A Study on Surfactant Adsorption Kinetics: Effect of Bulk Concentration

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Langmuir 2000, 16, 4573-4580

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A Study on Surfactant Adsorption Kinetics: Effect of Bulk Concentration on the Limiting Adsorption Rate Constant Chengdi Dong,† Ching-Tien Hsu,‡ Chin-Yuan Chiu,‡ and Shi-Yow Lin*,‡ Department of Marine Environmental Engineering, National Kaohsiung Institute of Marine Technology, 142 Hai-Chuan Road, Nan-Tzu Dist., Kaohsiung, 811 Taiwan, and Department of Chemical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Sec. 4, Taipei, 106 Taiwan Received July 26, 1999. In Final Form: February 1, 2000 A limiting adsorption rate constant βl was defined as the value at which the mixed diffusive-kinetic controlled surface tension profile is indistinguishable from the diffusion-controlled one (Hsu et al. J. Chem. Eng. Jpn. 1996, 29, 634). The value of βl depends on surfactant, bulk concentration, the shape of fluid interface, and the process (adsorption or desorption). In this work, the dependence of βl on surfactant and bulk concentration (C) was examined. First, the dependence of βl vs C of eight well-known surfactants was investigated. In the second part, the effects of equilibrium constant (surfactant activity), molecular interaction, and maximum surface concentration were systematically studied. The difference of βl for a planar and a spherical interface with 0.1 cm radius of curvature was also examined. A universal relationship in terms of dimensionless variables was obtained. Using this universal relationship, one needs only the model constants of adsorption isotherm of surfactant to evaluate the value of βl at any concentration and to know the dependence of βl as a function of surfactant concentration.

Introduction Surface tension relaxation of a surfactant solution is governed by either interfacial adsorption/desorption or bulk diffusion. Relaxation profiles for surfactants adsorbing onto a freshly created fluid interface are usually measured and compared with the model-predicted relaxation profiles, which are generated using the model constants obtained from the equilibrium data. If the relaxation profiles are found to be in agreement with that from a diffusion-controlled adsorption model with a reasonable value of diffusivity, the adsorption process is usually concluded to be of diffusion control and the diffusivity is then determined. Due to the limitation of the tensiometer, usually only a limited range of bulk concentration is studied, and the relaxation profiles at these bulk concentrations are utilized to determine the adsorption kinetics and diffusivity. Such apparent agreement in a small concentration range is insufficient evidence and may cause an incorrect conclusion if the surfactant has a shifting control mechanism. A concept on the adsorption kinetics of surfactants was explored and illustrated with C12E8 recently: the controlling mechanism of the adsorption process changes as a function of bulk concentration; it shifts from diffusion control at dilute concentration to mixed diffusive-kinetic control at more elevated bulk concentration.1-3 A similar behavior has also been observed for C12E6 and C10E8.4-7 * To whom correspondence should be addressed. Tel: 886-22737-6648. Fax: 886-2-2737-6644. E-mail: [email protected]. † National Kaohsiung Institute of Marine Technology. ‡ National Taiwan University of Science and Technology. (1) Hsu, C. T.; Chang C. H.; Lin, S. Y. J. Chem. Eng. Jpn. 1996, 29, 634. (2) Lin, S. Y.; Tsay, R. Y.; Lin, L. W.; Chen, S. I. Langmuir 1996, 12, 6530. (3) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13, 3191. (4) 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.

The shift in controlling mechanism is caused by a large variation on the limiting adsorption rate constant (βl) with a wide range of workable bulk concentrations. Here, βl is defined as the adsorption rate constant at which the mixed diffusive-kinetic controlled surface tension profile is indistinguishable from the diffusion-controlled one.1 The workable bulk concentration indicates the range of surfactant concentration that the reduction of surface tension and the equilibration time are both large enough for the dynamic surface tension measurement using a commercial tensiometer. The limiting adsorption (or desorption) rate constant can be obtained from the theoretical simulations once one knows (i) the adsorption isotherm which describes the adsorption behavior of surfactant, (ii) the model constants of the adsorption isotherm, and (iii) the diffusivity of surfactant molecules in solvent. Of course, a numerical simulation is needed for solving the coupled adsorption equation and the bulk diffusion equation (eqs 1 and 7 in the next section). The value of βl depends on surfactant, bulk concentration, the shape of fluid interface, and the process (adsorption or desorption). The aim of this work is to investigate the dependence of βl on surfactant and bulk concentration (C). First, the dependence of βl vs C of eight well-known surfactants was investigated. In the second part, a systematic study on the effects of equilibrium constant (surfactant activity), molecular interaction, and maximum surface concentration was performed. The difference of βl for a planar and a spherical interface with 0.1 cm radius of curvature was also examined. Theoretical Framework Mass Transfer in Bulk. The adsorption of surfactant molecules onto a freshly formed interface in a quiescent (5) Pan, R.; Green, J.; Maldarelli, C J. Colloid Interface Sci. 1998, 205, 213. (6) Chang, H. C.; Hsu, C. T.; Lin, S. Y. Langmuir 1998, 14, 2476. (7) Shao, M. J. Master’s Thesis, National Taiwan University of Science and Technology, 1999.

10.1021/la991001b CCC: $19.00 © 2000 American Chemical Society Published on Web 04/09/2000

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surfactant solution was modeled. It was considered only the case of one-dimensional bulk diffusion and the adsorption onto a spherical or planar interface from a bulk phase containing an initially uniform concentration of the surface active solute. The surfactant was assumed not to dissolve into the gas (or oil) phase of bubble (or drop). Convection was assumed to be negligible. The diffusion of surfactant in the aqueous bulk phase is described by Fick’s law. Case I. For a spherical fluid interface with a finite radius of curvature, r ) b, diffusion in bulk phase was assumed to be spherical symmetric for a spherical fluid interface.

clean planar surface. The extra term, (D/b)[C0t - ∫t0Cs(τ) dτ], in the right-hand side of eq 3 accounts for the interfacial curvature. Adsorption Equation. The mass transfer of surfactant molecules between sublayer and fluid interface is described by the sorption kinetics. The model used here assumed that adsorption/desorption was an activated process assumed to obey the following rate expression:10,11 (i) The adsorption rate is proportional to the subsurface concentration Cs and the available surface vacancy (1 - Γ/Γ∞). (ii) The desorption rate was proportional to the surface coverage Γ.

∂C D ∂ 2 ∂C r > b, t > 0 r ) 2 ∂r ∂r ∂t r

dΓ/dt ) β exp(-Ea/RT)Cs(Γ∞ - Γ) - R exp(-Ed/RT)Γ (7)

(

)

(1)

with the following initial and boundary conditions:

C(r, t) ) C0 r > b, t ) 0

(2a)

C(r, t) ) C0 r f ∞, t > 0

(2b)

dΓ/dt ) D(∂C/∂r) r ) b, t > 0

(2c)

Γ(t) ) Γb t ) 0

(2d)

Here r and t are the spherical radial coordinate and time, D denotes the diffusion coefficient, C(r, t) is the bulk concentrations, Γ(t) is the surface concentration, b is the bubble radius, C0 is the concentration far from the fluid interface, and Γb is the initial surface concentration and is equal to zero for surfactants adsorption onto a initially clean fluid interface. By the Laplace transform, the solution of the above set of equations can easily be formulated in terms of unknown subsurface concentration Cs(t) ) C(r ) b, t):8

∫0tCs(τ) dτ] + 2(D/π)1/2[C0t1/2 - ∫0xtCs(t - τ) dxτ]

Γ(t) ) Γb + (D/b)[C0t -

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. The activation energies of adsorption and desorption processes may be dependent on the surface concentration of surfactant molecules adsorbed onto the fluid interface. Three cases are considered in this work. Case A. To account for enhanced molecular interaction between the adsorbed molecules at increasing surface coverage, the activation energies were assumed to be Γ dependent and a power form was assumed:

Ea ) E0a + νaΓn

(8a)

Ed ) E0d + νdΓn

(8b)

Here E0a, E0d, νa, and νd are constants. With substitution of Ea and Ed (eq 8) into eq 7 and nondimensionlization, eq 7, in dimensionless form, becomes

(3)

dx/dτ ) Ka exp(-νa*xn)Cs*(1 - x) - Kd exp(-νd*xn)x (9)

(4)

where x ) Γ/Γ∞, τ ) tD/h2, h ) Γe/C0, Ka ) β exp(-E0a/RT)C0/(D/h2), Cs* ) Cs/C0, Kd ) R exp(-E0d/RT)/ (D/h2), νa* ) νaΓ∞n/RT, and νd* ) νdΓ∞n/RT. At equilibrium, the time rate of change of Γ vanishes and the adsorption isotherm that follows is given by

C(x, t) ) C0 x > 0, t ) 0

(5a)

C(x, t) ) C0 x f ∞, t > 0

(5b)

C Γ )x) Γ∞ C + a exp(Kxn)

dΓ/dt ) D(∂C/∂x) x ) 0, t > 0

(5c)

Γ(t) ) Γb t ) 0

(5d)

Case II. For a planar fluid interface (i.e., r ) ∞) 2

∂ C ∂C x > 0, t > 0 ) ∂t ∂x2

D

with the following initial and boundary conditions:

Here x is the planar coordinate and C(x, t) is the bulk concentration. By the Laplace transform, the solution of the above set of equations was formulated in terms of unknown subsurface concentration Cs(t) ) C(x ) 0, t):9

Γ(t) ) Γb + 2(D/π)1/2[C0t1/2 -

∫0xtCs(t - τ) dxτ]

(6)

Equation 6 with Γb ) 0 is the classical solution of Ward and Tordai9 for the unsteady diffusion toward an initially (8) Lin, S. Y. Ph.D. Dissertation, City University of New York, New York, 1991. Lu, T. L. Master’s Thesis, National Taiwan Institute of Technology, Taipei, 1993. (9) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453.

(10)

where K ) (νa - νd)Γ∞n/RT ) νa* - νd* accounts for the molecular interaction between the adsorbed surfactant molecules and equilibrium constant a ) (R/β) exp[(E0a E0d)/RT] represents the interfacial activity of surfactant molecules. Case B. If a linear dependence between the activation energies and the surface coverage is assumed to account for enhanced molecular interaction between the adsorbed molecules,12-16 eqs 8-10 become (10) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (11) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (12) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, U.K., 1973; Chapters 1 and 3. (13) Frumkin, A. Z. Phys. Chem. (Leipzig) 1925, 116, 466. (14) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (15) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (16) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1996, 182, 526.

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where νa* ) νaΓ∞/RT, νd* ) νdΓ∞/RT, and K ) (νa - νd)Γ∞/ RT ) νa* - νd*. Equation 13 is the Frumkin adsorption isotherm.13 Case C. When the molecular interaction between the surfactant molecules adsorbed at the fluid interface is negligible, Ea and Ed are independent of surface concentration. Equations 9 and 10 become

interface and a planar interface, respectively) and eq 10 or eq 13 or eq 15 (for the generalized Frumkin, Frumkin and Langmuir model, respectively). Equation 3 (or 6) describes the mass transfer between sublayer and bulk, and eq 10 (13, or 16) is the sorption kinetics between subsurface and interface. The dynamic surface tension γ(t) was then calculated from eq 16. Mixed Control. If the adsorption process is of mixed control, eq 9 or eq 12 or eq 14 (for the generalized Frumkin, Frumkin, and Langmuir model, respectively) instead of eq 10 (or eq 13 or 15) is solved coupled with eq 3 or eq 6 (for a spherical interface and a planar interface, respectively) to find out the surface concentration. Then the dynamic surface tension γ(t) was calculated from eq 16.

dx/dτ ) KaCs*(1 - x) - Kd x

(14)

βl of Eight Well-Known Surfactants

C Γ ) Γ∞ C + a

(15)

Ea ) E0a + νaΓ, Ed ) E0d + νdΓ

(11)

dx/dτ ) Ka exp(-νa*x)Cs*(1- x) - Kd exp(-νd*x)x (12) C Γ ) Γ∞ C + a exp(Kx)

(13)

where Ka ) β exp(-Ea/RT)C0/(D/h2), Kd ) R exp(-Ed/RT)/ (D/h2), and a ) (R/β) exp[(Ea - Ed)/RT]. Equation 15 is the Langmuir isotherm. For surfactant with cohesive intermolecular forces between the adsorbed molecules at the fluid interface, K is less than zero, and this cohesive force increases with surface coverage and lowers the desorption rate. A positive K indicates that the adsorption is anticooperative, and adsorption becomes more difficult as the surface becomes more covered. Numerical Solution. The theoretical framework that describes the unsteady bulk diffusion and interfacial adsorption of surfactant toward an initially clean fluid interface and its effect on the surface tension has been formulated previously,11,17 and therefore, only a brief outline is given here. The interface was treated as a sphere surrounded by an infinite, quiescent medium which at time t ) 0 contained a uniform concentration C0 of surfactant. When the curvature is zero, it becomes a planar fluid interface. The concentration of surfactant at the interface was assumed to be equal to be a constant initial surface concentration Γb. Γb ) 0 for a clean adsorption process, in which the interface was created suddenly. When the surfactant solution can be considered ideal, the Gibbs adsorption equation dγ ) - ΓRT dln C and the equilibrium isotherm (eq 10, 13, or 15) allow for the calculation of the surface tension explicitly in terms of Γ:

γ - γ0 ) Γ∞RT[ln(1 - x) - Knxn+1/(n+1)] generalized Frumkin model (16a) ) Γ∞RT[ln(1 - x) - Kx2/2] Frumkin model (16b) ) Γ∞RT[ln(1 - x)] Langmuir model

(16c)

Here x ) Γ/Γ∞ and γ0 is the surface tension of solvent. By the fitting of equilibrium data of the surface tension as a function of the bulk concentration using eqs 16 and 10 (or 13 for the Frumkin model and eq 15 for the Langmuir model), the model parameters (Γ∞, a, K, and n) accounting for the equilibrium state can be obtained. For the Langmuir model, K ) n ) 0; for the Frumkin model, n ) 1. Diffusion Control. When the adsorption process is controlled solely by bulk diffusion, the surface concentration was obtained by solving eq 3 or eq 6 (for a spherical (17) Lin, S. Y.; Lu, T. L.; Hwang, W. B. Langmuir 1995, 11, 555.

In this work, the dependence of the limiting adsorption rate constant of surfactant on bulk concentration is investigated in two manners: (i) Examine the βl for eight well-known surfactants; i.e., use the adsorption model (the Langmuir, Frumkin, and generalized Frumkin adsorption isotherm) and model constants (best-fitting the equilibrium surface tension data). (ii) Investigate βl systematically; i.e., assign a model (the Frumkin adsorption isotherm) and study the effect of maximum surface concentration (Γ∞), surfactant activity (a), molecular interaction (K), spherical curvature (1/b; with b ) 1 mm), and bulk concentration (C0) on βl. Surfactants. Poly(oxyethylene) nonionic surfactants (>99% purity) were purchased from Nikko, Japan. They are C10E8 (octaethylene glycol mono-n-decyl ether (C10H21(O CH2CH2)8OH), C12E8 (octaethylene glycol mono-n-dodecyl ether (C12H25(OCH2CH2)8O H), C12E6 (hexaethylene glycol mono-n-dodecyl ether (C12H25(OCH2CH2)6OH), and C12E4 (tetraethylene glycol mono-n-dodecyl ether (C12H25(OCH2CH2)4OH). 1-Octanol (>99.5% purity) was from Fluka Chemie. 1-Decanol (99+% purity) and nonionic surfactant Triton X-100 (CH3C(CH3)2CH2C(CH3)2C6H4(OCH2CH2)nOH, n ) 9-10) were from Aldrich Chemical Co. Diazinon (phosphorothioic acid, O,O-diethyl O-(6-methyl-2-(1-methylethyl)-4-pyrimidinyl) ether, 99% purity) was from Chem Service. These surfactants were used without modification. The water with which the aqueous solutions were made was purified via a Barnstead NANOpure water purification system, with the output water having a specific conductance of less than 0.057 µΩ-1/cm. Equilibrium Surface Tension. Pendant bubble tensiometry enhanced by video digitization was employed for the measurement of surface tension relaxation and the equilibrium tensions are extracted from the long-time asymptotes of the relaxation profiles. The video-enhanced pendant bubble tensiometry has been detailed in previous studies;10,18 therefore, only a brief description is given here. The system creates a silhouette of a pendant bubble, video images the silhouette, and digitizes the image. A collimated beam with constant light intensity passes through the pendant bubble and forms a silhouette of a bubble on a solid-state video camera. A pendant bubble with a diameter around 2 mm was generated in a surfactant aqueous solution, which was put inside a quartz cell. The quartz cell was enclosed in a thermostatic air chamber. All the data are measured at 25.0 ( 0.1 °C except Triton X-100 at 22.5 ( 0.5 °C. Edge location of the bubble is obtained from the digitized image. The surface tension is (18) Lin, S. Y.; Hwang, H. F. Langmuir 1994, 10, 4703.

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Table 1. Model Constants from the Best-Fit with the Equilibrium Surface Tension

1-octanol 1-decanol C10E8 C12E8 C10E8 C12E8 C12E6 C12E4 1-octanol 1-decanol TX-100 diazinon a

modela

1010Γ∞ (mol/cm2)

1010a (mol/cm3)

K

n

GF GF GF GF F F F F F F F L

6.125 6.946 3.424 5.272 3.070 2.668 4.533 4.633 5.910 6.334 3.519 4.352

18580 9700 0.213 0.0233 1.302 0.250 1.566 3.521 7653 917.5 4.50 294.

-3.023 -5.031 10.877 13.28 9.628 5.186 4.4816 1.8752 -2.661 -3.717 2.901 0

0.4289 0.2590 0.5556 0.5032 1 1 1 1 1 1 1

106D (cm2/s)

b (cm)

6.9 6.6 6.5 8.0 9.0 11.0 7.0 7.0 7.3 7.0 2.6 4.3

0.1 0.13 0.1 0.13 0.1 0.13 0.1 0.1 0.1 0.13 0.1 0.13

GF ) generalized Frumkin model; F ) Frumkin model; L ) Langmuir model.

Figure 1. Equilibrium surface tension and the theoretical profiles best-fitting the data using the Langmuir (L), Frumkin (F), and generalized Frumkin (GF) models for C10E8 (O), C12E4 (3), C12E6 (0), C12E8 (4), diazinon (]), Triton X-100 (g), 1-octanol (×), and 1-decanol (+).

obtained from the best fit between the edge coordinates and theoretical curve generated from the classical Laplace equation. The equilibrium surface tension data are plotted in Figure 1. The surface tension profiles predicted from the models and best-fitted the equilibrium data are also shown in Figure 1. The dashed curve is that from the Langmuir isotherm, solid curves are for the Frumkin isotherm, and dotted curves are for the generalized Frumkin isotherm. The adsorption isotherm and model constants that describes reasonably well the equilibrium surface tension are listed in Table 1. Listed in Table 1 also the radius at apex of the bubble in experiment and the diffusivity obtained from the dynamic surface tension data and utilized for the theoretical calculation of the limiting adsorption rate constant.2,6,7,10,17,19-21 Limiting Adsorption Rate Constant. The relaxations of surface tension for the adsorption of surfactant molecules onto a clean fluid interface are simulated for both diffusion-controlled and mixed-controlled processes. Figure 2 shows two representative examples of 1-decanol of the surface tension profiles as a function of time, in which adsorption rate constant β varies from infinite (a diffusion-controlled adsorption) to 106 (mixed-control) to 104 cm3 mol-1 s-1 (a kinetic-controlled process). The generalized Frumkin model with a diffusivity of 6.6 × 10-6 cm2/s is utilized for data shown in Figure 2.22 The relaxations of surface tension on different bulk concentration have the same general dependence on β. The diffusion(19) Hsu, C. T. Master’s Thesis, National Taiwan University of Science and Technology, 1999. (20) Lin, S. Y.; Wang W. J.; Hsu, C. T. Langmuir 1997, 13, 6211. (21) Lin, S. Y.; Lin, L. W.; Chang, H. C.; Ku, Y. J. Phys. Chem. 1996, 100, 16678. (22) Hsu, C. T.; Lin S. Y. J. Chin. Inst. Chem. Eng. 1998, 29, 1.

Figure 2. Surface tension relaxations for 1-decanol adsorption onto a clean air-water interface for a diffusion-controlled (dashed curve) and mixed-controlled (solid curves) using the generalized Frumkin model. Model constants are listed in Table 1. C0 ) (a) 0.2 and (b) 1.0 (10-7 mol/cm3), and D ) 6.6 × 10-6 cm2/s. β ) 104 (1), 105 (2), 106 (3), 4 × 106 (4), 107 (5), and 3.3 × 107 (6) cm3 mol-1 s-1.

limited curve shows the fastest relaxation; slower relaxations are obtained as β decreases because of the increasing kinetic barrier. The limiting value of β for which the mixedcontrolled curve is indistinguishable (a criteria of ∆γ < 0.1 mN/m with 0.1 < Cs* < 0.2 is set in this study) from the diffusion-control profile is defined as βl. The distance between the diffusion-limited curve and the curve of a particular value of β varies gently with bulk concentration C0. In other words, the distance between the curves of diffusion-control and of mixed-control with a specific β departs gradually with increasing C0. A limiting adsorption rate constant βl was defined as the value at which the mixed diffusive-kinetic controlled surface tension profile is indistinguishable from the diffusion-controlled one. The relationship between βl and C0, which obtained from the above simulations, is plotted in Figure 3 for these eight surfactants. No matter what adsorption model (the dashed, solid and dotted lines for the Langmuir, Frumkin, or generalized Frumkin model, respectively) is applied, βl increases linearly roughly on a log scale as surfactant concentration increases. The slope varies a little bit for different surfactants and for different adsorption isotherms.

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Figure 3. Limiting adsorption rate constant as a function of bulk concentration for different surfactants obtained from the Langmuir (dashed line), Frumkin (dotted line), and generalized Frumkin (solid line) adsorption isotherms. Symbols are the same as in Figure 1.

Figure 5. Limiting adsorption rate constant (a, in dimensional form; b, in dimensionless form) as a function of bulk concentration for different surfactants and different adsorption isotherms (Langmuir, Frumkin, and generalized Frumkin). Symbols are same the as in Figure 1. Best-fit line: y ) Rxω; R and ω are listed in Table 2. Figure 4. Representative example showing the possible controlling mechanism determined from βl and β of surfactant.

There exists four possible controlling mechanisms for the adsorption of surfactants onto a freshly created fluid interface: diffusion control, kinetic control, mixed control, and a shift from diffusion control at dilute concentration to mixed-control at more elevated concentration. If a correct adsorption model is applied, the adsorption rate constant of surfactant should be a constant, i.e., independent of bulk concentration. Figure 4 shows if the β of surfactant (line DC) is larger than the largest value of βl of that surfactant, then the adsorption process is of diffusion control. If the β of surfactant (line KC) is much smaller than βl, the process is of kinetic control. When the β of surfactant (line M) is only slightly smaller than the smallest βl, the process is mixed-controlled. However, when the β of surfactant (line S) is ranged between the smallest and the largest value of βl, the adsorption process is of diffusion control at dilute concentration but is mixed control at elevated concentration. Therefore, there exists a shift on the controlling mechanism as reported1-7 for C12E6, C12E8, and C10E8. For surfactants with cohesive adsorption behavior (K < 0), the workable concentration range is usually less than 2 orders of magnitude; therefore, the variation of βl during the workable concentration range is only 1 or 2 orders of magnitude. The workable concentration is one that the reduction of surface tension and the equilibration time are both large enough for the dynamic surface tension measurement using a commercial tensiometer. For surfactants with anticooperative adsorption behavior (K > 0), the workable concentration range could be up to 4- or 5-fold; therefore, the variation of βl during the workable concentration range may be larger than 4 order of magnitude. Since anticooperative surfactants have a larger βl variation during the workable concentration

range, there exists a higher possibility for them to have a shift on controlling mechanism. On the investigation of the surfactant adsorption kinetics, one needs to know the range of workable concentration and the order of the variation of βl. Of course, it is pretty fast and accurate for one to do the above theoretical simulation once one has obtained the equilibrium surface tension data and the model constants, that best-fit the equilibrium data using a proper adsorption isotherm. For those people who like to skip the simulation work, Figure 5 brings some help. The dependence of βl on C0/a shown in Figure 3 are replotted in Figure 5 by choosing four dimensional or dimensionless groups of βl. A universal relationship was obtained by introducing the surfactant activity, equilibrium surface concentration (or maximum surface concentration), and diffusivity for different surfactants and adsorption models. All four parameters plotted as a function of dimensionless bulk concentration (C0/a) show a linear dependence on a log scale: βl/a, the simplest parameter, and two dimensionless group (βlΓ2/ aD and βlΓ∞2/aD) all show a good linear dependence with C0/a, whereas βlΓ/a shows the best linear relationship with C0/a. The data indicate that equilibrium constant (or surfactant activity) a is the major parameter affect the limiting adsorption rate constant βl and βl depends on surface concentration Γ weakly. These linear relationships on a log scale can be described by a power law equation (y ) Rxω; x ) C0/a) well, and values of R and ω are listed in Table 2. Theoretical Study In the second part of this study, the effect of maximum surface concentration (Γ∞), surfactant activity (a), molecular interaction (K), spherical curvature (1/b), diffusivity (D), and bulk concentration (C0) on the limiting adsorption rate constant βl was studied systematically via a theoretical simulation. The Frumkin adsorption

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Table 2. Parameters (y ) rxω) that Best-Fit the Universal Curves Shown in Figures 5, 8, and 10 (x ) C0/a) y

R

ω

in Figure

βl/a βlΓ/a βlΓ2/aD βlΓ∞2/aD

1.1258 × 1015 cm6/(mol2‚s) 2.7170 × 105 cm4/(mol‚s) 11.619 49.742

1.1936 1.1874 1.1755 1.1269

5a 5a 5b 5b

βl/a βlΓ βlΓ/a βlΓ∞/a βlΓ∞2/aD

1.5945 × 1015 cm6/(mol2‚s) 1.1814 × 10-4 cm/s 1.5170 × 105 cm4/(mol‚s) 7.0491 × 105 cm4/(mol‚s) 51.764

1.0295 1.2017 1.1681 1.0508 1.0393

8 8 10a 10a 10b

Table 3. Parameters Used for the Theoretical Simulation 1010Γ∞ (mol/cm3)

1010a (mol/cm3)

5 5 5 3 5 7 5 5 5 5

1000 10 0.1 10 10 10 10 10 10 10

K

b (cm)

in Figure

symbola in figure

15 15 15 15 15 15 -3.5 0 10 15

∞ 0.1, ∞ 0.1, ∞ ∞ 0.1, ∞ ∞ 0.1, ∞ ∞ ∞ 0.1, ∞

7, 8, 10, 11 7, 8, 10, 11 7, 8, 10, 11 8-11 8-11 8-11 9-11 9-11 9-11 9-11

+ b, O 9, 0 × b, O g 2, 4 3 * b, O

Figure 7. Dependence of the limiting adsorption rate constant on surfactant activity (a), interfacial curvature (1/b), and surfactant concentration. a ) 10-7 (+), 10-9 (O, b), and 10-11 (0, 9) mol/cm3; b ) ∞ (open symbols) and 0.1 cm (closed symbols).

a Open symbols are for a planar interface, and closed symbols are for a spherical interface with b ) 0.1 cm.

Figure 8. Limiting adsorption rate constant (βl/a and βlΓ) as a function of bulk concentration for surfactants with different Γ∞, a, and b. Symbols are the same as in Figures 7 and 9.

Figure 6. Equilibrium surface tension as a function of bulk concentration for different maximum surface concentration Γ∞, surfactant activity a, and molecular interaction K predicted from the Frumkin adsorption isotherm. Γ∞, a, K, and symbols are listed in Table 3.

isotherm was utilized for the investigation in the following simulation. The model constants used in simulations were listed in Table 3, and the corresponding equilibrium surface tension profiles were plotted in Figure 6. Relaxations of surface tension for the adsorption of surfactants with different Γ∞, a, K, and b onto a clean fluid interface were simulated for both diffusion-controlled and mixed-controlled processes. Relaxation profiles, similar to Figure 2, have a same dependence on adsorption rate constant β. The diffusion-limited curve shows the fastest relaxation in surface tension; slower relaxations are obtained as β decreases because of the increasing kinetic barrier. The distance between the diffusion-limited curve and that for a particular value of β varies gently with bulk concentration C0. Surface tension profiles as a function of time like Figure 2 were plotted and used for the determination of βl. The limiting adsorption rate constant βl was obtained with a criteria of ∆γ < 0.1 mN/m with 0.1 < Cs* < 0.2, being indistinguishable between the diffusion-controlled and mixed-controlled profiles. Figure 7 shows the effect of a and b on βl. The dependence of βl with bulk concentration at three different a (10-7, 10-9, and 10-11 mol/cm3) was examined by setting other model constants as Γ∞ ) 5 × 10-10 mol/cm2, K ) 15, and

D ) 7 × 10-6 cm2/s. Two spherical curvatures (b ) 0.1 cm or ∞, a planar interface) were considered. For a planar fluid interface, βl increases roughly linear on a log scale as C0/a increases and as a increases 100 times βl also goes up for 100 time roughly. For a spherical interface with b ) 0.1 cm, a similar behavior takes place except at extremely dilute concentration where βl deviates from the linear dependence. This is due to the large adsorption depth h ()Γe/C0) at the dilute concentration. Figure 7 (the dashed curves) shows that a large deviation from the linear dependence on βl resulted as C0 is extremely low. If we divide βl by surfactant activity a, the data of βl on the above five lines for three different a and two different interfacial curvature merge into one line, as shown in Figure 8, except those points for b ) 0.1 cm at extremely dilute C0. In general, when h/b is larger than 1, βl starts to deviate from the line obtained from the planar interface. For example, h/b ) 4.8, 24, 64, 115, and 1.1 for point A-E in Figure 8, respectively. The effects of maximum surface concentration Γ∞ and molecular interaction K are shown in Figure 9. The dependence of βl with bulk concentration at a planar fluid interface for three different Γ∞ [3, 5, and 7 (10-10 mol/ cm2)] was investigated with other parameters being set as a ) 10-9 mol/cm3, K ) 15, b ) 0.1 cm or ∞, and D ) 7 × 10-6 cm2/s. For a smaller Γ∞, a slightly larger βl resulted. As discussed in the above paragraph, βl for b ) 0.1 cm and for a planar interface is very close since the adsorption depth (h/b ) 1.1 for point A and less than 1 for other data points on the dashed line) in this case is low. If we multiple βl by surface concentration Γ, the above data of βl for three different Γ∞ merge into one line, as shown in Figure 8. Shown also in Figure 9 is the effect of the molecular interaction K. The dependence of βl with bulk concentra-

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Figure 9. Influence of maximum surface concentration (Γ∞) and molecular interaction (K) on the limiting adsorption rate constant (βl). Symbols and model constants used are listed in Table 3.

Figure 11. Dimensionless limiting adsorption rate constant Kal as a function of C0/a with a criteria of (a) Cs* ) 0.15 and (b) ∆γ < 0.1 mN/m with 0.1 < Cs* < 0.2. Symbols and model constants used are listed in Table 3.

limiting adsorption rate constant βl and βl also depends weakly upon surface concentration Γ and molecular interaction K. The effect of spherical curvature may be significant or negligible, dependent on the adsorption depth (h/b). Listed in Table 2 are the parameters of R and ω, which describe perfectly the universal relationship with a power law equation (y ) Rxω). Note that only the Frumkin adsorption isotherm was applied in the theoretical simulation in this section. Discussion and Conclusions Figure 10. Limiting adsorption rate constant as a function of bulk concentration for surfactants with different Γ∞, K, a, D, and b for the Frumkin adsorption isotherm. Symbols are listed in Table 3. Best-fit line: y ) Rxω; R and ω are listed in Table 2.

tion for four different K values (-3.5, 0, 10, and 15) was studied for b ) 0.1 cm and ∞. Other model constants were set as Γ∞ ) 5 × 10-10 mol/cm2, a )1 × 10-9 mol/cm3, and D ) 7 × 10-6 cm2/s. The effect of K on βl is very weak and nearly negligible. The effect on βl due to spherical curvature (b ) 0.1 cm and ∞) is negligible at high concentration and is not negligible, but not very significant either, for the data at dilute concentration. h/b ) 0.6-11 for data points for K ) -3.5 (triangles) and 0.0002-1.1 for data points for K ) 15 (circles). To get the limiting adsorption rate constant quickly from the equilibrium data (with the model constants) while skipping the simulation work, one can use the universal relationship shown in Figure 8 or Figure 10. A universal relationship was obtained by introducing the (maximum) surface concentration, surfactant activity, molecular interaction parameter, and/or diffusion coefficient. Four dimensional (βl/a, βlΓ, βlΓ∞/a, βlΓ/a) and one dimensionless (βlΓ∞2/aD) parameters were plotted as a function of dimensionless bulk concentration (C0/a), and they all show a linear dependence on a log scale. The data indicate that surfactant activity a is the major parameter affecting the

The dependence of the limiting adsorption rate constant of surfactant on bulk concentration was investigated by evaluating the βl for eight well-known surfactants and by examining βl systematically. Three adsorption isotherms were applied in the first part of this study, and the Frumkin isotherm was used in the second part on investigating the effect of maximum surface concentration (Γ∞), surfactant activity (a), molecular interaction (K), spherical curvature (1/b), and bulk concentration (C0). A universal relationship in terms of either dimensionless or dimensional variables was obtained; therefore one can easily obtain the βl vs C0 dependence by just knowing the model constant (Γ∞, a, and/or K) of adsorption isotherm, and/or the surfactant diffusivity, and/or the interfacial curvature (1/b). It has been reported1 that, using Cs* ) 0.15 as a criteria for finding βl at which the diffusion-controlled profile is indistinguishable from the mixed-controlled profile, Kal [)βl exp(-E0a/RT)C0/(D/h2)] shows a roughly constant value, O(10), for different C12E8 concentrations. Note that the molecular interaction parameter K is positive (i.e., showing an anticooperative adsorption behavior) for C12E8. For a surfactant with a negative K (showing a cooperative adsorption behavior; e.g., 1-decanol), Kal is about 10 times larger than surfactants with K > 0. Besides, the Kal vs C0/a shows a different dependence: Kal increases with C0 at dilute concentration and then decreases with C0 at more elevated bulk concentration. Figure 11a shows the dependence for both cases, surfactants with a positive or

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negative K, when Cs* ) 0.15 is used as the criteria for finding βl. Figure 11a also indicates that Kal has the same behavior for different interfacial curvature (b ) 0.1 cm or a planar interface). In this work, ∆γ < 0.1 mN/m with 0.1 < Cs* < 0.2 were set as the criteria for finding βl since 0.1 mN/m is the experimental uncertainty for surface tension measurement. With this criteria, Cs* is larger than 0.15 at extremely dilute concentration, less than 0.15 at very high concentration, and roughly equal to 0.1 or 0.2 at the middle range. With this new criteria, Kal keeps increasing (but only slightly) at increasing C0 for a surfactant with K > 0. For a surfactant with K < 0, Kal is roughly the same as the surfactant with K > 0 but shows an abrupt increase at dilute concentration and then levels off slowly at increasing C0. Shown in Figure 11b is the relationship between Kal and C0/a by using this new criteria for a planar interface and for a spherical interface with b ) 0.1 cm. The study on βl is based on the Frumkin model and the Langmuir and generalized Frumkin models also show a similar βl vs C0 dependence. Recently, another formulation, the two states model,23-25 has been widely applied on surfactant adsorption kinetics. It will be interesting to see whether this model has a similar βl vs C0 dependence. The limiting adsorption rate constant investigated in this study is only for the process of surfactant molecules adsorbing onto a freshly created fluid interface. Note that for different mass transport process (for example, surfactants desorbing out of a suddenly compressed, therefore overcrowded, fluid interface), the dependence of βl vs C0/a is different. Shown in Figure 12 is an example, adapted from ref 22, illustrating the difference between (βl)ads, for the adsorption of 1-decanol molecules onto a freshly created spherical interface, and (βl)des, for the desorption out of a suddenly compressed interface. Here, (βl)ads and (βl)des are the limiting adsorption rate constant for the adsorption and desorption processes. The equations needed for simulating the desorption process are the same as the eqs 1-3 and 7 in this work; the simulation has been detailed in ref 22, and discussed here are only parts of the data for showing the different βl dependence for different mass transport processes. The example concerns the desorption of 1-decanol molecules out of a compressed, spherical air-water interface to a bulk phase, which is of initially uniform (23) Miller, R.; Aksenenko, E. V.; Liggieri, L.; Ravera, F.; Ferrari, M.; Fainerman, V. B. Langmuir 1999, 15, 1328. (24) Fainerman, V. B.; Miller, R.; Wustneck, R.; Makuevski, A. V. J. Phys. Chem. 1996, 100, 3054. (25) Fainerman, V. B.; Miller, R.; Wustneck, R. J. Phys. Chem. 1997, 101, 6479.

Dong et al.

Figure 12. Limiting adsorption rate constant as a function of bulk concentration for the adsorption process (b) and desorption process at three different compression ratios: Ab/Ae ) 0.80 (O), 0.90 (4), and 0.95 (0).

concentration. The interface with a surface area Ae and surface concentration Γe, in equilibrium with a uniform bulk concentration C0, was instantly compressed to a smaller area Ab and a higher surface concentration Γb (>Γe). The overcrowding of surfactant molecules at interface caused a desorption of the adsorbed surfactant molecules into the initially uniform bulk phase. The overcrowding was taken with different compression ratio (Ab/Ae) of 0.80, 0.90, and 0.95. The generalized Frumkin model and a diffusivity of 6.6 × 10-6 cm2/s were used for the theoretical simulation. The limiting adsorption rate constant for this desorption process (βl)des was obtained from the relaxation profiles of surface tensions for different bulk concentrations and shown as the dashed curves in Figure 12. At these three different compression, values of βl for the desorption process all increase dramatically as C0 becomes more elevated, which is quite different from the βl for the adsorption of surfactant molecules onto a freshly created fluid interface, (βl)ads. Therefore, there exists a higher possibility for 1-decanol to have a shifting control mechanism or a higher possibility for the desorption process than the adsorption one to be of mixed control. It is therefore concluded that the βl dependence is dependent upon the process of adsorption/desorption /expansion and also upon the compression/expansion ratio if a desorption or expansion process is considered. Currently we are in the process of investigating this point in our laboratory. Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grant NSC 84-2214-E-011-019). LA991001B