Adsorption and Desorption Kinetics of C12E4 on Perturbed Interfaces

Two processes for the mass transport of C12E4 in a quiescent surfactant solution were studied: (i) the desorption of C12E4 out of a suddenly compresse...
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Langmuir 2000, 16, 4846-4852

Adsorption and Desorption Kinetics of C12E4 on Perturbed Interfaces Ching-Tien Hsu, Ming-Jian Shao, Ya-Chi Lee, and Shi-Yow Lin* Department of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Sec. 4, Taipei, 106 Taiwan, Republic of China Received October 12, 1999. In Final Form: February 1, 2000 Two processes for the mass transport of C12E4 in a quiescent surfactant solution were studied: (i) the desorption of C12E4 out of a suddenly compressed interface; (ii) the adsorption of C12E4 onto a suddenly expanded air-water interface. A video-enhanced pendant bubble tensiometer was used for the measurement of surface tension relaxation due to the adsorption or desorption of surfactant molecules. Both processes were found to be diffusion-controlled at dilute concentration but diffusive-kinetic mixed-controlled at more elevated bulk concentration for C12E4. Rate constants of adsorption/desorption were evaluated from the comparison between the surface tension profiles and the theoretical profiles predicted from the Frumkin model. The values of the kinetic rate constants of C12E4 obtained from these two experiments are nearly the same. A theoretical simulation was also performed to confirm the shifting mechanism (J. Chem. Eng. Jpn. 1996, 29, 634) for C12E4 for both desorption and expansion processes.

Introduction It has been reported that the controlling mechanism of adsorption onto a freshly created air-water interface for water soluble nonionic surfactants C10E8, C12E8, and C12E6 is of diffusion control at dilute concentration whereas shifts to be of mixed control at more elevated concentration.1-6 The desorption process for C10E8 and C12E8 out of a suddenly compressed interface has been confirmed to be diffusive-kinetic mixed-controlled.4,7 However, the adsorption of C12E4 onto a freshly created air-water interface has been reported to be solely diffusion-controlled.8 Due to the large polar group, the poly(oxyethylene) surfactant shows a behavior significantly different from that of the surfactant molecules with a small polar head like 1-hexanol,9 1-octanol (C8E0),10-13 1-decanol (C10E0),14-16 and 1-decanoic acid.17 1-Octanol is diffusion-controlled for both adsorption and desorption processes, 1-decanol has a diffusion-controlled adsorption and a mixed-controlled desorption, and 1-decanoic acid is of mixed control for the * To whom correspondence should be addressed. Tel: 886-22737-6648. Fax: 886-2-2737-6644. E-mail: [email protected]. (1) Lin, S. Y.; Chang, H. C.; Chen, E. M. J. Chem. Eng. Jpn. 1996, 29, 634. (2) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (3) Pan, R.; Green, J.; Maldarelli, C. J. Colloid Interface Sci. 1998, 205, 213. (4) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Langmuir 1998, 14, 2476. (5) Liggieri, L.; Ferrari, M.; Massa, A.; Ravera, F. Colloids Surf. A 1999, in press. (6) Miller, R.; Aksenenko, E. V.; Liggieri, L.; Ravera, F.; Ferrari, M.; Fainerman, V. B. Langmuir 1999, in press. (7) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (8) Hsu, C. T.; Shao, M. J.; Lin, S. Y. Langmuir, in press. (9) Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 1996, 178, 168. (10) Chang, C. H.; Franses, E. I. Chem. Eng. Sci. 1994, 49, 313. (11) Chang, C. H.; Franses, E. I. Colloids Surf. A 1995, 100, 1. (12) Lin, S. Y.; Hwang, W. B.; Lu, T. L.Colloids Surf. A 1996, 114, 143. (13) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526. (14) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1994, 166, 73. (15) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (16) Lin, S. Y.; Wang W. J.; Hsu, C. T. Langmuir 1997, 13, 6211. (17) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637.

adsorption process. Here, adsorption means the adsorption of surfactants onto a fresh air-water interface, and desorption is the reequilibration of surfactants desorbing from an overcrowded air-water interface due to a sudden shrinkage of the air-water interface. The effect of EO number on the adsorption kinetics has been also reported by Liggieri et al.5 and Fainermann et al.18 By using a pendant bubble as a monolayer film balance, Pan19 and Kwok et al.20,21 shrank and/or expanded a pendant bubble to compress and/or dilute the surfactant monolayer for studying the surfactant equations of state. In this work, the adsorption and desorption processes of C12E4 onto and out of a suddenly perturbed air-water interface in a quiescent aqueous solution were examined. The relaxations of surface tension due to the adsorption and desorption of C12E4 were detected by a video-enhanced pendant bubble tensiometer. The surface tension relaxation profiles were then compared with the theoretical profiles predicted from the Frumkin model. Values of adsorption and desorption rate constants were evaluated from the comparison. Experimental Measurements 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. The value of the surface tension of air-water, using the pendant bubble technique described in the following, was 72.0 mN/m at 25.0 ( 0.1 °C. Pendant Bubble Apparatus. The dynamic surface tension for a suddenly perturbed (compression or expansion) air-water interface in a quiescent C12E4 solution was measured by a videoenhanced pendant bubble tensiometer. The system is similar to (18) Fainerman, V. B.; Miller, R.; Makuevski, A. V. Langmuir. 1995, 11, 3054. (19) Pan, R. A Study of Surface Equations of State and Transport Dynamics at the Air/Water Interface. Ph.D. Thesis, City University of New York, NY, 1996. (20) Kwok, D. Y.; Vollhardt, D.; Miller, R.; Li, D.; Neumann, A. W. Colloids Surf. A 1994, 88, 51. (21) Kwok, D. Y.; Tadros, B.; Deol, H.; Vollhardt, D.; Miller, R.; Cabrerizo-Vilchez, M. A.; Neumann, A. W. Langmuir 1996, 12, 1851.

10.1021/la991338d CCC: $19.00 © 2000 American Chemical Society Published on Web 04/22/2000

Kinetics of C12E4 on Perturbed Interfaces

Langmuir, Vol. 16, No. 11, 2000 4847

the one that investigated the adsorption onto a freshly created air-water interface of C12E4 aqueous solution; therefore, only a brief description is given here.7,8 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 was (0.1 K. A 16-gauge stainless steel inverted needle (0.047 in. i.d.; 0.065 in o.d.), which was connected to the normally closed port of a threeway miniature solenoid valve, was used for the bubble generation. The common port of the valve, controlled by a computer, was connected to a gastight Hamilton syringe placed in a syringe pump. The silhouette image was digitized into 480 lines × 512 pixels with 8-bit resolution. The edge was defined as the x or z position which corresponds to an intensity of 127.5.22 Adsorption. The experimental protocol was as follows: The quartz cell with inside diameter of 26 × 41 × 43 mm 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. Sequential digital images of the bubble were taken. Surface Compression. After the adsorption of C12E4 had reached the equilibrium state, the valve was opened for 0.11 s (controlled by a computer) while the syringe pump was off. A small part of air inside the bubble was allowed to pass through the solenoid valve, and the surface area of pendant bubble decreased abruptly around 10-16%. The images were recorded on a recorder during the shrinkage of bubble and also taken sequentially onto the computer. After the relaxation of the reequilibration process was complete, the images on computer and tape were processed, by the edge detection routine, to determine the bubble edge coordinates, bubble volume, bubble surface area, and surface tension. Four different bulk concentrations [C ) 1.0, 2.0, 4.0 and 6.0 (10-9 mol/cm3)] were chosen for the experiment of surface compression. At each concentration, the compression experiments were performed 4-7 times. Surface Expansion. For some pendant bubbles, after the adsorption had reached the equilibrium state, the air-water interface was fast expanded while the syringe pump and solenoid valve was energized and air was allowed to pass through the needle. While the surface area increased around 30 to 80%, the valve was closed. The surface concentration of surfactant molecules at the air-water interface was depleted due to the expansion of surface area of bubble. Therefore, an adsorption process was energized since the surface coverage was lower than the equilibrium value after the fast expansion. The relaxation of surface tension was monitored via the sequential digital images taken of the pendant bubble. Two different bulk concentrations [C ) 1.0 and 6.0 (10-9 mol/cm3)] were chosen for this reequilibration experiment of surface expansion. At each concentration, the expansion experiments were performed 3-5 times. After the experiments, the bubble images were processed in order to determine the edge coordinates of bubble, bubble volume, surface area of bubble, and surface tension. Relaxations of surface tension (γ) and surface area (A) were obtained. Surface Tension Calculation. The theoretical shape of the pendant bubble was derived according to the classical Laplace equation that relates the pressure difference across the curved fluid interface:23,24

Figure 1. Representative dynamic surface tensions and surface area of the pendant bubble for the reequilibration process due to a sudden compression of the interface for C12E4 aqueous solution. C0 ) 2.0 (a) and 6.0 (b) (10-9 mol/cm3). recast as a set of three first-order differential equations for the spatial positions x and z and turning angle φ of the interface as a function of the arc length s and then was integrated with boundary conditions x(0) ) z(0) ) φ(0) ) 0. As demonstrated in Lin et al.,25,26 the accuracy and reproducibility of the dynamic surface tension measurements obtained by this procedure are ca. 0.1 mN/m.

Experimental Results

Here 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 was

Consider first the desorption of C12E4 molecules from an overcrowded air-water interface to the bulk phase due to a suddenly shrinkage of the pendant bubble in a quiescent surfactant solution. Two representative relaxation profiles of surface tension (the circles) and surface area (the triangles) of pendant bubble are shown in Figure 1. For example, the surface tension decreased from the equilibrium value (47.24 mN/m for C ) 6.0 × 10-9 mol/ cm3, in Figure 1b) to a lower value (41.60 mN/m), corresponding to a higher surface coverage than the equilibrium one, in 0.167 s. The surface tension then increased and went back to the equilibrium tension in about 1000 s. The bubble surface area decreased 12% in 1/ 30 s and then kept a nearly constant value. Shown in Figure 1a is another desorption relaxation profile at C ) 2.0 × 10-9 mol/cm3. All the relaxation data showed a similar behavior: surface tension decreased abruptly from the equilibrium value to a lower one and then increased smoothly up to its equilibrium value after the abrupt falling. Figure 1 shows the dynamic surface tension profile starting from the equilibrium value (γe; at this moment, surface area was Ae and surface concentration was Γe), before the interface been perturbed, to the end of the relaxation. Data in Figure 1 were replotted in Figure 2, and the moment with the lowest surface tension value (γb; at this moment, surface area was Ab and surface coverage was Γb, for example, the point L at t ) 0.23 s in Figure 1b) was set to be the zero time for the convenience

(22) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (23) Huh, C.; Reed, R. L. J. Colloid Interface Sci. 1983, 91, 472. (24) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169.

(25) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159. (26) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf. A 1996, 114, 31.

γ[1/R1 + 1/R2] ) ∆P

(1)

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before the interface been expanded, to the end of the relaxation. Data in Figure 3 were replotted in Figure 4, and the moment with the highest surface tension value (γb; at this moment, surface area is Ab and surface coverage is Γb, corresponding to the end of expansion; for example, the point H at t ) 1.42 s in Figure 3b) was set to be the zero time for the convenience of the theoretical simulation. Table 2 presents two sets of surface properties (surface tension γ, surface area A, surface concentration Γ, and the amount of surfactants AΓ at the bubble surface) during the expansion of bubble that has been shown in Figure 3. The data in Table 2 (the column of AΓ, showing the amount of C12E4 molecules at the bubble surface) indicate that the adsorption during the ramp type area change was nearly negligible for the present system. Therefore, the adsorption during the expansion process was neglected in the following theoretical simulation. In each set of data, two Γ and AΓ, one from the prediction of the Frumkin model and the other from the experimental surface equation of state (γ vs Γ/Γref), are presented, and more discussion will be given in the Discussion section. Theoretical Framework Figure 2. Dynamic surface tensions and the theoretical predictions of mixed-controlled reequilibration of the Frumkin model for different adsorption rate constants [β1 ) β exp(E0a/RT)] for C12E4. C0 ) 2.0 (a) and 6.0 (b) (10-9 mol/cm3). DC denotes the diffusion-controlled curves.

of the theoretical calculation. Table 1 present two sets of surface properties (surface tension γ, surface area A, surface concentration Γ, and the amount of surfactants AΓ at the bubble surface) during the shrinkage of bubble that has been shown in Figure 1. The data in Table 1 (the column of AΓ, showing the amount of C12E4 molecules at the bubble surface) indicate that the desorption during the ramp type area change was insignificant for the present system. The desorption of C12E4 molecules during the compression process was therefore neglected in the following theoretical simulation. Two sets of Γ and AΓ, one from the prediction of the Frumkin model and the other from the experimental surface equation of state (γ vs Γ/Γref), are presented and more discussion will be given in the Discussion section. Consider next the adsorption of C12E4 molecules onto a depleted air-water interface due to a sudden expansion of the pendant bubble in a quiescent surfactant solution. Two representative relaxation profiles of surface tension (the circles) and surface area (the triangles) of pendant bubble are shown in Figure 3. For example, the surface tension increased from the equilibrium value (47.24 mN/m for C ) 6.0 × 10-9 mol/cm3, in Figure 3b) up to a higher value (56.80 mN/m), corresponding to a lower surface coverage than the equilibrium one, in 0.82 s. The surface tension increased during the expansion of bubble and then decreased and went back to the equilibrium tension in several hundred seconds. The surface area of pendant bubble increased 37% during the expansion and then kept a nearly constant value. Shown in Figure 3a is another desorption relaxation profile at C ) 1.0 × 10-9 mol/cm3. All the relaxation data showed a similar behavior: surface tension increased in about 1.4 s from the equilibrium value to a higher one, corresponding to the end of bubble expansion, and then decreased smoothly down to its equilibrium value. Figure 3 shows the dynamic surface tension profile starting from the equilibrium value (γe; at this moment, surface area was Ae and surface concentration was Γe),

Mass Transfer in Bulk. It was considered only the case of one-dimensional diffusion and the adsorption/ desorption onto/out of a spherical interface from a bulk phase containing an initially uniform bulk concentration of C12E4. The surfactant molecules were assumed not to dissolve into the gas phase of the bubble. Diffusion in bulk phase was assumed to be spherical symmetric, and convection was assumed to be negligible. The diffusion of surfactant in the bulk phase is described by Fick’s law

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

( )

(2)

with the following 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) ) Γb t ) 0

(3d)

Here 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, C0 is the concentration far from the bubble, and Γb is the initial surface concentration. By using the Laplace transform, the above set of equations can be solved in terms of unknown subsurface concentration Cs(t) ) C(r ) b, t):

∫0tCs(τ) dτ] + 2(D/π)1/2 [C0t1/2 - ∫0xtCs(t - τ) dxτ]

Γ(t) ) Γb + (D/b)[C0 t -

(4)

Adsorption Equations. To complete the solution for the surface concentration, the sorption kinetics must be specified:

dΓ/dt ) β exp(-Ea/RT)Cs(Γ∞ - Γ) - R exp(-Ed/RT)Γ (5) Here 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 concentra-

Kinetics of C12E4 on Perturbed Interfaces

Langmuir, Vol. 16, No. 11, 2000 4849

Table 1. Relaxations of Surface Properties during the Shrinkage of Bubble t (s)

γ (mN/m)

A (mm2)

Ai/Ae

1010Γa (mol/cm2) 10-9

-1/30 0 1/ 60 1/ 30 3/ 30 4/ 30 5/ 30 6/ 30

56.25 56.21c 56.13d 55.44d 52.55d 52.23e 52.37 52.28

22.25 22.24 22.16 22.05 19.79 19.80 19.80 19.79

Run 1 (C ) 2 × 1.00 1.00 1.00 0.99 0.89 0.89 0.89 0.89

-1/30 0 1/ 30 2/ 30 3/ 30 4/ 30 5/ 30 6/ 30 7/ 30

47.31 47.27c 47.01d 44.25d 42.61d 41.97d 41.60e 41.66 41.75

21.87 21.86 21.76 19.19 19.24 19.25 19.29 19.28 19.28

Run 2 (C ) 6 × 10-9 mol/cm3) 1.00 3.663 1.00 3.666 1.00 3.682 0.88 3.845 0.88 3.929 0.88 3.961 0.88 3.978 0.88 3.975 0.88 3.971

AiΓi/AeΓea

Γi/Γγ)60b

AiΓi/AeΓeb

1.00 1.00 1.00 1.02 0.99 1.00 1.00 1.00

1.176 1.178 1.182 1.213 1.345 1.360 1.354 1.357

1.00 1.00 1.00 1.02 1.02 1.03 1.02 1.03

1.00 1.00 1.00 0.92 0.94 0.95 0.96 0.96 0.96

1.579 1.581 1.592 1.719 1.793 1.821 1.837 1.834 1.831

1.00 1.00 1.00 0.95 1.00 1.01 1.03 1.02 1.02

mol/cm3)

2.925 2.930 2.938 3.007 3.273 3.300 3.289 3.296

a The surface coverage calculated from the value of surface tension using eq 8 with the Frumkin model. b The relative surface concentration calculated from γ by applying the relationship of γ vs Γ/Γγ)60, which was obtained from the fast expansion of pendant bubble. c The point right before the desorption process, corresponding to the equilibrium state. d The point during the shrinkage of bubble. e The point with the lowest surface tension, corresponding to the end of shrinkage and the beginning of the desorption process.

Figure 3. Representative dynamic surface tensions and surface area of the pendant bubble for the reequilibration process due to a fast expansion of the interface for C12E4 aqueous solution. C0 ) 1.0 (a) and 6.0 (b) (10-9 mol/cm3).

tion, T is the temperature, and R is the gas constant. To account for enhanced intermolecular interaction at increasing surface coverage, the activation energies were assumed to be Γ dependent:

Ea ) E0a + νaΓ Ed ) E0d + νdΓ

(6)

Here E0a, E0d, νa, and νd are constants. At equilibrium, the time rate of change of Γ vanished and the adsorption isotherm that follows was given by

Γ C )x) Γ∞ C + a exp(Kx)

Figure 4. Dynamic surface tensions and the theoretical predictions of mixed controlled reequilibration of the Frumkin model for different adsorption rate constants [β1 ) β exp(E0a/RT)] for C12E4. C0 ) 1.0 (a) and 6.0 (b) (10-9 mol/cm3). DC denotes the diffusion-controlled curves.

Here K ) (νa - νd)Γ∞/RT and a ) (R/β)exp[(E0a - E0d)/RT]. Equation 7 becomes the Langmuir adsorption isotherm when νa ) νd ) K ) 0. Numerical Solution. The theoretical framework that describes the unsteady bulk diffusion of surfactant toward a depletive interface and out of an overcrowded interface of a pendant bubble and its effect on the surface tension has been formulated previously.7,27 When the surfactant solution can be considered ideal, the Gibbs adsorption equation dγ ) - ΓRT dln C and the equilibrium isotherm (eq 7) allow for the calculation of the surface tension

(7) (27) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555.

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Table 2. Relaxations of Surface Properties during the Expansion of Bubble t (s)

γ (mN/m)

A (mm2)

Ai/Ae

1010Γa (mol/cm2) 10-9

0.39 0.50 0.61 0.71 0.82 0.93 0.99 1.10 1.21 1.32 1.43 1.87 2.25 2.69

60.89c 59.90d 61.17d 61.90d 62.40d 62.71d 63.71d 64.28d 64.68d 65.21d 65.65d 66.71e 66.66f 66.53f 66.55f

20.78 19.36 20.24 21.13 22.01 22.81 23.69 24.51 25.22 26.05 26.83 29.95 29.95 29.96 29.98

Run 3 (C ) 1 × 1.00 0.93 0.97 1.02 1.06 1.10 1.14 1.18 1.21 1.25 1.29 1.44 1.44 1.44 1.44

0.38 0.49 0.60 0.71 0.82 0.93 0.98 1.09 1.20 1.31 1.42 1.86 2.25 2.69

47.18c 43.89d 45.49d 47.31d 48.58d 50.01d 51.12d 52.35d 53.16d 54.25d 55.04d 56.80e 56.47f 56.15f 55.88f

19.90 17.73 18.54 19.37 20.21 20.98 21.79 22.59 23.29 24.14 24.94 27.21 27.24 27.25 27.25

Run 4 (C ) 6 × 10-9 mol/cm3) 1.00 3.671 0.89 3.864 0.93 3.775 0.97 3.663 1.02 3.578 1.06 3.476 1.10 3.390 1.14 3.290 1.17 3.221 1.21 3.122 1.25 3.046 1.37 2.867 1.37 2.902 1.37 2.935 1.37 2.963

AiΓi/AeΓea

Γi/Γγ)64b

AiΓi/AeΓeb

1.00 0.98 0.95 0.95 0.96 0.98 0.94 0.93 0.92 0.90 0.89 0.87 0.87 0.89 0.89

1.161 1.214 1.146 1.107 1.080 1.063 1.006 0.974 0.951 0.920 0.893 0.827 0.830 0.838 0.837

1.00 0.97 0.96 0.96 0.99 1.01 0.99 0.99 0.99 0.99 0.99 1.03 1.03 1.04 1.04

1.00 0.94 0.96 0.97 0.99 1.00 1.01 1.02 1.03 1.03 1.04 1.07 1.08 1.09 1.10

1.924 2.096 2.013 1.917 1.849 1.771 1.710 1.641 1.596 1.535 1.490 1.391 1.409 1.427 1.443

1.00 0.97 0.98 0.97 0.98 0.97 0.97 0.97 0.97 0.97 0.97 0.97 1.00 1.02 1.03

mol/cm3)

2.379 2.507 2.340 2.240 2.168 2.121 1.965 1.875 1.808 1.715 1.637 1.436 1.445 1.471 1.466

a The surface coverage calculated from the value of surface tension using eq 8 with the Frumkin model. b The relative surface concentration calculated from γ by applying the relationship of γ vs Γ/Γγ)64, which was obtained from the fast expansion of pendant bubble. c The point right before expansion, corresponding to the equilibrium state. d The point during the expansion of bubble. e The point with the highest surface tension, corresponding to the end of expansion and the beginning of reequilibration. f The point during the reequilibration process; surfactants adsorb onto the interface.

explicitly in terms of Γ:

γ - γ0 ) Γ∞RT[ln(1 - x) - Kx2/2]

(8)

Here x ) Γ/Γ∞ and γ0 is the surface tension for the clean interface. By the fitting of equilibrium data of the surface tension as a function of the bulk concentration using eqs 7 and 8, the equilibrium constants (molecular interaction K, surfactant activity a, and the maximum coverage Γ∞) can be obtained. When the adsorption/desorption process was controlled solely by bulk diffusion, the surface concentration could be obtained by solving eq 4, describing the mass transfer between sublayer and bulk, and eq 7, the sorption kinetics between subsurface and interface. If the adsorption process was of mixed control, eq 5 instead of eq 7 was solved coupled with eq 4 to find out the surface concentration. Then the dynamic surface tension γ(t) was calculated from eq 8. Shift in Controlling Mechanism. It has been reported1-4 that for poly(oxyethylene) nonionic surfactants C10E8, C12E8, and C12E6, there exists a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusion-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 C12E4 desorbed out of an initially overcrowded, spherical surface and C12E4 adsorbed onto a suddenly depleted air-water interface from a bulk phase of initially uniform concentration were performed. The ranges of bulk concentration considered, 5 × 10-10 to 7 × 10-9 mol/cm3 for compression perturbation and 3 × 10-10 to 4 × 10-8 mol/cm3 for expansion perturbation, are ones that the

reduction or increase of surface tension and the relaxation time are both 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 that, adapted from ref 8, obtained from the best fit between the equilibrium surface tension data and the model prediction of the Frumkin adsorption isotherm: Γ∞ ) 4.883 ×10-10 mol/cm2; a ) 3.521 ×10-10 mol/cm3; K )1.875. The effect of choosing a different νa has been discussed in a previous article.7 Comparisons of Dynamic Data and Theoretical Profiles Compression. The adsorption of C12E4 onto a freshly created air-water interface has been shown to be of diffusion control, and the value of diffusion coefficient has been obtained in a previous study (D ) 6.4 × 10-6 cm2/s), where the Frumkin model was utilized to simulate the mass transport of C12E4 in water.8 This diffusivity was used in the following calculation. For this reequilibration process, an initially equilibrium-established air-water interface was suddenly compressed; the surface tension relaxation profiles are shown in Figure 2. If this reequilibration process was assumed to be diffusion-controlled, the diffusion-controlled relaxation profiles predicted by the Frumkin model with diffusivity D ) 6.4 × 10-6 cm2/s are shown in Figure 2 as the dashed curves. It was found that the diffusion-limited relaxation profiles obtained from the Frumkin model predicted the dynamic surface tension data very well for the runs at C ) 1.0, 2.0, and 4.0 (10-9 mol/cm3) (e.g., Figure 2a), however, departed significantly from the dynamic data at C ) 6.0 × 10-9 mol/cm3 (e.g.,

Kinetics of C12E4 on Perturbed Interfaces

Figure 5. Values of adsorption rate constant β1 [)β exp(E0a/RT)] evaluated from the reequilibration process of C12E4 due to a sudden compression (squares) and expansion (circles) of a pendant bubble for C0 ) 6.0 × 10-9 mol/cm3.

Figure 2b). Therefore, this reequilibration process is of diffusion control at C e 4.0 ×10-9 mol/cm3 and is diffusivekinetic mixed-controlled at more evaluated concentration (C g 6.0 ×10-9 mol/cm3). Theoretical relaxation profiles, using the Frumkin model, with a finite adsorption rate constant [β1 ) β exp(-E0a/RT)] were calculated and plotted in Figure 2 (the solid curves). From the comparison between the dynamic surface tension data and the model-predicted profiles, it was found that the desorption rate constant (β1) is about the same for different experimental runs, 1.5 × 107 and 2.0 × 107 cm3/(mol‚s) as shown in Figure 5 (squares). Expansion. Two relaxation profiles of surface tension due to the adsorption of C12E4 onto a fast expanded airwater interface are shown in Figure 4. If this reequilibration process was assumed to be of diffusion control, the diffusion-controlled relaxation profiles predicted by the Frumkin model with D ) 6.4 × 10-6 cm2/s are shown in Figure 4 as the dashed curves. The comparison in Figure 4 indicated the diffusion-limited relaxation profiles obtained from the Frumkin model predict the dynamic surface tension data reasonably well for the runs at C ) 1.0 × 10-9 mol/cm3 (Figure 4a), however, depart significantly from the dynamic data at C ) 6.0 × 10-9 mol/cm3 (Figure 4b). Therefore, this reequilibration process is of diffusion control at C ) 1.0 × 10-9 mol/cm3 and is diffusivekinetic mixed-controlled at more evaluated concentration (C ) 6.0 × 10-9 mol/cm3). Theoretical relaxation profiles, using the Frumkin model, with a finite adsorption rate constant (β1) were calculated and plotted in Figure 4 (the solid curves). From the comparison between the dynamic surface tension data and the model-predicted profiles, it was found that β1 is about the same for different experimental runs, ranging between 6 × 106 and 1 × 107 cm3/(mol‚s), as shown in Figure 5 (circles). β1, obtained from the expansion experiment for C12E4 adsorption onto a suddenly depleted interface, is a little bit lower than, but close to, that from the compression experiment for C12E4. Simulation. To confirm the shift of the controlling mechanism for C12E4 on the reequilibration processes due to the compression and expansion of the air-water interface, a theoretical simulation was performed. The Frumkin model, with model constants Γ∞ ) 4.883 × 10-10 mol/cm2, a ) 3.521 × 10-10 mol/cm3, and K ) 1.875 that best-fit with the equilibrium surface tension, with a diffusivity of D ) 6.4 × 10-6 cm2/s was utilized.8 The theoretical simulation procedure has been detailed before;1 therefore, only the simulation resulted is presented here. Figure 6 shows the relationship of βl vs C (surfactant concentration). Here, βl is the lower limit of β1, with which

Langmuir, Vol. 16, No. 11, 2000 4851

Figure 6. Limiting adsorption rate constant βl (cm3 mol-1 s-1) as a function of bulk concentration C0 for the adsorption of C12E4 onto a fresh interface (curve A) and the reequilibration processes due to a sudden compression (curve C) and expansion (curve E) of a pendant bubble. The broken box indicates the adsorption rate constant β1 of C12E4 evaluated from the reequilibration experiment.

the surface tension relaxation profile is indistinguishable from the diffusion-controlled one. Curves C (squares) and E (circles) are for this reequilibration process due to the compression (12% of surface area) and fast expansion (50% of surface area) of air-water interface, respectively. The limiting adsorption rate constant βl increases from 4 × 106 to 7 × 108 cm3/(mol‚s) for curve C for the reequilibration due to a sudden compression of the interface. βl increases from 2 × 106 to 2 × 108 cm3/(mol‚s) for curve E for the reequilibration due to a fast expansion of the interface. There does exist a possibility for both reequilibration processes to be diffusion-controlled at dilute concentration and shift to a diffusive-kinetic mixedcontrolled process at more elevated concentration. Values of β1 evaluated from two reequilibration processes in this work (see Figure 5) are plotted in Figure 6 as a broken rectangular box at the experimental concentration range of C12E4. It indicates clearly that cures C and E cross the broken box. This is evidence of a shifting mechanism for these two equilibration processes. Curve A (the triangles) represents βl for the adsorption of C12E4 onto a freshly created air-water interface. βl increases from 1 × 106 to 2 × 107 cm3/(mol‚s) at increasing C12E4 concentration. Theoretically, there exists also a possibility for this clean adsorption process to be of diffusion control at dilute concentration and shifts to a diffusive-kinetic mixed-controlled process at more elevated concentration. However, the value of β1 evaluated from these two reequilibration experiments is about 1.1 × 107 cm3/(mol‚s). It is above or too close to curve A to see the mixed-controlled effect for the clean adsorption process. Discussion and Conclusions If C12E4 does follow what the Frumkin model predicts, the mass transport of C12E4 in a quiescent surfactant solution on a perturbed air-water interface is confirmed to be diffusion-controlled at dilute concentration and diffusive-kinetic mixed-controlled at more elevated bulk concentration. Rate constants of adsorption and desorption were evaluated from the comparison between the dynamic surface tension data and the theoretical profiles of the Frumkin model: β exp(-E0a/RT) ) 1.1 × 107 cm3/(mol‚s) and R exp(-E0d/RT) ) 3.9 × 10-3 s-1 at 25 °C. For some of the surface tension relaxation profiles, there exists an overshoot in data at the end of the relaxation, as shown in Figure 2b, for the reequilibration process due

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to a surface compression. We do not know exact the reason for the overshoot of surface tension, but this is probably due to the imperfect assumption of a spherical shape of the pendant bubble with a radius of apex. The actual bulk diffusion may be larger, especially at large time, than what the theory predicts because of the smaller radius of curvature for the interface near the neck region of the pendant bubble. In the bubble expansion process, there exists an initial negative excursion in dynamic surface tension. When the solenoid valve is open, the volume of air of the pendant bubble decreases for a very short time before the bubble grows up. Therefore, both the surface area and surface tension go down for a very short time before they go up (see the data ranging between 0.2 and 0.4 s in Figure 3. In the above theoretical simulation, it was assumed the reequilibration processes begin at the moment with the lowest (point L in Figure 1b) or the highest (point H in Figure 3b) surface tension after the air-water interface been perturbed. In other words, the desorption or adsorption during the compression or expansion was assumed to be negligible. Detailed in Tables 1and 2 are four sets of experimental runs of surface properties (γ, A, Γ, and AΓ at the bubble surface) during the shrinkage or expansion of bubble. Examine the data in the column of AiΓi/AeΓe (the amount of C12E4 molecules at the bubble surface) in detail. Consider first the data predicted from the Frumkin model: AiΓi during the perturbation varied 2 and 8% for the runs of bubble shrinkage and 10 or 13% for the runs of bubble expansion. Although the Frumkin adsorption isotherm describes nearly exactly the equilibrium surface tension, it does not predict perfectly the slope (i.e., the surface concentration) of the γ vs ln C curve. Figure 7 shows that the Frumkin isotherm does not describe well the experimental data of surface equation of state (γ vs Γ/Γref). If one examine the data of AiΓi/AeΓe calculated from the data of surface equation of state, it is found the amount of C12E4 molecules at the bubble surface during the bubble shrinkage or expansion varies only 3 or 4% at most. Therefore, the adsorption or desorption during the expansion and compression was ignored in this study.

Hsu et al.

Figure 7. Comparison between the experimental data and the profiles predicted from the Langmuir (L, Γ∞ ) 2.668 × 10-10 mol/cm2 and a ) 0.250 × 10-10 mol/cm3), Frumkin (F), and generalized Frumkin (GF) adsorption isotherms for a surface equation of state.

The Frumkin model assumes a linear relationship between activation energy of adsorption/desorption processes (Ea/Ed) and surface converge Γ. Instead of the linearity, the generalized Frumkin model assumes a power law on Γ. The generalized Frumkin model was also applied to fit the equilibrium surface tension data. It is hard to tell which one predicts better the equilibrium surface tension data since both models predict the equilibrium data nearly perfectly. The curve of γ vs Γ/Γref data, from an experiment of fast expanding an equilibrium-established air-water interface, offers a second data set for the examination of the adsorption models. The generalized Frumkin isotherm, with Γ∞ ) 4.533 × 10-10 mol/cm2, a ) 5.166 × 10-10 mol/cm3, K ) 1.168, and n ) 2.39, seems fitting the γ - Γ/Γref data much better than the Frumkin model (see Figure 7). Currently we are in the process of verifying this point for CmEn surfactants in our laboratory. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 89-2214-E-011-021). LA991338D