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Langmuir 1997, 13, 6204-6210
Comments on the Adsorption Isotherm and Determination of Adsorption Kinetics Ching-Tien Hsu,† Chien-Hsiang Chang,‡ and Shi-Yow Lin*,† Department of Chemical Engineering, National Taiwan University of Science and Technology, 43, Keelung Road, Sec. 4, Taipei, 106 Taiwan, Republic of China, and Department of Chemical Engineering, National Cheng Kung University, Tainan, 701 Taiwan, Republic of China Received June 10, 1997X The equilibrium and dynamic surface tension data are widely utilized to investigate the information of intermolecular interactions between the adsorbed surfactant molecules, the adsorption kinetics, and the mass transfer coefficients. In this paper, a concept is explored: 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. This idea is illustrated theoretically for clean interface adsorption on a pendant bubble or drop using the Langmuir and Frumkin adsorption models. Two types of surfactant, one with cooperative adsorption behavior and the other with anticooperative adsorption behavior, are discussed. This concept is further demonstrated using the relaxation data of surface tension of 1-octanol.
1. Introduction Adsorption/desorption of soluble surface active material onto or out of a fluid interface can envisage three consecutive steps: (i) surfactant molecules diffuse and/or convect between a deeper layer of solution and the subsurface layer immediately adjacent to the fluid interface; (ii) surfactant molecules adsorb and/or desorb between the sublayer and fluid interface; (iii) adsorbed surfactant molecules rearrange at the fluid interface. The diffusion is driven by the concentration gradient, adsorption/desorption is driven by the chemical potential of the molecule, and rearrangement may be caused by reorientation, complex formation, phase transition, or formation of a two-dimensional micelle structure. For nonpolymeric molecules, rearrangement is a very fast process, and the dynamic surface phenomenon is governed by the first two steps. The adsorption kinetics of soluble surfactants is primarily investigated through the measurement of surface tension. The equilibrium surface/interfacial tension data are used to determine the reduction of surface tension, the surface concentration at different bulk concentration, the critical micelle concentration, and information on intermolecular interaction of the adsorbed surfactant molecules. Many adsorption isotherms and equations of state have been reported to describe the above information. For example, adsorption isotherms of Henry, Langmuir, Volmer, Frumkin, and van der Waals are all well-known models.1-4 Recently, Lin et al.5 reported that the intermolecular interaction is significant for many surfactants, and although the Frumkin isotherm describes the equilibrium surface tension data of 1-decanol well, it cannot * Author to whom correspondence should be addressed: telephone, 886-2-737-6648; fax, 886-2-737-6644; e-mail, ling@ ch.ntust.edu.tw. † National Taiwan University of Science and Technology. ‡ National Cheng Kung University. X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, 1973; Chapters 1 and 3. (2) Lucassen-Reynders, E. H. In Progress in Surface and Membrane Science; Cadenhead, D. A., Danielle, J. F., Eds.; Academic Press: New York, 1976; Vol. 10, p 253. (3) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995. (4) Chang, C. H.; Franses, E. I. Colloids Surf., A 1995, 100, 1. (5) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055.
S0743-7463(97)00613-6 CCC: $14.00
predict the dynamic surface tension data correctly. A generalized Frumkin model and a phase transition model were therefore proposed. Lunkenheimer and Hirte6 proposed an adsorption isotherm consisting of two distinct regions. A new isotherm for molecules forming twodimensional aggregates in the adsorbed layer was proposed by Fainerman et al.7-9 The equilibrium surface tension can be measured by many methods,10,11 for example, the Wilhelmy plate or du Nouy ring, drop weight or volume, maximum bubble pressure, pendant bubble, and video-enhanced plate methods.12 Usually, only a limited range of bulk concentration C0 is studied, and the equilibrium tension data are found to be in agreement with the Langmuir or Frumkin adsorption isotherm. The model constants are commonly determined by minimizing the difference between experimental surface tensions and the model predictions. When the rearrangement is very fast, the surface tension relaxation is governed by both adsorption and diffusion processes. Dynamic surface tension profile of the adsorption or desorption process at an expanded, or compressed, or freshly created interface is then measured. Some examples of the measuring technique include oscillating jet,13 expanding or compressing on a Langmuir trough,14,15 oscillating bubble,16 growing drop,17 and pendant bubble methods.18,19 If the relaxation profiles are found to be in (6) Lunkenheimer, K.; Hirte, R. J. Phys. Chem. 1992, 96, 8683. (7) Fainerman, V. B.; Miller, R. Langmuir 1996, 12, 6011. (8) Fainerman, V. B.; Miller, R.; Wustneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 7669. (9) Fainerman, V. B.; Vollhardt, D.; Melzer, V. J. Phys. Chem. 1996, 100, 15478. (10) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1990. (11) Franses, E. I.; Basaran O. A.; Chang, C. H. Curr. Opin. Colloid Interface Sci. 1996, 1, 297. (12) Tsay, R. Y.; Yan, S. C.; Lin, S. Y. Rev. Sci. Instrum. 1995, 66, 5065. (13) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1989, 127, 97. (14) Baret, J. F.; Bois, A. G.; Casalta, L.; Dupin, J. J.; Firpo, J. L.; Gonella, J.; Melinon, J. P. J. Colloid Interface Sci. 1975, 53, 50. (15) Van hunsel, J.; Vollhardt, D.; Joos, P. Langmuir 1989, 5, 528. (16) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526. (17) MacLeod, C. A.; Radke, C. J. J. Colloid Interface Sci. 1993, 160, 435. (18) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (19) Lin, S. Y.; Hwang, H. F. Langmuir 1994, 10, 4703.
© 1997 American Chemical Society
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agreement with a diffusion-controlled adsorption model, the value of diffusion coefficient is then determined by minimizing the difference between dynamic surface tension data and the mass transfer model predictions. If the surface tension relaxes slower than that predicted by the diffusion-controlled adsorption model with a reasonable value of diffusivity, say about 5 × 10-6 cm2/s, a mixed diffusive-kinetic- or a kinetic-controlled process is then concluded. Thus, the rate constant of adsorption/ desorption can be determined from the best fit between the dynamic surface tension data and the model-predicted relaxation profiles. In this study, it is proposed 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 kinetics and/or on the evaluation of diffusivity from best fitting the dynamic surface tension data. That is, 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 application of limited range of equilibrium surface tension data with high surface pressure only 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 of diffusivity based upon equilibrium surface tension probed over a limited concentration range. This idea is developed by first considering the equilibrium surface tension data of 1-octanol and poly(oxyethylene) nonionic surfactant C10E8. The Frumkin adsorption isotherm fits these two sets of equilibrium tension data well over the whole concentration range, whereas the Langmuir isotherm fits them poor if the whole concentration range is considered. If only a limited range of concentrations is considered, for example, only the high concentration range in which surface pressure is larger than 10 mN/m, the data are in agreement with the prediction of the Langmuir isotherm. The Frumkin and the Langmuir models, which predict well the equilibrium data at high concentration range, are then utilized to simulate the surface tension relaxation at different bulk concentrations. A misleading conclusion on the adsorption kinetics or on the evaluation of diffusivity due to using the Langmuir model is then illustrated. 2. Governing Mass Transfer Equation Bulk Diffusion. The adsorption of surfactant onto the interface of a freshly formed pendant bubble in a quiescent surfactant solution is modeled. We shall consider only the case of one-dimensional diffusion and adsorption onto 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 considered to be spherical symmetric and convection effects are negligible. The diffusion of surfactant in the bulk phase is described by Fick’s law
∂C D ∂ 2 ∂C r ) 2 ∂r ∂r ∂t r
(
)
(r > b, t > 0)
(1)
with the following initial and boundary conditions
C(r, t) ) C0
(r > b, t ) 0)
C(r, t) ) C0
(r f ∞, t > 0)
(2)
dΓ/dt ) D (∂C/∂r) Γ)0
(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 (b ) 1 mm in this study), and C0 is the concentration far from the bubble. By use of 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) )
D [C t b 0
∫t0 Cs(τ) dτ] + 2
xDπ [C xt - ∫ 0
xt
0
Cs(t - τ) dxτ] (3)
Adsorption Isotherm. The Frumkin adsorption kinetics is used to describe the adsorption-desorption process 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 pre-exponential 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. The adsorption of n-alcohols on a solution-air interface has been shown to have a negative value of k.5,16,20,21 Polyethylene glycol alkyl ether RO(CH2CH2O)nH has been reported to have a positive k.22,23 A positive value of k indicates that the adsorption is anticooperative, and adsorption becomes more difficult as the surface becomes more crowded. If the surfactant solution is considered ideal, the Gibbs adsorption equation dγ ) -ΓRT d ln C and the equilibrium isotherm (eq 6) allow for the calculation of the surface tension explicitly in terms of surface concentration (20) Fainerman, V. H.; Lylyk, S. V. Colloid J. USSR 1982, 44, 538. (21) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (22) 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. (23) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530.
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Figure 1. Equilibrium surface tension (mN/m) for air/1octanol-aqueous solution from ref 25 and the theoretical predictions of the Langmuir (dashed line, using the close points only) and Frumkin (solid line) adsorption isotherms.
Figure 2. Equilibrium surface tension (mN/m) for air/C10E8aqueous solution from ref 26 and the theoretical predictions of the Langmuir (dashed line) and Frumkin (solid line) adsorption isotherms.
γ - γ0 ) Γ∞RT[ln(1 - x) - kx2/2]
Table 1. Constants of Optimal Fit for Octanol and C10E8 Aqueous Solutions
(7)
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 are solved (eqs 3 and 6 or eqs 3 and 4) numerically, the technique used is a modification of that used by Miller and Kretzschmar.24 The method on the integration and calculation has been detailed in ref 18. 3. Results Equilibrium Data and Model Predictions. The equilibrium surface tensions of 1-octanol and C10E8 aqueous solutions in refs 25 and 26 are used in this study and shown in Figures 1 and 2. All experiments are performed at 25.0 ( 0.1 °C, and the accuracy and reproducibility of the dynamic surface tension measurements obtained by using the video-enhanced pendant bubble tensiometer are ≈0.1 mN/m.27,28 The best-fit curves using the Frumkin adsorption isotherm are also presented (24) Miller, R.; Kretzschmar, G. Colloid Polym. Sci. 1980, 258, 85. (25) Lin, S. Y.; Wang; W. J.; Hus, C. T. Langmuir 1997, 13, 6211. (26) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Submitted to Langmuir for publication. (27) Lin, S. Y.; Chen, L. J.; Xyu, J. W.; Wang, W. J. Langmuir 1995, 11, 4159.
octanol
C10E8
modela
Γ∞ (mol/cm2)
a (mol/cm3)
k
L1 F L2
7.90 × 5.64 × 10-10 1.09 × 10-5
4.288 × 7.246 × 10-7 9.159 × 10-3
-2.72
L F
1.804 × 10-10 3.070 × 10-10
5.437 × 10-10 1.302 × 10-10
10-10
10-7
9.63
a
L1, the Langmuir isotherm using data of γ e 66.3 mN/m; L2, the Langmuir isotherm using data of γ > 55 mN/m; F, the Frumkin isotherm.
as the solid lines in Figures 1 and 2. The Langmuir model fits poorly the 1-octanol data.25 To demonstrate the proposed idea in this study, only parts of the equilibrium data, shown as the closed circles in Figure 1 that the surface tension is lower than 66.3 mN/m, are used on the best-fit with the Langmuir model. The model constants, shown as model L1 in Table 1, are obtained by adjustment so as to minimize the error between the model predictions and experimental values. The best-fit curve for C10E8 data using the Langmuir isotherm is presented as the dashed line in Figure 2. The more exact agreement of the Frumkin model indicates that the intermolecular interaction is significant for the adsorbed 1-octanol and C10E8 molecules at the air-water interface as the surface coverage is high. The Frumkin model predicts a negative k value for octanol and a positive k value for C10E8. The negative k indicates a cooperative adsorption for octanol, whereas the positive k tells that the adsorption of C10E8 is anticooperative and (28) Lin, S. Y.; Wang, W. J.; Lin, L. W.; Chen, L. J. Colloids Surf., A 1996, 114, 31.
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Figure 4. Relaxation of surface tension of C10E8 for clean interface adsorption predicted from the Langmuir (L) and Frumkin (F) adsorption models for a diffusion-controlled process. C0 ) 5 × 10-9 (a) and 4 × 10-7 mol/cm3 (b).
Figure 3. Relaxation of surface tension of 1-octanol for clean interface adsorption predicted from the Langmuir (L) and Frumkin (F) adsorption models for a diffusion-controlled process. C0 ) 4 × 10-7 (a), 5 × 10-7 (b), and 16 × 10-7 mol/cm3 (c).
adsorption becomes more difficult as the surface becomes more covered by C10E8. Figure 1 indicates that the model prediction of the Langmuir isotherm and the data at high concentrations are in good agreement, and the Frumkin model predicts the equilibrium surface tension well in the whole concentration range. At dilute bulk concentration of octanol, there exists a significant deviation and the Langmuir model predicts a lower value of surface tension. Figure 2 tells that the Frumkin model predicts also well the C10E8 equilibrium data in the whole concentration range. However, the Langmuir model predicts a higher value of surface tension for C10E8 at dilute concentration. These significant deviations at dilute concentration for octanol and C10E8 contribute greatly on the misinterpretation of the adsorption kinetics and will be discussed below. Relaxations of Surface Tension. A series of simulations in which surfactants adsorbed onto an initially clean, spherical surface from a bulk phase of initially
uniform concentration are performed. Two adsorption isotherms (the Langmuir and Frumkin) are applied for investigating the adsorption kinetics of surfactants. The model constants used here are those listed in Table 1, which are obtained from the best fit of the equilibrium tension data being described in the above section. In this study, two surfactants (1-octanol, with a cooperative adsorption behavior, and C10E8, which is anticooperative) are picked as model compounds. The adsorption processes for both surfactants are assumed to be diffusion-controlled first, and some representative relaxation profiles of surface tension are predicted and shown in Figures 3 and 4 for 1-octanol and C10E8, respectively. The diffusion coefficient of octanol used here is 7.3 × 10-6 cm2/s, which is obtained from ref 25 when the Frumkin isotherm is utilized. Figure 3 indicates that the Langmuir model predicts a faster relaxation (the dashed lines) than the Frumkin model (the solid lines) at dilute concentration (Figure 3a,b). At elevated concentration, the relaxation profile of the Langmuir model relaxes faster at the beginning and then slower at the end of relaxation profile (Figure 3c). The deviations in surface tension between the relaxation profiles predicted by these two models (the Langmuir and Frumkin) are significant. If the relaxation of surface tension of octanol does follow that predicted by the Frumkin model, the Langmuir model then fails clearly in predicting the dynamic surface tension with the same value of diffusion coefficient, D ) 7.3 × 10-6 cm2/s here. The dotted lines in Figure 3 are also the diffusioncontrolled relaxation profiles predicted by the Langmuir model, but with a lower D value. It is surprised that the dotted lines fit the data generated by the Frumkin model well at dilute concentration. This implies that if one uses
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Figure 5. Deviation of diffusion coefficient as a function of bulk concentration of 1-octanol (a) and C10E8 (b) obtained by using the Langmuir adsorption model. The dashed lines represent the correct values of diffusion coefficient.
parts of the equilibrium surface tension data and a simpler adsorption model (the equilibrium data with γ e 66.3 mN/m and the Langmuir isotherm in this study) 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 in agreement with the model predictions, whereas the value of diffusion coefficient is incorrect. A lower value is obtained from the example of octanol. The deviation in diffusivity is a function of surfactants (i.e., the energy of intermolecular interaction indicated by parameter k) and bulk concentration, and that for 1-octanol is shown in Figure 5a. C10E8 has an anticooperative adsorption behavior. A diffusivity of 9.0 × 10-6 cm2/s is obtained for C10E8, from ref 26, when the Frumkin adsorption isotherm is used, and therefore D ) 9.0 × 10-6 cm2/s is assumed for the following calculation. The relaxation profiles predicted by the Langmuir (the dashed lines) and Frumkin (the solid lines) models are shown in Figure 4. The Langmuir model predicts a slower relaxation in surface tension at dilute concentration (see Figure 4a) and at the beginning of the relaxation profile at elevated concentration (Figure 4b). At the end of the relaxation profile at elevated concentration, the Langmuir model predicts a faster relaxation (Figure 4b). The deviation in surface tension between the relaxation profiles predicted by these two models is also significant. If the Langmuir model is again utilized to model the equilibrium and dynamic surface tension data of C10E8, a reasonable fit on the equilibrium data results (Figure 2) and it predicts the dynamic data well at dilute concentration (shown as the dotted lines in Figure 4). Whereas, the value of diffusion coefficient is not reasonable, and a higher value of 2.0 × 10-5 cm2/s
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Figure 6. Relaxation of subsurface concentration (Cs*, for the Langmuir model) and the comparison between the generated dynamic surface tension of diffusion-controlled (DC) clean adsorption and the relaxation profiles of mixed diffusivekinetic-controlled adsorption using the Langmuir adsorption model. β exp(Ea0/RT) ) 105 (a), 106 (b), 107 (c), and 108 cm3/ (mol‚s) (d). F ) Frumkin; L ) Langmuir. C0 ) 5 × 10-7 mol/cm3.
(Figure 5) is obtained from the example of C10E8. The dependence of diffusion coefficient on bulk concentration is shown in Figure 5b. Figure 5 shows the results of theoretical simulations on the deviation of diffusion coefficient when the Langmuir model is used to interpret the equilibrium and dynamic surface tension data and a diffusion-controlled clean adsorption process is assumed. At dilute concentration, the relaxation profiles of the Langmuir model are in good agreement with the generated relaxation profiles from the Frumkin model, but the values of D decrease from 7.3 × 10-6 to 2.3 × 10-6 cm2/s for octanol and increase from 9.0 × 10-6 to 2.0 × 10-5 cm2/s for C10E8. At elevated bulk concentration, the agreement between the Langmuir relaxation profiles and the Frumkin ones is poor, but still acceptable. Diffusion coefficients, obtained from the best fit between the relaxation profiles, deviate about 10% for both octanol and C10E8. Since the diffusion coefficient of octanol at diluted concentration is low (2.3 × 10-6 cm2/s), one may conclude that the resistance of the kinetic adsorption is not negligible. Figure 6 shows the comparison between the generated relaxation profiles (assuming that the Frumkin model predicts exactly the surface tension relaxation of octanol with D ) 7.3 × 10-6 cm2/s) and the relaxation profiles of the mixed-controlled adsorption process (D ) 7.3 × 10-6 cm2/s also), which is predicted by the Langmuir model. The data in Figure 6 tell that the relaxation profile with a finite adsorption rate constant β* ()β exp(Ea0/RT)) fits the generated profile well for t > 0.1 s. The corresponding subsurface concentration profiles are plot-
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Figure 8. Equilibrium surface tension (mN/m) for air/1octanol-aqueous solution from ref 25 and the theoretical predictions of the Langmuir (dashed line, using data with γ > 55 mN/m) and Frumkin (solid line) adsorption isotherms.
data well, whereas the dynamic data are far away from the diffusion-controlled profile predicted by the Langmuir model with D ) 7.3 × 10-6 cm2/s. Figure 7b shows that the diffusion-controlled relaxation profile predicted by the Langmuir model with D ) 2.0 × 10-6 cm2/s is in good agreement with the dynamic data. The data of Table 3 in ref 25 also clearly indicate that the diffusion coefficients estimated from the best fit by using the Langmuir model vary significantly (from 0.3 × 10-6 to 5.0 × 10-6 cm2/s) from dilute bulk concentration to elevated one. While the clean adsorption process has been verified25 to be a diffusion-controlled process, Figure 7c shows the mixedcontrolled relaxation profile predicted by the Langmuir model with D ) 7.3 × 10-6 cm2/s and β*) 6 × 106 cm3/ (mol‚s) is also in agreement with the dynamic data. 4. Discussion and Conclusions
Figure 7. Dynamic surface tensions (mN/m) for clean adsorption of 1-octanol aqueous solutions for C0 ) 4 × 10-7 mol/cm3, the relaxation profiles of diffusion-controlled adsorption (a and b, upper and middle) predicted by the Langmuir (L) and Frumkin (F) adsorption model, and that of mixed-controlled adsorption (c, lower) predicted by the Langmuir model with D ) 7.3 × 10-6 cm2/s. DC denotes the diffusion-limited curves.
ted. Clearly, the profile with β* ) 107 cm3/(mol‚s) is far from the diffusion-controlled one and it is of mixed control. The relaxation profiles at different dilute bulk concentrations are examined and β* is around 107 cm3/(mol‚s) for different concentrations. Dynamic Surface Tension of 1-Octanol. The relaxation data of surface tension in ref 25 are utilized to demonstrate the idea proposed in this article. The clean adsorption process is verified in ref 25 to be a diffusioncontrolled process, and D ) 7.3 × 10-6 cm2/s as the Frumkin adsorption isotherm is utilized. Figure 7 shows the comparison between the dynamic surface tension data and the relaxation profiles predicted from the Langmuir and Frumkin models. Figure 7a indicates that the diffusion-controlled relaxation profile with D ) 7.3 × 10-6 cm2/s predicted by the Frumkin model fits the dynamic
From the above comparison between the dynamic surface tension data of 1-octanol and the relaxation profiles predicted by the Langmuir model, one may conclude that the clean adsorption is a mixed diffusive-kineticcontrolled process (since the fit in Figure 7c is pretty good) or conclude that the diffusion coefficient is 2.0 × 10-6 cm2/s (the fit in Figure 7b is nearly perfect). All these discrepancies are simply due to using a limited range of equilibrium surface tension data on determining the adsorption isotherm and the corresponding model constants. The above theoretical simulation tells also that for surfactants with cohesive forces between the adsorbed molecules at the air-water interface, one may underestimate the diffusion coefficient when the clean adsorption is considered to be a diffusion-controlled process. Whereas, for surfactants with anticooperative adsorption behavior, one may overestimate the diffusivity. All the incorrect predictions (lower or higher value) on diffusivity for octanol and C10E8 are simply due to the inaccurate prediction on equilibrium surface tension at dilute bulk concentration. Figure 1b shows that the Langmuir isotherm predicts a lower value of surface tension compared with the experimental data. Therefore, the Langmuir model predicts a relaxation profile with lower tension (i.e., a faster relaxation) as shown in parts a and b of Figure 3. The relaxation profile in Figure 3c has a lower tension at the beginning and a higher tension (i.e., a slower relaxation) at the end of the adsorption. The first lower tension is from the underprediction in equilibrium surface tension in region A of Figure 1 at dilute concentration, while the higher tension at the end of the
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dynamic adsorption is from the overprediction in region B at the elevated bulk concentration. Similarly, Figure 2b indicates that the Langmuir isotherm predicts a higher value of surface tension at dilute bulk concentrations. Therefore, the Langmuir model predicts a relaxation profile with higher tension as shown in Figure 4a. The relaxation profile in Figure 4b has a higher tension at the beginning and a lower tension at the end of the adsorption. The first higher tension is from the overprediction in equilibrium surface tension in region A of Figure 2 at dilute concentration, while the lower tension at the end of the dynamic adsorption is from the underprediction in region B at the elevated bulk concentration. Therefore, it is concluded that using only the range of equilibrium surface tension with high surface pressures to determine the adsorption isotherm can cause a serious mistake on further interpreting the dynamic surface tension data. Care must be taken on choosing the adsorption isotherm in order to avoid the mistakes in determining the adsorption kinetics and in evaluating the diffusivity from best fitting the dynamic surface tension data. The theoretical simulation is also performed for the case of only using the range of the equilibrium data with low
Hsu et al.
surface pressure. Figure 8 shows the equilibrium data of 1-octanol and the profile of best fitting the data with γ > 55 mN/m by using the Langmuir adsorption isotherm. The model constants are listed in Table 1. Figure 8 indicates that the Langmuir model cannot predict the near-cusp behavior in the γ-ln C curve of 1-octanol. There is no surprise at this poor fitting since the Langmuir adsorption isotherm assumes no intermolecular interaction between the adsorbed surfactant molecules at the fluid interface. It has been reported5 that the cusp or near-cusp behavior in the γ-ln C curve is evidence of the development of strong attractive energies between surfaceadsorbed molecules at increasing surface coverage. The agreement between the dynamic surface tension data and the relaxation profiles of surface tension using the model constants from Figure 8 is poor also. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 84-2214-E-011-019). LA970613Y
