A Study of Surfactant Adsorption Kinetics: Effect of Intermolecular

A concept on the adsorption kinetics of surfactants was explored and illustrated with 1-octanol and C12E8 recently (Langmuir 1997, 13, 6204): using on...
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Langmuir 1999, 15, 1952-1959

A Study of Surfactant Adsorption Kinetics: Effect of Intermolecular Interaction between Adsorbed Molecules Ching-Tien Hsu,† Chien-Hsiang Chang,‡ and Shi-Yow Lin*,† Departments of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Sec. 4, Taipei, 106 Taiwan, and National Cheng Kung University, Tainan, 701 Taiwan Received June 4, 1998. In Final Form: December 16, 1998 A concept on the adsorption kinetics of surfactants was explored and illustrated with 1-octanol and C12E8 recently (Langmuir 1997, 13, 6204): using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a serious mistake on the determination of adsorption kinetics and/or on the evaluation of diffusion coefficient from the dynamic surface tension data. To learn the effect of the intermolecular interaction between adsorbed molecules, a systematic theoretical study is performed for surfactants adsorbing onto a clean planar air-water interface. Data from the theoretical simulation indicate that the stronger are the surfactants interactions (i.e., with a larger absolute K value), the greater are the deviations in apparent diffusivity or in sorption rate constants obtained from a Langmuir analysis. Besides, a larger deviation on diffusivity results at more dilute concentration or when a smaller range of surface tension data are utilized to determine the model constants.

Introduction Adsorption kinetics and the mass transfer coefficients are usually studied through the measurement of equilibrium surface tension and surface tension relaxation profiles at different surfactant concentrations. Equilibrium surface tension data indicate the reduction of surface tension and the surface concentration at different surfactant concentrations and the information on intermolecular interaction of the adsorbed surfactant molecules. The equilibrium tension data are usually fitted with the adsorption isotherm, for example, the Langmuir or Frumkin model, to determine the model constants. Due to the limitation of the tensiometer, usually only a limited range of surfactant concentration is measured, and therefore only the surface tension data with high surface pressure are applied for the determination of adsorption isotherm and the model constants. The surface tension relaxation is governed by adsorption and/or diffusion processes as the rearrangement of surfactant molecules at the fluid interface is very fast. Dynamic surface tension profiles for surfactants adsorbing onto a freshly created interface are usually measured and compared with the model-predicted relaxation profiles, which are generated using the model constants from the equilibrium data. If the relaxation profiles are found to be in agreement with a diffusion-controlled adsorption model, the diffusivity is then determined. If the surface tension relaxes slower than that predicted by the diffusioncontrolled adsorption model with a reasonable value of diffusivity, a mixed diffusive-kinetic or a kineticcontrolled process is then concluded. Then, rate constants of adsorption/desorption are determined from the best fit between the dynamic surface tension data and the modelpredicted profiles. Recently, it has been reported that the intermolecular interactions between the adsorbed surfactant molecules are significant for many surfactants.1-16 The inter* To whom correspondence should be addressed. Tel: 886-22737-6648. Fax: 886-2-2737-6644. E-mail: [email protected]. † National Taiwan University of Science and Technology. ‡ National Cheng Kung University.

molecular interactions can be estimated from equilibrium surface tension data. It is important to evaluate accurately the intermolecular interactions since this intermolecular force plays an important role in investigating the adsorption kinetics of surfactants. A new concept has been explored and illustrated with 1-octanol and C12E8 in a previous study: using only a limited range of equilibrium data of surface tension to determine the adsorption isotherm can cause a serious mistake in the determination of adsorption kinetics and/or on the evaluation of diffusion coefficient from the dynamic surface tension data.17 The aim of this work is to examine the concept systematically, especially the effect of the intermolecular interaction between adsorbed molecules. Surfactants with cooperative or anticooperative adsorption behavior are investigated. The equilibrium surface tension profile and the tension relaxation for surfactant adsorbing onto a freshly created air-water planar interface are simulated using the Langmuir and Frumkin adsorption equations. Governing Mass Transfer Equation Bulk Diffusion. The adsorption of surfactant onto a freshly formed planar interface in a quiescent surfactant (1) Miller, R. Colloid Polym. Sci. 1981, 259, 375. (2) Fainerman, V. H.; Lylyk, S. V. Colloid J. USSR 1982, 44, 538. (3) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (4) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (5) Lunkenheimer, K.; Hirte, R. J. Phys. Chem. 1992, 96, 8683. (6) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (7) Chang, C. H.; Franses, E. I. Chem. Eng. Sci. 1994, 49, 313. (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. (10) Lin, S. Y.; Hwang, W. B.; Lu, T. L. Colloids Surf. A 1996, 114, 143. (11) Lin, S. Y.; Lin, L. W.; Chang, H. C.; Ku, Y. J. Phys. Chem. 1996, 100, 16678. (12) Pan, R. Ph.D. Dissertation, City University of New York, New York, 1996. (13) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (14) Lin, S. Y.; Wang W. J.; Hsu, C. T. Langmuir 1997, 13, 6211. (15) Pan, R.; Green, J.; Maldarelli, C J. Colloid Interface Sci. 1998, 205, 213. (16) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Langmuir 1998, 14, 2476. (17) Hsu, C. T.; Chang C. H.; Lin, S. Y. Langmuir 1997, 13, 6204.

10.1021/la980656u CCC: $18.00 © 1999 American Chemical Society Published on Web 02/19/1999

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solution is modeled. We shall consider only the case of one-dimensional diffusion and adsorption onto a planar interface. It is assumed that the bulk phase contains an initially uniform concentration of the surface active solute, which does not dissolve into the gas phase of the bubble. The convection effects are assumed to be negligible. The diffusion of surfactant in the bulk phase is described by the Fick’s law:

value of K,2,4,6,7,14,17 whereas poly(oxyethylene glycol alkyl ether), RO(CH2CH2O)nH, has a positive K.12,13,15,19 If the surfactant solution is considered ideal, the Gibbs adsorption equation dγ ) - ΓRT dln C and the equilibrium isotherm (eq 6) allow for the calculation of the surface tension explicitly in terms of surface concentration:

γ - γ0 ) Γ∞ RT[ln(1 - x) - Kx2/2]

(7)

2

D

∂ C ∂C ) (x > 0, t > 0) ∂t ∂x2

(1)

with the initial and boundary conditions

C(x,t) ) C0 (x > 0, t ) 0) C(x,t) ) C0 (x f ∞, t > 0) dΓ/dt ) D(∂C/∂x) (x ) 0, t > 0) Γ ) 0 (t ) 0)

(2)

where x and t are the distance away from the interface and time, D denotes the diffusion coefficient, C(x,t) is the bulk concentration, Γ is the surface concentration, and C0 is the concentration far from the bubble. By using the Laplace transform, the solution of the above set of equations can easily be formulated in terms of unknown subsurface concentration Cs(t) ) C(x ) 0, t):

Γ(t) ) 2(D/π)1/2[C0xt -

∫0xtCs(t - τ) dτ1/2]

(3)

Adsorption Isotherm. The Frumkin adsorption kinetics is used to describe the adsorption-desorption process of surfactant molecules between the interfacial sublayer and the interface itself:

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 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 interactions at increasing surface coverage, the activation energies are assumed to be proportional to surface concentration

Ea ) Ea0 + νaΓ Ed ) Ed0 + νdΓ

(5)

where Ea0, Ed0, νa, and νd are constants. At equilibrium, the time rate of change of Γ vanishes and the adsorption isotherm that follows is given by

Γ/Γ∞ ) x ) C/(C + a exp(Kx))

(6)

where K ) (νa - νd)Γ∞ /RT and a ) R/β exp[(Ea0 - Ed0)/RT]. Equation 6 becomes the Langmuir adsorption isotherm when νa ) νd ) K ) 0. The presence of cohesive intermolecular forces which increase with surface coverage and lower the desorption rate (relative to that of adsorption) 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 crowded. The adsorption of n-alcohols on a aqueous solution-air interface has been shown to have a negative

where x ) Γ/Γ∞ and γ0 is the surface tension of pure water. Numerical Solution. When the adsorption process is controlled solely by bulk diffusion, the surface concentration can be obtained by solving eq 3, describing the mass transfer between sublayer and bulk, and eq 6, the sorption kinetics between sublayer and interface. If the adsorption process is of mixed control, eq 4 instead of eq 6 is solved coupled with eq 3 to find out the surface concentration. Then the dynamic surface tension γ(t) is calculated from eq 7. When these two equations (eqs 3 and 6 or eqs 3 and 4) are solved numerically, the technique used is a modification of that used by Miller and Kretzschmar.20 The method on the integration and calculation has been detailed in ref 21. Results Equilibrium Surface Tension. An expression for the surface tension as a function of bulk concentration C0 can be obtained by numerically solving eq 6 to obtain Γ/Γ∞ and then substituting this result into eq 7. In the following calculation, maximum surface concentration (Γ∞), surfactant activity, temperature, and surface tension of pure water (γ0) are assumed be to 5 × 10-10 mol/cm2, 1 × 10-9 mol/cm3, 25 °C, and 72.0 mN/m, respectively. Figure 1 plots the surface tension as a function of the logarithm of C0 for different values of K. Γ is the equilibrium surface concentration for the surfactant solution with bulk concentration C0. Recall that K ) 0 indicates the Langmuir adsorption, a negative K indicates the presence of cohesive intermolecular forces, and a positive K indicates the adsorption is anticooperative. Figure 1 shows that as K decreases, especially approaching the limiting value -4, the curve begins to develop a cusplike behavior in which the slope (and therefore the surface concentration, ΓRT ) -dγ/dln C) increases rapidly in absolute value with only a very small change in C0/a. This presents the effect of intermolecular attraction.4 The cohesive energies also lower the surface pressure, and hence, the increased adsorption does not lower the surface tension that significantly. After the concentration is raised to values nearing Γ∞, the first term ln(1 - Γ/Γ∞) in eq 7 dominates, and the surface tension decreases rapidly. Therefore, a cusplike behavior is formed. For surfactant with a positive K, the adsorption is anticooperative. Adsorption becomes more difficult as the surface becomes more crowded; therefore, a smaller equilibrium surface concentration is resulted at the same value of C0/a. Whereas, at the same surface coverage, the surfactant with positive K value has a higher surface pressure. These specific properties are shown in the insert figures in Figure 1. To demonstrate the proposed idea, using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a serious mistake, it (18) Hsu, C. T.; Lin S. Y. J. Chin. Inst. Chem. Eng. 1998, 29, 1. (19) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (20) Miller, R.; Kretzschmar, G. Colloid Polym. Sci. 1980, 258, 85. (21) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785.

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Figure 1. Equilibrium surface properties as a function of the cohesive interaction of K: surface tension as a function of C0/a, surface concentration (Γ/Γ∞) as a function of C0/a (upper right), and surface pressure (π ) (γ0 - γ)/RTΓ∞) as a function of surface area A [)(NΓ)-1, Å2/molecule] (lower left).

Figure 2. Eight picked data points (case no. 1, 25 e γ e 65 mN/m) from the equilibrium surface tensions of Figure 1 and the best-fit theoretical profiles using the Langmuir adsorption isotherm (the solid curves). Parts of the data (between 50 and 70 mN/m) in (a) and (b) are enlarged and shown in (c) and (d). The dashed curves in (c) and (d) indicate the surface tension profiles from the Frumkin adsorption isotherm.

is assumed in this study that the Frumkin model describes perfectly the equilibrium surface tension of surfactants. Surfactants are assumed to be either with cooperative (for example, K ) -1, -2, -3, or -3.5) or with anticooperative (for example, K ) 1, 5, 10, or 15) adsorption behavior. Therefore, the equilibrium data shown in Figure 1 are exactly their equilibrium profiles. In most cases, one measures the equilibrium surface tension only for surfactant solutions of high bulk concentration and the Langmuir adsorption isotherm fits the equilibrium data well. Two cases are studied in this work: (i) One has equilibrium surface tension data (γ) ranging between 25 and 65 mN/m. (ii) One has γ ranging between 25 and 55 mN/m. Therefore, only parts of the equilibrium data (eight data points picked from each profile) in Figure 1 are used on the best fit with the Langmuir model. The model constants, listed in Table 1, are obtained by adjustment so as to minimize the error between the model predictions and eight data points. The best-fit profiles (the solid curves)

Table 1. Frumkin Model Constants for the Equilibrium Surface Tension Profiles in Figure 1 and Model Constants of Optimal Fit of the Langmuir Model for the Eight Picked Data Shown in Figure 2a Langmuir case no. 1, 25 < γ < 65 mN/m

Frumkin

Langmuir case no. 2, 25 < γ < 55 mN/m

Γ∞

a

K

Γ∞

a

Γ∞

a

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1 5 10 15 -1 -2 -3 -3.5

4.62 3.45 2.67 2.27 5.35 5.65 5.86 5.96

1.222 2.081 3.279 5.052 0.771 0.560 0.383 0.314

4.73 3.62 2.92 2.49 5.22 5.37 5.46 5.49

1.326 2.586 5.241 9.220 0.712 0.479 0.309 0.245

a

Γ∞: 10-10 mol/cm2. a: 10-9 mol/cm3.

and the eight picked data points (the symbols) are shown in Figure 2 for case no. 1 and in Figure 3 for case no. 2. In general, the Langmuir model fits the equilibrium

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Figure 3. Eight picked data points (case no. 2, 25 e γ e 55 mN/m) from the equilibrium surface tensions of Figure 1 and the best-fit theoretical profiles using the Langmuir adsorption isotherm (the solid curves). Parts of the data (between 45 and 65 mN/m) in (a) and (b) are enlarged and shown in (c) and (d). The dashed curves in (c) and (d) indicate the surface tension profiles from the Frumkin adsorption isotherm.

data of the cooperative or anticooperative surfactants reasonable well. The fitting was examined more closely, especially at the region for low bulk concentration. Figures 2 and 3 indicate that for surfactant having stronger cooperation (with a more negative K value) or anticooperation (with a larger positive K value), the Langmuir model describes worse the equilibrium surface tension. The worst fitting usually occurs at low concentration region (as shown in Figures 2c,d, and 3c,d, the dashed curves are the profiles from the Frumkin model). Although the Langmuir model in case 2 has a better prediction for those eight data points (from the comparison between Figures 2a, 3a, 2b, and 3b), it actually predicts worse the actual behavior of surfactant at low concentration, compared with that for case 1 (the comparison between Figures 2c, 3c, 2d, and 3d) at the same range of low bulk concentration. This can be easily verified by examining the deviations between the solid and dashed curves at the region for γ around 60 or 65 mN/m in Figures 2c, 3c, 2d, and 3d. This comparison will be verified again in the next section. Diffusion-Controlled Surface Tension Relaxations. To demonstrate the effect of applying only a limited range of equilibrium surface tension data on investigating the adsorption kinetics of surfactants, a series of theoretical simulations are performed. Consider only the case that surfactants adsorbed onto an initially clean, planar surface from a bulk phase initially with uniform concentration. Two adsorption isotherms (the Langmuir and Frumkin) are applied for investigating the adsorption kinetics of surfactants. The predictions from the Frumkin model represent the exact behavior of surfactants. Since the Langmuir adsorption isotherm describes the equilibrium data (the eight points from Figure 1) reasonable well, the Langmuir model is picked to simulate the relaxation behavior of surfactants. This is just like the common case, using a simple model that fits the equilibrium data reasonable well to describe the adsorption kinetics. The model constants used on simulating the relaxation profiles

of surface tension are what were obtained from the above section and are listed in Table 1. A diffusivity of 7.0 × 10-6 cm2/s is assumed in the following calculations. The adsorption process is assumed to be diffusioncontrolled first, and some representative relaxation profiles of surface tension are shown in Figure 4 for case no. 1, using the equilibrium data between 25 and 65 mN/ m. Consider first the cooperative surfactants. The Langmuir model always predicts faster relaxations (the dashed curves) than the Frumkin model (i.e., the exact relaxations, shown as the solid curves) at dilute concentration, with surface pressure (γ0 - γ) around 15 mN/m. Figure 4A shows the cases for K ) -2 and -3.5 at dilute concentration. At elevated concentration, the relaxation profiles of the Langmuir model relax faster at the beginning and then slower at the end of relaxation profiles (Figure 4B for K ) -2 and -3.5). The deviations in surface tension between the relaxation profiles that are predicted by these two models (the Langmuir and Frumkin) are significant. If the relaxations on surface tension for cooperative surfactants do follow what is predicted by the Frumkin model, the Langmuir model then fails clearly in predicting the dynamic profiles of surface tension with the same value of diffusion coefficient, D ) 7.0 × 10-6 cm2/s, here. The dotted curves in Figure 4A,B are also the diffusioncontrolled relaxation profiles predicted by the Langmuir model but with a lower diffusivity. It is surprising that the dotted curves fit the exact relaxation profiles (data generated by the Frumkin model) reasonable well at dilute concentration for surfactants with K > -3. Shown in Figure 4A is the case for K ) -2, and curve c of D ) 3.8 × 10-6 cm2/s fits the exact profile well. This implies that if one uses parts of the equilibrium surface tension data and a simpler adsorption model (the equilibrium data with γ e 65 mN/m and the Langmuir isotherm in this case) to calculate the model constants, and applies this information to model the dynamic surface tension data, one may find out that both sets of equilibrium and dynamic data are

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Figure 4. Relaxation of surface tension for clean interface adsorption predicted from the Langmuir (L; dashed curves) and Frumkin (F; solid curves) adsorption models for a diffusion-controlled process: C0 ) 5.5 × 10-10 (A1), 1.3 × 10-9 (A2), 5.0 × 10-9 (B1), 1.1 × 10-8 (B2), 1.3 × 10-8 (C1), 7.0 × 10-8 (C2), 2.5 × 10-7 (D1), and 7.0 × 10-6 (D2) mol/cm3.

in agreement with the model predictions. Whereas, the value of the diffusion coefficient is underestimated. For surfactants with strong cohesive energies between the adsorbed molecules, the Langmuir model is not able to describe the entire relaxation profile well even with a lower diffusivity. Although the Langmuir is not good for the entire relaxation profile, it does fit the profile at small or long time well with a lower diffusivity. For example, the curve b of D ) 3.0 × 10-6 cm2/s for K ) -3.5 at dilute concentration shown in Figure 4A fits the exact profile well at the regions of short and long time. Consider next the anticooperative surfactants, with a positive model constant K. The Langmuir model always predicts slower relaxations (the dashed curves) than the Frumkin model (the solid curves) at dilute concentration, with surface pressure (γ0 - γ) around 15 mN/m. Figure 4C shows the cases for K ) 5 and 15 at dilute concentration. At elevated concentration, the relaxation profiles of the Langmuir model relax slower at the beginning and then faster at the end of relaxation profiles (Figure 4D). Again, the deviations in surface tension between the relaxation profiles that are predicted by the Langmuir and Frumkin models are significant. If the relaxations on surface tension for anticooperative surfactants do follow what is predicted by the Frumkin model, the Langmuir model then fails clearly in predicting the dynamic profiles of surface tension with the same value of diffusion coefficient, D ) 7.0 × 10-6 cm2/s, here. The dotted curves in Figure 4C,D are also the diffusion-controlled relaxation profiles predicted by the Langmuir model but with a higher diffusivity. The dotted curves fit the exact relaxation profiles (generated by the Frumkin model) pretty well at dilute concentration for surfactants with K < 15. Shown in Figure 4C are the cases for K ) 5 and 15, where curve a of D ) 13.0 × 10-6 cm2/s and curve b of D ) 18.0 × 10-6 cm2/s fit the exact profiles well. This implies that if one uses parts of the equilibrium surface tension data and a simpler adsorption model (the equilibrium data with γ e 65 mN/m and the

Langmuir isotherm here) to calculate the model constants, and applies this information to model the dynamic surface tension data, one may find that both sets of equilibrium and dynamic data are in agreement with the model predictions. Whereas, the value of the diffusion coefficient is overestimated. The deviation of underestimating or overestimating the diffusivity is a function of surfactants (i.e., the cohesive energy and surfactant activity, which are indicated by model parameters K and a) and bulk concentration (C0). Figure 5 shows the dependence of the deviation of diffusivity on C0/a (dimensionless bulk concentration) and K. The results for case no. 1 (with equilibrium γ between 25 and 65 mN/m) are shown as the solid lines for surfactants with different cohesive energy. The results for case no. 2 (with γ between 25 and 55 mN/m) are shown as the dashed lines. Figure 5a indicates that a larger deviation is resulted for surfactant with a larger positive K value (stronger anticooperation). Figure 5b shows that a slightly larger deviation is resulted for surfactants with a more negative K value (stronger cooperation). For both cooperative and anticooperative surfactants, a larger deviation is resulted at more dilute concentration. For the cases we picked in this study, the diffusivity decreases from 7 × 10-6 to 1.3 × 10-6 cm2/s for cooperative surfactant with K ) -3.5, and D increases from 7 × 10-6 to 1.8 × 10-5 cm2/s for anticooperative surfactant with K ) 15. All the incorrect predictions (a lower or higher value) on diffusivity for surfactants with cooperation or anticooperation are simply due to the inaccurate prediction on equilibrium surface tension at dilute bulk concentration. Figures 2d and 3d indicate that the Langmuir adsorption isotherm predicts a lower value of surface tension compared with the exact surface tension profiles for cooperative surfactants. Therefore, the Langmuir model predicts relaxation profiles with lower tension (i.e., a faster relaxation) as shown in Figure 4A. The relaxation profiles in Figure 4B have lower tensions at the beginning and

Surfactant Adsorption Kinetics

Figure 5. Deviations of diffusion coefficient as a function of bulk concentration for anticooperative surfactants (a) and cooperative surfactants (b) obtained by using the Langmuir adsorption model. The solid curves represent the results for case no. 1, and the dashed curves represent the results for case no. 2. The correct values of diffusivity is 7.0 × 10-6 cm2/s.

higher tensions (i.e., a slower relaxation) at the end of the adsorption. The lower tension at the beginning is from the underprediction in equilibrium surface tension at dilute concentration, while the higher tension at the end of the dynamic adsorption is from the overprediction in equilibrium surface tension at the elevated bulk concentration. Similarly, Figures 2c and 3c indicate that the Langmuir isotherm predicts a higher value of surface tension at dilute bulk concentrations for anticooperative surfactants. Therefore, the Langmuir model predicts relaxation profiles with higher tension as shown in Figure 4C. The relaxation profile in Figure 4D has higher tensions at the beginning and lower tensions at the end of the adsorption. The higher tension at the beginning is from the overprediction in equilibrium surface tension at dilute concentration, while the lower tension at the end of the dynamic adsorption is from the underprediction in equilibrium surface tension at the elevated bulk concentration. Mixed-Controlled Surface Tension Relaxations. For surfactant with cohesive energy, a lower diffusivity is obtained from profiles of best-fitting the dynamic surface tension curves. In other words, the resistance of the mass transport process is higher than that of the diffusioncontrolled process. One may conclude that the resistance of the kinetic adsorption is not negligible. Figure 6 shows the comparison between the exact relaxation profiles of diffusion-control (predicted by the Frumkin model with D)7.0 × 10-6 cm2/s) and the relaxation profiles of mixedcontrolled adsorption process, predicted by the Langmuir model with D ) 7.0 × 10-6 cm2/s. The data in Figure 6 indicate that the relaxation profile with a finite adsorption rate constant β* ()β exp(Ea0/RT)) fits the exact profile well for γ < 66 mN/m in Figure 6a (K ) -2, C0 ) 1.3 × 10-9 mol/cm3) and for γ < 68 mN/m in Figure 6b (K ) -3.5, C0 ) 5.5 × 10-10 mol/cm3). Clearly, the profiles with

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Figure 6. Comparison between the exact dynamic surface tension profiles of diffusion-controlled (DC) clean adsorption (solid curves; from the Frumkin model) and the relaxation profiles of mixed diffusive-kinetic controlled adsorption (dot curves; using the Langmuir model): (a) K ) -2, C0 ) 1.3 × 10-9 mol/cm3; (b) K ) -3.5, C0 ) 5.5 × 10-10 mol/cm3. Adsorption rate constant of the Langmuir model β* ) β exp(Ea0/RT) [cm3/ (mol‚s)], and Ka ) β exp(Ea0/RT)C0h2/D. F ) Frumkin, and L ) Langmuir.

Figure 7. Limiting adsorption rate constant Kal [cm3/(mol‚s)] and the adsorption rate constant Ka with which the mixedcontrolled relaxation profile from the Langmuir model best fits the diffusion-controlled curves generated by the Frumkin model: solid lines for case no. 1 (25 e γ e 65 mN/m); dashed lines for case no. 2 (25 e γ e 55 mN/m).

β* ) 105 cm3/(mol‚s) in Figure 6a and 104 cm3/(mol‚s) in Figure 6b are far from diffusion-controlled ones and are of mixed-control. The relaxation profiles at different bulk concentrations for cooperative surfactants with different cohesive interaction K are examined, and the dimensionless adsorption rate constant Ka ()β exp(Ea0/RT)C0h2/D, detailed in Discussion section) of best-fitting the diffusioncontrolled curves are plotted in Figure 7 (solid line for case no. 1 and dashed line for case no. 2). The horizontal lines at the top of Figure 7 (solid line for case no. 1 and dashed line for case no. 2) indicate the limiting Ka value (marked as Kal) at which the mixed-controlled surface tension curve is indistinguishable from the diffusioncontrolled one. Data in Figures 5b and 7 also indicate that surfactants with stronger cooperation (i.e., with a larger absolute K value) have a slightly larger deviation on

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diffusivity (see Figure 5b) or a larger deviation between Ka and Kal (see Figure 7). A larger deviation on diffusivity and between Ka and Kal are also resulted at more dilute concentration. Figure 7 shows clearly that the Ka of best-fitting the diffusion-controlled curves are far from diffusion-controlled ones for 1-2 order of magnitude. In other words, one may make a mistake and conclude that the diffusioncontrolled adsorption process, described perfectly by the Frumkin model, is of mixed-control if a limited range of equilibrium surface tension data and a simpler adsorption isotherm (the Langmuir model in this study) are applied to determine the adsorption isotherm and to interpret the dynamic surface tension profiles. Discussion and Conclusions The equilibrium data are the underpinnings of the dynamic analysis. Careful attention should be paid to obtaining these data in the low surface coverage (small change in surface tension) region. Regardless of the final surface concentration, all dynamic adsorption studies which start with a freshly formed interface pass through the low surface concentration region as the interface equilibrates. Thus, the dynamic surface tension reduction is also influenced by this low concentration behavior. Failure to define it clearly causes errors in the dynamic surface tension analysis. This is the focus of this paper. In this study, this concept is illustrated by using an incomplete set of equilibrium data which does not contain the range 72-65 or 72-55 mN/m. From the above simulation and the examples explored in ref 17, it can be concluded that using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a mistake on the determination of adsorption isotherm. Usually, a simpler model is chosen from the limited range of equilibrium data. When this simple adsorption model is utilized to determine the adsorption kinetics and/or to evaluate the diffusivity of surfactant from the dynamic surface tension data, the surface tension relaxation may (i) show agreement with a diffusion-controlled model but in fact the value of diffusion coefficient is incorrect, or (ii) show agreement with a mixed diffusive-kinetic controlled model, but in fact the process is a diffusion-controlled one. These mistakes are simply due to the use of only limited range of equilibrium surface tension data with high surface pressure on determining the adsorption isotherm and the corresponding model constants. Therefore, care must be taken to avoid drawing incorrect conclusions for the mass transfer kinetics and incorrect value on diffusivity based upon equilibrium surface tension probed over a limited concentration range. Data in Figure 4 indicate that the entire relaxation profile of the surface tension data is required for determining the diffusivity of surfactant molecules. If only parts of the relaxation profiles are utilized, for example, the data at long or short time, a larger deviation on diffusivity maybe resulted. The underestimation or overestimation on diffusivity is dependent on (i) the bulk concentration, (ii) the data at long or short time, and (iii) the intermolecular interaction (K > 0 or K < 0). Data in Figure 5 indicate that a larger deviation on diffusivity is resulted when a smaller range of equilibrium surface tension data are utilized for determining the model constant or at more dilute concentration. When a same range of equilibrium surface tension is utilized, the surfactant with a larger absolute K value (stronger cooperation for K < 0 or stronger anticooperation for K > 0) has a larger deviation on diffusivity.

Hsu et al.

Figure 8. Representative example for the data shown in Figure 6b, showing the comparison between the exact dynamic surface tension of diffusion-controlled adsorption (solid curve; from the Frumkin model for a surfactant with K ) -3.5, C0 ) 5.5 × 10-8 mol/cm3, and surfactant activity a ) 1 × 10-7 mol/cm3) and the relaxation profiles of mixed diffusive-kinetic controlled adsorption (dot curves; using the Langmuir model with C0 ) 5.5 × 10-8 mol/cm3, Γ∞ ) 5.96 × 10-10 mol/cm2, and a ) 0.314 × 10-7 mol/cm3). Adsorption rate constant of the Langmuir model β* ) β exp(Ea0/RT) [cm3/(mol‚s)], and Ka ) β exp(Ea0/RT)C0h2/ D. F ) Frumkin, and L ) Langmuir.

In the above simulations, it is assumed that the adsorption kinetics of surfactants can be described exactly by the Frumkin model and two of the three model parameters are assigned as maximum surface concentration Γ∞ ) 5 × 10-10 mol/cm2 and surfactant activity a ) 1 × 10-9 mol/cm3. Equation 7 indicates that surface pressure (γ0 - γ) is proportional to maximum surface concentration. Therefore, choosing a different Γ∞ will enlarge or condense surface pressure and therefore change the value of surface tension. However, the concepts discussed in this study about the effect of the intermolecular interaction on the determination of adsorption kinetics and all the simulation results (Figures 1-7) still hold. For surfactants with different surfactant activity, one will have exactly the same results on that shown in Figures 1-3, 5, and 7 by choosing different surfactant concentrations. Relaxation profiles of surface tension shown in Figures 4 and 6 will show exactly the same tendency but at different times since a dimensional time is used in these figures. Figure 8 shows a representative example for the data shown in Figure 6b for a surfactant with a different surfactant activity (a ) 1 × 10-7 mol/cm3). The surface tension profiles in Figure 6b and 8 have the exactly same relaxation except with different dimensional time scale and with different values of dimensional adsorption rate constants. When the dimensionless parameters are utilized, Figures 6b and 8 will be exactly same. This can be seen by the nondimensionalized Langmuir adsorption equation (eq 4 with νa ) νd ) K ) 0)

dx/dτ ) Ka Cs*(1 - x) - Kd x

(8)

where x ) Γ/Γ∞, τ ) tD/h2, h ) Γe/C0 (Γe is the equilibrium surface concentration at C0), Cs* ) Cs/C0, Ka ) β exp(Ea0/RT)C0h2/D, and Kd ) R exp(Ed0/RT)h2/D. This can also be verified from the exactly same value of Ka for the profiles shown in Figures 6b and 8. Although the equilibrium surface tension data relate the surface tension and bulk concentration of surfactant solution, the surface concentration in the present approach is obtained indirectly from the slope of γ-ln C curve by the Gibbs adsorption equation. Different models may describe well the equilibrium tension data but predict very different surface concentration curve. The γ-Γ dependence

Surfactant Adsorption Kinetics

Figure 9. Surface tension/surface concentration dependence for data shown in Figure 2. Solid curves indicate the γ-Γ relationships for surfactants described by the Frumkin isotherm, and dashed curves are the γ-Γ relationships predicted by the Langmuir isotherm which best-fitted the eight data points shown in Figure 2. F ) Frumkin, L ) Langmuir.

for the data in Figure 2 are plotted and shown in Figure 9. The solid curves indicate the γ-Γ relationships for

Langmuir, Vol. 15, No. 6, 1999 1959

surfactants with intermolecular interactions (described by the Frumkin model), and the dashed curves are the γ-Γ relationships predicted from the Langmuir isotherm (assuming no intermolecular interactions between the adsorbed molecules) which best-fitted the eight data points shown in Figure 2. Although the Langmuir model predicts the γ-ln C data reasonably well (as shown in Figure 2), the Langmuir model does predict a very different γ-Γ dependence. Recall that it is assumed in this study the Frumkin isotherm describes the surfactants behavior exactly; therefore, the Langmuir model fails to predicts the γ-Γ dependence for surfactants with intermolecular interactions. Recently, Pan12 proposed a new approach to solve the difficulty in locating the slope on the γ-ln C curve in order to find the surface concentration. Using the pendant bubble as a monolayer film, a equilibrium relationship between γ-Γ is measured directly. Pan also concluded that the surfactant interaction on the fluid interface is significant. The above theoretical simulation about the effect of the intermolecular interaction between adsorbed molecules on the adsorption kinetics is performed for the adsorption of surfactants onto a clean planar air-water interface. For a small spherical or pendant drop/bubble, the curvature effect of drop or bubble on the dynamic surface tension profiles is significant. Currently we are in the process in our laboratory to study this effect on the issue discussed in this study. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grand NSC 84-2214-E-011-019). LA980656U