Adsorption Kinetics of C10E8 at the Air− Water Interface

The adsorption of C10E8 onto a clean air-water interface and the desorption out of ... water interface and desorbing out of a compressed air-water int...
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Langmuir 1998, 14, 2476-2484

Adsorption Kinetics of C10E8 at the Air-Water Interface Hong-Chi Chang, Ching-Tien Hsu, 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 August 15, 1997. In Final Form: February 23, 1998 The adsorption of C10E8 onto a clean air-water interface and the desorption out of an overcrowded interface due to a sudden shrinkage of a pendant bubble in a quiescent surfactant solution are studied. Video-enhanced pendant bubble tensiometry is employed for the measurement of the relaxation in surface tension. Relaxation profiles of surface tension for C10E8 molecules absorbing onto a freshly created airwater interface and desorbing out of a compressed air-water interface are obtained. The adsorption of C10E8 is found to be anticooperative from the equilibrium surface tension data compared with the prediction of the Frumkin model. The controlling mechanism of the adsorption process changes as a function of bulk concentration; it shifts from diffusion control at dilute concentration to mixed diffusive-kinetic control at more elevated bulk concentration. It is also confirmed that the desorption of C10E8 out of a compressed interface is a mixed diffusive-kinetic controlled process. Comparison is made for the entire relaxation period of the tension data and the model predictions. Values of the diffusivity and the adsorption/desorption rate constants of C10E8 are calculated from these dynamic surface tension profiles. The values of the kinetic rate constants obtained from the desorption experiment are the same as that obtained from the clean adsorption experiment.

1. Introduction The transport of bulk soluble surfactant between bulk solution and fluid interface includes bulk diffusion and adsorption/desorption processes. It was reported recently that the controlling mechanism of this mass transfer for C12E8 and C12E6 can change as a function of bulk concentration from diffusion to mixed diffusive-kinetic control.1-3 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. The desorption of C12E8 out of an overcrowded interface due to a sudden shrinkage in a quiescent surfactant solution is also found to be mixed diffusive-kinetic controlled.4 Values of mass transfer coefficient, such as the diffusivity and the adsorption/ desorption rate constants of surfactant molecules, are then obtained from the dynamic surface tension profiles of the adsorption and desorption processes. The adsorptions of C12E8 and C12E6 were found to be anticooperative, i.e., the adsorption becomes more difficult as the surface becomes more crowded. Nonionic surfactants Triton X-100 were found to have a similar anticooperative adsorption.5 More surfactants, for example, n-alcohol,6-9 decanoic acid,10,11 and diazinon,12 have co* To whom correspondence should be addressed. Tel: 886-22737-6648. Fax: 886-2-2737-6644. E-mail: [email protected]. (1) 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: IL, 1996; pp 23-47. (2) Lin, S. Y.; Chang, H. C.; Chen, E. M. J. Chem. Eng. Jpn. 1996, 29, 634. (3) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (4) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (5) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (6) Fainerman, V. H.; Lylyk, S. V. Kolloidn. Zh. 1982, 44, 598. (7) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1994, 10, 3442. (8) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (9) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526.

hesive intermolecular forces between the adsorbed molecules; i.e., the desorption becomes more difficult as the surface concentration increases. One may wonder if all the polyoxy(ethylene) nonionic surfactants have anticooperative adsorption and have a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusive-kinetic control at more elevated bulk concentration. In this study, the adsorption kinetics of another polyoxy(ethylene) nonionic surfactant C10E8 is examined. Video-enhanced pendant bubble tensiometry is employed for the measurement of the relaxation in surface tension. The adsorption of C10E8 onto a clean airwater interface and the desorption out of an overcrowded interface due to a sudden shrinkage of a pendant bubble in a quiescent surfactant solution are studied. An outline of this paper is as follows. Section 2 describes briefly the pendant bubble experimental technique. The adsorption/desorption relaxation profiles for C10E8 onto a clean interface and out of a compressed air-water interface are detailed in section 3. The theoretical framework for the mass transfer process of surfactant molecules and the numerical solution procedure is given in section 4. In section 5, the experimental relaxation profiles are compared with theoretical solutions, which leads to computation of the diffusion coefficient and sorptive rate constants. The paper ends with a conclusion and discussion section. 2. Experimental Measurements Materials. Nonionic surfactant C10E8 (octaethylene glycol mono n-decyl ether (C10H25(OCH2CH2)8OH) of greater than 99% purity purchased from Nikko (Tokyo, Japan) was used without modification. Acetone (HPLC grade) used to verify the measurement of surface tension was obtained from Fisher Scientific Co. 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 (10) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (11) Lunkenheimer, K.; Hirte, R. J. Phys. Chem. 1992, 96, 8683. (12) Lin, S. Y.; Lin, L. W.; Chang, H. C.; Ku, Y. J. Phys. Chem. 1996, 100, 16678.

S0743-7463(97)00923-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/11/1998

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0.057 µΩ-1/cm. The values of the surface tension of air-water and air-acetone, using the pendant bubble technique 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 C10E8 and the equilibrium tensions. The apparatus is similar to what investigates C12E8; therefore only a brief description is given here.3,8 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 C10E8 aqueous solution, which was put inside a quartz cell. The quartz cell was enclosed in a thermostatic air chamber, and the temperature stability of this thermostat is better than (0.02 K.13 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 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 x or z position that corresponds to an intensity of 127.5.14 Experimental Procedure. The experimental protocol was as follows: the quartz cell of 26 × 41 × 43 mm inside diameter was initially filled with the C10E8 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 time required to create an air bubble of this size is about 0.6 s. 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 valve was closed and the bubble was formed, sequential digital images were then taken of the bubble, first at intervals of approximately 0.1 s, and then later in intervals on the order of seconds. 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 10-15%. 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, bubble volume, bubble surface area, and 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 tape (horizontal resolution: 330 lines), which is worse than that of either the image digitizer (512 lines) or CCD camera (610 lines). Surface Tension Calculations. 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/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 (13) Lin, S. Y.; Hwang, H. F. Langmuir 1994, 10, 4703. (14) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (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.

Figure 1. Representative dynamic surface tensions (mN/m) for adsorption of C10E8 onto a clean air-water interface for C0 ) (1) 0.20, (2) 0.50, (3) 1.0, (4) 2.0, (5) 4.0, (6) 6.0, (7) 10.0, and (8) 30.0 [10-9 mol/cm3]. as a function of the arc length s and then integrated by using a Runge-Kutta scheme17 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 location 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,17 and from the optimal values of R0 and B the tension can be computed. As demonstrated in Lin et al.,18,19 the accuracy and reproducibility of the dynamic surface tension measurements obtained by this procedure are ≈0.1 mN/m.

3. Experimental Results Relaxation in the surface tension due to adsorption of C10E8 onto a clean air-water interface was measured. Data were recorded up to more than 1 h from the moment (referenced as t ) 0) at which one-half of the bubble volume is generated during the bubble formation. Shown in Figure 1 are representative dynamic surface tension profiles (for one selected bubble at each bulk concentration) of C10E8 aqueous solutions at eight different bulk concentrations, C ) 0.20, 0.50, 1.0, 2.0, 4.0, 6.0, 10.0, 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 C10E8 aqueous solutions at the aqueous-air interface were extracted from the longtime asymptotes in Figures 1 and 2 and are plotted as the circles in Figure 3. The re-equilibration process due to C10E8 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 triangles) of the pendant bubble are shown in Figure 4. For example, the surface tension decreases from the equilibrium value (59.67 mN/m for C ) 1.0 ×10-8 mol/cm3, in Figure 4b) to a lower value (57.89 (17) Carnahan, B.; Luther, H. A.; Wilkes, J. O. Applied Numerical Methods; Wiley: New York, 1969. (18) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159. (19) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf. A 1996, 114, 31.

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Figure 3. Equilibrium surface tension (mN/m) for air-C10E8 aqueous solution and the theoretical predictions of the Langmuir (L), Frumkin (F), and generalized Frumkin (GF) adsorption isotherms.

Figure 2. Experimental values of the dynamic surface tensions (mN/m) for adsorption of C10E8 onto a clean air-water interface and the theoretical predictions of diffusion control for the Langmuir (L), Frumkin (F), and generalized Frumkin (GF) models for C0 ) (1) 0.20, (2) 1.0, (3) 4.0, (4) 6.0, (5) 10.0, and (6) 30.0 [10-9 mol/cm3]. Each symbol represents a separate bubble.

mN/m), corresponding to a surface coverage higher than the equilibrium one, in 0.13 s. The surface tension then increases and goes back to the equilibrium tension in several hundred seconds. The bubble surface area decreases 12% in 1/20 s and then keeps a nearly constant value for several hundred seconds. Shown in Figure 4a is another desorption relaxation profile at C ) 6.0 × 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 a different percentage (10-15% in this study). Note that the moment at which surface tension is still at the equilibrium value (γe; at this moment, the surface area is

Figure 4. Representative dynamic surface tensions (mN/m) and surface area of the pendant bubble for reequilibration of C10E8 aqueous solutions for C0 ) 0.60 (a) and 1.0 (b) [10-8 mol/ cm3].

Ae and surface coverage is Γe) is referenced as the zero time in Figure 4. Data in Figure 4 are replotted later in Figure 8, and the moment with the lowest surface tension value (γb; at this moment, the 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 (see Table 1) and can therefore be neglected. The complete models for interfacial relaxations after area changes of different types was discussed recently by Dukhin et al.20

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Table 1. Relaxations of Surface Tension, Surface Area, and the Amount of C10E8 Molecules at the Interface during the Shrinkage of the Bubble (C ) 1 × 10-8 (mol/ cm3) and γe ) 59.67 mN/m) t (s) -2/30 -1/30 0 1/60 3/60 3/30 4/30 5/30 6/30

γ A Γa × 1010 AΓ × 109 (mN/m) (mm2) (mol/cm2) (mol) Ai/Ae AiΓi/AeΓe 59.71 59.67 59.68b 58.49c 58.27c 58.18c 57.89d 57.96 58.11

23.66 23.66 23.63 23.41 20.85 20.86 20.92 20.92 20.92

1.372 1.375 1.374 1.463 1.479 1.486 1.507 1.501 1.490

3.25 3.25 3.25 3.43 3.08 3.10 3.15 3.14 3.12

1.00 1.00 1.00 0.99 0.88 0.88 0.89 0.89 0.89

1.00 1.00 1.00 1.06 0.95 0.96 0.97 0.97 0.96

a The surface coverage calculated from the value of surface tension using eq 9 with the generalized Frumkin model. b The point right before the desorption process, corresponding to the equilibrium state. c The point during the shrinkage of bubble. d The point with the lowest surface tension, corresponding to the end of shrinkage and the beginning of the desorption process.

4. Theoretical Framework Mass Transfer in Bulk. The adsorption of surfactant molecules onto the interface of a freshly formed pendant bubble in a quiescent surfactant solution and the desorption out of a compressed interface are modeled. We shall consider only the case of one-dimensional diffusion and adsorption (desorption) onto (out of) a spherical interface from (onto) 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:

D ∂ 2∂C ∂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)

(3a)

C(r,t) ) C0

(r f ∞, t > 0)

(3b)

dΓ/dt ) D(∂C/∂r) Γ(t) ) Γb

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

(3c) (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 easily be formulated in terms of unknown subsurface concentration Cs(t) ) C(r)b,t):

Γ(t) ) Γb +

D [C t b 0

∫0t Cs(τ) dτ] + 2xDπ [C0xt ∫0xtCs(t - τ) dxτ]

(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 assumed to obey the (20) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995.

following rate expression:21 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 is the temperature, and R is 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:

Ea ) E°a + νaΓn Ed ) E°d + νdΓn

(6)

where E°a, E°d, νa and νd are constants. Equation 5 in nondimensional form becomes

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

C Γ )x) Γ∞ C + a exp(kxn)

(8)

where k ) (νa - νd)Γn∞ /RT and a ) (R/β) exp[(E°a E°d)/RT]. Equation 8 becomes the Frumkin isotherm12,22,23 when n ) 1 and 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,5,7 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. Γb ) 0 for a clean adsorption process, in which the bubble was created suddenly. Although there is a small amount of surfactant present at the air-water interface before bubble growth, the surface area of the 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 (21) Aveyard, R.; Haydon, D. A. A Introduction to the Principles of Surface chemistry; Cambridge University Press: Cambridge 1973; Chapters 1 and 3. (22) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (23) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189.

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Table 2. Constants of Optimal Fit of C10E8 Aqueous Solution L F GF

Γ∞ (10-10 mol/cm2)

a (10-10 mol/cm3)

k

n

1.804 3.070 3.424

5.437 1.302 0.213

9.628 10.877

1 0.556

equilibrium isotherm (eq 8) allow for the calculation of the surface tension explicitly in terms of Γ:

γ - γ0 ) Γ∝RT[ln(1 - x) - knxn+1/(n + 1)]

(9)

where x ) Γ/Γ∝ and γ0 is the clean surface tension. By fitting equilibrium data of the surface tension as a function of the bulk concentration using eqs 8 and 9, one can obtain the equilibrium constants k, a, and n and the maximum coverage Γ∝. 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 the 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. Shift in Controlling Mechanism. It has been reported1-3 that for polyoxy(ethylene) nonionic surfactant 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 if there is a similar shift in controlling mechanism for C10E8, a series of simulations in which surfactant adsorbed onto an initially clean, spherical surface from a bulk phase of initially uniform concentration are performed. The range of bulk concentration considered, 1-50 000 times C0/a, is one in which the reduction of surface tension and the relaxation time are both large enough for the dynamic surface tension measurement of C10E8. The generalized Frumkin model with ν* a ) 0 was picked for the simulation; the model constants (Γ∝, a, k, and n) utilized are listed in Table 2. The effect of choosing a different ν* a has been discussed in a previous article.4 The nondimensional relaxation in surface tension Θ(τ), Θ(τ) ) (γ - γe)/(γ0 - γ), is a function of C0/a and Ka. Simulations of Θ as a function of τ for different values of C0/a are performed, in which Ka is varied from diffusion control (Ka f ∞) to mixed control (Ka ) O(1 or 10-1)) to sorption kinetic control (Ka ) O(10-2)). The relaxations of Θ(τ) have the same general dependence on Ka, similar to what is presented in ref 2. The diffusion-limited curve shows the fastest relaxation; slower relaxations are observed as Ka decreases because of the increasing kinetic barrier. The distance between the diffusion-limited curve and that for a particular value of Ka varies little with C0/a. The limiting value of Ka for which the mixed-controlled curve is indistinguishable from diffusion-control profiles is defined as Kal. A signature of the mix-control profile 2 is a minimum in the C* s ()Cs/C0) profile. The minimum value of C* approaches unity as K decreases, and for the a s curves corresponding to Kal at each C0/a, the minimum C*s is around 0.15. 5. Comparisons of Experimental Relaxation Curves and Theoretical Profiles Adsorption. Relaxation profiles of surface tension, due to C10E8 adsorption onto a freshly created air-water interface by using the pendant bubble tensiometry, were

shown in Figure 1. The equilibrium surface tensions, extracted from the long-time asymptotes of the dynamic tension profiles, were plotted in Figure 3. Also presented in Figure 3 are the comparison between the C10E8 equilibrium data and the best fit from the adsorption isotherms of the Langmuir, Frumkin, and generalized Frumkin models. The model constants, as shown in Table 2, are obtained by adjustment so as to minimize the error between the model predictions and experimental values. The Frumkin and generalized Frumkin models fit the equilibrium data better than the Langmuir model. The more exact agreement of the Frumkin or generalized Frumkin model indicates clearly that the intermolecular interactions between the adsorbed C10E8 molecules are significant. Note that both the Frumkin and generalized Frumkin models predict a positive k value from the equilibrium surface tension data. This positive k indicates that the adsorption is anticooperative, and adsorption becomes more difficult as the surface becomes more covered. A similar result has also been obtained for polyethylene nonionic surfactants C12E8 and C12E6.1-3 A different result, negative k value, has been reported for n-alcohols6-9 and decanoic acid,10 which shows cooperative adsorption. The generalized Frumkin model predicts the equilibrium surface tension better than the Frumkin model, especially at a concentration range between 10-8 and 10-7 mol/cm3. There is a significant deviation, around 0.2 mN/ m, between the equilibrium data and the theoretical predictions of the Frumkin model. This 0.2 mN/m deviation makes the Frumkin model fail to predict the relaxation profiles of surface tension in the desorption process. Diffusion-Controlled Model. If the adsorption of C10E8 onto a clean interface was assumed to be diffusion controlled, Figure 1 may be used to determine the diffusivity of C10E8 molecules in the aqueous phase. The model constants Γ∝, a, k, and n (shown in Table 2) 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 Langmuir isotherm is poor. This poor prediction in dynamic surface tension is 2-fold: the poor prediction on the equilibrium data and the poor relaxation tendency of surface tension with time. The poor prediction on the equilibrium data, as shown in Figure 3, makes the theoretical relaxation profiles reach different values of surface tension at large time. The Frumkin and generalized Frumkin numerical profiles fit the experimental data much better than the Langmuir model. The values of diffusion coefficient, determined from the best fit between the dynamic tension data and the theoretical prediction, are plotted in Figure 5 for the Langmuir, Frumkin, and generalized Frumkin models. Data in Figure 5 indicate that the values of diffusion coefficient from all these three models vary with bulk concentration significantly. This implies that the assumption applied in this paragraph, the mass transfer process of C10E8 molecules onto a clean air-water interface is of diffusion control, may fail. Both the Frumkin and generalized Frumkin models predict higher values of diffusivity at dilute bulk concentrations and lower values of diffusivity when the bulk concentration is higher than 1 × 10-9 mol/cm3. The decrease in diffusivity with

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Figure 5. Values of diffusion coefficient (D, 10-6 cm2/s) from the best fit between the clean adsorption relaxation and the model predictions for the (L) Langmuir, (F) Frumkin, and (GF) generalized Frumkin models as a function of bulk concentrations.

Figure 6. Limiting adsorption rate constant βl (cm3 mol-1 s-1) and Kal as a function of dimensionless bulk concentration C0/a. D ) 6.5 ×10-6 cm2/s and ν*a ) 0.

increasing bulk concentration, similar to that in Figure 4 of ref 3, is evidence of a shift in controlling mechanism. Mixed-Controlled Model. The limiting value of dimensionless adsorption rate constant (Kal) for which the mixed-control curve is indistinguishable from diffusioncontrol profiles is obtained and is equal to a nearly constant value of 10. The corresponding values of adsorption rate constant βl ()β exp( - E°a/RT)) is plotted in Figure 6 as a function of dimensionless bulk concentration (C0/a). Figure 6 indicates that, if the value of β exp( - E°a/RT) of C10E8 ranges between 2 × 106 and 5 × 108 cm3/(mol‚s), C10E8 is of diffusion control at dilute concentration and of diffusive-kinetic control at more elevated bulk concentration. The concept of a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusionkinetic control at more elevated bulk concentration is adopted in the following calculation. A diffusivity of 6.5 × 10-6 cm2/s, which is the value obtained from the best fit between dynamic tension data and the theoretical prediction profiles of diffusion control at dilute bulk concentration of C10E8, is utilized and a mixed (diffusivekinetic) control simulation is performed using the generalized Frumkin adsorption model. Figure 7 shows the comparisons of experimental relaxation curves of surface

Figure 7. Experimental values of the dynamic surface tension (mN/m) for adsorption of C10E8 and the theoretical predictions of mixed controlled adsorption for different adsorption rate constants (a ) 105, b ) 106, c ) 107, d ) 108, e ) 2.5 × 106, and f ) 5 × 106 cm3/(mol‚s)) of the generalized Frumkin model; C0 ) 0.20 (a), 4.0 (b), and 10.0 (c) [10-9 mol/cm3]. Dashed lines denote the diffusion-limited curves. D ) 6.5 × 10-6 cm2/s and ν* a ) 0.

tension and the theoretical profiles of both diffusion control and mixed control. At low bulk concentration (for example, for C0 ) 2.0 × 10-10 mol/cm3 in Figure 7a), the experimental relaxation data of surface tension are still nearly indistinguishable from the diffusion-controlled profile predicted from the model. A lower bound on the adsorption rate constant, β exp( - E°a/RT) ) 2.6 × 106 cm3/(mol‚s), is

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Table 3. Values of Adsorption/Desorption Rate Constants for C10E8 (νa ) 0) Frumkin C (10-10 mol/cm3) 10-6βFa 104RFb Ka 1.0 2.0 5.0 10.0 20.0 40.0 60.0 100.0 300.0 av

4.2 4.8 5.7 6.9 2.8 4.8 7.8 11.0 19.0 7.4

5.5 6.3 7.4 9.0 3.6 6.3 10.2 14.3 24.7 9.7

10.1 1314 9.4 610 7.6 198 6.5 84.5 1.8 11.6 2.0 6.5 2.5 5.4 2.5 3.3 2.0 0.9 4.9 248

GF C (10-10 mol/cm3) 10-6βGa 105RGb Ka 1.0 2.0 5.0 10.0 20.0 40.0 60.0 100.0 300.0 av

2.2 2.6 3.5 4.5 2.3 4.0 10.0 13.0 20.0 6.9

4.7 5.5 7.5 9.6 4.9 8.5 21.3 27.7 42.6 14.7

Langmuir 102Kd 10-6βLc 103RLd KaL 4.2 4.8 5.7 6.9 2.8 4.8 7.8 11.0 19.0 7.4

KdL

2.4 10.1 56.5 4.0 9.4 39.2 8.4 7.6 22.5 16.1 6.5 15.2 10.5 1.8 3.3 29.0 2.0 3.0 62.5 2.5 3.3 126 2.5 2.9 474 2.0 1.6 81.5 4.9 16.4 Langmuir

103Kd

6.5 1380 5.9 628 5.3 226 4.9 104 1.7 18.3 2.0 10.8 4.0 14.1 3.8 8.1 2.9 2.0 4.1 265.7

10-6βLc 103RLd KaL 2.2 2.6 3.5 4.5 2.3 4.0 10.0 13.0 20.0 6.9

1.5 2.8 6.8 13.9 11.4 31.8 105 194 630 111

6.5 5.9 5.3 4.9 1.7 2.0 4.0 3.8 2.9 4.1

KdL 44.2 31.3 20.6 15.0 4.3 4.0 7.0 5.6 3.0 15.0

a β or β ) β exp( - E°/RT) [cm3/(mol‚s)]. b R or R ) R exp F G F G a ( - E°d/RT) [s-1]. c βL ) [cm3/(mol‚s)]. d RL ) [s-1].

obtained for C0 ) 2.0 × 10-10 mol/cm3 from the theoretical simulation shown in Figure 7a. At more elevated bulk concentrations (for example, for C0 ) 4.0 × 10-9 and 1.0 × 10-8 mol/cm3 in Figure 7b,c), the experimental relaxation data depart significantly from the diffusion-limited curves. The mixed-controlled curves with finite values of adsorption rate constant fit the experimental data well in each of the elevated bulk concentrations. The best-fit diffusion coefficient obtained from a diffusion controlled analysis has been shown in Figure 5. This apparent diffusivity decreases from 6.5 × 10-6 to 4.3 × 10-6 cm2/s as the bulk concentration increases when the generalized Frumkin model is used. When the Frumkin model is utilized, the apparent diffusivity decreases from 9.5 × 10-6 to 5.3 × 10-6 cm2/s. This change is larger than a typical change in diffusivity with bulk concentration. It is more likely that the resistance for the adsorption/desorption processes for C10E8 varies significantly with surface coverage. For example, Figure 6 indicates that the limiting adsorption rate constant βl for C10E8 varies from 106 to 109 cm3/(mol‚s) as the bulk concentration increases. When the adsorption rate constant β exp( - E°a/RT) ranges in this region, there exists a shift in controlling mechanism from diffusion control at dilute concentration to mixed diffusion-kinetic control at more elevated bulk concentration. The adsorption/desorption rate constants obtained, using the generalized Frumkin model, from the above calculations are shown in Table 3. The values of kinetic rate constant vary with bulk concentration, but the dependence is weak. The average values of the kinetic rate constant are β exp( - E°a/RT) ) 6.9 × 106 cm3/(mol‚s) and R exp( - E°d/RT) ) 1.5 × 10-4 s-1. The dimensionless adsorption/desorption rate constants Ka, and Kd in eq 7 are also calculated and shown in Table 3. At dilute concentration, Ka is around 6 and is very close to the limiting adsorption rate constant Kal, shown in Figure 5, from the theoretical simulations. At more elevated concentration, Ka decreases and approaches a nearly constant value of 2. Desorption. Two relaxation profiles of surface tension, due to C10E8 desorption out of an overcrowded interface,

Figure 8. Experimental values of the dynamic surface tension (mN/m) for reequilibration of C10E8 and the theoretical predictions of mixed-controlled reequilibration for different adsorption rate constants of the generalized Frumkin model for C0 ) 6.0 (a) and 10.0 (b) [10-9 mol/cm3]. β exp( - E°a/RT) ) (a) 105, (b) 106, (c) 107, (d) 108, and (e) 1.3 × 107 cm3/(mol‚s). DC (dashed line) denotes diffusion-limited curves. D ) 6.5 × 10-6 cm2/s and ν* a ) 0.

were plotted in Figure 4. The value of the diffusion coefficient of C10E8 in water has been obtained from the adsorption experiments (D ) 6.5 × 10-6 cm2/s when using the generalized 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. If this process is assumed to be diffusion controlled, the diffusion-controlled relaxation profiles by using the generalized Frumkin model with the diffusion coefficient D ) 6.5 × 10-6 cm2/s are shown in Figure 8. It is clear that the desorption relaxation profiles depart significantly from the diffusionlimited curves (the dashed curves). Therefore, the reequilibration process is assumed mixed controlled. Theoretical relaxation profiles, using the generalized Frumkin model and assuming νa ) 0, with finite adsorption rate constant are calculated and plotted in Figure 8 (the solid curves). From the data and the model predictions, it is found that the value of adsorption rate β exp( - E°a/RT) is 1.0 × 107 and 1.3 × 107 cm3/(mol‚s) for C ) 6.0 × 10-9 and 1.0 × 10-8 mol/cm3, respectively. The corresponding desorption rate constant R exp( - E°d/RT) is 2.1 × 10-4 and 2.8 × 10-4 s-1. These values of sorption kinetic constants obtained from the reequilibration experiment are exactly the same as those obtained from the clean adsorption experiment, as shown in Table 3.

Adsorption Kinetics of C10E8

Langmuir, Vol. 14, No. 9, 1998 2483

6. 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 was proposed and verified for C12E8 and C12E6 recently.1-3 The idea was adapted for the measurement of diffusion coefficient and adsorption/desorption rate constants using a clean adsorption process. The values of adsorption/desorption rate constants of C12E8 are also examined in a previous study using a reequilibration process.4 In this study, another polyoxy(ethylene) nonionic surfactant C10E8 was adapted for examining the shift in controlling mechanism. There does exist a shift in the controlling mechanism from diffusion control to mixed control at increasing bulk concentration on the adsorption of C10E8 onto a clean air-water interface. Values of diffusivity and adsorption/desorption rate constants are then obtained from the best fit between the dynamic surface tension data and the theoretical profiles of model prediction. The reequilibration experiment, in which molecules desorb out of an overcrowded interface due to a sudden shrinkage of a pendant bubble in a quiescent surfactant solution, was also performed. Adsorption/desorption rate constants are calculated from the relaxation profiles in surface tension of the reequilibration process. The values of kinetic rate constants obtained from these two different processes, clean adsorption and reequilibration, are exactly same. Contamination of the trace surface-active impurity may play an important role in the adsorption/desorption process. Due to their high surface activity they may remain at the surface after the compression. Therefore there exists a significant difference in equilibrium surface tension after compression and dilation.24,25 In this study, the impurity effect is examined via the compression experiment. Since the surface tension always goes back to, or reaches closely (with a deviation less than 0.1 mN/ m, as shown in Figure 4), the equilibrium surface tension after the monolayer is compressed for the most runs, the impurity effect is believed to be insignificant in the concentration range of this study. The depletion of surfactant in bulk due to the adsorption of C10E8 onto the air-water interface is concerned. At extremely low surfactant bulk concentration the depletion of surfactant from bulk due to adsorption may be significant. In this study, the pendant bubble with a diameter of 2-2.6 mm was generated in a quartz cell of 26 × 41 × 43 mm inside diameter. The amount of C10E8 adsorbed onto the air-water interface is 0.81%, 0.16%, 0.025%, and 0.003% of the C10E8 in bulk at C ) 1 × 10-10, 1 × 10-9, 1 × 10-8, and 1 × 10-7 mol/cm3, respectively. The lowest bulk concentration used for the tension measurement in this study is 1.0 × 10-10 mol/cm3. Therefore, the surfactant-depletion problem is insignificant in this study. The reequilibration process is found to be mixed diffusive-kinetic controlled, assuming D ) 6.5 × 10-6 cm2/s and using the generalized Frumkin model. Figure 9 shows the fit between the reequilibration data and the theoretical profiles from the generalized Frumkin mode when the reequilibration process is assumed to be of diffusion control. The diffusion-controlled relaxation curves with D ) 6.5, 3.0, and 1.0 [10-6 cm2/s] all fit the data poorly. Therefore it further confirmed that the reequilibration process is of mixed diffusive-kinetic control. (24) Lunkenheimer, K.; Miller, R. J. Colloid Interface Sci. 1987, 120, 176. (25) Kretzschmar, G.; Miller, R. Adv. Colloid Interface Sci. 1991, 36, 65.

Figure 9. Experimental values of the dynamic surface tension (mN/m) for reequilibration of C10E8 (C0 ) 1.0 × 10-8 mol/cm3) and the theoretical profiles of diffusion control of the generalized Frumkin model for different diffusivities. D ) 1.0 (a), 3.0 (b), and (c) 6.5 (c) [10-6 cm2/s].

Figure 10. Experimental values of the dynamic surface tension (mN/m) for adsorption of C10E8 and the theoretical predictions of (a) diffusion control for the Frumkin model for C0 ) 4.0 (1) and 10.0 (2) [10-9 mol/cm3] and of (b) mixed-controlled adsorption for different adsorption rate constants (a ) 105, b ) 106, c ) 4.8 × 106, d ) 107, and e ) 108 cm3/(mol‚s)) of the Frumkin model for C0 ) 4.0 × 10-9 mol/cm3. Dashed lines denote the diffusion-limited curves.

The Frumkin model assumes a linear relationship between activation energy of the adsorption/desorption processes Ea (Ed) and the surface converge Γ, while the generalized Frumkin model assumes a power law on Γ.

2484 Langmuir, Vol. 14, No. 9, 1998

In fact, the Frumkin adsorption model fits the dynamic surface tension profiles for C10E8 adsorbing onto a clean air-water interface as well as the generalized Frumkin model. Figure 10a shows two representative dynamic tension data and the diffusion-controlled best-fit profiles using the Frumkin model. The dependence of the diffusion coefficient on bulk concentration, when the adsorption is assumed to be a diffusion-controlled process, has been shown in Figure 5. The value of diffusivity at dilute concentration is around 9.0 × 10-6 cm2/s, which seems a little high compared with the diffusivity obtained using the generalized Frumkin model. The idea of shift in controlling mechanism is also examined by applying the Frumkin adsorption model. Figure 10b shows the comparison between the adsorption data for curve 1 in Figure 10a and the theoretical predictions of mixed diffusivekinetic controlled model with different values of adsorption rate constant. The kinetic rate constants are calculated in a similar way and are shown in Table 3. The average values of the kinetic rate constant β exp( - E°a/RT) ) 7.4 × 106 cm3/(mol‚s) and R exp( - E°d/RT) ) 9.7 × 10-4 s-1. The dimensionless kinetic rate constants Ka, and Kd in eq 7 are also shown. A higher Ka (or Kd) is obtained at dilute concentration, and at more elevated concentrations, Ka (or Kd) decreases and approaches a nearly constant value.

Chang et al.

The model used in this study is the generalized Frumkin model. If the intermolecular interaction between the adsorbed molecules is negligible, eq 5 becomes the Langmuir adsorption equation

dΓ/dt ) βLCs(Γ∞ - Γ) - RLΓ

(10)

or in dimensionless form

dx/dτ ) KaLC*s(1 - x) - KdLx

(11)

The rate constants βL, RL, KaL, and KdL correspond to β n exp(-Ea/RT), R exp(-Ed/RT), Ka exp( - ν* ax ), and Kd n exp( - ν*dx ), in the Frumkin or generalized Frumkin adsorption equation. Values of βL, RL, KaL, and KdL when ν*a ) 0 are calculated from the data in this study and listed in Table 3. The rate constants of the Langmuir adsorption equation calculated from the Frumkin model agree well with that from the generalized Frumkin model. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 84-2214-E-011-019). LA970923G