Langmuir 2000, 16, 1211-1215
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Study on Surfactant Adsorption Kinetics: Effects of Interfacial Curvature and Molecular Interaction 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, and Department of Chemical Engineering, National Cheng Kung University, Tainan, 701 Taiwan Received March 10, 1999. In Final Form: September 23, 1999 A concept on the surfactant adsorption kinetics was explored and illustrated with 1-octanol and C12E8 (Langmuir, 1997, 13, 6204): using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a serious mistake in the determination of adsorption kinetics and in the evaluation of diffusion coefficient from the dynamic surface tension data. A systematic theoretical study was also reported recently for surfactants adsorbing onto a clean planar air-water interface (Langmuir, 1999, 15, 1952). To learn the effect of curvature of the fluid interface on this issue, a theoretical study is performed for surfactants adsorbing onto a clean spherical air-water interface with different curvature. Data from this simulation indicate that (a) for any interfacial curvature, the stronger the surfactant interactions (i.e., with a larger absolute K value), the greater the deviation in apparent diffusivity (D) or in sorption rate constants (ka, kd) obtained from a Langmuir analysis; (b) at the same K, the larger the interfacial curvature, the smaller the deviation in D or in ka and kd; and (c) a larger deviation in diffusivity results at more dilute concentrations.
Introduction The molecular interactions between the adsorbed surfactant molecules are of importance and proven to be significant for many surfactants.1-9 Recently, a systematic theoretical study was reported on surfactant adsorption kinetics: using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a serious mistake in the determination of adsorption kinetics and in the evaluation of diffusion coefficient from the dynamic surface tension data.10 For a planar fluid interface, it is concluded that (a) the stronger the surfactant interactions (i.e., with a larger absolute K value), the greater the deviations in apparent diffusivity or in sorption rate constants obtained from a Langmuir analysis; and (b) a larger deviation on diffusivity results at more dilute concentration, or when a smaller range of surface tension data is utilized to determine the model constants. Investigations on surfactant adsorption kinetics are usually performed at fluid interfaces of different curvature. The aim of this work is to examine theoretically the effect of interfacial curvature, coupled with the molecular interactions between adsorbed surfactant molecules, on the above issue on surfactant adsorption kinetics. 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 spherical or planar air-water interface are simulated using the Langmuir and Frumkin adsorption equations. Governing Mass Transfer Equation The governing mass transfer equations of bulk diffusion, adsorption isotherm, and the numerical procedure reported in refs 9 and 10 are adapted and used in this study, except now fluid interfaces with different curvature are studied. Therefore, only a brief description is given here. Bulk Diffusion. The adsorption of surfactant molecules onto a freshly formed spherical interface (with curvature b-1 or radius b) in a quiescent surfactant solution is modeled. Only the case of one-dimensional diffusion and adsorption onto a planar or spherical interface is considered. 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 Fick’s law:4,6
∂C D ∂ 2∂C (r > b, t > 0) r ) ∂r ∂t r2 ∂r
( )
(1)
with the following initial and boundary conditions * 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. (1) Miller, R. Colloid Polym. Sci 1981, 259, 375. (2) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (3) Lunkenheimer, K.; Hirte, R. J. Phys. Chem. 1992, 96, 8683. (4) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555. (5) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526. (6) Lin, S. Y.; Wang W. J.; Hsu, C. T. Langmuir 1997, 13, 6211. (7) Pan, R.; Green, J.; Maldarelli, C J. Colloid Interface Sci. 1998, 205, 213. (8) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Langmuir 1998, 14, 2476. (9) Hsu, C. T.; Chang C. H.; Lin, S. Y. Langmuir 1997, 13, 6204. (10) Hsu, C. T.; Chang C. H.; Lin, S. Y. Langmuir 1999, 15, 1952.
C(r, t) ) C0 (r > b, t ) 0) C(r, t) ) C0 (r f ∞, t > 0) dΓ/dt ) D(∂C/∂r) (r ) b, t > 0) Γ(t) ) 0 (t ) 0)
(2)
where r and t are the spherical radial coordinate and time, D denotes the diffusion coefficient, C(r,t) is the bulk concentration, Γ(t) is the surface concentration, b is the bubble radius, and C0 is the concentration far from the
10.1021/la9902845 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/04/1999
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bubble. By using the Laplace transform, the solution of the above set of equations can be formulated in terms of the unknown subsurface concentration Cs(t) ) C(r ) b,t);
∫0t Cs(τ) dτ] + 2(D/π)1/2 [C0 t1/2 - ∫0xt Cs(t - τ) dxt]
Γ(t) ) (D/b)[C0 t -
(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 coverages, the activation energies are assumed to be proportional to surface concentration:9
Ea ) Ea0 + νa Γ Ed ) Ed0 + νd Γ
(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 ) Cs/[Cs + a exp(K x)]
(6)
where K ) (νa - νd)Γ∞/RT, indicating the molecular interaction between the adsorbed surfactant molecules, and a ) R/β exp[(Ea0 - Ed0)/RT], representing the surfactant activity. Equation 6 becomes the Langmuir adsorption isotherm when νa ) νd ) K ) 0. If the surfactant solution is considered ideal, the Gibbs adsorption equation dγ ) - ΓRTdln 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)
where x ) Γ/Γ∞, and γ0 is the surface tension of pure water. 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. Results Solid curves in Figure 1 show the surface tension as a function of the logarithm of C0 for different surfactant interaction parameter K. The γ-C0 relation is obtained from eqs 6 and 7 with following assumptions: Γ∞ ) 5 × 10-10 mol/cm2, a (surfactant activity) ) 1 × 10-9 mol/cm3, T ) 25 °C, and γ0 ) 72.0 mN/m. To investigate the curvature effect on the issue, using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a serious mistake; it is assumed that the Frumkin model describes perfectly the
Figure 1. Equilibrium surface tension as a function of the cohesive interaction of K and C0/a (solid curves); the eight picked data points (symbols); and the best-fit theoretical profiles using the Langmuir adsorption isotherm (dashed curves). Insets show the relations of γ versus Γ and adsorption depth versus C0/a. Table 1. Frumkin Model Constants for Equilibrium Surface Tension Profiles in Figure 1 and Model Constants of Optimal Fit of the Langmuir Model for Eight Picked Data Shown in Figure 2 Frumkin
Langmuir
Γ∞
a
K
Γ∞
a
5.0 5.0 5.0 5.0
1.0 1.0 1.0 1.0
5 15 -2 -3.5
3.45 2.27 5.65 5.96
2.081 5.052 0.560 0.314
Γ∞: (10-10 mol/cm2). a: (10-9 mol/cm3).
equilibrium surface tensions of surfactants. Surfactants are assumed to be either with cooperative (for example, K ) -2 or -3.5) or with anticooperative (for example, K ) 5 or 15) adsorption behavior. The equilibrium data shown in Figure 1 (the solid curves) 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 usually fits the equilibrium data reasonably well. Therefore, only parts of the equilibrium data (eight data points, with roughly equal ∆γ, picked from each profile of γ ranging between 25 and 65 mN/m) are used on the best fit with the Langmuir model. The model constants, listed in Table 1, are obtained by adjustment to minimize the error between the model predictions and the eight data points. The eight picked data points (the symbols) and the best-fit profiles (the dashed curves) are also shown in Figure 1. Note that one may pick 16 or 32 data points with equal ∆γ; however, the same model constants will result if they are all ranged between 25 and 65 mN/m. In general (for -3.5 < K < 15, discussed in this work), the Langmuir model fits the equilibrium data of the cooperative or anticooperative surfactants reasonably well, especially for γ < 60 mN/m. The fittings at low bulk concentration region indicate that for surfactant having stronger cooperation (with a more negative K value) or stronger anticooperation (with a larger positive K value), the Langmuir model describes the equilibrium surface tensions poorly. Because of the lack of picked or experimental data with γ > 65 mN/m for the above best fit, the worst fitting usually occurs at the low concentration region (with a small surface pressure). To demonstrate the curvature effect on applying only a limited range of equilibrium surface tension data in investigating the adsorption kinetics of surfactants, a series of theoretical simulations is performed. Only the case that surfactants adsorbed onto an initially clean, spherical or planar surface from a bulk phase initially with a uniform concentration is considered. Two adsorption isotherms (the Langmuir and Frumkin) are applied
Surfactant Adsorption Kinetics
Figure 2. Relaxation of surface tension for clean interface adsorption predicted from the Langmuir (L; dashed curves) and Frumkin (F; solid curves) models for a diffusion-controlled process. C0 ) 1.3 × 10-9 (A) and 1.1 × 10-8 (B) mol/cm3.
for investigating the adsorption kinetics of surfactants. The predictions from the Frumkin model represent the exact adsorption behavior of surfactants. Because the Langmuir adsorption isotherm describes the equilibrium data (the eight points for each case shown in Figure 1) reasonably 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 reasonably well to describe the adsorption kinetics. The model constants used on simulating the relaxation profiles of surface tension are those 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. Some representative relaxation profiles of surface tension are shown in Figure 2 for a cooperative surfactant with K ) -2. For both spherical and planar interfaces, 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 concentrations. At elevated concentrations, the relaxation profiles of the Langmuir model relax faster at the beginning, then slower at the end of relaxation profiles (Figure 2B). 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 the trend 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 Figures 2 are the diffusion-controlled 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) reasonably well at dilute concentrations. Shown in Figure 2A for K ) -2, curve a (planar interface) of D ) 3.8 × 10-6 cm2/s and curve b (spherical interface with radius r ) 0.1 cm) of D ) 5.0 × 10-6 cm2/s
Langmuir, Vol. 16, No. 3, 2000 1213
both fit the exact profiles reasonably 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 for K ) -2) 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. However, the value of diffusion coefficient is underestimated in this case. For both planar and spherical interfaces, the Langmuir model is not able to describe the entire relaxation profiles well even with a lower diffusivity for surfactants with strong cohesive energies between the adsorbed molecules. The Langmuir model can only fit the profile either at short or at long times well with a lower diffusivity. Consider next the surfactants with anticooperative adsorption behavior, which has a positive model constant K. For interfaces with different radii of curvature (0.01, 0.1, 0.3, 1, and ∞ cm studied in this work), similar results are obtained, as shown in Figure 4c and d of ref 10 for a planar interface. The Langmuir model always predicts slower relaxations than the Frumkin model at dilute concentrations. At elevated concentrations, the relaxation profiles of the Langmuir model relax more slowly at the beginning, then faster at the end of relaxation profiles. 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 the curve 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 diffusioncontrolled relaxation profile with a higher diffusivity predicted by the Langmuir model can predict the exact surface tension profile pretty well at dilute concentrations. 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. However, the value of the diffusion coefficient is overestimated. The deviation on diffusion coefficient from the underestimation or overestimation of D is a function of surfactant properties (i.e., the cohesive energy and surfactant activity, which are indicated by model parameters K and a), curvature of fluid interface (1/b), and bulk concentration (C0). Table 2 lists the dependence of the deviation of diffusivity on dimensionless bulk concentration (C0/a), radius of curvature b, and molecular interaction K. The data indicate that a slightly larger deviation results for surfactants with a more negative K value (stronger cooperation) or for surfactants with a larger positive K value (stronger anticooperation). For the fluid interface with a larger curvature (smaller b), a smaller deviation results for both types of surfactants. Besides, a larger deviation results at more dilute concentrations. To show the tendency of deviation, parts of the results for cooperative surfactants are plotted in Figure 3. Why are the deviations lower when the curvature is higher (i.e., a smaller b)? This can be seen from the adsorption depth (h ) Γe/C0) of the surfactant mass transport process. Consider the mass transport process for a surfactant (with a specific K) at the bulk concentration C0. The adsorption depth is a constant since Γe is fixed (eq
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Table 2. Values of Diffusion Coefficient for Curves Predicted from Langmuir Model Best-Fitting Exact Surface Tension Profiles r (cm) K -2
C0/a
1 1.3 1.7 11 -3.5 0.5 0.55 4 5.5 5 10 13 180 250 10 55 70 5500 7000
hF (cm)
hL (cm)
0.01
0.1
0.3
1
∞
0.42 0.34 0.07 0.045 0.93 0.85 0.12 0.09 0.024 0.020 0.002 0.0016 0.0029 0.0024 5 × 10-5 4 × 10-5
0.36 0.30 0.075 0.049 0.73 0.69 0.14 0.10 0.029 0.023 0.0019 0.0014 0.0038 0.0030 4 × 10-5 3 × 10-5
4.8 5.2 6.5 6.7 3.5 3.8 6 6 11 10.5 7.5 7.5 16 15 8 8
4.5 5 6 6.5 3 3.5 6 6.2 12.5 12 8 7.5 19 15 8.5 8
4.2 4.5 6 6 3 3 6 6 12.5 12 7.5 8 19 18 8.5 8.5
3.8 4 6 6 2.3 2.8 6 6 13.3 13 8 8.5 19 18 8.5 8.5
3.5 3.8 6.2 6.5 1.3 1.5 5 5.5 13.3 13 8 8 19 18 8 8
hF and hL are the adsorption depth calculated from the Frumkin and Langmuir adsorption isotherms, respectively.
Figure 4. Comparison between the exact surface tension profiles of diffusion-controlled (DC) adsorption onto a clean interface and the relaxation profiles of mixed diffusive-kineticcontrolled adsorption (dot curves) using the Langmuir model: K ) -2, C0 ) 1.3 × 10-9 mol/cm3 for a planner interface (a) and a spherical interface of r ) 0.1 cm (b).
Figure 3. (a) Deviations of diffusion coefficient as a function of bulk concentration obtained by using the Langmuir model; (b) adsorption rate constant Ka with which the mixed-controlled relaxation profile from the Langmuir model best-fits the diffusion-controlled curves generated by the Frumkin model.
6). When the curvature is higher, h/b becomes larger, and therefore the curvature effect on the surfactant mass transfer increases. This means the bulk diffusion of surfactant speeds up because of the geometric effect of a spherical fluid interface. This increase of surfactant transport compensates the underestimation of diffusivity for cohesive surfactant molecules; therefore a smaller deviation results. The inset in Figure 1 shows the dependence of adsorption depth as a function of bulk concentration. Note that the range of C0/a for cohesive surfactants studied in this work is from 0.5 to 11 (see Table 2) and that for anticohesive surfactants is from 10 to 7000. The value of h does have a significant change for C0/a at O(1), and the change for C0/a larger than O(102) is nearly negligible (as shown in Table 2). Therefore, the curvature effect for an anticohesive surfactant with C0/a ranging between10 and 7000 is very minor, just like the data shown in Table 2 for the case of K > 0.
When the surfactant interactions are weaker (i.e., as K is closer to zero), the deviation due to the use of a simpler adsorption isotherm (the Langmuir model in this work) to describe the true adsorption behavior (the Frumkin model in this work, indicating the existence of surfactant interaction) is smaller. Therefore, when only a limited range of equilibrium surface tension data is utilized to determine the adsorption isotherm and to evaluate the diffusion coefficient, a smaller deviation results. For surfactants with cohesive adsorption behavior, 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 diffusion-controlled process. One may conclude that the resistance of the kinetic adsorption is not negligible. Figure 4 shows an example (K ) -2, C0 ) 1.3 × 10-9 mol/cm3) of the comparison between the exact relaxation profiles of diffusion-controlled (predicted by the Frumkin model with D ) 7.0 × 10-6 cm2/s) and mixedcontrolled (predicted by the Langmuir model with D ) 7.0 × 10-6 cm2/s) adsorption processes. The data in Figure 4 indicate that the relaxation profile with a finite adsorption rate constant β* [)βexp(Ea0/RT)] fits the exact profile well for γ < 66 mN/m in Figure 4a for a planar interface and fits the whole exact profile reasonably well in Figure 4b for a spherical interface with r ) 0.1 cm. Clearly, the profiles with β* ) 105 cm3/(mol s) in Figure 4a and 106 cm3/(mol s) in Figure 4b are significantly away from the diffusion-controlled ones and are of mixed control. The relaxation profiles at different bulk concentrations and at different interfacial curvature for cooperative surfactants with different cohesive interaction parameter K are examined and the dimensionless adsorption rate constant Ka [) β exp(Ea0/RT) C0h2/D] of best-fitting the diffusioncontrolled curves is plotted in Figure 3b. The horizontal region at the top of Figure 3b indicates the limiting Ka value (marked as Kal) at which the mixed-controlled surface tension curve is indistinguishable from the diffusion-controlled one. Data in Figure 3 also indicate that (a) surfactants with stronger cooperation (i.e., with a larger
Surfactant Adsorption Kinetics
absolute K value) have a slightly larger deviation on diffusivity or a larger deviation between Ka and Kal, and (b) larger deviations on diffusivity and between Ka and Kal also result at more dilute concentrations or for the fluid interface with a smaller curvature (larger b). Figure 3b indicates that the values of Ka of best-fitting the diffusion-controlled curves are far from the diffusioncontrolled ones for one to two orders of magnitude for planar interfaces, whereas the deviations are less than one order of magnitude for a spherical interface of b ) 0.1 cm. Discussion and Conclusions The above simulations indicate, no matter what the interfacial curvature is, that using only a limited range of equilibrium surface tension data to determine the adsorption isotherm can cause a mistake in the determination of adsorption kinetics and in the evaluation of diffusion coefficient. 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 or to evaluate the diffusivity of a surfactant from the dynamic surface tension data, the surface tension relaxation may (a) show agreement with a diffusion-controlled model, but in fact the value of
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diffusion coefficient is incorrect; or (b) 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 a limited range of equilibrium surface tension data with high surface pressures in determining the adsorption isotherm and the corresponding model constants. No matter what the interfacial curvature is, the entire relaxation profile of the surface tension 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 times, a larger deviation on diffusivity may result. The underestimation or overestimation on diffusivity is dependent on (a) the interfacial curvature, (b) the bulk concentration, (c) the data at long or short times, and (d) the surfactant interaction. A smaller deviation results when the fluid interface has a larger curvature (smaller b), which is due to an increase on bulk diffusion from the more important curvature effect. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (grant NSC 84-2214-E-011-019). LA9902845
