine the equilibrium adsorption isotherm (eq A.6), which, in turn, is used to relate the instantaneous surfactant sub-surface concentration, Cs(t), to the instantaneous surfactant surface concentration, Γ(t) (see eq A.7), and the equation of state (eq A.8), which, in turn, is used to relate the instantaneous surfactant surface concentration, Γ(t), to the instantaneous surface tension, γ(t) (see eq A.9). The existing methods to test the accuracy of the input ESTC model involve:38 (i) comparing the ESTC model predictions with the equilibrium surface tension measurements, and (ii) comparing the predictions of the EOS corresponding to the ESTC model to surface-expansion measurements. Note that the equilibrium surface tension measurements relate γe to Cbe, and the surfaceexpansion measurements relate γe to the normalized equilibrium surfactant surface concentration, Γe/Γref, where Γref is a reference value of the surfactant surface concentration. Recall that, typically, the equilibrium surface tension measurements are used to fit the (47) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley Interscience: New York, 1989.

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ESTC model, and the surface-expansion measurements are used to test the fitted ESTC model. In refs 36-38, it was demonstrated that (1) using only the equilibrium surface tension measurements to fit the ESTC model can lead to significantly different predictions of the equation of state, and (2) the surface-expansion measurements can be extremely useful to validate the accuracy of the fitted ESTC model. Keeping all of the above in mind, we use both the equilibrium surface tension measurements and the surface-expansion measurements to identify the ESTC models of C12E4 and C12E6. The following two steps were followed to identify the ESTC models of C12E4 and C12E6. Step 1 involves using the surface-expansion measurements to fit a polynomial relating Γe/Γref to γe:

Γe/Γref ) p(γe)

(A.11)

such that the resulting polynomial passes through the point (Γ ) 0, γe ) γw/a ) 72.0 mN/m), corresponding to T ) 298 K.7,10 Step 2 involves using the Gibbs adsorption equation (eq A.10) to determine the corresponding ESTC model. Specifically, applying the Gibbs adsorption equation to the polynomial EOS in eq A.11 yields:

p(γe)Γref ) -

1 dγe RT d ln C e

(A.12)

b

Rearranging eq A.12 then yields:

d ln Cbe ) -

[ ]

dγe 1 RT Γref p(γe)

(A.13)

Integrating eq A.13 between the limits (Cbr, γr) and (Cbe, γe) yields:

ln Cbe ) ln Cbr -

1 RT Γref

dγ

∫γγ p(γee) e

r

(A.14)

where γr is an arbitrary reference (r) value for γe, and Cbr is the surfactant bulk solution concentration corresponding to γe ) γr. Note that (i) the ESTC behavior in eq A.14 needs to approach the Henry’s law limit as Cbe f 0, and (ii) γr cannot be chosen to be equal to γw/a because in that case, Cbr ) 0 (corresponding to the pure water solution), and the value of ln Cbr diverges. In addition, note that choosing γr