Adsorption Kinetics of C12E4 at the Air− Water Interface: Adsorption

Langmuir 2000, 16, 3187-3194

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Adsorption Kinetics of C12E4 at the Air-Water Interface: Adsorption onto a Fresh Interface Ching-Tien Hsu, Ming-Jian Shao, and Shi-Yow Lin* Department of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei, 106 Taiwan, Republic of China Received September 17, 1999. In Final Form: January 5, 2000 The adsorption of C12E4 onto a fresh air-water interface was investigated by using video-enhanced pendant bubble tensiometry. From the comparison between the equilibrium surface tension data and the theoretical relaxation profiles predicted by the Frumkin adsorption isotherm, the adsorption process was found to be anticooperative. Dynamic surface tension data for C12E4 molecules absorbing onto a freshly created air-water interface for different bulk concentrations were used for the determination of the controlling mechanism and the evaluation of diffusivity. Comparison was made for the entire relaxation period of the surface tension data and the model predictions. It is concluded that the adsorption process is of diffusion control and the diffusion coefficient is 6.4 × 10-6 cm2/s. The lower limit of the adsorption rate constant of C12E4 were obtained from the theoretical simulation. Besides, the pendant bubble, at which the interface had reached the equilibrium state, was expanded rapidly and a relationship between surface tension (γ) and surface area (A) was obtained. A curve relating γ and relative surface concentration Γ/Γref was obtained from the γ-A data and then used to examine the adsorption isotherm utilized in this study.

1. Introduction Three consecutive steps are involved in the mass transport of a bulk soluble surfactant between the bulk phase of solution and the fluid interface. They are bulk diffusion and/or convection, adsorption and/or desorption, and rearrangement of surfactant molecules at the interface. For nonpolymeric molecules, the rearrangement is very fast. The convection is negligible for the mass transport taking place in a quiescent surfactant solution. Lucassen and Giles1 reported that the adsorptions of soluble nonionic surfactants C12E3, C12E6, and C14E6 are controlled only by bulk diffusion from the dilational modulus experiment. However, it was reported recently by Eastoe et al. that the adsorption onto a clean airwater interface of C10E4, C10E5, and C12En (n ) 5-8) is diffusive-kinetic-controlled.2 Maldarelli’s team found that C10E8, C12E8, and C12E6 are of diffusion control at dilute concentration but shift to mixed control at more elevated concentration.3-9 In other words, the surface tension relaxations show agreement with a diffusion-controlled model at dilute concentrations but exhibit a mixed diffusive-kinetic-controlled transport as the concentration

is brought to more elevated values. Besides, the adsorptions of C10E8, C12E8, and C12E6 were found to be anticooperative (the adsorption becomes more difficult as the surface becomes more crowded), whereas the adsorption of 1-decanol (C10E0) and 1-octanol (C8E0) are shown to be cooperative.10-14 The aim of the work is to examine the adsorption kinetics of C12E4 to see whether all of the poly(oxyethylene) nonionic surfactants have anticooperative adsorption and have a shift in the controlling mechanism for surfactant molecule adsorption onto a freshly created air-water interface. Video-enhanced pendant bubble tensiometry was employed for the measurement of the equilibrium and dynamic surface tension. Besides, the air-water interface, being an equilibrium state, was expanded by increasing the volume of the pendant bubble. Data profiles between surface tension (γ) and surface area (A) were obtained for different expansion rates. A γ vs Γ/Γref relationship with nearly no surfactant molecules adsorbing onto the expanding interface during the expansion process was obtained from the γ-A curve. The γ-Γ/Γref data were then used to examine the adsorption isotherm utilized in this work.

* To whom correspondence should be addressed. Tel: 886-22737-6648. Fax: 886-2-2737-6644. E-mail: [email protected].

2. Experimental Measurements

(1) Lucassen, J.; Giles, D. J. Chem. Soc., Faraday Trans. 1975, 71, 217. (2) Eastoe, J.; Dalton, J. S.; Rogueda, P. G.; Crooks, E. R.; Pitt, A. R.; Simister, E. A. J. Colloid Interface Sci. 1997, 188, 423. (3) Pan, R.; Maldarelli, C.; Ennis, B.; Green, J. Diffusive-Kinetic Adsorption of a Polyethoxylated Surfactant to the Air/Water Interface. In Dynamic Properties of Interfaces and Association Structures; Pollai, V., Shah, D. O., Eds.; AOCS Press: Champaign, IL, 1996; pp 23-47. (4) Lin, S. Y.; Chang, H. C.; Chen, E. M. J. Chem. Eng. Jpn. 1996, 29, 634. (5) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (6) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (7) Pan, R.; Green, J.; Maldarelli, C. J. Colloid Interface Sci. 1998, 205, 213. (8) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Langmuir 1998, 14, 2476. (9) Liggieri, L.; Ferrari, M.; Massa, A.; Ravera, F. Colloids Surf. A 1999, in press.

Materials. Nonionic surfactant C12E4 (tetraethylene glycol mono-n-dodecyl ether (C12H25(OCH2CH2)4OH) of greater than 99% purity purchased from Nikko (Tokyo, Japan) was used without modification. The water with which the aqueous solutions were made was purified via a Barnstead NANOpure water purification system, with the output water having a specific conductance of less than 0.057 µΩ-1/cm. Acetone (HPLC grade) used to verify the measurement of surface tension was obtained from Fisher Scientific Co. The values of the surface tension of air-water and air-acetone, using the pendant bubble technique (10) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (11) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (12) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526. (13) Chang, C. H.; Franses, E. I. Chem. Eng. Sci. 1994, 49, 313. (14) Lin, S. Y.; Wang, W. J.; Hsu, C. T. Langmuir 1997, 13, 6211.

10.1021/la9912444 CCC: $19.00 © 2000 American Chemical Society Published on Web 03/02/2000

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described in the following, were 72.0 and 23.1 mN/m, respectively at 25.0 ( 0.1 °C. Pendant Bubble Apparatus. Pendant bubble tensiometry enhanced by video digitization was employed for the measurement of dynamic surface tension of C12E4 and the equilibrium tensions. The system is similar to what investigates C12E8 and C10E8; therefore, only a brief description is given here.5,8 The system creates a silhouette of a pendant bubble, video-images the silhouette, and digitizes the image. A collimated beam with constant light intensity passes through the pendant bubble and forms a silhouette of a bubble on a solid-state video camera. The pendant bubble was generated in a C12E4 aqueous solution, which was put inside a quartz cell. The quartz cell was enclosed in a thermostatic air chamber, and the temperature stability of surfactant solution is (0.1 K. A 16-gauge stainless steel inverted needle (0.047 in. i.d. and 0.065 in o.d.), which was connected to the normally closed port of a three-way miniature solenoid valve, was used for the bubble generation. The common port of the valve, controlled by a computer, is connected to a gastight Hamilton syringe placed in a syringe pump. The silhouette image was digitized into 480 lines × 512 pixels with a level of gray with 8-bit resolution. The edge is defined as the x1 or x2 position, which corresponds to an intensity of 127.5.15 Adsorption onto a Fresh Surface. The experimental protocol was as follows: the quartz cell of 26 × 41 × 43 mm inside diameter was initially filled with the C12E4 aqueous solution, and the bubble-forming needle was positioned in the cell in the path of the collimated light beam. The solenoid valve was energized and the gas was allowed to pass through the needle, thereby forming a bubble of air. The valve was then closed when the bubble achieved a diameter of approximately 2 mm. The bubble so created is one of constant mass. After the solenoid valve was closed and the bubble was formed, sequential digital images of the bubble were then taken, first at intervals of approximately 0.1 s and then later in intervals on the order of seconds. After the relaxation of the adsorption is complete, the images are processed to determine the bubble edge coordinates, bubble volume, bubble surface area, and surface tension. Surface Expansion. After the adsorption has reached the equilibrium state, the air-water interface can be expanded once the syringe pump and solenoid valve was energized and the gas was allowed to pass through the needle. Sequential digital images of the bubble were then taken to the computer and recorded on the tape since the valve was opened. Three different bulk concentrations were chosen for the experiment of surface expansion. At each concentration, the expansion experiment is performed 9 or 10 times at different expansion rates, from 0.1 to 11 mm2/s. After the experiments, the bubble images were processed to determine the bubble edge coordinates, bubble volume, bubble surface area, and surface tension. Relationships of surface tension (γ) vs time (t) and γ vs surface area (A) were obtained. It is found that there exists a unique γ vs relative surface area (A/A ref) for those runs with large expansion rate (dA/dt), and therefore, there is nearly no surfactant molecule adsorbing onto the expanding interface during the expansion process. A unique curve relating γ and relative surface concentration (Γ/Γref) are then obtained. Surface Tension Calculation. The theoretical shape of the pendant bubble is derived according to the classical Laplace equation that relates the pressure difference across the curved fluid interface:16,17

γ[1/R1 + 1/R2] ) ∆P

(1)

where γ is the surface tension, R1 and R2 are the two principal radii of curvature of the surface, and ∆P is the pressure difference across the interface. For the pendant bubble geometry, eq 1 can be recast as a set of three first-order differential equations for the spatial positions x1 and x2 and turning angle φ of the interface as a function of the arc length s and then integrated by using a (15) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (16) Huh, C.; Reed, R. L. J. Colloid Interface Sci. 1983, 91, 472. (17) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169.

Runge-Kutta scheme18 with boundary conditions x1(0) ) x2(0) ) φ(0) ) 0. An objective function is defined as the sum of squares of the normal distance between the measured points and the calculated curve obtained from eq 1. The objective function depends on four unknown variables: the actual locations of the apex (X10 and X20), the radius of curvature at the apex (R0), and the capillary constant (B ) ∆FgR02/γ). The surface tension is obtained from the best fit between the theoretical curve and the data points by minimizing the objective function. Minimization equations are solved by directly applying the Newton-Raphson method,18 and from the optimal values of R0 and B, the tension can be computed. As demonstrated in Lin et al.,19,20 the accuracy and reproducibility of the dynamic surface tension measurements obtained by this procedure are ≈0.1 mN/m.

3. Theoretical Framework Mass Transfer in Bulk. The adsorption of surfactant molecules onto the interface of a freshly created pendant bubble in a quiescent surfactant solution is modeled. It is considered only the case of one-dimensional diffusion and adsorption onto a spherical interface from a bulk phase containing an initially uniform bulk concentration of the surface-active solute. The surfactant is assumed not to dissolve into the gas phase of the bubble. Diffusion in the bulk phase is assumed to be spherical symmetric, and convection is assumed to be negligible. The diffusion of surfactant in the bulk phase is described by Fick’s law:

∂C D ∂ 2 ∂C (r > b, t > 0) r ) 2 ∂r ∂r ∂t r

(

)

(2)

with the initial and boundary conditions

C(r,t) ) C0 (r > b, t ) 0)

(3a)

C(r,t) ) C0 (r f ∞, t > 0)

(3b)

dΓ/dt ) D(∂C/∂r) (r ) b, t > 0)

(3c)

Γ(t) ) 0 (t ) 0)

(3d)

where r and t are the spherical radial coordinate and time, D denotes the diffusion coefficient, C(r,t) is the bulk concentration, Γ(t) is the surface concentration, b is the bubble radius, and C0 is the concentration far from the bubble. By the Laplace transform, the solution of the above set of equations can be easily formulated in terms of unknown subsurface concentration Cs(t) ) C(r)b,t);

Γ(t) ) (D/b)[C0t -

∫t0Cs(τ) dτ] + 2(D/π)1/2[C0t1/2 ∫t0 Cs(t-τ) dxτ] 1/2

(4)

Adsorption Equations. To complete the solution for the surface concentration, the sorption kinetics must be specified. The model used here assumes that adsorption/ desorption is an activated process which is assumed to obey the following rate expression:21,22 the adsorption rate is proportional to the subsurface concentration Cs and the available surface vacancy (1 - Γ/Γ∞), and the desorption rate is proportional to the surface coverage Γ (18) Carnahan, B.; Luther, H. A.; Wilkes, J. O. Applied Numerical Methods; Wiley: New York, 1969. (19) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159. (20) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf. A 1996, 114, 31. (21) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: New York, 1995. (22) Aveyard, R.; Haydon, D. A. A Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, U.K., 1973; Chapters 1 and 3.

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dΓ/dt ) β exp(-Ea/RT)Cs(Γ∞ - Γ) - R exp(-Ed/RT)Γ (5) where β, R, Ea(Γ), and Ed(Γ) are the preexponential factors and the energies of activation for adsorption and desorption, respectively. Γ∞ is the maximum surface concentration, T the temperature, and R the gas constant. To account for enhanced intermolecular interaction at increasing surface coverage, the activation energies are assumed to be Γ dependent

Ea ) E0a + νaΓ Ed ) E0a + νdΓ

(6)

where E0a, E0d, νa, and νd are constants. Equation 5 in nondimensional form becomes

dx/dτ ) Ka exp(-νa*x)Cs*(1 - x) - Kd exp(-νd*x)x

(7)

where x ) Γ/Γ∞, τ ) tD/h2, h ) Γe/C0, Ka ) β exp(E0a/RT)C0/(D/h2), Cs* ) Cs/C0, Kd ) R exp(-E0d/RT)/(D/h2), νa* ) νaΓ∞/RT, νd* ) νdΓ∞/RT, and K ) νa* - νd*. At equilibrium, the time rate of change of Γ vanishes and the Frumkin adsorption isotherm that follows is given by

C Γ )x) Γ∞ C + a exp(Kx)

(8)

where K ) (νa - νd)Γ∞/RT and a ) (R/β) exp[(E0a - E0d)/RT]. Equation 8 becomes the Langmuir adsorption isotherm when νa ) νd ) K ) 0. The presence of cohesive intermolecular forces, which increase with surface coverage and which lower the desorption rate, is described by K < 0. A positive value of K indicates that the adsorption is anticooperative and adsorption becomes more difficult as the surface becomes more covered. Numerical Solution. The theoretical framework that describes the unsteady bulk diffusion of surfactant toward an initially clean pendant bubble and its effect on the surface tension has been formulated previously,4,10 and therefore only a brief outline is given here. The pendant bubble is treated as a sphere surrounded by an infinite, quiescent medium which at time t ) 0 contains a uniform concentration C0 of surfactant. The concentration of surfactant on the bubble surface is assumed to be equal to a constant initial surface concentration Γb. Γb ) 0 for a clean adsorption process, in which the bubble was created suddenly. Although, there is a small amount of surfactant which presents at the air/water interface before bubble growth. The surface area of pendant bubble increases roughly 1000 times during the rapid bubble growth and depletes this preadsorbed surfactant to a negligible surface coverage. When the surfactant solution can be considered ideal, the Gibbs adsorption equation dγ ) -ΓRT d ln C and the equilibrium isotherm (eq 8) allow for the calculation of the surface tension explicitly in terms of Γ:

γ - γ0 ) Γ∞RT[ln(1 - x) - Kx2/2]

(9)

where x ) Γ/Γ∞ and γ0 is the surface tension for the clean interface. When equilibrium data of the surface tension are fitted as a function of the bulk concentration using eqs 8 and 9, the equilibrium constants K, a, and the maximum coverage Γ∞ can be obtained.

Figure 1. Representative dynamic surface tensions for adsorption of C12E4 onto a fresh air-water interface for C0 ) (1) 0.60, (2) 1.0, (3) 1.3, (4) 2.0, (5) 4.0, (6) 6.0, (7) 10.0, (8) 15.0, (9) 20.0, and (10) 30.0 [10-9 mol/cm3].

When the adsorption process is controlled solely by bulk diffusion, the surface concentration can be obtained by solving eq 4, describing the mass transfer between sublayer and bulk, and eq 8, the sorption kinetics between subsurface and interface. If the adsorption process is of mixed control, eq 5 instead of eq 8 is solved coupled with eq 4 to find out the surface concentration. Then the dynamic surface tension γ(t) was calculated from eq 9. 4. Comparisons of Experimental Data and Theoretical Profiles Surface Tension Relaxation. Relaxation in the surface tension due to adsorption of C12E4 onto a clean air-water interface was measured. Data were recorded up to several hours for solutions of dilute concentration and 1 h for high concentration ones from the moment (referenced as t ) 0) at which half of the bubble volume has been generated during the bubble formation. Shown in Figure 1 are representative dynamic surface tension profiles (for one selected bubble at each bulk concentration) of C12E4 aqueous solutions at 10 different bulk concentrations, C ) 0.60, 1.0, 1.3, 2.0, 4.0, 6.0, 10.0,15.0, 20.0, and 30.0 [10-9 mol/cm3]. The reproducibility of these profiles is demonstrated in Figure 2, where the results of several pendant bubbles at six concentrations are given. The equilibrium surface tensions for C12E4 aqueous solutions at the air-water interface were extracted from the longtime asymptotes in Figures 1 and 2 and are plotted as the circles in Figure 3. Presented in Figure 3 also is the comparison between the C12E4 equilibrium data and the best fit from the adsorption isotherms of the Langmuir and Frumkin. The model constants, as shown in Table 1, are obtained by adjustment to minimize the error between the model predictions and experimental data. The Langmuir and Frumkin models both fit the equilibrium surface tension data very well. However, these two isotherms do predict quite different γ vs Γ relationships, shown as the inset in Figure 3, and this will result in different dynamic surface tension profiles and values of diffusion coefficient obtained using these two models. Note that the Frumkin adsorption isotherm predicts a positive K value from the equilibrium data. This positive K value indicates that the adsorption is anticooperative, and adsorption becomes more difficult as the surface becomes more covered. A similar result has

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Figure 3. Equilibrium surface tension for an air-C12E4 aqueous solution and the theoretical predictions of the Langmuir (L) and Frumkin (F) adsorption isotherms. The inset shows the relationship between surface tension and surface concentration predicted by the Langmuir and Frumkin models. Table 1. Model Constants of Optimal Fit of C12E4 Aqueous Solution

C12E4 C12E4 C12E8

modela

Γ∞ × 1010 (mol/cm2)

a × 1010 (mol/cm3)

F L F

4.663 3.905 2.668

3.521 4.660 0.2501

K

SEb

ref

1.875

0.79 2.24

this work this work 5, 6

5.186

F ) Frumkin, L ) Langmuir. SE ) summation error between equilibrium data and the best-fit profile from adsorption isotherm ∑i (∆γi)2, (mN/m)2. a

Figure 2. Dynamic experimental surface tensions for adsorption of C12E4 onto a clean air-water interface and the theoretical predictions of diffusion-control for the Langmuir (L) and Frumkin (F) models for C0 ) (1) 1.0, (2) 2.0, (3) 6.0, (4) 10.0, (5) 15.0, and (6) 30.0 [10-9 mol/cm3]. Each symbol represents a separate bubble.

b

surfactant activity a, and molecular interaction K used in obtaining the computed profiles represent the optimal fits of the γ - ln C equilibrium data. Numerical profiles were computed by adjusting the diffusion coefficient individually for each of the different bulk concentrations to achieve the best agreement with the data. As shown in Figure 2, the agreement between the theoretical relaxation profile obtained by using the Frumkin model is only slightly better than that by the Langmuir model. This is due to the fact that the value of K for C12E4 is only slightly greater than zero and the molecular interaction is not very significant. The values of diffusion coefficient, determined from the best fit between the dynamic tension data and the theoretical prediction, are plotted in Figure 4 for the Langmuir and Frumkin models. Data in Figure 4, for using both models, show a nearly constant diffusivity at various bulk concentrations but different values, D ) 6.4 and 7.4 [10-6 cm2/s], for the Frumkin and Langmuir model, respectively. The value of diffusivity is also close to what the Stokes-Einstein equation predicts. The Stokes-Einstein equation,25 based on a model in which a solute sphere is considered to move through a continuum of the solvent, is

also been obtained for polyethylene nonionic surfactants C10E8, C12E8, and C12E6.3-9 A different result, a negative K value, has also been reported for 1-decanol,10-12 1-octanol,13,14 7-tetradecyn-6,9-diol,23 and decanoic acid,24 which show cooperative adsorption. Diffusion Control. If the adsorption of C12E4 onto a fresh air-water interface is assumed to be diffusion controlled, Figure 1 may be used to determine the diffusivity of C12E4 molecules in the aqueous phase. The model constants of maximum surface concentration Γ∞,

where k is Boltzmann’s constant, T is temperature, RA is the sphere radius, and µB is the viscosity of the pure solvent. RA is estimated from the molecular weight (M ) 362.55 g/mol) and density (F ) 0.946 g/cm3 at 20 °C). The volume of each C12E4 is 4πRA3/3 ()M/FN), where N is

(23) Ferri, J. K.; Stebe, K. J. J. Colloid Interface Sci. 1999, 209, 1. (24) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637.

(25) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960; Chapter 16.

DAB ) kT/(6πRAµB)

(10)

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Figure 4. Values of the diffusion coefficient (D, 10-6 cm2/s) from the best fit between surface tension data and the model predictions for the Langmuir (L) and Frumkin (F) models as a function of bulk concentrations.

Avogadro’s number. DAB ) 4.1 × 10-6 cm2/s for C12E4 from eq 10. Hayduk and Buckley26 reported DAB for linear molecules to be some 30% greater than that for substances of spherical molecules. Therefore, it is concluded that the mass transport of C12E4 molecules onto a clean air-water interface is a diffusion-controlled process. This is different from those for other polyethylene nonionic surfactants C10E8, C12E8, and C12E6.3-9 For the latter three surfactants, values of diffusion coefficient all show a decrease at increasing bulk concentration and there exists a shift in the controlling mechanism. Limiting Adsorption Rate Constant. It has been reported3-9 that, for poly(oxyethylene) nonionic surfactant C10E8, C12E8, and C12E6, there exists a shift in the controlling mechanism from diffusion control at dilute concentration to mixed diffusive-kinetic control at more elevated bulk concentration. To examine the possibility for C12E4 to have a shifting control mechanism, a series of simulations in which surfactant adsorbed onto an initially clean, spherical interface from a bulk phase of initially uniform concentration were performed. The range of bulk concentration considered, 1 × 10-10-4 × 10-8 mol/ cm3, is one in which both the reduction of surface tension and the relaxation time are large enough for the dynamic surface tension measurement of C12E4. The Frumkin model with νa* ) 0 was picked for the simulation; the model constants (Γ∞, a, and K) utilized are those listed in Table 1. The effect of choosing a different νa* has been discussed in a previous paper.6 The relaxation in surface tension (γ) is a function of C0 and β1 ()β exp(-E0a/RT)). Simulations of γ as a function of time for different C0 are performed, in which β1 is varied from diffusion control [β1 or Ka f ∞] to mixed control [β1 ) O(105 or 106) or Ka ) O(1 or 10-1)] to sorption kinetic control [β1 < 104 or Ka < O(10-2)]. Figure 5 shows two representative surface tension relaxation profiles for C0 ) 1 × 10-10 and 4 × 10-8 mol/cm3. The relaxations of γ have the same general dependence on β1. The diffusionlimited curve shows the fastest relaxation; slower relaxations are observed as β1 decreases because of the increasing kinetic barrier. The distance between the diffusion-limited curve and that for a particular value of (26) Hayduk, W.; Buckley, W. D. Chem. Eng. Sci. 1972, 27, 1997.

Figure 5. Influence of the bulk concentration on the surface tension relaxation for clean interface adsorption of diffusion control (dashed curves) and of mixed control (solid curves). C0 ) (a) 1 × 10-10 and (b) 4 × 10-8 mol/cm3. β exp(-E0a/RT) ) (1) 104, (2) 105, (3) 106, (4) 107, (5) 2.5 × 105, and (6) 2.0 × 107 cm3/(mol.s).

β1 varies gradually with C0. The limiting value of β1 for which the mixed-controlled curve is indistinguishable from diffusion-controlled profiles is defined as βl. Lower Limit of β1. The limiting value of dimensionless adsorption rate constant (Kal) for which the mixedcontrolled curve is indistinguishable from diffusioncontrolled profiles is obtained and plotted in Figure 6. The corresponding values of adsorption rate constant βl ()β exp(-E0a/RT)) is also plotted in Figure 6 as a function of bulk concentration. Figure 6 indicates that, if the value of β1 of C12E4 is ranging between 5 × 105 and 3 × 107 cm3/(mol‚s), C12E4 is of diffusion control at dilute concentration and of diffusive-kinetic control at more elevated bulk concentration. In the above section, it has been concluded that the mass transport of C12E4 onto a clean air-water interface is of diffusion control. Therefore, all we can conclude from the theoretical simulation is that the adsorption rate constant β1 for C12E4 must be larger than 3 × 107 cm3/(mol‚s) or the desorption rate constant R1 ()R exp(-E0d/RT)) is equal to or larger than 1.1 × 10-2 s-1. These two numbers are also the lower limit of the sorption rate constant (β1 and R1) of the Frumkin model for C12E4. Surface Expansion. To examine which adsorption isotherm predicts better the adsorption behavior of C12E4, an expansion experiment was performed. The pendant bubble, at which the interface has reached the equilibrium

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Figure 6. Limiting adsorption rate constant βl (cm3 mol-1 s-1) and Kal as a function of bulk concentration C0 and the lower limit of the adsorption rate constant (the dashed line) of C12E4 predicted by the Frumkin model. D ) 6.4 × 10-6 cm2/s and νa* ) 0.

Figure 7. Experimental data of surface area (A) vs time (t) and surface tension (γ) vs time for the expansion experiment. C0 ) 4 × 10-9 mol/cm3. Each symbol represents a separate run with different expansion rate (dA/dt).

state, was expanded at different expansion rates and at different bulk concentrations. Figure 7 shows representative relaxation profiles of surface tension and surface area of pendant bubble during interface expansion at C ) 4 × 10-9 mol/cm3. The expansion rate of the interface is obtained from the slope of A vs t relationship (dA/dt) shown

Hsu et al.

Figure 8. Representative surface tension as a function of the surface area obtained from data in Figure 7.

in Figure 7b. The data in Figure 7 are replotted in Figure 8, showing the relationship of γ vs A. All data profiles show a similar tendency: surface tension increases with increasing surface area. With closer examination, one can find that the profile for a lower expansion rate, for example, the circles with 0.26 mm2/s, has a smaller rate of change in surface tension. It is clearer when we replot the surface tension profiles as a function of relative surface area (A/Aγ)50), in which the surface area at γ ) 50 mN/m is picked as the reference point for data profiles at C ) 4 × 10-9 mol/cm3. A similar process is done for the data at 1 × 10-9 and 6 × 10-9 mol/cm3. Figure 9 shows the relaxation profiles of surface tension as a function of the relative surface area. Surface areas at γ ) 60 and 42 mN/m are picked for data at 1 × 10-9 and 6 × 10-9 mol/cm3, respectively. Data in Figure 9b for C ) 4 × 10-9 mol/cm3 show clearly that when the expansion rate is higher than 6 mm2/s, there is a unique γ-A/Aγ)50 profile, while at a lower expansion rate, a larger deviation from the unique profile resulted. After expansion, the air-water interface with a 0.26 mm2/s expansion rate (the circles in Figure 9b) has the lowest value of surface tension at the same relative surface area A/Aγ)50. This indicates that there are a significant amount of surfactant molecules adsorbing onto the interface during the interface-expansion process if the expansion rate is too low. In the case of Figure 9b for C ) 4 × 10-9 mol/cm3, the limiting expansion rate is about 6 mm2/s in order for the process to have a negligible adsorption of surfactant molecules during the interface-expansion period for interfacial area increasing from 14 to 32 mm2. The limiting expansion rate is, of course, dependent on the bulk concentration and the working range of the surface area. For a wide working range of the interfacial area, a higher limiting expansion rate is needed. A lower bulk concentration should have a lower limiting expansion rate, as shown in Figure 9 (a and c). At C ) 1 × 10-9 mol/cm3, when the expansion rate is higher than 0.8 mm2/s, all data profiles show a unique γ-A/Aγ)60 dependence. At higher bulk concentration (C ) 6 × 10-9 mol/cm3, in Figure 9c), only the relaxation data with dA/dt larger than 7.6 mm2/s show a unique γ-A/Aγ)42 profile. The unique data profiles in Figure 9 (a-c) are recalculated and merged into a unique curve. Shown in Figure 10 are the γ-A/Aγ)64, γ-Γ/Γγ)64, and surface pressure (π ) γ0 - γ)-A/Aγ)64 relationships for C12E4 from the interface-expansion experiment. If the adsorption of surfactant molecules during the short expanding period

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Figure 10. Comparison between the experimental data and the profiles predicted from the Langmuir (solid curves) and Frumkin (dashed curves) adsorption isotherms for surface tension vs relative surface area (a), surface tension vs relative surface concentration (b), and surface pressure vs relative surface area (c, the inset).

Figure 9. Influence of the expansion rate (dA/dt, mm2/s) on the relationship between the surface tension and the relative surface area (A/Aref) for the expansion experiment at C0 ) (a) 1, (b) 4, and (c) 6 [10-9 mol/cm3].

is negligible, this unique profile represents the equilibrium surface property and can be utilized for examination of the accuracy of the adsorption isotherm applied to describe the equilibrium and dynamic surface tension data. Shown in Figure 10 are also the γ-A/Aref, γ-Γ/Γref, and π-A/Aref dependences predicted from the Langmuir (the solid curves) and Frumkin (the dashed curves) models. The Frumkin model does predict the equilibrium surface property from the interface-expansion experiment much better than the Langmuir model. However, both models predict the data only reasonably, not perfectly, well. At the region for γ > 65 and γ < 47 mN/m, the Frumkin isotherm predicts a lower surface tension, whereas at the middle surface tension region, it predict a higher surface tension. In brief, the Frumkin isotherm predicts a poor γ vs Γ dependence for C12E4, even though it predicts an excellent γ vs log(C) relationship, as shown in Figure 3. Therefore, the interface-expansion experiment does provide the second relationship for choosing the right

adsorption isotherm. A good isotherm has to follow both the γ-log(C) and γ-Γ equilibrium data from experimental measurement. Because the effect of interaction between the adsorbed molecular is not significant (i.e., molecular interaction parameter K is only slightly larger than zero), the generalized Frumkin model10 does not give a better prediction on both the equilibrium and dynamic surface tension relaxations than the Frumkin model. A new model is needed. A molecular reorientation model has been proposed recently;27,28 Pan using a molecular interaction model28 and Lin et al. proposing a phase transition model10,30 may be the candidates, but further examination is needed. 5. Discussion and Conclusions The adsorption of C12E4 onto a clean air-water interface is found to be anticooperative and of diffusion control. A diffusivity of 6.4 × 10-6 cm2/s is obtained from the bestfit (27) Fainerman, V. B.; Miller, R.; Wustneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 7669. (28) Liggieri, L.; Ferrari, M.; Massa, A.; Ravera, F. Colloids Surf. A 1999, in press. (29) Pan, R. A Study of Surface Equations of State and Transport Dynamics at the Air/Water Interface. Ph.D. Thesis, City University New York, New York, New York, 1996. (30) Lin, S. Y.; Hwang, W. B.; Lu, T. L.Colloids Surf. A 1996, 114, 131; 1996, 114, 143.

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between the dynamic surface tension data and the theoretical profiles predicted from the Frumkin model. Lower limits of the adsorption rate constant, 3 × 107 cm3/ (mol‚s) for β exp(-E0a/RT), and of the desorption rate constant, 1.1 × 10-2 s-1 for R exp(-E0d/RT), are obtained from the theoretical simulation. Data profiles relating γ and relative surface concentration Γ/Γref and relating γ and A/Aref are obtained from an interface-expansion experiment. The Frumkin adsorption isotherm predicts an excellent γ vs log(C) relationship but predicts the γ vs Γ dependence only reasonably well. More effort is needed for finding a better adsorption isotherm to fit more exactly the γ-log(C) and γ-Γ/Γref relationships and to predict more exactly the dynamic surface tension profiles. Both the Langmuir and Frumkin adsorption isotherms predict the equilibrium surface tension data, γ vs log(C), very well, but the fit on the dynamic surface tension profiles is not perfect. For the dynamic data profiles at low bulk concentration (for example, curves 1-3 in Figure 2), the Frumkin model predicts a lower surface tension at γ > 65 mN/m and a higher value at γ < 65 mN/m. For the dynamic data profiles at high bulk concentration (for example, curves 5 and 6 in Figure 2), the Frumkin model predicts a higher surface tension at γ > 47 mN/m and a

Hsu et al.

lower tension at γ < 47 mN/m. These deviations may cause trouble and error on evaluating the diffusion coefficient. Note that this deviation is from the inaccurate prediction on the γ-Γ/Γref data profile. Because the Frumkin adsorption isotherm predicts a lower surface tension at γ > 65 mN/m and γ < 47 mN/m (Figure 10), this lower prediction on γ on these two regions causes the above lower tension prediction of the dynamic data profiles on the region of γ > 65 mN/m (see curve 1) and γ < 47 mN/m (see curve 6 if Figure 2). The higher tension prediction on the region of 65 < γ < 47 mN/m in the γ-Γ/Γref data profile makes the surface tension predicted by the Frumkin model always have a higher surface tension at this tension region on the dynamic surface tension profiles. This deviation behavior is shown clearly in Figure 2. Currently we are in the process of looking at a better adsorption isotherm for predicting both the γ-log(C) and γ-Γ/Γref data more precisely in our laboratory. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 84-2214-E-011-019). LA9912444