Adsorption Kinetics of C12E8 at the Air−Water Interface: Desorption

Langmuir 1997, 13, 3191-3197

3191

Adsorption Kinetics of C12E8 at the Air-Water Interface: Desorption from a Compressed Interface Ruey-Yug Tsay Institute of Biomedical Engineering, National Yang-Ming University, Taipei, 112, Taiwan, R.O.C.

Shi-Yow Lin* and Lung-Wei Lin Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, 106 Taiwan, R.O.C.

Shou-I Chen Union Chemical Laboratory, Industrial Technology Research Institute, Hsinchu, 300 Taiwan, R.O.C. Received October 18, 1996X The desorption of C12E8 out of an overcrowded interface due to the sudden shrinkage of a pendant bubble in a quiescent surfactant solution is studied. A video-enhanced pendant bubble tensiometry is utilized for the measurement of the relaxation in surface tension due to the desorption of surfactant. The desorption process is found to be diffusive-kinetic mixed controlled. Rate constants of adsorption/desorption are computed by comparing these tension profiles with numerical solutions, which consider both bulk diffusion and kinetic desorption processes. The values of the kinetic rate constants of C12E8 obtained from the desorption experiment are nearly the same as that obtained from the clean adsorption study (Lin, S. Y., et al., Langmuir 1996, 12, 6530). The concept that there exists a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusion-kinetic control at more elevated bulk concentration for C12E8 is therefore confirmed.

1. Introduction The surface tension relaxation at interfaces between surfactant solutions and air (or immiscible fluids) is dominated by the rate that surfactant adsorbs in a twostep process: surfactant in the sublayer adsorbs, which, in turn, establishes a diffusive flux from the bulk.1-13 These rates are commonly determined by minimizing the difference between experimental surface tension vs time profiles and mass transfer model predictions. Usually, only a limited range of bulk concentrations C0 is studied, and the profiles are found to be in agreement with a diffusion-control model. It is argued in our previous study11 that such apparent agreement is insufficient evidence to disregard the role * Author to whom correspondence should be addressed. Fax: 886-2-737-6644. Tel: 886-2-737-6648. E-mail: [email protected]. Address: Department of Chemical Engineering, National Taiwan Institute of Technology, 43, Keelung Road, Section 4, Taipei 106, Taiwan. X Abstract published in Advance ACS Abstracts, May 15, 1997. (1) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (2) van den Temple, M.; Lucassen-Reynders, E. H. Adv. Colloid Interface Sci. 1983, 18, 281. (3) Bleys. G.; Joos, P. J. Phys. Chem. 1985, 89, 1027. (4) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (5) Kretzschmar, G.; Miller, R. Adv. Colloid Interface Sci. 1991, 36, 65. (6) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (7) Johnson, D. O.; Stebe K. J. J. Colloid Interface Sci 1994, 168, 21. (8) Chang. C. H.; Franses. E. I. Colloids Surf. A 1995, 100, 1. (9) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (10) Johnson, D. O.; Stebe K. J. J. Colloid Interface Sci. 1996, 182, 526. (11) Lin, S. Y.; Chang, H. C.; Chen, E. M. J. Chem. Eng. Jpn. 1996, 29, 634. (12) Lin, S. Y.; Tsay, R. Y.; Lin. L. W.; Chen S. I. Langmuir 1996, 12, 6530. (13) 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; Pillai, V., Shah, D. O., Eds; AOCS Press: Champaign, IL, 1996; pp 23-47.

S0743-7463(96)01008-6 CCC: $14.00

of the adsorption/desorption kinetics in the adsorption process. A concept of a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusion-kinetic control at more elevated bulk concentration was proposed. This idea was applied to the measurement of diffusion coefficient and adsorption/ desorption rate constants using a clean adsorption process.12,13 The shift and the values of adsorption/ desorption rate constants is examined in this study by investigating a reequilibration process, which C12E8 molecules desorb out of an overcrowded interface due to a sudden shrinkage of a pendant bubble in a quiescent surfactant solution. The outline of this paper is as follows. Section 2 describes briefly the pendant bubble experimental technique and details the adsorption relaxation profiles for C12E8 out of an overcrowded interface. The theoretical framework for the surfactant mass transfer process and the numerical solution procedure are given in section 3. In section 4, the experimental relaxation profiles are compared with theoretical solutions, which leads to computation of the sorptive rate constants. The paper ends with a conclusion and discussion section. 2. Experimental Measurements Materials. Nonionic surfactant C12E8 (octaethylene glycol mono n-dodecyl ether (C12H25(OCH2CH2)8OH) 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 less than 0.057 µmho/cm (µS/cm). Apparatus. A video-enhanced pendant bubble tensiometer was employed for measuring the relaxation in surface tension for the desorption process at 25 °C. The system used is the same

© 1997 American Chemical Society

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as that in the adsorption studies,4,12 and therefore only a brief description is given here. The pendant bubble system has a halogen light source and a lens system which creates a collimated beam with constant light intensity. The light passes through the pendant bubble, which is generated in a C12E8 aqueous solution inside a quartz cell of 26 × 41 × 43 mm inside dimensions, and forms a silhouette of the bubble on a solid state video camera. The quartz cell is enclosed in a thermostatic air chamber and the temperature stability of this thermostat is better than (0.02 K for temperature ranging from 10 to 30 °C.14 A 16 gauge stainless steel inverted needle (0.047 in. i.d.; 0.065 in o.d.) is connected to the normally closed port of a three-way miniature solenoid valve and used for the bubble generation. The valve is connected to a gas-tight syringe which is placed in a syringe pump and the valve is controlled by a computer. The image is digitized into 480 lines × 512 pixels with a level of gray with eight-bit resolution. An edge detection routine was devised to locate the interface contour from the digitized image. The edge is defined as the x or z position which corresponds to an intensity of 127.5.4 The quartz cell is initially filled with solution and the inverted needle is positioned in the cell in the path of the collimated light beam. The solenoid valve is energized and the gas is allowed to pass through the needle, thereby forming a bubble of air on the tip of needle. The valve is then closed when the bubble attained a diameter of approximately 2.5 mm. The bubble so created is one of constant mass. The change in volume, as the surface tension relaxes during the adsorption of surfactants onto the clean interface, is only a few percent over a few hours. After the solenoid is closed and the bubble is formed, sequential digital images are taken of the bubble. After the relaxation of clean adsorption is complete, the valve is opened for 0.11 s (controlled by a computer) while the syringe pump is off. A small part of the gas inside the bubble is allowed to pass through the solenoid valve and the surface area of gas bubble decreases around 20%. The images are recorded on a recorder during this shrinkage process and also taken sequentially onto the computer. After the relaxation of the desorption is complete, the images on tape are processed, by the edge detection routine, to determine the bubble edge coordinates and the surface tension. There is a nearly constant deviation of surface tension, about 0.7 mN/m, between the images directly on computer and those saved on tape. This variation is probably due to the different resolution of the tape (horizontal resolution 330 lines), which is worse than that of either image digitizer (512 lines) or CCD camera (610 lines). 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:15,16

γ

(

)

1 1 + ) ∆P R1 R2

(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 x and z and turning angle φ of the interface as a function of the arc length s, and then integrated. 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 location of the apex, the radius of curvature at the apex, and the capillary constant. The surface tension is obtained from the best fit between the theoretical curve and the data points by minimizing the objective function. The accuracy and reproducibility of the dynamic surface tension measurements, in this study, obtained by this procedure are ≈0.1 mN/m.17,18 (14) Lin, S. Y.; Hwang, H. F. Langmuir 1994, 10, 4703. (15) Huh, C.; Reed, R. L. J. Colloid Interface Sci. 1983, 91, 472. (16) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (17) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159. (18) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf. A 1996, 114, 31.

Figure 1. Representative dynamic surface tensions (mN/m) and surface area of pendant bubble for reequilibration of C12E8 aqueous solutions for C0 ) 6.0 (a) and 7.32 (b) (10-9 mol/cm3). Experimental Results. The reequilibration process due to C12E8 desorption from a suddenly compressed air-water interface was measured and the images were recorded. Representative relaxation profiles (the circles) of surface tension and surface area (the diamonds) of pendant bubble are shown in Figure 1. For example, the surface tension decreases from the equilibrium value (50.9 mN/m for C ) 6.0 × 10-9 mol/cm3, in Figure 1a) to a lower value (45.0 mN/m), corresponding to a surface coverage higher than the equilibrium one, in 0.1 s. The surface tension then increases and goes back to the equilibrium tension in several hundred seconds. The bubble surface area, decreases 19% in 1/15 s and then keeps a nearly constant value for a few hundred seconds. Shown in Figure 1b is another desorption relaxation profile at C ) 7.32 × 10-9 mol/cm3. All the relaxation data show a similar behavior: surface tension decreases abruptly from the equilibrium value and then increases smoothly up to its equilibrium value after the abrupt falling. Each run is compressed with different percentage (15-20% as shown in column Ab/Ae in Table 1; Ae and Ab denote the bubble surface area before and right after the compression, respectively). Note that the moment in which surface tension begins to deviate from the equilibrium value (γe; at this moment, surface area is Ae and surface coverage is Γe) is referenced as the zero time in Figure 1. The tension data in Figure 1 are re-plotted in Figure 2, and the moment with the lowest surface tension value (γb; at this moment, surface area is Ab and surface coverage is Γb) is set to be the zero time for the convenience of the theoretical calculation. Any desorption during the ramp type area change is insignificant for the present system and can therefore be neglected. The complete model for interfacial relaxations after area changes of different types was discussed recently by Dukhin et al.19

3. Theoretical Framework Governing Mass Transfer Equations. The desorption of surfactant out of an overcrowded interface due to a sudden shrinkage of the surface area of the pendant (19) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: New York, 1995.

Adsorption Kinetics of C12E8

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transform, the solution of the preceding set of equations can easily be formulated in terms of unknown subsurface concentration Cs(t) ) C(r ) b,t):

Γ(t) ) Γb +

D [C t b 0

∫0tCs(τ) dτ] + 2 xDπ [C0 xt ∫0xt Cs(t - τ) dxτ]

(4)

Adsorption Isotherms and Equations of State. To complete the solution for the surface concentration, the sorption kinetics must be specified. The model used here assumes, as in the adsorption study for C12E8,12 that adsorption/desorption is an activated process assumed to obey the following rate expression: 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

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 and a power form is assumed: Figure 2. Experimental values of the dynamic surface tensions (mN/m) for reequilibration of C12E8 and the theoretical predictions of mixed-controlled reequilibration for different adsorption rate constant of the generalized Frumkin model for C0 ) (a) 6.0 and (b) 7.32 (10-9 mol/cm3). β exp(E0a/RT) ) (a) 109, (b) 1010, (c) 1011, (d) 1012, (e) 3 × 1010, and (f) 5 × 1010 cm3/(mol.s). DC (dashed line) denotes diffusion-limited curves. D ) 8.0 × 10-6 cm2/s.

bubble in a quiescent surfactant solution is modeled. We shall consider only the case of desorption out of an overcrowded spherical interface and one-dimensional diffusion to 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 bulk phase is assumed to be spherically 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 ) 2 ∂r ∂r ∂t r

( )

(r > b, t > 0)

(2)

with the following initial and boundary conditions

C(r,t) ) C0

(r > b, t ) 0)

C(r,t) ) C0

(r f ∞, t > 0)

dΓ(t) ∂C )D dt ∂r

( )

Γ(t) ) Γb

(r ) b, t > 0)

(3)

(t ) 0)

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, C0 is the concentration far from the bubble, and Γb is the initial surface coverage. By the Laplace

Ea ) E0a + vaΓn Ed ) E0d + vdΓn

(6)

where E0a, E0d, va and vd are constants. Equation 5 in nondimensional form becomes

dx/dτ ) Ka exp(-v*axn) C*s(1 - x) - Kd exp(-v*dxn) x (7) where x ) Γ/Γ∞, τ ) tD/h2, h ) Γe/C0, Ka ) β exp(-E0a/ RT)C0/(D/h2), C*s ) Cs/C0, Kd ) R exp(-E0d/RT)/(D/h2), v*a ) n vaΓn∞/RT, v* d ) vdΓ∞/RT, and k ) v* a - v* d. At equilibrium, the time rate of change of Γ vanishes and the adsorption isotherm that follows is given by

Γ C )x) Γ∞ a exp(kxn) + C

(8)

where x ) Γ/Γ∞ and a ) (R/β)exp[(E0a - E0d)/RT]. Equation 8 becomes the Frumkin isotherm20-22 when n ) 1 and the Langmuir adsorption isotherm when va ) vd ) k ) 0. According to the result in our previous study,12 on the adsorption process that C12E8 molecules adsorb onto a clean air-water interface, both Frumkin and generalized Frumkin models predict a positive k value for C12E8 from the equilibrium tension data. This positive k indicates that the adsorption of C12E8 is anticooperative, and adsorption becomes more difficult as the surface becomes more covered. To simplify the theoretical simulation, two specific cases for the value of sorption rate constants, v* a ) 0 or v* d ) 0, are assumed for the calculation in this study. The effect of choosing v* d value is discussed in the Discussion and Conclusions section. (20) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (21) Borwankar R. P.; Wasan D. T. Chem. Eng. Sci. 1983, 38, 1637. (22) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189.

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Table 1. Values of Adsorption-Desorption Rate Constants from the Desorption Experiments v*a ) 0 C (10-9 mol/cm3) 6.0 (run 1) 6.0 (run 2) 7.32 (run 1) 7.32 (run 2) 7.32 (run 3) 7.32 (run 4) 7.32 (run 5)

d

γe/γb (mN/m) 50.9/45.0 50.9/46.3 49.8/44.2 49.7/44.7 50.0/41.7 49.9/43.2 49.7/43.0 average

Ab./Ae

AbΓb/AeΓea

0.86 0.81 0.85 0.81 0.85 0.80 0.81

1.00 0.96 0.98 0.95 0.98 1.00 0.98

GFb 5 c 10 R* 10-6β*d 1.0 1.0 0.8 1.0 1.2 1.5 1.0 1.1

4.3 4.3 3.4 4.3 5.2 6.4 4.3 4.6

v*d ) 0 Fb

104R*

10-7β*

3.8 5.0 3.8 5.0 3.8 5.0 3.8 4.3

1.5 2.0 1.5 2.0 1.5 2.0 1.5 1.7

GF 102R* 10-10β* 7.0 7.0 4.7 9.3 11.6 14.0 9.3 9.1

3.0 3.0 2.0 4.0 5.0 6.0 4.0 3.9

F 102R*

10-9β*

2.5 3.3 2.5 3.8 3.3 5.0 2.5 3.3

1.0 1.3 1.0 1.5 1.3 2.0 1.0 1.3

a Γ calculated from the generalized Frumkin model. b GF: generalized Frumkin model. F: Frumkin model. c R* ) R exp(-E0/RT) (s-1). d β* ) β exp(-E0a/RT) (cm3 mol-1 s-1).

Figure 3. Model predictions of mixed-controlled reequilibration for different adsorption rate constant of the generalize Frumkin model. The broken lines denote that for v*a ) 0; the solid lines denote that for v* d ) 0; DC (dashed line) denotes diffusion-limited curves. β exp(E0a/RT) ) (a) 109, (b) 1010, (c) 1011, (d) 1012, (e) 105, (f) 106, (g) 107, and (h) 108 cm3/(mol‚s).

Numerical Solution Procedure. The theoretical framework that describes the unsteady bulk diffusion of surfactant out of an overcrowded interface has been formulated previously,6,9 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 be a constant initial surface concentration Γb. For this desorption process, the air-water interface was suddenly compressed for a small percentage of the surface area, and Γb is assumed to be equal to the surface coverage corresponding to the point with the lowest surface tension value, for example, the point L at t ) 0.1 s and 0.067 s in Figure 1, a and b. So in Figure 2, point L, plotted at the position of t ) 0.01 s, is referenced as the zero time corresponding to the beginning of the desorption process. When the surfactant solution can be considered ideal, the Gibbs adsorption equation dγ ) -Γ RT 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) -

kn n+1 x n+1

]

mol/cm3, k ) 13.228, and n ) 0.5032 when the generalized Frumkin adsorption isotherm is utilized. Γ∞ ) 2.668 × 10-10 mol/cm2, a ) 2.501 × 10-11 mol/cm3, and k ) 5.186 when the Frumkin isotherm is used. When the desorption 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 desorption 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. The technique used for solving these two equations (eqs 4 and 9 or eqs 4 and 5) numerically is a modification of that used by Miller and Kretzschmar.23 The method on the integration and calculation has been detailed in ref 4. When the desorption process is of mixed control, eq 5 (or eq 7 in dimensionless form) is solved coupled with eq 4 to find out the surface concentration. While the value of k ()v*a - v*d) is obtained from the best fit between the equilibrium surface tension and the model predictions (eqs 4 and 9), no data is available for finding out the value of v*a or v*d. The relaxation profiles of surface tension are dependent upon the value of v* a. One possible way to obtain v*a value is from the comparison between the desorption data and the model prediction for different v*a. If one v*a fits the data superior to others clearly, one may conclude that it is the right value of v* a. The relaxation curves of mixed diffusion-kinetic control for different v*a were calculated for the generalized Frumkin model. Figure 3 shows the comparison for two special cases: the broken curves are the model prediction for different β exp(-E0a/RT) of v* a ) 0 (i.e., v* d ) -k ) -13.23); the solid lines are that of v* d ) 0 (i.e., v* a ) k ) 13.23). In the first case, a zero v* a represents that the activation energy of adsorption is independent on surface coverage, and the negative v* d represents that desorption rate increases as the surface concentration increases. The second case, a zero v* d and a positive v*a represent that the activation energy of desorption is independent on surface coverage and the adsorption slows down as the surface concentration increases. The data in Figure 3 shows that the relaxation profile of C12E8 varies very little when v*a changes from 0 to 13.23. In order to simplify the theoretical simulation, only two cases, v*a) 0 and v* d ) 0, were picked for the following calculation.

(9)

where x ) Γ/Γ∞ and γ0 is the clean surface tension. The model constants (Γ∞, a, k, and n) obtained from a previous study,12 by fitting the equilibrium tension data and the model prediction (eqs 8 and 9), are utilized in this study. For C12E8, Γ∞ ) 5.28 × 10-10 mol/cm2, a ) 2.329 × 10-12

4. Comparisons of Experimental Relaxation Curves and Theoretical Profiles Relaxation in the surface tension due to C12E8 desorption out of an overcrowded interface was measured. The value (23) Miller, R.; Kretzschmar, G. Colloid Polym. Sci. 1980, 258, 85.

Adsorption Kinetics of C12E8

Figure 4. Experimental values of the dynamic surface tensions (mN/m) for reequilibration of C12E8 and the theoretical predictions of mixed-controlled reequilibration for different adsorption rate constants of the Frumkin model for C0 ) (a) 6.0 and (b) 7.32 (10-9 mol/cm3). β exp(E0a/RT) ) (a) 107, (b) 108, (c) 109, (d) 1010, (e) 1011, and (f) 2 × 109 cm3/(mol.s). DC (dashed line) denotes diffusion-limited curves. D ) 1.1 x10-5 cm2/s.

of diffusion coefficient of C12E8 in water has been obtained in a previous study12 (D ) 8.0 × 10-6 cm2/s using the generalized Frumkin model, and D ) 1.1 × 10-5 cm2/s for the Frumkin model) and is used in the following calculation. For the reequilibration process in this study, an initially equilibrium-established air-water interface is suddenly compressed and two surface tension relaxation profiles are shown in Figure 1. If this process was assumed to be diffusion controlled, the diffusion-controlled relaxation profiles by using the generalized Frumkin model with the diffusion coefficient D ) 8.0 × 10-6 cm2/s are shown in Figure 2. It is clear that the desorption relaxation profiles depart significantly from the diffusion limited curves (the dashed lines). So, the reequilibration process is assumed mixed (diffusive-kinetic) controlled. Theoretical relaxation profiles, using the generalized Frumkin model and assuming v* d ) 0, with finite adsorption rate constant are calculated and plotted in Figure 2 (the solid curves). From the data and the model predictions, it is found that the desorption rate constant is about the same for different experimental runs, and the value of β exp(-E0a/RT) is averaged as 3.9 × 1010 cm3/(mol‚s). Listed in Table 1 are the values of adsorption/ desorption rate constants from different runs of the reequilibration experiment for two different bulk concentrations. The average value of desorption rate constant R exp(-E0d/RT) is 0.091 s-1. If v* a ) 0 is assumed, β exp(-E0a/RT) ) 4.6 × 106 cm3/(mol‚s) and R exp(-E0d/RT) ) 1.1 × 10-5 s-1. This values of sorption kinetic constants, obtained from the reequilibration experiment for C12E8 desorption out of an overcrowded interface due to a sudden

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Figure 5. Values of adsorption/desorption rate constants from the best fit between the adsorption/desorption relaxations and the model prediction for the (F) Frumkin and (GF) generalized Frumkin model as a function of bulk concentrations: (4) adsorption rate constant from adsorption experiment; (O) desorption rate constant from adsorption data; (]) adsorption rate constant from desorption experiment; (0) the desorption rate constant from sorption experiment.

shrinkage of pendant bubble, are nearly the same as that obtained from the clean adsorption experiment in ref 12. A similar calculation is also done for using the Frumkin adsorption model. Figure 4 shows a comparison between the data, the same set that is shown in Figure 2, and the prediction profiles of mixed control from the Frumkin model. The diffusion coefficient used here for the theoretical calculation is 1.1 × 10-5 cm2/s. The values of adsorption/desorption rate constants for Frumkin model are also listed in Table 1. The value of β exp(-E0a/RT) is 1.3 × 109 and 1.7 × 107 cm3/(mol‚s) and the value of R exp(-E0d/RT) is 0.033 and 4.3 × 10-4 s-1 for v* d ) 0 and v* a ) 0, correspondingly. Again, the values of sorption kinetic constants from this study are very close to that from the clean adsorption experiment in ref 12. Figure 5 compares the values of sorption kinetic constants from the reequilibration process with v* a ) 0, desorption out of an overcrowded interface due to a sudden shrinkage of the pendant bubble, and that from the clean adsorption process, adsorption onto a freshly created air-water interface. Both perturbed processes take place in a quiescent surfactant solution, and the relaxations of surface tension are monitored and used for calculating the kinetic constants. The values of kinetic rate constant obtained from the clean adsorption process vary slightly with bulk concentration, and the averages are β exp(-E0a/RT) ) 3.8 × 106 cm3/(mol‚s) and R exp(E0d/RT) ) 8.9 × 10-6 s-1 for the generalized Frumkin model. The average from this study is β exp(-E0a/RT) ) 4.6 × 106 cm3/(mol‚s) and R exp(-E0d/RT) ) 10.7 × 10-6 s-1. A similar result is also obtained using the Frumkin model: β exp(-E0a/RT) ) 1.2 × 107 cm3/(mol‚s) and R exp(-E0d/RT) ) 2.9 × 10-4 s-1 from clean adsorption data,

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

γ A Γa × 1010 AΓ (mN/m) (mm2) (mol/cm2) (10-9 mol) Ai/Ab AiΓi/AbΓb

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

50.90 50.89 50.86b 47.54c 47.02c 44.97d 45.10 45.47

(a) C ) 6.0 × 10-9 mol/cm3 22.34 2.10 4.688 22.33 2.10 4.687 22.29 2.10 4.683 20.15 2.32 4.683 18.19 2.36 4.289 18.16 2.49 4.518 18.15 2.48 4.500 18.16 2.46 4.460

1.00 1.00 1.00 0.90 0.82 0.81 0.81 0.81

1.00 1.00 1.00 1.00 0.92 0.96 0.96 0.95

-1/30 0 1/60 2/60 3/60 4/60 6/60

49.98 49.89b 49.71c 48.36c 43.88c 43.21d 43.83

(b) C ) 7.32 × 10-9 mol/cm3 22.69 2.16 4.902 22.65 2.17 4.909 22.60 2.18 4.926 18.76 2.27 4.259 18.83 2.56 4.811 18.42 2.60 4.783 18.41 2.59 4.711

1.00 1.00 1.00 0.83 0.91 0.81 0.81

1.00 1.00 1.01 0.87 0.98 0.98 0.96

a The surface coverage calculated from the value of surface tension using eq 8, and a generalized Frumkin model is utilized for the data shown here. b The point right before desorption process, corresponding to the equilibrium state. c The points during the shrinkage of bubble. d The point with the lowest surface tension, corresponding to the end of shrinkage and the beginning of desorption process.

β exp(-E0a/RT) ) 1.7 × 107 cm3/(mol‚s) and R exp(E0d/RT) ) 4.3 × 10-4 s-1 from reequilibration data. 5. Discussion and Conclusions A concept that there exists a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusive-kinetic control at more elevated bulk concentration for C12E8 was proposed in a previous study.12 This idea was verified and adapted for measuring values of diffusion coefficient and adsorption/desorption rate constants using a clean adsorption process.11,12 The values of adsorption/desorption rate constants are examined in this study using a reequilibration process, which C12E8 molecules desorb out of an overcrowded interface due to a sudden shrinkage of a pendant bubble in a quiescent surfactant solution. The values of kinetic rate constants obtained from these two different processes, clean adsorption and reequilibration, are nearly same. Surface-active impurity is an important issue in the study of adsorption mechanism. Due to their high surface activity they may remain at the surface for a long time after compression of air-water interface. Therefore, there exists a significant difference in equilibrium surface tension after compression and dilation.5,24 The impurity problem is considered in this study. Since the surface tension always goes back to, or reaches closely (with a deviation about 0.1 mN/m; as shown in Figure 1), the equilibrium surface tension after compression, the impurity effect is believed to be insignificant for the process in the concentration range of this study. The model used in this study is the generalized Frumkin and the Frumkin models. If the intermolecular interaction between the adsorbed molecules is negligible, eq 5 becomes a Langmuir adsorption equation

dΓ/dt ) βLCs(Γ∞ - Γ) - RLΓ

(10)

The rate constants βL and RL correspond to β exp(-Ea/ RT) and R exp(-Ed/RT) in the generalized Frumkin (24) Lunkenheimer, K.; Miller, R. J. Colloid Interface Sci. 1987, 120, 176.

Figure 6. Experimental values of the dynamic surface tensions (mN/m) for reequilibration of C12E8 (C0 ) 6.0 × 10-9 mol/cm3) and the theoretical profiles of diffusion control of (GF) generalized Frumkin and (F) Frumkin models for different diffusivity. D ) (a) 0.3, (b) 0.8, (c) 2.0, (d) 8.0, (e) 0.1, (f) 1.0, and (g) 11.0 (10-6 cm2/s).

adsorption equation, and β exp(-Ea/RT) ) 4.6 × 106 cm3/ mol‚s) (as v*a ) 0) or 8.6 × 106 cm3/(mol‚s) (as v* d ) 0), R -1 (as v* ) exp(-Ed/RT ) 0.050 s-1 (as v* a ) 0) or 0.091 s d 0) calculated from the data in this study. The video-enhanced pendant bubble tensiometry is an effective tool for studying the reequilibration process, which is due to a suddenly compression on the adsorbed monolayer at the fluid interface. With the aid of a video recorder, it is possible to monitor the surface tension, bubble volume, and surface area of bubble during the entire period, with 1/30 s interval, of the reequilibration process. Some images during the shrinkage are also split, to be 1/60 s interval, when the shrinkage proceeds too fast. The surface concentration Γi(t) and the amount (AiΓi) of C12E8 molecules at the interface during the reequilibration relaxation are also calculated from eq 9. Table 2 shows two representative data sets of surface tension, surface area, surface concentration, and the amount of C12E8 molecules during the shrinkage. According to the rule of mass balance, the total amount of C12E8 molecules at the interface should be a constant (if no C12E8 molecule desorbed out of interface during this shrinkage) or decreasing only a little bit (if there are some C12E8 molecules desorbed out during this shrinkage). The data in Table 2 tells that the amount of C12E8 molecules at the interface varied 4% (for Table 2a) and 2% (for Table 2b) at the end of the shrinkage (the point with the lowest surface tension). For those points during the shrinkage, the deviation may be larger. This large deviation is due to the hardware limitation: the image acquired is not from the exact same moment for the CCD camera used in this study. The image acquired represents an average of 1/30 or 1/60 s (if it is split into two images) interval, and the tension and surface area obtained are the average

Adsorption Kinetics of C12E8

during this 1/30 or 1/60 s. When the shrinkage proceeds too fast, the images during the shrinkage have a larger deviation, being compared with what is acquired before or after the shrinkage. Diffusion coefficients of 8.0 × 10-6 (for generalized Frumkin model) and 1.1 × 10-5 cm2/s (for Frumkin model) used in this study are from the result of ref 12 of clean adsorption. The reequilibration profiles are then found to be mixed diffusive-kinetic controlled. Figure 6 shows the fit between the reequilibration data and the model prediction profiles when the reequilibration process is assumed to be of diffusion control. The diffusion-controlled relaxation curves of the Frumkin and generalized Frumkin model fit the data well, except that the values of diffusion coefficient are low. With the diffusion-control assumption, the diffusion coefficient obtained is 2.0 × 10-6 cm2/s from the generalized Frumkin model, and 1.0 × 10-6 cm2/s from the Frumkin model. If the diffusivity is 1 or 2 × 10-6 cm2/s, it is impossible to interpret the relaxation data from the clean adsorption in the previous study. Therefore, it

Langmuir, Vol. 13, No. 12, 1997 3197

is more reasonable to assume that the reequilibration process is mixed diffusive-kinetic controlled. A shift in controlling mechanism has been confirmed by Pan et al.13 and Lin, et al.12 with increasing the bulk concentration. Such kind of shift in mechanism from diffusion control to mixed control can also possibly be attained by making the surface compaction large. To further confirm the shift in controlling mechanism occurring either for elevated bulk concentration or for large surface compaction, currently we are in the process of theoretical calculation and experimental measurement of the relaxations of surface tension for C12E8 and other nonionic polyoxyethylene surfactants. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grand NSC 84-2214-E-011-019) and by the Union Chemical Lab, ITRI. LA961008I