Modeling of the Adsorption Kinetics of Surfactants at the LiquidFluid

Langmuir 2004, 20, 2503-2511

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Modeling of the Adsorption Kinetics of Surfactants at the Liquid-Fluid Interface of a Pendant Drop Chaodong Yang and Yongan Gu* Petroleum Technology Research Centre (PTRC), Faculty of Engineering, University of Regina, Regina, Saskatchewan, S4S 0A2 Canada Received October 27, 2003. In Final Form: January 7, 2004 This paper presents a theoretical model for simulating the adsorption kinetics of a surfactant at the liquid-fluid interface of a pendant drop. The diffusion equation is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the time-dependent surfactant concentration distributions inside the pendant drop and inside the syringe needle that is used to form the pendant drop. With the obtained bulk surfactant concentration distributions, the adsorption at the interface is determined by using the conservation law of mass. It should be noted that the theoretical model developed in this study considers the actual geometry of the pendant drop, the depletion process of the surfactant inside the pendant drop, and the mass transfer of the surfactant from the syringe needle to the pendant drop. The present pendant-drop model is applied to study the adsorption kinetics of surfactant C10E8 (octaethylene glycol mono n-decyl ether) at the water-air interface of a pendant drop. The numerical results show that the Ward and Tordai equation, which was derived for adsorption from a semi-infinite surfactant solution to a planar interface, is unsuitable for interpreting the dynamic surface or interfacial tension data measured by using the pendant-drop-shape techniques, especially at low initial surfactant concentrations. The spherical-drop model, which assumes the pendant drop to be a perfectly spherical drop with the same drop volume, can be used to interpret the dynamic surface or interfacial tension data for pendant drops either with high initial surfactant concentrations or with low initial surfactant concentrations in short adsorption durations only. For pendant drops with low initial surfactant concentrations in long adsorption durations, the theoretical model developed in this study is strongly recommended.

1. Introduction Surfactant adsorption at liquid-fluid interfaces is an important phenomenon in many fundamental studies and practical applications, such as oil recovery, food processing, and the pharmaceutical industry.1-3 Initially, the adsorption of a surfactant at a fresh interface is assumed to be zero. Then the adsorption process starts up and induces a diffusion process in the neighboring bulk phase. The adsorption continues until an equilibrium state is reached, corresponding to the vanishing of the diffusion process in the bulk phase. The adsorption of surfactants at a liquid-fluid interface is commonly studied by means of the dynamic surface or interfacial tension measurements. As a surfactant adsorbs at an interface, the interfacial tension between the two phases reduces. Therefore, the adsorption kinetics of the surfactant can be revealed by the dynamic interfacial tension reduction. In the literature, various drop methods, such as the drop-volume, drop-shape, and drop-pressure methods, have been developed to measure the dynamic surface or interfacial tensions.1,2 With the development of modern computer-aided image acquisition and analysis techniques, the pendant-drop-shape techniques4-7 become an accurate standard method for measuring the dynamic surface or interfacial tensions. This experimental method is based on the fact that the shape of a pendant drop is controlled by the surface or interfacial tension and the * To whom correspondence should be addressed. Tel.: 1 (306) 585-4630. Fax: 1 (306) 585-4855. E-mail: [email protected]. (1) Dukhin, S. S.; Kretzschmar, G. K.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995. (2) Joos, P.; Fainerman, V. B.; Loglio, G.; Lucassen-Reynders, E. H.; Miller, R.; Petrov, P. Dynamic Surface Phenomena; VSP: Utrecht, The Netherlands, 1999. (3) Ferri, J. K.; Stebe, K. J. Adv. Colloid Interface Sci. 2000, 85, 61. (4) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (5) Faour, G.; Grimaldi, M.; Richou, J.; Bois, A. J. Colloid Interface Sci. 1996, 181, 385.

gravity. In the past decade, the pendant-drop-shape techniques have been applied to study the adsorption kinetics of a variety of surfactants, such as proteins,8-10 alkylphosphine oxides,11 C10E8 (octaethylene glycol mono n-decyl ether),12-14 nonionic cellulose derivatives,15 alkylpyridinium chlorides,16 asphaltenes,17 and bovine lipid extract surfactant.18 Various adsorption models have been developed to analyze and interpret the measured dynamic surface or interfacial tension data. In their pioneering theoretical work on adsorption kinetics, Ward and Tordai19 solved the problem of adsorption from a semi-infinite surfactant solution to a planar interface. Later, a number of theoretical and numerical studies focused on improving Ward and Tordai’s theory. For example, Mysels20 derived the general solution of diffusion-controlled adsorption at a spherical interface by the superposition method. Bor(6) Song, B.; Springer, J. J. Colloid Interface Sci. 1996, 184, 64. (7) Rı´o, O. I. D.; Neumann, A. W. J. Colloid Interface Sci. 1997, 196, 136. (8) Tripp, B. C.; Magda, J. J.; Andrade, J. D. J. Colloid Interface Sci. 1995, 173, 16. (9) Beverung, C. J.; Radke, C. J.; Blanch, H. W. Biophys. Chem. 1999, 81, 59. (10) Miller, R.; Fainerman, V. B.; Wu¨stneck, R.; Kra¨gel, J.; Trukhin, D. V. Colloids Surf., A 1998, 131, 225. (11) Ferrari, M.; Liggieri, L.; Ravera, F.; Amodio, C.; Miller, R. J. Colloid Interface Sci. 1997, 186, 40. (12) Ferrari, M.; Liggieri, L.; Ravera, F. J. Phys. Chem. B 1998, 102, 10521. (13) Chang, H.; Hsu, C.; Lin, S. Langmuir 1998, 14, 2476. (14) Makievski, A. V.; Loglio, G.; Kra¨gel, J.; Miller, R.; Fainerman, V. B.; Neumann, A. W. J. Phys. Chem. B 1999, 103, 9557. (15) Persson, B.; Nilsson, S.; Bergman, R. J. Colloid Interface Sci. 1999, 218, 433. (16) Semmler, A.; Kohler, H. J. Colloid Interface Sci. 1999, 218, 137. (17) Bauget, F.; Langevin, D.; Lenormand, R. J. Colloid Interface Sci. 2001, 239, 501. (18) Lu, J. J.; Yu, L. M. Y.; Cheung, W. W. Y.; Policova, Z.; Li, D.; Hair, M. L.; Neumann, A. W. Colloids Surf., B 2003, 29, 119. (19) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (20) Mysels, K. J. J. Phys. Chem. 1982, 86, 4648.

10.1021/la0360097 CCC: $27.50 © 2004 American Chemical Society Published on Web 02/14/2004

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wankar and Wasan21 developed a general model of surfactant adsorption at a gas-liquid surface, which takes account of both the diffusion in the bulk phase and the barrier to adsorption. This model was solved numerically for the case of the Frumkin adsorption isotherm. Fainerman and Miller22 considered the effect of a nonequilibrium surface layer on the dynamic surface tension in a diffusion-controlled adsorption model. An approximate solution was obtained to interpret the dynamic surface tensions of short-chain alcohols at the water-air interface. Liggieri et al.23 presented a theoretical model for describing the adsorption of surfactants at a spherical interface with mass transport across the interface, which separates two finite bulk phases. The finite difference method was used to numerically solve the adsorption model. Filippov and Filippova24 developed a theory to describe the adsorption kinetics at a spherical interface in the consideration of the molecular diffusion in both bulk phases. Analytical solutions were achieved for several finite systems, such as a spherical liquid drop, a spherical liquid drop surrounded by another semi-infinite immiscible liquid phase, and a spherical liquid drop surrounded by a spherical shell of another immiscible liquid. In the past, the dynamic surface or interfacial tension data measured by the pendant-drop-shape techniques were usually interpreted by applying either Ward and Tordai’s theory, which is called the planar-interface model here, or the surfactant adsorption model for a sphericaldrop interface, which is termed as the spherical-drop model for simplicity. However, the actual interface of a pendant drop is neither planar nor spherical. In fact, the pendantdrop-shape techniques require that a well-deformed pendant drop should be used so as to accurately measure the surface or interfacial tension. In addition, because pendant drops used in the measurements are usually rather small, the depletion of the surfactant inside the pendant drop due to the adsorption at the interface sometimes cannot be neglected if the initial surfactant concentration is low. This means that the surfactant concentration in the bulk phase of the pendant drop is nonuniform and that the approximation of a semi-infinite surfactant solution is inapplicable in this case. Furthermore, the depletion of the surfactant inside the pendant drop induces the mass transfer of the surfactant from the syringe needle that is used to form the pendant drop to the pendant drop. Obviously, the spherical-drop model, which considers the pendant drop as a perfectly spherical drop with the same drop volume, does not consider this mass transfer. Semmler and Kohler16 showed that the dynamic surface or interfacial tensions might be strongly affected by diffusion limitations (i.e., limited surfactant reservoir of the pendant drop). Similar concern has also been raised by Miller et al.,10 Ferrari et al.,12 Makievski et al.,14 and Svitova et al.25 These studies indicate that the actual geometry of the pendant drop may play an important role in the dynamic surface or interfacial tension phenomenon. Thus, it is desired to study the adsorption kinetics of surfactants at the liquid-fluid interface of a pendant drop, taking its real geometry into account. In this paper, a comprehensive theoretical model is developed to describe the adsorption kinetics of surfactants (21) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1983, 38, 1637. (22) Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 1996, 178, 168. (23) Liggieri, L.; Ravera, F.; Ferrari, M.; Passerone, A.; Miller, R. J. Colloid Interface Sci. 1997, 186, 46. (24) Filippov, L. K.; Filippova, N. L. J. Colloid Interface Sci. 1997, 187, 352. (25) Svitova, T. F.; Wetherbee, M. J.; Radke, C. J. J. Colloid Interface Sci. 2003, 261, 170.

Yang and Gu

Figure 1. Schematic of an axisymmetric pendant drop of a surfactant solution in an immiscible fluid depicted in the cylindrical coordinate system (r, z).

at the liquid-fluid interface of a pendant liquid drop. This model considers the real geometry of the pendant drop, the depletion of the surfactant inside the pendant drop due to the adsorption at the interface, and the mass transfer of the surfactant from the syringe needle to the pendant drop. The theoretical model is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the transient surfactant concentration distributions inside the pendant drop and inside the syringe needle, the surfactant adsorption at the interface, and the dynamic surface or interfacial tension as well. Finally, the developed model together with its numerical scheme is applied to study the adsorption kinetics of C10E8 at the water-air interface of a pendant drop. The calculated dynamic surface tensions for the actual pendant drops are compared with the results obtained from the Ward and Tordai equation (i.e., the planar-interface model) and those from the spherical-drop model, respectively. 2. Theory Figure 1 shows an axisymmetric pendant drop of a surfactant solution in a quiescent immiscible fluid, forming from the tip of a cylindrical syringe needle. The outer radius of the syringe needle is rn, the wall thickness of the needle is n, and the height of the syringe needle is Hn. The boundary formed between the syringe needle and the surfactant solution and the cutting plane at the top of the syringe needle are denoted by Φn. The surface of the pendant drop, that is, the surfactant solution-fluid interface, is represented by Φint, and the subsurface of the pendant drop is denoted by Φsub. The latter is an imaginary layer and lies immediately underneath the former. The surfactant concentration at the solution-fluid interface Φint is expressed in terms of the number of surfactant molecules at the interface per unit surface area, that is, adsorption Γ. The surfactant concentration at the subsurface Φsub is expressed by the number of surfactant molecules per unit volume, that is, bulk surfactant concentration c. In this study, the ambient fluid phase in Figure 1 can be either a gas or another immiscible liquid in which the

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Langmuir, Vol. 20, No. 6, 2004 2505

surfactant is insoluble. Therefore, there is no mass transfer of surfactant across the surfactant solution-fluid interface. It is also assumed that the pendant drop remains invariant once formed. During the adsorption of the surfactant at the solution-fluid interface, in fact, the interfacial tension reduces gradually. This interfacial tension reduction will cause the shape of the pendant drop to change. The pendant drop during the adsorption process is not stationary. This makes the adsorption and diffusion process become an extremely complicated moving boundary problem. In general, however, the shape change of the pendant drop due to the slow adsorption and diffusion processes is rather slow. Hence, in this study, the influence of the shape change of the pendant drop on adsorption is neglected. The adsorption occurs as if the pendant drop is stationary once formed. It should be pointed out that, nevertheless, the dynamic interfacial tension is considered to change with time due to the adsorption of surfactants at the interface. In the Cartesian coordinate system, the molecular diffusion of the surfactant inside the pendant drop is a three-dimensional problem. Noting the axisymmetry of the pendant drop, however, it is more convenient to choose the cylindrical coordinate system (r, z), where r denotes the radial coordinate and z stands for the axial coordinate as shown in Figure 1. The origin of the cylindrical coordinate system lies at the apex of the pendant drop. Thus, the diffusion process becomes an unsteady twodimensional problem in the chosen coordinate system. The diffusion equation to describe the bulk surfactant concentration distributions can be expressed as

[

]

∂c ∂2c 1 ∂ ∂c )D r + 2 , (r, z) ∈ Ω, t > 0 ∂t r ∂r ∂r ∂z

( )

(2a)

c(r, z, t)|t)0 ) 0, (r, z) ∈ Φsub

(2b)

Γ(r, z, t)|t)0 ) 0, (r, z) ∈ Φint

(2c)

where c0 is the initial uniform bulk surfactant concentration and Γ(r, z, t) is the adsorption of the surfactant at the interface of the pendant drop. Because the syringe needle is impermeable to the surfactant, the corresponding nonpenetrating boundary condition is applied: r

z

(r, z) ∈ Φn

(∂r∂cn + ∂z∂cn )| r

z

)(r,z)∈Φsub

|

∂Γ(r, z, t) (3b) ∂t (r,z)∈Φint

Furthermore, the term on the right-hand side of eq 3b can be generally related to the exchange of surfactant between the interface and the subsurface:21,27

|

∂Γ(r, z, t) ∂t

(r,z)∈Φint

) φ[c(r, z, t)|(r,z)∈Φsub, Γ(r, z, t)|(r,z)∈Φint] (4)

where φ is a function of the subsurface concentration and the adsorption at the interface. For diffusion-controlled adsorption, the explicit forms of the function φ can be derived from the specific adsorption isotherms.26 For mixed diffusion-kinetic-controlled adsorption, Danov et al.27 gave the explicit forms of the function φ for different adsorption isotherms. 3. Numerical Method To solve the bulk diffusion equation in eq 1 subject to the initial conditions in eqs 2a-c and the boundary conditions in eqs 3a,b together with eq 4, it is more convenient to express all the equations in a dimensionless form. These equations can be nondimensionalized by introducing the following dimensionless variables:

C)

c(r, z, t)|t)0 ) c0, (r, z) ∈ Ω ∪ Φn

(∂r∂cn + ∂z∂cn ) ) 0,

D

(1)

where c is the surfactant concentration in the bulk phase of the pendant drop, D is the diffusion coefficient of the surfactant in the surfactant solution, and Ω represents the bulk phase of the surfactant solution inside the pendant drop and inside the syringe needle. Regarding the initial conditions for eq 1, an initial clean interface and uniform bulk surfactant concentration distributions inside the pendant drop19-24 are assumed. The clean-interface assumption requires that the pendant drop be formed instantaneously. However, this requirement may not be satisfied in practice and, thus, the pendant-drop-shape techniques are generally unsuitable for studying the dynamic surface or interfacial tension phenomenon that occurs in a very short period (e.g., 1 s).2,26 Under the assumptions just mentioned, the corresponding initial conditions are given by

D

z components of the outward unit vectors normal to the surfactant solution-fluid interface. By applying the conservation law of mass, the boundary condition at the subsurface of the pendant drop is obtained as follows.

z Dt r c , R) , Z) , τ) 2 c0 h h h

(5)

Here, C is the dimensionless surfactant concentration inside the pendant drop; R and Z are the dimensionless radial and axial coordinates, respectively; h is the adsorption depth,3 h ) Γeq/c0, and Γeq is the equilibrium adsorption corresponding to the initial bulk concentration c0; and τ is the dimensionless time. With these dimensionless variables, eqs 1-3b become

∂2C ∂C 1 ∂ ∂C ) R + 2, (R, Z) ∈ Ω, τ > 0 ∂τ R ∂R ∂R ∂Z

(6)

C(R, Z, τ)|τ)0 ) 1, (R, Z) ∈ Ω ∪ Φn

(7a)

C(R, Z, τ)|τ)0 ) 0, (R, Z) ∈ Φsub

(7b)

( )

|

Γ(R, Z, τ) Γm

) 0, (R, Z) ∈ Φint

(7c)

∂C ∂C n + n ) 0, (R, Z) ∈ Φn ∂R R ∂Z Z

(8a)

|

τ)0

|

Γm ∂(Γ/Γm) ∂C ∂C n + n )∂R R ∂Z Z (R,Z)∈Φsub Γeq ∂τ (R, Z)∈Φint

(8b)

where nR and nZ are the direction cosines, that is, R and Z components of the outward unit vectors normal to the surfactant solution-fluid interface, and Γm is defined as

(3a)

where nr and nz are the direction cosines, that is, r and

(26) Eastoe, J.; Dalton, J. S. Adv. Colloid Interface Sci. 2000, 85, 103. (27) Danov, K. D.; Valkovska, D. S.; Kralchevsky, P. A. J. Colloid Interface Sci. 2002, 251, 18.

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Yang and Gu

element and the shape functions can be found elsewhere.28 The corresponding Galerkin residual for eq 6 is equal to

RSD(e)(R, Z, τ) )

(

)

∂C ˜ (e) ∂2C ˜ (e) ∂C ˜ (e) 1 ∂ R ∂τ R ∂R ∂R ∂Z2

(10)

Application of the Galerkin minimization principle28 to the residual RSD(e)(R, Z, τ) in conjunction with some mathematical manipulations results in the following element equation in a matrix form:

{

}

dC(e)(τ) + [K](e){C(e)(τ)} ) {F(τ)}(e) dτ

[H](e)

(11)

where

H(e) ij )

Figure 2. Triangular mesh for a pendant drop, where drop volume V ) 20 mm3, syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, syringe needle height Hn ) 2.50 mm, water-air surface tension γ ) 72 mJ/m2, and g ) 9.81 m/s2 (the mesh of 1357 elements and 752 nodes is chosen for graphical reasons).

the maximum adsorption at the surfactant solution-fluid interface and used to nondimensionalize the surfactant adsorption Γ(R, Z, τ). The dimensionless form of eq 4 can be derived if the explicit expressions of the function φ are given.26,27 The finite element methods are especially effective in solving diffusion equations in complex geometries. In this study, the semidiscrete Galerkin finite element method is applied to solve the diffusion equation subject to its corresponding initial and boundary conditions. As illustrated in Figure 2, an unstructured triangular mesh is used in the spatial discretization of the computational domain to provide an excellent approximation of the curved boundary of the pendant drop. To obtain more accurate surfactant concentration distributions near the interface and higher computational efficiency, a finer mesh is used in regions close to the interface, whereas a coarser mesh is used in regions far away from the drop boundary. For graphical reasons, Figure 2 shows the triangular mesh with 1357 elements and 752 nodes. It should be noted that the results presented in this paper are obtained for various triangular meshes with approximately 5500 elements and 3000 nodes. Typical dimensional mesh sizes for elements near the interface are around 30 µm. The finite element solution of the diffusion equation in eq 6 is obtained by using the semidiscrete Galerkin approximation:28

K(e) ij )

∫∫(e)φ(e)i φ(e)j dR dZ,

(

∫∫(e)

F(e) i )

i ) 1, 2, 3; j ) 1, 2, 3 (12a)

(

)

(e) (e) (e) ∂φ(e) ∂φ(e) 1 ∂φj i ∂φj i ∂φj + - φ(e) dR dZ, ∂R ∂R ∂Z ∂Z R i ∂R i ) 1, 2, 3; j ) 1, 2, 3 (12b) (e)

(e)

˜ ∂C ˜ (e) φ(e) φ(e) n(e) ∫(e) ∂C∂R i nR + ∂Z i Z

)

dS, i ) 1, 2, 3 (12c)

(e) Here, n(e) R and nZ are the direction cosines, that is, R and Z components of the outward unit vectors normal to the element boundary; S ) s/h is the dimensionless coordinate along the element boundary. When the element equations for all the triangular elements are assembled altogether, the resultant system equation can be written in a matrix form:

[H]

{ }

dC(τ) + [K]{C(τ)} ) {F(τ)} dτ

(13)

where [H] is called the capacity matrix, [K] is termed the stiffness matrix, and {F(τ)} is referred to as the load vector. By applying the θ method,29 the temporal discretization of the spatially discretized equation in eq 13 yields the following equation:

[

]

1 [H] + θ[K] {C}n ) (1 - θ){F}n-1 + θ{F}n + ∆τn 1 [H] - (1 - θ)[K] {C}n-1 (14) ∆τn

[

]

where C ˜ (e)(R, Z, τ) is the approximation for the dimensionless surfactant concentration in the triangular element (e); C(e) j (τ), j ) 1, 2, and 3, are the dimensionless surfactant concentrations at the three nodes of the triangular element; and φ(e) j (R, Z), j ) 1, 2, and 3, are the so-called shape functions. In this work, the linear interpolation is applied to approximate the solution in each

where θ is termed as the weighting factor and subscripts n and n - 1 represent the numerical solutions at the time steps n and n - 1, respectively. Equation 14 is the desired recurrence relation, which represents a set of linear algebraic equations. In principle, the weighting factor θ used in eq 14 can be chosen from 0 to 1. The effect of this factor on the performance of the recurrence relation has been thoroughly discussed by Burnett.29 Generally, when 1/2 e θ e 1, the θ method is unconditionally stable, whereas the algorithm becomes conditionally stable if 0 e θ < 1/2. The value of θ will also affect the oscillation of the solution. Numerical experiments29 suggested that θ ) 2/3 is near the optimum value, though it is preferable to choose θ close or equal to unity when the unsteady-state solution

(28) Wait, R.; Mitchell, A. R. Finite Element Analysis and Applications; John Wiley & Sons: Chichester, U.K., 1985.

(29) Burnett, D. S. Finite Element Analysis: From Concepts to Applications; Addison-Wesley Publishing Company: Reading, MA, 1987.

3

C ˜ (e)(R, Z, τ) )

(e) C(e) ∑ j (τ) φj (R, Z) j)1

(9)

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Langmuir, Vol. 20, No. 6, 2004 2507

is close to the steady-state solution. Therefore, in this study, the weighting factor θ ) 2/3 is used at the beginning of the adsorption process, and θ ) 1.0 is used when the adsorption at the interface proceeds rather slowly. Equation 14 is solved by employing the Gauss-Seidel iterative method30 to yield the dimensionless surfactant concentration distributions inside the pendant drop. The dimensional surfactant concentration distributions can be readily obtained by using the definitions given in eq 5. In general, the surfactant adsorption at the interface of a pendant drop can be calculated from eq 4. Then the dynamic surface or interfacial tension can be calculated by using the corresponding surface equation of state for a specific adsorption isotherm.26 4. Results and Discussion In this section, the theoretical model just developed and its numerical scheme are applied to study the adsorption kinetics of nonionic surfactant C10E8 at the water-air interface of a pendant drop. In the literature, the equilibrium surface tension data for the C10E8 aqueous solution-air system were fitted by using the Langmuir adsorption isotherm, the Frumkin adsorption isotherm, and the generalized Frumkin adsorption isotherm13 as well as the two-state adsorption model12. In this study, the Langmuir adsorption isotherm is used:

Γ c ) Γm aL + c

(15)

where Γm is the maximum adsorption at the surfactant solution-air interface and aL is the Langmuir adsorption constant and represents the bulk surfactant concentration at which 50% of Γm is reached.11 The Langmuir adsorption constant can be regarded as a measure of the affinity of the surfactant for the interface. The larger aL, the less affinity the surfactant has for the interface. According to Chang et al.,13 the maximum adsorption is equal to Γm ) 1.8 × 10-10 mol/cm2 and the Langmuir adsorption constant is found to be aL ) 5.44 × 10-10 mol/cm3 for the C10E8 aqueous solution-air system. At low C10E8 concentrations, the adsorption kinetics at the C10E8 aqueous solution-air interface was found to be diffusion-controlled.12-14 Therefore, in this study, the relation between the adsorption at the interface and the bulk surfactant concentration at the subsurface can be described by the Langmuir adsorption isotherm. Noting that Γeq/Γm ) c0/(aL + c0) and substituting eq 15 into eq 8b and simplifying it yields

1 + CL ∂C ∂C ∂C n + n ), (R,Z) ∈ Φsub (16) ∂R R ∂Z Z (1 + CLC)2 ∂τ Here, CL is the reduced initial bulk surfactant concentration or the so-called adsorption number,3 CL ) c0/aL. The corresponding surface equation of state for the Langmuir adsorption isotherm is expressed by the Langmuir-Szyszkowski relation:26

[

γ(t) ) γ0 + RTΓm ln 1 -

]

Γ(t) Γm

(17)

where R is the universal gas constant, T is the absolute temperature, and γ0 is the surface or interfacial tension (30) Mathews, J. H. Numerical Methods for Computer Science, Engineering, and Mathematics; Prentice Hall: Englewood Cliffs, NJ, 1987.

of the solvent with the ambient fluid. For the water-air system, its surface tension is chosen as γ0 ≈ 72 mJ/m2 at 25 °C. It is seen from eqs 3b and 4 that the time-dependent adsorption Γ at the interface of the pendant drop can vary with the position at the interface. In this paper, an average value of the adsorption over the entire interface is used to calculate the dynamic surface tension γ(t) from eq 17. It can be seen from eq 16 that the initial bulk C10E8 concentration c0 has to be specified to determine the reduced initial bulk concentration CL. Here, four typical initial bulk C10E8 concentrations c0) 2.72, 5.44, 10.88, and 27.20 × 10-9 mol/cm3 are used.12,13 With the given maximum adsorption Γm and the Langmuir adsorption constant aL, the equilibrium adsorption corresponding to the initial bulk concentration c0 can be determined from Γeq ) Γmc0/(aL + c0). The corresponding adsorption depths h ) Γeq/c0 are equal to 0.551, 0.301, 0.157, and 0.065 mm, and the corresponding adsorption numbers CL ) c0/aL are 5, 10, 20, and 50, respectively. For most surfactants,3 the diffusion coefficients are in the range of 1 × 10-6 to 10 × 10-6 cm2/s. Here, D ) 5 × 10-6 cm2/s is chosen as the diffusivity of C10E8 in water.13 The geometry of a pendant drop can be described by its drop volume V, the outer radius rn of the syringe needle, and the wall thickness n of the syringe needle. In this work, the following typical values are chosen in the numerical simulations: the drop volume V ) 10, 20, and 30 mm3 and the outer radius of the syringe needle rn ) 0.75, 1.00, and 1.25 mm. The wall thickness of the syringe needle is chosen as n ) 0.25 mm. In comparison with various experimental data published in the literature,8,14-16,18 these values should cover most cases of practical interest. With the given surface tension γ0, water-air density difference, drop volume V, and outer radius of the syringe needle rn, the complete interfacial profile of the pendant drop can be determined by numerically solving the Laplace equation of capillarity.4 To specify the computational domain, the height of the syringe needle Hn is approximated by the diffusion penetration depth,2 that is, Hn ∝ (πDt)1/2, which quantifies how far the diffusion extends along the syringe needle. Because the dynamic surface tension behavior within about 1 h is of interest in this study, Hn ) 2.50 mm is chosen in the numerical simulations. 4.1. Comparison of the Pendant-Drop Model with the Planar-Interface Model and the Spherical-Drop Model. In this section, a typical case is chosen to show the dimensionless C10E8 concentration distributions inside the pendant drop. More importantly, the calculated surface coverage Γ/Γm and dynamic surface tension γ(t) of C10E8 at the water-air interface of a pendant drop are compared with the results obtained from the Ward and Tordai equation (i.e., the planar-interface model) and those from the spherical-drop model, respectively. As mentioned previously, the Ward and Tordai equation was derived for adsorption from a semi-infinite surfactant solution to a planar interface. The spherical-drop model considers the pendant drop as a perfectly spherical drop with the same drop volume. In this typical case, the volume of the pendant drop is chosen as V ) 20 mm3 and the outer radius of the syringe needle is taken as rn ) 1.00 mm. For diffusion-controlled adsorption from a semi-infinite surfactant solution to a planar interface, the timedependent adsorption Γ(t) is described by the famous Ward and Tordai19 equation:

Γ(t) )

x4Dπ [c xt - ∫ 0

xt

0

csub(t′) dxt - t′]

(18)

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Langmuir, Vol. 20, No. 6, 2004

Figure 3. Dimensionless C10E8 concentration contours inside the pendant drop and inside the syringe needle, where the drop volume V ) 20 mm3, syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, syringe needle height Hn ) 2.50 mm, and initial bulk concentration c0 ) 2.72 × 10-9 mol/cm3 at time (a) t ) 600 s and (b) t ) 900 s.

where t′ is a dummy variable of integration and csub is the bulk surfactant concentration in the subsurface. This equation together with the Langmuir adsorption isotherm in eq 15 can be solved numerically to obtain the adsorption Γ(t) of the surfactant at the interface. Then the dynamic surface tension γ(t) can be determined from the Langmuir-Szyszkowski relation in eq 17. For diffusion-controlled adsorption to a perfectly spherical-drop interface, the adsorption Γ(t) is obtained by applying the analytical solution derived by Filippov and Filippova24 for an equivalent spherical drop, which has the drop radius of 1.68 mm and the drop volume of V ) 20 mm3 in the present case. It is worthwhile noting that the C10E8 solution inside this equivalent spherical drop is completely isolated by the surrounding solution-air interface so that there is not any mass transfer of C10E8 from the syringe needle to the spherical drop in the spherical-drop model. The corresponding dimensionless C10E8 concentration contours inside the pendant drop and inside the syringe needle are plotted in Figure 3a at t ) 600 s and in Figure 3b at t ) 900 s, respectively. Obviously, as the surfactant inside the pendant drop is depleted as a result of the adsorption of surfactant at the interface, the surfactant inside the syringe needle gradually diffuses into the pendant drop. Because the direction of this mass transfer of C10E8 is from the upper syringe needle to the lower pendant drop in the present case, the adsorption of the surfactant at the upper interface is higher than that at the lower interface. Therefore, there exists a surface concentration gradient on the drop interface. This gradient will cause a surface tension gradient and a surface flow of the surfactant. In this study, it is assumed that the surface flow is so fast that the average surface coverage can be used to calculate the dynamic surface tension. The average surface coverage Γ/Γm of the interface is found to be 0.547 at t ) 600 s and 0.588 at t ) 900 s. Figure 4 shows the calculated average surface coverage Γ/Γm for the pendant-drop model (drop volume V ) 20

Yang and Gu

Figure 4. Comparison of the average surface coverage Γ/Γm for C10E8 solutions versus time at two different initial bulk concentrations c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3 for the pendant-drop model (drop volume V ) 20 mm3, syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm), the planar-interface model, and the spherical-drop model (drop volume V ) 20 mm3).

Figure 5. Comparison of the dynamic surface tension γ(t) at the C10E8 aqueous solution-air interface versus time at two different initial bulk concentrations c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3 for the pendant-drop model (drop volume V ) 20 mm3, syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm), the planar-interface model, and the sphericaldrop model (drop volume V ) 20 mm3).

mm3, outer radius of the syringe needle rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, syringe needle height Hn ) 2.50 mm), the planar-interface model, and the spherical-drop model (drop volume V ) 20 mm3) at two initial bulk surfactant concentrations c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3. The corresponding dynamic surface tension γ(t) versus time curves are shown in Figure 5. It is clearly seen from these two figures that the Ward and Tordai equation (i.e., the planar-interface model) always overestimates the surface coverage Γ/Γm and, thus, underestimates the dynamic surface tension γ(t). For example, at c0 ) 2.72 × 10-9 mol/cm3, the difference between the surface tension predicted from the pendant-drop model and that predicted from the planar-

Adsorption Kinetics of Surfactants

interface model is about 1.6 mJ/m2 and 2.5 mJ/m2 at t ) 600 s and t ) 3600 s, respectively. Basically, there are two reasons for the large discrepancy. First, a semi-infinite surfactant solution is assumed in the Ward and Tordai equation, which means that the bulk C10E8 concentration always remains at the initial bulk C10E8 concentration c0. In both the pendant-drop model and the spherical-drop model, however, the depletion of the surfactant in the bulk phase caused by the adsorption of the surfactant at the drop interface is considered. Thus, the bulk surfactant concentration decreases with time in both models. Second, the outward diffusion field in the radial direction inside a spherical drop or a pendant drop diverges and, thus, decelerates the diffusion process.31 As a result, the dynamic surface tension reduces more slowly. It should be noted that, in such a case, if the Ward and Tordai equation (i.e., the planar-interface model) is still used to interpret the dynamic surface tension data, a much smaller diffusion coefficient is resulted or an erroneous conclusion that the adsorption process is not diffusion-controlled can be made. In comparison with the results for the pendant-drop model and the spherical-drop model shown in Figures 4 and 5, it is found that the spherical-drop model always slightly underestimates the surface coverage and overestimates the dynamic surface tension. This is because, in the spherical-drop model, there is no mass transfer of C10E8 from the syringe needle to the spherical drop, though this mass transfer is considered in the pendant-drop model. Therefore, the depletion of the surfactant in the bulk phase for the spherical-drop model is faster than that for the pendant-drop model. Consequently, the bulk surfactant concentration inside the spherical drop is lower and, thus, the adsorption of surfactant at the spherical interface is lower. This leads to a higher dynamic surface tension. In general, it is expected that the differences between these two models become larger at even longer times. In summary, the spherical-drop model is not recommended for describing the adsorption of the surfactant at the interface of a pendant drop at low initial bulk surfactant concentrations in long adsorption durations. Figure 6 shows similar discrepancies among the calculated dynamic surface tensions for these three adsorption models at two higher initial bulk surfactant concentrations c0 ) 10.88 × 10-9 mol/cm3 and c0 ) 27.20 × 10-9 mol/cm3, respectively. A comparison of Figure 5 with Figure 6 indicates that the differences between the dynamic surface tensions predicted from the pendantdrop model and those from the spherical-drop model become much smaller at higher initial bulk surfactant concentrations, whereas the planar-interface model always considerably underestimates the dynamic surface tension. Therefore, the spherical-drop model is suitable for interpreting the dynamic surface tension data measured by using the pendant-drop-shape techniques at high initial bulk surfactant concentrations. 4.2. Effect of Drop Volume. To examine the effect of the drop volume on the adsorption kinetics of the C10E8 solution at the water-air interface, numerical simulations are conducted for pendant drops formed from the same syringe needle (outer radius rn ) 1.00 mm, wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm) but with different drop volumes (V ) 10, 20, and 30 mm3). The calculated dynamic surface tensions are shown in Figure 7 for two initial bulk surfactant concentrations c0 (31) Miller, R.; Fainerman, V. B.; Makievski, A. V.; Kra¨gel, J.; Grigoriev, D. O.; Kazakov, V. N.; Sinyachenko, O. V. Adv. Colloid Interface Sci. 2000, 86, 39.

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Figure 6. Comparison of the dynamic surface tension γ(t) at the C10E8 aqueous solution-air interface versus time at two higher initial bulk concentrations c0 ) 10.88 × 10-9 mol/cm3 and c0 ) 27.20 × 10-9 mol/cm3 for the pendant-drop model (drop volume V ) 20 mm3, syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm), the planar-interface model, and the sphericaldrop model (drop volume V ) 20 mm3).

Figure 7. Comparison of the dynamic surface tension γ(t) at the C10E8 aqueous solution-air interface for two different initial bulk concentrations c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3 exhibiting the effect of the drop volume V of the pendant drop (syringe needle radius rn ) 1.00 mm, syringe needle wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm).

) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3. It is clearly shown in this figure that the adsorption kinetics is strongly affected by the drop volume. At short times (e.g., t < 300 s), the predicted dynamic surface tensions are rather close for different drop volumes. The physical explanation of this phenomenon is as follows. At short times, the surfactant molecules adsorbed at the interface mainly comes from the adjacent regions. Therefore, the geometrical effect of the pendant drop and the depletion of the surfactant in the bulk phase can be neglected at the beginning. As time proceeds, nevertheless, the differences among the predicted dynamic surface tensions for different drop volumes become larger. Generally, the smaller the pendant drop, the higher the surface tension if it is compared at the same time. This is because, at large times,

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Figure 8. Comparison of the dynamic surface tension γ(t) at the C10E8 aqueous solution-air interface for two different initial bulk concentrations c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3 exhibiting the effect of the radius rn of the syringe needle used to form the pendant drop (drop volume V ) 20 mm3, syringe needle wall thickness n ) 0.25 mm, and syringe needle height Hn ) 2.50 mm).

the bulk diffusion of C10E8 (i.e., the depletion of the surfactant in the bulk phase) caused by the adsorption of the surfactant at the interface extends to almost the entire pendant drop. Thus, the geometrical effect and the depletion of the surfactant in the bulk phase become pronounced. In particular, for smaller drops, more significantly reduced bulk surfactant concentrations lead to higher surface tensions at the same time in comparison with those for larger drops. In addition, it should be noted that, although the depletion of the surfactant in the bulk phase will induce the mass transfer from the syringe needle to the pendant drop, this mass transfer process is too slow to compensate the surfactant loss due to the adsorption at the interface. Further numerical results for two higher initial bulk concentrations c0 ) 10.88 × 10-9 mol/cm3 and c0 ) 27.20 × 10-9 mol/cm3 indicate that, as expected, the effect of the drop volume on the adsorption kinetics becomes smaller at higher initial concentrations. This is due to the fact that, as the initial surfactant concentration increases, the depletion of the surfactant in the bulk phase during the adsorption process becomes less important. In summary, it can be concluded that the effect of the drop volume on the adsorption kinetics should be considered in interpreting the dynamic surface or interfacial tension data in long durations (t > 300 s), particularly at low initial bulk surfactant concentrations. 4.3. Effect of Syringe Needle Size. The effect of the syringe needle size on the adsorption kinetics of C10E8 at the water-air interface is studied by predicting the dynamic surface tensions for pendant drops with the same volume V ) 20 mm3 but formed from three respective syringe needles (rn ) 0.75, 1.00, and 1.25 mm), whereas n ) 0.25 mm and Hn ) 2.50 mm. Figure 8 shows the predicted dynamic surface tensions at two different initial bulk concentrations, c0 ) 2.72 × 10-9 mol/cm3 and c0 ) 5.44 × 10-9 mol/cm3. It can be seen from this figure that, at short times (e.g., t < 600 s), the predicted surface tensions are almost the same for different syringe needle sizes. The reason for this behavior is already given in the preceding section. At long times, the larger the syringe needle size, the lower the surface tension, similar to the effect of the drop volume. This is due to the following two

Yang and Gu

reasons. First, the ratio of the water-air interfacial area to the volume of the pendant drop is smaller for a larger syringe needle. Therefore, the depletion of the surfactant inside the pendant drop is smaller for a pendant drop formed from a larger syringe needle. Second, a larger syringe needle means quicker mass transfer from the syringe needle to the pendant drop because the mass transfer flux is proportional to the cross-sectional area of the syringe needle. Thus, there is more adsorption of surfactant at the interface in this case. In addition, further numerical results for two higher initial bulk surfactant concentrations c0 ) 10.88 × 10-9 mol/cm3 and c0 ) 27.20 × 10-9 mol/cm3 show that the effect of the syringe needle size on the adsorption kinetics becomes smaller at higher initial concentrations. Hence, it is suggested that the effect of the syringe needle size on the adsorption kinetics should also be taken into account in interpreting the dynamic surface or interfacial tension data in long durations (t > 600 s), especially at low initial bulk surfactant concentrations. 5. Conclusions In this paper, a theoretical model has been developed to describe the adsorption kinetics of a surfactant at the liquid-fluid interface of a pendant drop. This model is solved numerically by applying the semidiscrete Galerkin finite element method. The distinct feature of this model is that it considers the real geometry of the pendant drop, the depletion of the surfactant inside the pendant drop, and the mass transfer of the surfactant from the syringe needle to the pendant drop. The developed model is applied to study the adsorption kinetics of C10E8 at the water-air interface of a pendant drop. The dynamic surface tensions calculated from the present pendant-drop model are compared with the results obtained from the Ward and Tordai equation (i.e., the planar-interface model) and those from the spherical-drop model. The comparison shows that the Ward and Tordai equation, which was derived for adsorption from a semiinfinite surfactant solution to a planar interface, predicts a much lower surface tension than the pendant-drop model and the spherical-drop model. In general, the planarinterface model is unsuitable for interpreting the dynamic surface or interfacial tension data measured by the pendant-drop-shape techniques, especially at low initial bulk surfactant concentrations. The spherical-drop model, which considers the pendant drop as a perfectly spherical drop with the same drop volume, however, can be used to interpret the dynamic surface or interfacial tension data for pendant drops either with high initial surfactant concentrations or with low initial bulk surfactant concentrations in short adsorption durations only. For pendant drops with low initial bulk surfactant concentrations in long adsorption durations, the spherical-drop model always overestimates the dynamic surface tension. Therefore, the pendant-drop model is strongly recommended. Moreover, the effects of the drop volume and syringe needle size on the adsorption kinetics at the interface of a pendant drop are also studied by varying the governing parameters in the practical ranges. It is found that the smaller the pendant drop, the higher the surface tension compared at the same time. The size of the syringe needle also has some effect on the surfactant adsorption kinetics at the interface of a pendant drop. Generally, the smaller the syringe needle, the higher the surface tension compared at the same time. In summary, it is concluded that the effects of the drop volume and syringe needle size on the adsorption kinetics should be

Adsorption Kinetics of Surfactants

considered in interpreting the dynamic surface or interfacial tension data in long adsorption durations (t > 300 s for the drop volume and t > 600 s for the syringe needle size), particularly at low initial bulk surfactant concentrations. Finally, it is worthwhile to emphasize that only the pendant-drop model developed in this study can consider the effect of the syringe needle size.

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Acknowledgment. The authors acknowledge the research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Innovation Fund from the Petroleum Technology Research Centre (PTRC) at the University of Regina to Y.Gu. LA0360097