Adsorption Kinetics of 1-Octanol at the Air− Water Interface

The adsorption of 1-octanol at an air−water interface is studied theoretically and experimentally. A video-enhanced pendant bubble tensiometry is ut...
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Langmuir 1997, 13, 6211-6218

6211

Adsorption Kinetics of 1-Octanol at the Air-Water Interface Shi-Yow Lin,* Wei-Jiunn Wang, and Ching-Tien Hsu Department of Chemical Engineering, National Taiwan University of Science and Technology, 43, Keelung Road, Sec. 4, Taipei, 106 Taiwan, Republic of China Received June 10, 1997X The adsorption of 1-octanol at an air-water interface is studied theoretically and experimentally. A video-enhanced pendant bubble tensiometry is utilized for the measurement of relaxation of surface tension. Two types of processes were investigated: the adsorption onto an initially clean air-water interface and the desorption out of a suddenly compressed interface which is originally at equilibrium. A theoretical simulation using the equilibrium surface tension is performed and it is concluded that there is no shift in controlling mechanism for the adsorption of 1-octanol molecules from an aqueous phase onto an initially clean air-water surface. The adsorption of 1-octanol onto a clean air-water interface is verified experimentally to be a diffusion-controlled process. A diffusion coefficient was computed by comparing these adsorption profiles with numerical solutions of bulk surfactant diffusion equation and the generalized Frumkin adsorption model. The re-equilibration of a compressed interface agrees well with a diffusioncontrolled process. Lower bounds on the kinetic constants for the sorption process are inferred for octanol by comparing numerical solutions of mixed diffusion and surface kinetic transfer with the desorption relaxation data.

1. Introduction Surfactant molecules with long, slender hydrocarbon chains and small polar groups are subject to strong, attractive van der Waals forces when surface crowding causes interchain contact.1-4 It has been reported that the cohesive forces between 1-decanol molecules adsorbed onto an air-water interface play an important role in the sorption kinetics. Although the adsorption of decanol molecules onto a clean air-water interface is found to be diffusion-controlled, the desorption process, due to a sudden compression for the adsorbed decanol monolayer, becomes mixed (diffusive-kinetic) controlled. It is believed that, right after this sudden compression, decanol molecules become overcrowded at the interface and have a lower desorption rate due to the intermolecular cohesive forces. The aim of this paper is to investigate the sorption kinetics of 1-octanol, another surfactant molecule with long, slender hydrocarbon chains and small polar groups. The adsorption kinetics of 1-octanol has been studied by many research groups.1,5-12 In 1960, it was found that the short time approximation equation6 (Γ ) 2(D/π)1/2C0t1/2) could fit the dynamic surface tension data well. Therefore, it was concluded that the adsorption of 1-octanol in aqueous phase onto an initially clean interface is of diffusion control. A value of the diffusion coefficient (D ) 4.9 or 6.0 × 10-6 cm2/s) was obtained using the short time approximation equation by Defay and co-workers.7,8 Using the oscillating jet technique and using the von Szyszkowski equation (π ) RTΓ∞ ln(1 + C/a); a is the * Author to whom correspondence should be addressed: telephone, 886-2-737-6648; fax, 886-2-737-6644; e-mail, [email protected]. edu.tw. X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Fainerman, V. H.; Lylyk, S. V. Kolloid. Zh. 1982, 44, 598. (2) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (3) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (4) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526. (5) Hommelen, J. R. J. Colloid Sci. 1959, 14, 385. (6) Hansen, R. S. J. Colloid Sci. 1961, 16, 453. (7) Defay, R.; Hommelen, J. R. J. Colloid Sci. 1959, 14, 411. (8) Defay, R.; Petre, G. In Surface and Colloid Science; Matijevic, E., Ed,; Wiley: New York, 1971; Vol. 3, pp 27. (9) Bleys, G.; Joos, P. J. Phys. Chem. 1985, 89, 1027. (10) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (11) Chang, C. H.; Franses, E. I. Chem. Eng. Sci. 1994, 49, 313. (12) Chang, C. H.; Franses, E. I. Colloids Surf., A 1995, 100, 1.

S0743-7463(97)00614-8 CCC: $14.00

Langmuir-Szyszkowski constant) to describe the equilibrium surface tension of 1-octanol, Bleys and Joos9 reported that 1-octanol has a diffusion-controlled adsorption. Chang and Franses,10-12 using a pulsating bubble surfactometer and a modified Langmuir-Hinshelwood equation, concluded that the adsorption of 1-octanol is mixed controlled. On the other hand, Fainerman and Lylyk1 claimed that the intermolecular attraction between the adsorbed octanol molecules was important, the Frumkin adsorption isotherm was applied to describe the equilibrium surface tension, and a negative k value was reported. In this study, two processes, adsorption onto a clean interface and desorption due to a sudden compression for the adsorbed monolayer, are studied to investigate the adsorption kinetics of 1-octanol. In order to clarify the transport mechanism of octanol molecule in water and to examine the role of the intermolecular cohesive forces between adsorbed 1-octanol molecules, a technique with excellent accuracy on the dynamic surface tension is necessary. The pendant bubble tensiometer enhanced by video-image digitization, having an accuracy of (0.1 mN/m on the dynamic surface tension measurement, is utilized for the measurement of relaxation of surface tension. The outline of this paper is as follows. Section 2 describes the theoretical framework for the mass transfer process of surfactant molecules and the numerical solution procedure. The pendant bubble experimental technique and the relaxation profiles of surface tension for 1-octanol onto or away from an air-water interface are given in section 3. A theoretical simulation, checking the shift in controlling mechanism, from the equilibrium surface tension data is also detailed in section 3. In section 4, the experimental relaxation profiles are compared with theoretical predictions, which leads to computation of the diffusion coefficient and a lower bound of sorptive rate constants. The paper ends with a conclusion and discussion section. 2. Theoretical Framework Mass Transfer Equation. The adsorption of surfactant onto the interface of a freshly formed pendant bubble and the desorption of surfactant away from an overcrowded air-water interface in a quiescent surfactant © 1997 American Chemical Society

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solution are modeled. We shall consider only the case of one-dimensional diffusion and adsorption onto (or desorption away from) a spherical interface. The bulk phase is assumed to contain an initially uniform bulk concentration of the surface active solute, which does not 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 ) ∂r ∂t r2 ∂r

(

)

(r > b, t > 0)

(1)

with the following initial and boundary conditions

C(r,t) ) C0

(r > b, t ) 0)

C(r,t) ) C0

(R f ∞, t > 0)

dΓ/dt ) D(∂C/∂r) Γ ) Γb

(2)

(r ) b, t > 0) (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 is the surface concentration, b is the bubble radius, C0 is the concentration far from the bubble, and Γb is the initial surface converge. By 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)

D Γ(t) ) Γb + [C0t b

∫0 Cs(τ) dτ] + t

x

2

D [C xt π 0

∫0

xt

Cs(t - τ) d xτ] (3)

Adsorption Equation. 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 following rate expression:13 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 (4) where β, R, Ea(Γ), and Ed(Γ) are the pre-exponential 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

Ea ) Ea0 + νaΓn

Figure 1. Sketch of the energies involved in the adsorption/ desorption of surfactant molecules between the sublayer and the fluid interface.

at the interface. Constants νa and νd indicate the dependence between the activation energies of adsorption and desorption, respectively, on the surface concentration. The values of νa and νd may be positive or negative and are dependent upon the properties of surfactants. Equation 4 in nondimensional form becomes

dx/dτ ) Ka exp(-νa*xn) Cs*(1 - x) - Kd exp(-νd*xn) x (6) where x ) Γ/Γ∞, τ ) tD/h2, h ) Γe/C0, Ka ) β exp(-Ea0/ RT)C0/(D/h2), Cs* ) Cs/C0, Kd ) R exp(-Ed0/RT)/(D/h2), νa* ) νaΓ∞n/RT, νd* ) νdΓ∞n/RT, and k ) νd* - νd*. 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)

(7)

where k ) (νa - νd)Γ∞n/RT and a ) (R/β) exp[(Ea0 - Ed0)/ RT]. Equation 7 becomes the Frumkin adsorption isotherm14-18 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 (relative to that of adsorption) is described by k < 0. The adsorption of 1-alcohols on an aqueous solution-air interface has been shown to have a negative value of k.1-4 Polyethylene glycol alkyl ether RO(CH2CH2O)nH has been reported having a positive k.19-21 A positive value of k indicates that the adsorption is anticooperative, and adsorption becomes more difficult as the surface becomes more covered. When the surfactant solution can be considered ideal, the Gibbs adsorption equation dγ ) -ΓRT d ln C and the equilibrium isotherm (eq 7) allow for the calculation of the surface tension explicitly in terms of

[

γ - γ0 ) Γ∞RT ln(1 - x) -

kn n+1 x n+1

]

(8)

where x ) Γ/Γ∞, and γ0 is the clean surface tension. Numerical Solution. When the adsorption process is controlled solely by bulk diffusion, the surface concentra-

(5)

Ed ) Ed0 + νdΓn where Ea0, Ed0, νa, and νd are constants. Figure 1 shows the sketch of the energies involved in the molecule transfer of the adsorption/desorption process. The energies of molecules in the activated transition state and in the interface are dependent on the concentration of surfactant (13) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, 1973; Chapters 1 and 3.

(14) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (15) Miller, R. Colloid Polym. Sci. 1981, 259, 375. (16) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (17) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1994, 166, 73. (18) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995; Chapter 4. (19) Pan, R.; Maldarelli, C.; Ennis, B.; Green, J. Diffusive-Kinetic Adsorption of a Polyethoxylated Surfactant to the Air/Water Interface. In Dynamic Properties of Interfaces and Association Structures; Pollai, V., Shah, D. O., Eds.; AOCS Press: Champaign, IL, 1996; pp 23-47. (20) Lin, S. Y.; Chang, H. C.; Chen, E. M. J. Chem. Eng. Jpn. 1996, 29, 634. (21) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530.

Adsorption Kinetics of 1-Octanol

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tion can be obtained by solving eq 3, describing the mass transfer between sublayer and bulk, and eq 7, the sorption kinetics between subsurface and interface. If the adsorption process is of mixed control, eq 4 instead of eq 7 is solved coupled with eq 3 to find out the surface concentration. Then the dynamic surface tension γ(t) was calculated from eq 8. When these two equations (eqs 3 and 7 or eqs 3 and 4) are solved numerically, the technique used is a modification of that used by Miller and Kretzschmar.22 First, the convolution integral of Cs in eq 3 is integrated by first partitioning the interval (0, t) into intervals of size δ in time and integrating separately each interval by assuming a linear relation in time for Cs. After integration, eq 3 is of form

Cs(tn) ) b1Γ(tn) + b2

(9)

where tn ) nδ and the constants b1 and b2 are a function of the concentrations Cs(0), Cs(δ), ..., Cs(tn-1). Equation 3 is then solved numerically by first separating the variables and then integrating from x(tn-1) to x(tn) using Simpson’s 1 /3rd rule.23 In this numerical integration, Cs is assumed to be a constant value during the small integration time step δ and is set equal to the average value of Cs(tn-1) and Cs(tn). By using Newton’s method, 23 a value for x(tn) is found for which the surface concentration integral is equal to δ. The dynamic surface tension at time tn is obtained from eq 8 once the surface concentration x(tn) is known. The Dependence on νa. When the adsorption/desorption process is of mixed control, eq 4 (or eq 6 in dimensionless form) is solved coupled with eq 3 to find out the surface concentration. While the value of k ()νa* νd*) is obtained from the best-fit between the equilibrium surface tension and the model predictions (eqs 7 and 8), no data are available for finding out the value of νa* or νd*. The relaxation profiles of surface tension are dependent upon the value of νa*. One possible way to obtain the νa* value is from the comparison between the adsorption or desorption relaxation data and the model predictions for different νa*. If one νa* fits the data clearly superior to others, one may conclude that it is the right value of νa*. The interaction parameter k for 1-octanol is less than zero, which can correspond to the scenario that (i) νa* < 0, νd* ) 0, (ii) νa* ) 0, νd* > 0, and (iii) νa* - νa* < 0, neither equal to zero. For 1-octanol, it is more likely that the energy barrier to desorption becomes increasingly important because of intermolecular cohesion. Therefore, although option 3 is the more likely scenario, option 2 was picked for the following calculation to simplify the theoretical simulation. A zero νa* represents that the activation energy of adsorption is independent on surface coverage, and a positive νd* represents that desorption rate slows down as the surface concentration increases. 3. Experimental Measurements Materials. 1-Octanol (purity >99.5%) was obtained from Fluka Chemie and 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 less than 0.057 µS/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. (22) Miller, R.; Kretzschmar, G. Colloid Polym. Sci. 1980, 258, 85. (23) Carnahan, B.; Luther, H. A. Wilkes, J. O. Applied Numerical Methods; Wiley: New York, 1969.

Apparatus. Pendant bubble tensiometry enhanced by video digitization was employed for the measurement of the 1-octanol surface tension relaxation and the equilibrium tensions. 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 an octanol 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 (0.1 K.24 A 22-gauge stainless steel inverted needle (0.016 in. i.d.; 0.028 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 gas-tight 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 experimental protocol was as follows: the quartz cell of 26 × 41 × 43 mm inside diameter was initially filled with the octanol 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 0.07 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 1 h. 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. The images were recorded on a recorder during the formation of bubble. The images on tape were processed to determine the bubble edge coordinates, bubble volume, bubble surface area, and the surface tension for studying the formation of bubble. There is a nearly constant deviation of surface tension, about 0.7 mN/m, between the images directly onto 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 image digitizer (512 lines) or CCD camera (610 lines). The re-equilibration experiments are performed as follows: After the relaxation of clean adsorption was complete, the valve was opened for 0.11 s (controlled by computer) while the syringe pump was off. A small part of the gas inside the bubble was allowed to pass through the solenoid valve and the surface area of the gas bubble decreases around 20%. The images were recorded on a recorder during this process and also taken sequentially onto the computer. After the relaxation of the desorption was complete, the images on both tape and computer were processed to determine the bubble edge coordinates and the surface tension. The theoretical shape of the pendant bubble is derived according to the classical Laplace equation that relates the pressure difference across the curved fluid interface25,26

γ[1/R1 + 1/R2] ) ∆P

(10)

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 10 can be recast as a set of three first-order differential equations for the spatial positions x1 and x2 and turning angle φ of the interface as a function of the arc length s, and then integrated by using a Runge-Kutta scheme23 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 10. 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 (24) Lin, S. Y.; Hwang, H. F. Langmuir 1994, 10, 4703. (25) Huh, C.; Reed, R. L. J. Colloid Interface Sci. 1983, 91, 472. (26) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169.

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Figure 2. Equilibrium surface tension (mN/m) for air/1-octanol aqueous solution (O, data from pendant bubble; 4, data from the Wilhelmy plate tensiometer) and the theoretical predictions of the Langmuir (L), Frumkin (F), and generalized Frumkin (GF) adsorption isotherms. Table 1. Constants of Optimal Fit of Octanol Aqueous Solution

L F GF

Γ∞ (10-10 mol/cm2)

a (10-7 mol/cm3)

k

n

8.958 5.910 6.125

5.565 7.653 18.58

-2.661 -3.023

1 0.4289

the data points by minimizing the objective function. Minimization equations are solved by directly applying the NewtonRaphson method,23 and from the optimal values of R0 and B the tension can be computed. As demonstrated by Lin et al.,27,28 the accuracy and reproducibility of the dynamic surface tension measurements obtained by this procedure are (0.1 mN/m. Equilibrium Data. The equilibrium surface tensions for octanol aqueous solutions at the air-water interface were extracted from the long-time asymptotes of the surface tension relaxation at different bulk concentrations. All measurements were performed at 25.0 ( 0.1 °C. The data from the pendant bubble method and those from a Wilhelmy plate surface tensiometer (CBVP-A3, Face) are both shown in Figure 2. The comparisons between the equilibrium surface tension and the best fit from the adsorption isotherms of the Langmuir, Frumkin, and generalized Frumkin are also presented in Figure 2. The model constants, as shown in Table 1, are obtained by adjustment so as to minimize the error between the model predictions and experimental values. The more exact agreement of the Frumkin and the generalized Frumkin models indicates that the relatively long, slender hydrocarbon chain of the 1-octanol molecule can give rise to strong intermolecular attraction as the surface coverage is high. Note that both the Frumkin and the generalized Frumkin models predict a negative k value from the equilibrium surface tension data.

4. Theoretical Simulation from Equilibrium Data It has been reported19,21 that for poly(oxyethylene) 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 1-octanol, a series of simulations in which surfactant adsorbed onto an initially clean, spherical surface from a bulk phase of initially uniform concentration are performed. Relaxation in surface tension is presented nondimensionally in terms (27) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159. (28) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf., A 1996, 114, 31.

Figure 3. Influence of bulk concentration on the dynamic surface tension for clean interface adsorption under the conditions of kinetic-diffusive control and C0/a ) 0.1 (a, upper) and 1 (b, lower). The dashed curves denote the diffusion control profile. Ka ) 0.01 (a), 0.1 (b), 1 (c), 10 (d), 22 (e), and 100 (f). The generalized Frumkin model with νa* ) 0 is chosen for the simulation.

of the variable θ(τ), θ ) (γ - γe)/(γ0 - γe). The range of bulk concentration considered, 0.03 to 1.6 times C0/a, is one that the reduction of surface tension and the relaxation time are both large enough for the dynamic surface tension measurement of octanol. The generalized Frumkin model with νa* ) 0 was picked for the simulation, the model constants (Γ∞, a, k, and n) utilized are those listed in Table 1. The nondimensional relaxation θ(τ) is a function of C0/a and Ka (Kd, the dimensionless characteristic desorption rate, is given by Ka/(C0/a)). Simulations of θ as a function of τ for different values of C0/a are presented in a set of two figures (Figure 3 a,b), in which Ka is varied from diffusion control (Ka f ∞) to mixed control (e.g., Ka ) 1, curve c in Figure 3) to sorption kinetic control (e.g., Ka )0.01, curve a in Figure 3). The relaxations in Figure 3 have the same general dependence on Ka. The diffusionlimited curve shows the fastest relaxation; slower relaxations are observed as Ka decreases because of the increasing kinetic barrier. (Note that decreasing Ka also decreases Kd since Kd ) Ka/(C0/a) and C0/a is fixed). The distance between the diffusion-limited curve and that for a particular value of Ka varies little with C0/a. As C0/a varies, the distance between the diffusion and mixedcontrolled curves is almost the same for fixed Ka. The limiting value of Ka for which the mixed-controlled curve is indistinguishable from diffusion-control profiles is defined as Kal. The relaxation of subsurface concentration for the curves presented in Figure 3b for C0/a ) 1 is examined more

Adsorption Kinetics of 1-Octanol

Figure 4. Relaxation of subsurface concentration in dimensionless form corresponding to the relaxation profiles in Figure 4b for C0/a ) 1. Ka ) 0.01 (a), 0.1 (b), 1 (c), 10 (d), 22 (e), and 100 (f). The dashed curves denotes the diffusion-controlled relaxation.

Figure 5. The limiting desorption rate constant βl and Kal as a function of C0/a for νa* ) 0. Above the curve, the mass transport is of diffusion-control; below the curve, it is mixed controlled.

closely; the curves are shown in Figure 4. The dimensionless subsurface concentration (Cs* ) Cs/C0) for diffusion-controlled adsorption increases monotonically from 0 to 1 at a dimensionless time of about 1 (the dashed line in Figure 4). A signature of the mix-control profile is a minimum in the Cs* profile (for example, curve d in Figure 4). The minimum value of Cs* approaches unity as Ka decreases, indicating a shift to complete kinetic control of the adsorption process. Note that for the curves corresponding to Kal at each C0/a, the minimum Cs* is around 0.15. Figure 5 shows the limiting adsorption rate constant Kal and βl ()Kal(D/h2)/C0), obtained from the simulations shown in Figures 3 and 4, as a function of C0/a. The data show that for νa* ) 0, Kal varies only slightly with bulk concentration. It increases first from 10 to 100, then comes down to 10 at higher bulk concentrations. The corresponding adsorption rate constant βl keeps nearly constant. This constant βl indicates that there is no shift in controlling mechanism for the adsorption of 1-octanol molecules from an aqueous bulk phase onto an initially clean air-water surface. 5. Comparisons of Experimental Data and Theoretical Profiles Dynamic Surface Tension Data. Relaxations in the surface tension due to adsorption of octanol were measured at the air-water interface. Data were recorded up to more

Langmuir, Vol. 13, No. 23, 1997 6215

Figure 6. Representative dynamic surface tensions (mN/m) for clean adsorption of 1-octanol aqueous solutions for C0 ) 2 (a), 3 (b), 4 (c), 5 (d), and 8 (e) (10-7 mol/cm3).

than 1 h from the moment (referenced as t ) 0) at which two-thirds of the bubble surface area is generated during the bubble formation. Shown in Figure 6 are representative dynamic surface tension profiles (for one selected bubble at each bulk concentration) of octanol aqueous solutions at five different bulk concentrations, C ) 2, 3, 4, 5, and 8 × 10-7 mol/cm3. The reproducibility of these profiles is demonstrated in Figure 7, where the results of several pendant bubbles at four concentrations are given. After the establishment of equilibrium, adsorbed octanol monolayers at the air-water interface are compressed by shrinking the air bubble. The relaxations of surface tension and surface area are monitored during the shrinkage. On the desorption experiments of octanol monolayer, it is difficult to get a complete set of relaxation profiles of surface tension. More is detailed in the discussion section. The data shown in Figure 8 are one of the several complete runs where the bubble volumes did not vary too much during the desorption process. In this desorption experiment, the surface tension decreases from the equilibrium value (66.4 mN/m for C ) 2 × 10-7 mol/cm3) to a lower value (63.1 mN/m), corresponding to a surface coverage higher than the equilibrium one, in 0.067 s. The surface tension then increases and goes back to the equilibrium value in a few hundred seconds. The bubble surface area, also shown in Figure 8, decreases 22% in 0.1 s and then keeps a nearly constant value for a few hundred seconds. Table 2 shows the changes of surface area and the mass balance during the shrinkage in columns Ai/Ae and AiΓi/AeΓe, where Ae and Γe denote the bubble surface area and surface concentration before the compression, respectively. The moment in which surface tension begins to deviate from the equilibrium value (γe, at this moment, the surface area is Ae and the surface coverage is Γe) is referred as the zero time. The data in column AiΓi/AeΓe of Table 2 indicate that the amount of surfactant molecules at the air-water interface keeps a nearly constant during the shrinkage. The relaxation data are replotted and compared with the model predictions in Figure 9. For the convenience of theoretical calculation, the moment with the lowest surface tension value (γb, at this moment, the surface area is Ab and the surface converge is Γb; marked at point L in Figure 8) is set to be the zero time in Figure 9. Comparison. If the adsorption of octanol onto a clean interface was assumed to be diffusion controlled, the data in Figure 6 may be used to determine a diffusion coefficient for the octanol molecules in the aqueous phase. The pendant bubble is treated as a sphere surrounded by an

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Lin et al.

Figure 7. Experimental values of the dynamic surface tensions (mN/m) for adsorption of 1-octanol onto a clean air-water interface and the theoretical predictions of the diffusion control process for the Langmuir (L), Frumkin (F), and generalized Frumkin (GF) models for C0 ) 2 (a), 3 (b), 4 (c), and 5 (d) (10-7 mol/cm3). Each symbol represents a separate bubble. Table 2. Relaxations of Surface Tension, Surface Area, and the Amount of Octanol Molecules at the Interface during the Shrinkage of the Bubble t (s)

γ (mN/m)

A (mm2)

Γa × 10-10 (mol/cm2)

AΓ × 109 (mol)

Ai/Ae

AiΓi/AeΓe

0 1/ 60 2/ 30 3/ 30

66.49b 66.36c 63.12d 63.36

17.62 17.59 13.98 13.69

3.10 3.15 4.07 4.01

5.465 5.543 5.686 5.495

1.00 1.00 0.79 0.78

1.00 1.01 1.04 1.01

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

Figure 8. Representative dynamic surface tension (mN/m; 4) and surface area (mm2; 0) of pendant bubble for reequilibration of 1-octanol aqueous solutions for C0 ) 10-7 mol/cm3.

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 an initial surface concentration Γb. Γb ) 0 for a clean adsorption process, in which the bubble was created suddenly (less than 0.08 s). There is a small amount of surfactant present at the air-water interface before bubble growth. While the presence of this adsorbed surfactant was shown to be significant in the growing drop technique,29 its effect on the pendant bubble method (29) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1993, 160, 435.

is insignificant because the surface area of the pendant bubble increases roughly 1000 times during the rapid bubble growth, which depletes the preadsorbed surfactants to a negligible surface converge. For the desorption process, the air-water interface was suddenly compressed for around 20-25% of the surface area. 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 ) 1/15 s in Figure 8. So in Figure 9, 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. The model constants (Γ∞, a, k, and n) 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

Adsorption Kinetics of 1-Octanol

Langmuir, Vol. 13, No. 23, 1997 6217

Figure 9. Experimental values of the dynamic surface tension (mN/m) for re-equilibration of 1-octanol solution and the theoretical predictions of diffusion-controlled re-equilibration (dashed line, DC) and mixed controlled ones (solid lines) for different desorption rate constants of the generalized Frumkin model. β exp(-Ea°/RT)) 105 (a), 106 (b), 107 (c), and 108 (d) cm3/(mol‚s); C0 ) 2 × 10-7 mol/cm3.

Figure 10. Relaxation data of surface tension (mN/m) for clean adsorption of 1-octanol and the model predictions of a mixed controlled process at different adsorption rate constant β exp(-Ea°/RT)) 105 (a), 106 (b), 107 (c), and 108 (d) cm3/(mol‚s). C0)2 × 10-7 mol/cm3, D ) 8.0 × 10-6 cm2/s, νa* ) 0, and the generalized Frumkin model is used. The dashed curve (DC) denotes the diffusion-limited profile.

Table 3. Diffusion Coefficients of Octanol D (10-6 cm2/s) L

F

GF

0.3

1.3 2.5

3.0 4.0

3.0 4.0 5.0 8.0

1.0 2.3 3.5 5.0

6.0 7.3 9.0 7.0

6.0 6.5 8.0 7.0

ava

3.0

7.3

6.9

C

a

mol/cm3)

1.5 2.0

(10-7

The averaged values of diffusivity for C0 g 3.0 × 10-7 mol/cm3.

the best agreement with the data. As shown in Figure 7, the agreement between the theoretical relaxation profile obtained by using the Langmuir isotherm is poor at low bulk concentrations. The Frumkin numerical profiles fit the experimental data reasonable well. The agreement between the data and the numerical relaxations constructed from the generalized Frumkin model is superior to that of the Langmuir and Frumkin models. The values of diffusion coefficient, determined from the best-fit between the dynamic tension data and the theoretical predictions, are listed in Table 3 for the Langmuir, Frumkin, and generalized Frumkin models. The averaged values of diffusion coefficient D ) 6.9 ( 1.0, 7.3, and 3.0 for the generalized Frumkin, Frumkin, and Langmuir models, respectively, for C0 g 3.0 × 10-7 mol/cm3. Lin et al.30 have shown that the imperfect fit at low bulk concentration on the semilog diagram of surface tension versus bulk concentration may cause a serious underestimate or overestimate on diffusivity. All three adsorption isotherms used in this study predict a lower surface tension at low bulk concentration; therefore a underestimate of diffusivity is expected from the fit of dynamic surface tension data. This is why the first two diffusivity data in Table 3 at low concentration are not taken to average the diffusivity. Since the value of diffusion coefficient obtained using the generalized Frumkin model is a nearly constant value, it seems to be reasonable assuming the mass transfer process of 1-octanol is diffusion-controlled. More simulations and comparisons are performed for a mixed diffusive-kinetic controlled clean adsorption process. It is assumed that there exists a shift in

Figure 11. Values of adsorption rate constant β exp(-Ea°/RT) obtained from the best fit between the dynamic surface tension data of octanol onto a clean interface and the model predictions by assuming that there exists a shift in controlling mechanism from diffusion control at high bulk concentration to mixed diffusive-kinetic control at dilute concentration. The generalized Frumkin model and D ) 8.0 × 10-6 cm2/s are used for the calculation.

controlling mechanism from diffusion control at high bulk concentration to mixed diffusive-kinetic control at dilute concentration. The generalized Frumkin model with νa* ) 0 and different diffusion coefficients, D ) 8 and 12 × 10-6 cm2/s, are picked for the calculation. A representative fitting between the adsorption data and the relaxation profiles for C ) 2 × 10-6 mol/cm3 predicted from a mixed controlled model is shown in Figure 10. The model prediction fits the dynamic surface tension data reasonably well, but the fit is worse than that with the diffusioncontrolled profiles as shown in Figure 7a. Besides, the adsorption rate constant β* ) β exp(-Ea°/RT) obtained from the mixed controlled model varies strongly with the bulk concentration. Figure 11 shows the values of adsorption rate constant β* for the case of D ) 8 × 10-6 cm2/s. The value of adsorption rate constant varies more than 2 orders of magnitude in a narrow range of bulk (30) Hus, C. T.; Chang; C. H.; Lin, S. Y. Langmuir 1997, 13, 6204.

6218 Langmuir, Vol. 13, No. 23, 1997

Figure 12. Representative relaxation of the bubble surface area on the formation of a pendant bubble on the tip of a 22 gauge needle.

concentration for D ) 8 × 10-6 cm2/s. For D ) 12 × 10-6 cm2/s, it varies around 1 order of magnitude at the same bulk concentration range, but the fit becomes worse. This strong variation or poor fitting indicates that the clean adsorption of octanol is highly unlikely a mixed diffusivekinetic-controlled process. Consider next the reequilibration process. An initially equilibrium-established 1-octanol monolayer at the airwater interface is suddenly compressed, and the surface tension relaxation profiles are shown in Figures 8 and 9. If this process is again assumed to be diffusion controlled, the diffusion-controlled relaxation profile by using the generalized Frumkin model with the diffusion coefficient obtained from the adsorption, onto a clean interface, process is shown in Figure 9 (the dashed lines). Clearly, the desorption relaxation data are pretty close to the diffusion-limited curve. Relaxation profiles with finite desorption rate constant β* are also calculated. Figure 9 shows that no curve with finite β* fits the desorption data better than the diffusion-limited one. Therefore it is concluded that the desorption process is of diffusion control, and the adsorption rate constant β exp(-Ea°/RT) of octanol has a lower bound 1 × 109 cm3 mol-1 s-1. A theoretical simulation with finite adsorption rate constant was also done for the clean adsorption process of octanol. Figure 3a shows that the relaxation profile of clean adsorption with β exp(-Ea°/RT) >1 × 109 cm3 mol-1 s-1 (see Figure 5 at C0/a ) 0.1) is indistinguishable with that of diffusion control for C0 ) 1.86 × 10-7 mol/cm3. Similar results are obtained for other bulk concentrations and the dependence of βl on bulk concentration is shown in Figure 5. 6. Discussion and Conclusions The adsorption of 1-octanol onto a clean air-water interface was previously assumed to be a diffusion-

Lin et al.

controlled process, and the surface tension relaxation data were then used to determine a diffusion coefficient for the octanol molecules in the aqueous phase. The desorption process of octanol was found to be diffusion controlled also. This is quite different from the sorption kinetics of 1-decanol, which has a diffusion-controlled clean adsorption process and a mixed (diffusive-kinetic) controlled desorption.3 The sorption kinetics of 1-octanol is similar to that obtained for diazinon in a previous study.31 For diazinon, the clean adsorption process is diffusion controlled and the theoretical diffusion-controlled curves fit the desorption data pretty well also at different bulk concentrations. For the poly(oxyethylene) nonionic surfactants 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 concentrations.19-21 The bubble-generation system is of computer control. In this study, it took about 0.07 ( 0.01 s to form an air bubble with around 2 mm diameter on the tip of a 22gauge stainless inverted needle with 90° bevel. Figure 12 shows a representative curve of the increase of surface area of the pendant bubble during bubble formation. To identify the age of the air-water surface, we defined the moment at which two-thirds of the surface area is generated during the bubble formation as t ) 0. Adsorbed octanol monolayers, after the establishment of equilibrium at the air-water interface, are compressed by shrinking the air bubble. Similar desorption experiments have also been performed for 1-decanol, polyoxylene surfactants C12E8, C10E8 and diazinon.3,31-33 In this study, a 22-gauge needle is used for the octanol desorption experiment. More than 20 runs were performed, but only several complete sets of relaxation data were obtained. It is found that for the octanol monolayer (i) most of the pendant bubbles cannot stay on the tip of the needle for a long time (longer than several hundred seconds) with a constant bubble volume and (ii) the surface area of bubble usually decreases more than 20%, which is too large to be a small perturbation of the adsorbed monolayer. The reason why there is trouble controlling the desorption experiment of the octanol monolayer, compared with that for decanol, C12E8, C10E8, and diazinon, is still not clear so far. It is probably due to the small diameter of needle used. A needle with larger diameter is recommended for those trying to perform a similar experiment. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 84-2214-E-011-019). LA970614Q (31) Lin, S. Y.; Lin, L. W.; Chang, H. C.; Ku, Y. J. Phys. Chem. 1996, 100, 16678. (32) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (33) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Submitted for publication in Langmuir.