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Langmuir 2004, 20, 4436-4445
Adsorption of Ionic Surfactants at an Expanding Air-Water Interface Dimitrina S. Valkovska,† Gemma C. Shearman,† Colin D. Bain,*,† Richard C. Darton,‡ and Julian Eastoe§ Department of Chemistry, University of Oxford, Mansfield Road, Oxford OX1 3TA, U.K., Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K., and School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K. Received September 17, 2003. In Final Form: March 16, 2004 A quantitative model for the kinetics of adsorption of ionic surfactants to an expanding liquid surface is presented for surfactant concentrations below and above the critical micelle concentration (cmc). For surfactant concentrations below the cmc, the electrostatic double layer is accounted for explicitly in the adsorption isotherm. An overflowing cylinder (OFC) was used to create nonequilibrium liquid surfaces under steady-state conditions. Experimental measurements of the surface excess for solutions of cationic surfactants CH3(CH2)n-1N+(CH3)3 Br- (CnTAB, n ) 12, 14, 16) and the anionic fluorocarbon surfactant sodium bis(1H,1H-nonafluoropentyl)-2-sulfosuccinate (di-CF4) in the OFC are in excellent agreement with the theoretical predictions for diffusion-controlled adsorption for all concentrations studied below the cmc. For surfactant concentrations above cmc, the diffusion of micelles and monomers are handled separately under the assumption of fast micellar breakdown. This simplified model gives excellent agreement for the system C14TAB + 0.1 M NaBr above the cmc. Agreement between theory and experiment for C16TAB + 0.1 M NaBr is less good. A plausible explanation for the discrepancy is that micellar breakdown is no longer fast on the time scale of the OFC (ca. 0.1 s).
Introduction The interfacial behavior of surfactant solutions away from equilibrium is commonplace in a wide range of natural and industrial processes, including, for example, foaming, emulsification, detergency, and coating.1-3 As a consequence, numerous experimental techniques have been developed for studying dynamics of adsorption from surfactant solutions.4 In most cases the relaxation of the surface tension, σdyn(t), with time is measured. When the surface of the solution is expanded (contracted), the adsorption (desorption) of surfactant molecules can be broken down into two processes: (i) diffusion of the molecules from the bulk to the subsurface and (ii) transfer of the molecule from the subsurface to the adsorbed state in the monolayer. The first process is governed by the diffusion, convection, and migration in the bulk phase and the second by the free energy barrier on the pathway of the surfactant molecule from the subsurface to the surface. The most common model for adsorption is based on the assumption of local thermodynamic equilibrium between the subsurface and the surface (diffusion-controlled adsorption). In an idealized experiment in which a fresh interface is created instantaneously in an otherwise stagnant solution, the dynamic surface excess, Γdyn(t), is given by the well-known equation of Ward and Tordai5
Γdyn(t) ) Γ0 + 2
xDπ [c xt - ∫ c (t - τ) dxτ] t
∞
0
s
(1)
where Γdyn(t) is the dynamic surface excess, c∞ is the bulk concentration, and cs is the subsurface concentration.6 * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Chemistry, University of Oxford. ‡ Department of Engineering Science, University of Oxford. § School of Chemistry, University of Bristol. (1) Clint, J. H. Surfactant Aggregation; Blackie: Glasgow, 1992.
If it is assumed that the equilibrium equation of state σ(Γ) also holds under dynamic conditions, Γdyn(t) may be deduced from σdyn(t) and hence compared with the WardTordai equation.7 The equation of state is normally derived from an experimental equilibrium adsorption isotherm, σ(c∞), and the Gibbs equation, though direct methods of measuring Γ, such as neutron reflectivity,8 are gaining currency. There are three practical difficulties with determining adsorption kinetics by means of the Ward-Tordai equation. First, the initial condition of an instantaneously created water surface cannot be realized in practice. For adsorption that is sufficiently slow that the time taken to create a fresh surface can be neglected, the exclusion of natural convection is difficult. Second, the state of the surface at any time t is determined by the entire history of the surface from time t ) 0 through the integral on the right-hand side of eq 1. Any error at short times introduced by the initial conditions or by a breakdown in the assumption of diffusion-controlled adsorption will therefore affect measurements at all longer times. Third, while the assumption of diffusion control is always valid at (2) Karsa, D. R. Industrial Applications of Surfactants 2; Royal Society of Chemistry: Cambridge, UK, 1990. (3) Dickinson, E., Walstra, P., Eds. Food Colloids and Polymers; Royal Society of Chemistry: Cambridge, UK, 1993. (4) Dukhin, S. S.; Kretzschmar, G.; Miller, R. In Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995; Chapter 5. Frances, E. I.; Chang, H. C. Colloid Surf. 1995, 100, 1. (5) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (6) For ionic surfactants, cs needs to be interpreted as the bulk concentration at the edge of the electrostatic double layer (which we will label cλ), since eq 1 contains no electrical terms. (7) This assumption is not strictly valid since it neglects the free energy that is dissipated irreversibly in the nonequilibrium process as surfactant molecules diffuse down the concentration gradient from the bulk to the surface (Diamant, H.; Andelman, D. J. Phys. Chem. 1996, 100, 13732). In addition, where the monolayer undergoes slow rearrangement, such as in the denaturation of adsorbed proteins, the dynamic and equilibrium equations of state will clearly differ. (8) Lu, J. R.; Thomas, R. K.; Penfold, J. Adv. Colloid Interface Sci. 2000, 84, 143.
10.1021/la035739b CCC: $27.50 © 2004 American Chemical Society Published on Web 04/21/2004
Adsorption of Ionic Surfactants
sufficiently long times, it will always break down at sufficiently short times within the Ward-Tordai formalism, since the mass-transfer rate diverges as t-1/2; at sufficiently high mass-transfer rates, the adsorption rate will be controlled by passage over a kinetic barrier. This barrier can have many origins: an electrostatic barrier in the electrical double layer, the creation of a void within a monolayer to make room for an adsorbing molecule, the reorganization of molecules within the monolayer, or (above the cmc) slow micellar breakdown. Depending on the time scale of the experiment, the adsorption process can be purely diffusion-controlled, mixed barrier-diffusioncontrolled, or purely barrier-controlled.9 The drawbacks of the Ward-Tordai approach may be circumvented by the continuous (often periodic) expansion or contraction of a surface originally at equilibrium. Such methods are useful but are suitable only for small deviations from equilibrium and relatively slow adsorption processes. An alternative strategy is to create a fresh surface continuously such that the fluid dynamical system is at steady state. Approaches include liquid jets,10 overflowing cylinders11-16 and funnels,17 and moving barriers.18 A common feature of these methods is that the adsorption equations become time-independent, but at the cost of an additional convection term. The problem is thus only simplified if the hydrodynamics are well-understood (which is not the case, for example, in an oscillating jet19). In some of these approaches, notably the overflowing cylinder and expanding trough, the rate of creation of fresh surface is approximately constant over the free surface (except very near the boundaries) with the result that convection in the plane of the interface can be neglected. An important consequence is that the properties of the surface are then dependent only on the adsorption kinetics at fixed bulk concentration and surface expansion rate, θ ) d ln A/dt (where A is the area of a surface element). θ-1 is the analogue of time in a non-steady-state experiment, but the “history” implicit in adsorption experiments under stagnant conditions is absent under steady-state conditions with forced convection. (We note that this simplification does not hold generally when θ is a function of surface position, since tangential convection then “mixes” surfaces with different ages.) (9) Joos, J. Dynamic Surface Phenomena; VSP BV: The Netherlands, 1999. Kralchevsky, P. A.; Danov, K. D.; Denkov, N. D. In Handbook of Surface and Colloid Chemistry, 2nd ed.; Birdi, K. S., Ed.; CRC Press: Boca Raton, FL, 2002; Chapter 5. (10) Davies, J. T.; Makepeace, R. W. AIChE J. 1978, 24, 524. Jobert, P. P.; Leblond, J. J Colloid Interface Sci. 1979, 68, 478. Hansen, R. S. J. Phys. Chem. 1964, 68, 2012. Battal, T.; Bain, C. D.; Weiss, M.; Darton, R. C. J Colloid Interface Sci. 2003, 263, 250. (11) Padday, J. F. Proc. Int. Congr. Surf. Act. 1957, 1. BerginkMartens, D. J. M.; Bos, H. J.; Prins, A.; Schulte, B. C. J. Colloid Interface Sci. 1990, 138, 1. Bergink-Martens, D. J. M.; Bos, H. J.; Prins, A. J. Colloid Interface Sci. 1994, 165, 221. (12) Manning-Benson, S.; Bain, C. D.; Darton, R. C. J. Colloid Interface Sci. 1997, 189, 109. (13) Manning-Benson, S.; Bain, C. D.; Darton, R. C.; Sharpe, D.; Eastoe, J.; Reynolds, P. Langmuir 1997, 13, 5808. (14) Manning-Benson, S.; Parker, S. R. W.; Bain, C. D.; Darton, R. C.; Penfold, J. Langmuir 1998, 14, 990. (15) Bain, C. D.; Manning-Benson, S.; Darton, R. C. J. Colloid Interface Sci. 2000, 229, 247. (16) Breward, C. J. W.; Darton, R. C.; Howell, P. D.; Ockendon, J. R. Chem. Eng. Sci. 2001, 56, 2867. Howell, P. D.; Breward, C. J. W. J. Fluid Mech. 2003, 474, 275. (17) Joos, P.; De Keyser, P. The overflowing funnel as a method for measuring surface dilational properties. Europhysics Conference Abstracts, Madrid, 1980; p 156. Schank, P. R.; Scriven, L. E. Dynamic surface tension by the overflowing cell. Annual Meeting AIChE, New Orleans, 1988. (18) Van Voorst Vader, F.; Erkens, Th. F.; van den Tempel, M. Trans. Faraday Soc. 1964, 60, 1170. Joos, P.; van Uffelen, M. J. Colloid Interface Sci. 1993, 155, 271. (19) Noskov, B. A. Adv. Colloid Interface Sci. 1996, 69, 63.
Langmuir, Vol. 20, No. 11, 2004 4437
In this paper, we employ an overflowing cylinder (OFC) to create a steady-state surface that is far from equilibrium.12 A surfactant solution is pumped vertically upward through the cylinder and then overflows the rim. The free liquid surface expands radially from a stagnation point at the center of the cylinder. In the presence of surfactants, a surface tension gradient is created that may increase the surface velocity by an order of magnitude compared to pure water. The radial acceleration leads to a surface expansion rate, θ, that is approximately uniform over the central region of the OFC. The time scale for adsorption, θ-1 ) 0.1-1 s. The reduction in the surface excess of surfactant as a result of surface expansion is balanced by diffusion and convection of surfactants from the bulk. Experimentally, we can measure the velocity profile, the surface excess, and the surface tension, all as a function of radial position, r.12-15 The important experimental parameters in this study are the surface excess at r ) 0 and the surface velocity, from which θ is calculated. The fluid dynamics of the OFC are described briefly in Section 1, below. In Section 2, we derive a relationship between surface excess, surface expansion rate, and bulk concentration in the OFC for ionic surfactants at concentrations below the cmc, taking account of the electrostatic double layer (EDL) and counterion bounding, but with some simplifying assumptions. In Section 3, we describe the adsorption isotherm employed for ionic surfactants. In Section 4, we compare the theoretical predictions for the dynamic surface excess, Γdyn, with the experimental data for a homologous series of cationic surfactants, CH3(CH2)n-1N+(CH3)3Br- (n ) 12, 14, 16; abbreviated CnTAB) and an anionic fluorocarbon surfactant, sodium bis(1H,1H-nonafluoropentyl)-2-sulfosuccinate (abbreviated di-CF4), and show that the adsorption kinetics of these surfactants is diffusion-controlled on the time scale provided by the OFC. We show the importance of the correct determination of the equilibrium surfactant isotherms, Γ(c), for proper interpretation of the dynamic processes. When the total surfactant concentration exceeds the cmc, the concentration of monomers remains roughly constant and the excess surfactant exists in the form of micelles. It is generally assumed that (highly charged) micelles of ionic surfactants do not adsorb at the airwater interface due to electrostatic repulsion by previously adsorbed monomers. The presence of micelles does, however, affect the adsorption kinetics in two ways. First, micelles diffuse at a different rate from monomers, altering the mass-transport rate to the surface. Second, micelles need to break down in the diffusion layer to release monomers that can adsorb to the surface. Micellar breakdown kinetics may limit the supply of monomer and thus slow the rate of adsorption. Above the cmc, we can therefore expect that σdyn(t) and Γdyn(t) will depend on the rate of diffusion of monomers and micelles, on the rate at which micelles break down, and on the rate of surface expansion. In Section 5, we develop a model for adsorption of ionic surfactants above the cmc under the assumption that micellar breakdown is rapid on the time scale of the OFC (i.e., monomers and micelles are locally in thermodynamic equilibrium). Finally, in Section 6, we compare the theoretical predictions with experimental data for C16TAB and C14TAB in excess electrolyte above the cmc of the surfactant. 1. Hydrodynamics of the OFC A schematic picture of the overflowing cylinder is presented in Figure 1. Surfactant solution is pumped
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The index k ) 1 corresponds to surface-active ions, 2 to counterions, and 3 to co-ions. Radial convection is absent by symmetry at r ) 0 (and is in fact negligible except for r ∼ R). Radial diffusion can be safely neglected since the diffusion layer thickness (D/θ)1/2 , R. In eq 3 ck, zk, and Dk are the concentration, valency, and diffusion coefficient, respectively, of ion k and φ ) eΦ/kBT is the dimensionless electric potential (e, kB, and T have their usual meanings). The electric potential is related to the ionic distribution through the Poisson equation,
d2φ
)-
dz2
e2
3
∑ zkck k Tk)1
(4)
B
where is the dielectric permittivity of the solvent. At steady state, the decrease of the surface excess, Γk, due to the expansion of the surface is balanced by diffusion and migration from the bulk to the surface. The resulting boundary condition at z ) 0 is Figure 1. Schematic diagram of the overflowing cylinder.
slowly up through a circular cylinder with radius R. The free surface at the top of the cylinder is nearly flat. At the free surface, the liquid flows radially outward from a stagnation point at the center and then overflows the rim. The velocity field inside the cylinder is uniform and the flow is assumed to be steady and radially symmetric. The stagnation point defines the origin, (r, z) ) (0, 0), of a cylindrical coordinate system: hence, υr(0,0) ) 0 where υr is the radial velocity. A detailed description of the hydrodynamics of the OFC can be found elsewhere.15,16 The ratio of the thicknesses of the hydrodynamic and diffusive boundary layers is given by xSc, where the Schmidt number Sc ) ν/D ∼ O(104); ν is the kinematic viscosity of the solvent and D the diffusion coefficient of the surfactant. We seek the steady-state solution of the diffusion problem near the center of the cylinder where r , R. In this region, the surface velocity is a linear function of the radial distance, r; therefore, the surface expansion rate, θ ) r-1 d(rυr)/dr is independent of r.15 Moreover, because xSc is so large, we can take the radial velocity to be constant inside the diffusion layer. Therefore, the solution of the continuity equation ∇‚υ ) 0 together with the kinematic boundary condition υz ) 0 at z ) 0 gives
υr ) θr/2, υz ) -θz
(2)
The local flow field at the surface of the OFC is the same as that at the surface of a Langmuir trough under constant dilation.18 2. Mass-Transport Equations for Ionic Surfactants below the cmc We consider an aqueous solution of a symmetrical ionic surfactant in the presence of an additional background electrolyte with a common counterion. The bulk concentrations of the surfactant and salt (outside the diffusion layer) are c1,∞ and c3,∞, respectively. We seek the solution of the mass-transport equations at the center of the cylinder, r ) 0, in the absence of a kinetic barrier. Under steady-state conditions the transport of the surface-active ions, counterions, and co-ions is governed by the convection-diffusion-migration equation:
(
)
dck d dck dφ ) Dk + zkck -θz dz dz dz dz
k ) 1,2,3
(3)
θΓk ) Dk
(
)
dck dφ + zkck dz dz
k ) 1,2
(5a)
Since co-ions do not adsorb, Γ3 ) 0 and the boundary condition for the co-ions at z ) 0 reads
(
0 ) D3
)
dc3 dφ + z3c3 dz dz
(5b)
Since the solution as a whole is electroneutral, the electric field at the surface can be related to the surface charge density by Gauss’s law:
( ) dφ dz
)-
e2
2
∑ zkΓk
(6)
kBTk)1
z)0
We assume that during the process of adsorption the subsurface and the surface are in local thermodynamic equilibrium. Therefore, the relationships between the surface excesses, Γk, and the subsurface concentrations, ck,s, are given by the adsorption isotherms, Γk (Γi*k, ci,s). A compendium of useful adsorption isotherms for the surface-active ions and counterions, and of equations of state for different models, can be found in Kralchevsky et al.20 The system of equations ((3)-(6)) together with the appropriate adsorption isotherms for ions 1 and 2 describe the diffusion problem in the OFC. To solve the problem, it is convenient first to nondimensionalize the above equations. We introduce the following dimensionless variables,
zj ) z/Lel, dk ) Dk/D, cjk ) ck/ct, γ )
Γk Lelct
(7)
where the Debye screening length Lel ) 1/κ )
x(kBT)/(2e2ct) characterizes the equilibrium thickness
of the electrostatic double layer and ct ) c1,∞ + c3,∞ is the total electrolyte concentration. With the definitions in (7), eqs 3-6 transform into
(
dcjk d dcjk dφ ) dk + zkcjk -2δ2zj dzj dzj dzj dzj
)
k ) 1,2,3
(8)
(20) Kralchevsky, P. A.; Danov, K. D.; Broze, G.; Mehreteab, A. Langmuir 1999, 15, 2351.
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Langmuir, Vol. 20, No. 11, 2004 4439
d2φ 1 ) (cj2 - cj1 - cj3) dzj2 2
(
2δ2γk ) dk
dcjk dφ + zkcjk dzj dzj
(
0 ) d3
)
at zj ) 0, k ) 1,2
dcj3 dφ + cj3 dzj dzj
)
at zj ) 0
dφ 1 ) (γ - γ1) at zj ) 0 dzj 2 2
(9)
(10a) (10b) (11)
In eqs 8 and 10 the parameter δ ) Lel/Ldif, where Ldif ) x2D/θ, gives the ratio of the thicknesses of the electrostatic and diffusion layers. For a low molecular weight surfactant adsorbing at the surface of the OFC, typical parameters are D ∼ 5 × 10-10 m2 s-1, θ ∼ 10 s-1, 1/κ ∼ 10-8 m, and δ2 ∼ 10-6 , 1. This scaling allows us to split the problem into two regions: (a) outer region, zj g zjλ: solution is charge neutral, the electrostatic potential φ f 0, and the distribution of ions is determined by convection and diffusion; (b) inner region, zj e zjλ, convection is negligible compared to electro-migration and the distribution of ions is governed by the electrostatic interactions. The boundary between the two regions is the edge of the electric double layer (EDL). The subscript λ indicates that the values of the parameters are taken at that boundary. (a) Outer Region. In the outer region (zj g zjλ) we set the electric potential φ ) 0. The Poisson equation then reduces to a simple statement of charge neutrality.
cj2 ) cj1 + cj3 for zj g zjλ
(12)
It is not strictly correct to state that the potential φ ) 0 in the outer region. Since the counterions will, in general, have a higher diffusion coefficient than the surfactant ions, a space charge will develop and a weak electric field will be set up that accelerates the surfactant ions and retards the counterions until the flux of the two ions toward the surface is identical. This migration field can be neglected if the self-diffusion coefficients dk are replaced by an effective diffusion coefficient deff.21 Equation 8 then takes the following form in the outer region:
( )
dcjk d dcjk ) deff -2zj dzj dzj dzj
cjk ) cjk,∞ - (cjk,∞ - cjk,λ)
erfc(zj/x deff) erfc(zjλ/xdeff)
k ) 1,2,3
(15)
where cjk,λ are the concentrations at the boundary between the diffusion and electrostatic double layers, z ) zλ. (b) Inner Region: zj e zj λ. The convective term on the left-hand side of eq 8 equals the difference between the diffusion and migration terms on the right-hand side. Since δ2 , 1, the convective term is much smaller than the diffusion and migration terms and can be neglected when computing the ion distributions (but not when calculating the mass balance since the mass balance is determined by the small difference between two large fluxes, see eq 20.) Integrating eq 8 subject to the boundary condition (10) then yields
(
2δ2γk ) dk
dcjk dφ + zkcjk dzj dzj
)
k ) 1,2,3
(16)
throughout the inner region. Even though δ2 , 1, one cannot automatically neglect the left-hand side of eq 16 for the surfactant ions because γ1 is typically quite large, O(102-103), and both cj1 and (dcj1/dzj) may be small near z ) 0 if the surface potential is several times kBT. If one can neglect the flux on the left-hand side of eq 16 (and we will justify this assumption later), the solution to eqs 9-11, together with the adsorption isotherm, is equivalent to calculating the equilibrium ion and potential distribution within the EDL. The ions then obey the Boltzmann distribution:
cjk ) cjk,λ exp(-zkφ) k ) 1,2,3
(17)
From eq 17 and the boundary condition φ f 0 and dφ/dz f 0 at z ) zλ, the distribution of the electric potential within the inner region takes the usual Gouy-Chapman form,
tanh
()
()
φs φ ) tanh exp(-zjxcj2,λ) 4 4
(18)
and the electroneutrality condition acquires the form,
k ) 1,2,3
(13)
For ionic solutions without added salt the effective diffusion coefficient is exactly deff ) 2d1d2/(d1 + d2).21 In the general case with added background electrolyte, no simple expression for deff exists. However, in the case where the added electrolyte is in large excess (c2, c3 . c1), the co-ion and counterion distributions respond almost instantaneously to maintain charge neutrality and the migration field is consequently extremely weak. In this case, one can set deff ) d1 for the surfactant ion in eq 13 and treat the co-ion and counterion concentrations as constant at their bulk values throughout the outer region. The boundary conditions for concentrations of the different species at infinity are
cj1,∞ )
The solutions of eq 13, together with the boundary conditions (14), are as follows,
c1,∞ ) η, cj2,∞ ) 1, cj3,∞ ) 1 - η ct
(14)
(21) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; John Wiley: New York, 1993; p 45.
γ1 - γ2 ) 4xcj2,λsinh
() φs 2
(19)
If we integrate eq 8 from c ) 0 to ∞ (now retaining the convective term) and use the boundary condition in eq 10, we arrive at an expression for conservation of mass of the different ions along the cylinder:
γk +
∫0zj
λ
(cjk - cjk,∞) dzj +
∫zj∞ (cjk - cjk,∞) dzj ) 0
1 δ
λ
k ) 1,2,3 (20)
The first term in eq 20 is the amount of ions adsorbed at the surface, the second is the amount distributed within the EDL, and the third is the quantity within the diffusion layer. If we specify the adsorption isotherms for surfaceactive ions and counterions, the system of equations (15), (17), (18), (19), and (20) completely describes the problem for the surface excesses Γ1 and Γ2 for a given surface expansion rate θ. In the particular case of the solution of a nonionic surfactant, the second term in eq 20 is zero as the boundary
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between the diffusion layer and EDL coincides with the subsurface: zjλ ) 0. Then the distribution (15) for the surface-active ion (component “1”) together with the mass balance (20) gives the solution of the problem obtained previously.15,16 In dimensional form it reads:
Γ)
(c x2D πθ
∞
- c s)
(21)
3. Adsorption Isotherms The mathematical model described in the previous section is valid for every type of adsorption isotherm. The simplest isotherms, such as Henry and Langmuir, do not account for the interactions between the adsorbed species. These isotherms are rarely adequate for describing ionic surfactants. Consequently, we will use a thermodynamic model for adsorption of ionic surfactants that includes lateral interactions between surfactant molecules and accounts for counterion binding. Kralchevesky et al.20 have argued that the most appropriate isotherms are the van der Waals isotherm for the surface-active ions and the Stern isotherm for counterions. The corresponding relationships between subsurface concentration and surface excess are20
(1 + KStc2,s)K1c1,s )
(
)
Γ1 Γ1 2βΓ1 exp Γ∞ - Γ1 Γ∞ - Γ1 kBT
(22) KStc2,s Γ2 ) Γ1 1 + KStc2,s
(23)
where Γ∞ is the saturation adsorption, the constant β is related to the energy of interaction between two adsorbed molecules, K1 accounts for the standard free energy of adsorption of surface-active ions, and KSt is the Stern constant for the surfactant. To determine the four unknown parameters unambiguously, it is highly desirable to fit simultaneously adsorption isotherms of the surfactant in the absence and in the presence of added electrolyte. Equations 22 and 23 close the system of equations (15) and (17)-(20) for the determination of the dynamic surface excess in the overflowing cylinder, provided that the surface expansion rate, θ, is known. The set of equations at r ) 0 does not alone allow θ to be determined a priori. In Figure 2 we plot experimental values for the equilibrium surface tension as a function of bulk surfactant concentration for C16TAB, C14TAB, C12TAB, C12TAB + 0.2 M NaBr, and the fluorinated surfactant di-CF4. The solid lines represent the best fits to the van der Waals model. The equation of state for the van der Waals model is
{(
)
βΓ12 Γ∞Γ1 + σ ) σ0 - kBT Γ∞ - Γ1 kBT
( ( ) )}
φs 8c2,∞ cosh -1 κ 2
(24)
The last term on the right in eq 24 represents the contribution of the diffuse part of the electric double layer to the surface tension in the case of a symmetric 1:1 electrolyte. The best-fit parameters of the isotherms in eqs 22 and 23 are listed in Table 1. These parameters are valid for the surfactant systems with or without added electrolyte.
Figure 2. Surfactant adsorption isotherms: C16TAB (0), C14TAB (2), C12TAB(O), C12TAB + 0.2 M NaBr (b), di-CF4 (3). Solid lines represent best fits to the van der Waals adsorption isotherm. Data for C16TAB are from ref 15, for C12TAB from ref 22, and for di-CF4 from ref 23. Data for C14TAB were obtained by the du Nou¨y ring method as described in ref 24. Table 1. Parameters of Fits of the van der Waals Adsorption Isotherm to the Data in Figure 1 system
K1 (m3 mol-1)
KSt (m3 mol-1)
Γ∞ (10-6 mol m-2)
2βΓ∞/kBT
C16TAB C14TAB C12TAB di-CF4
4.87 × 103 8.13 × 102 33.20 4.59 × 105
7.45 × 10-3 2.40 × 10-3 2.42 × 10-3 1.81 × 10-2
4.2 4.5 4.4 3.3
1.0 1.0 2.1 -4.8
Although we have written all the mass-transport equations and adsorption isotherms in terms of concentrations, for solutions with high ionic strength, one has to replace concentrations, ci, with activities, ai. For solutions of C14TAB, C12TAB, and C12TAB + 0.2 M NaBr the ionic strength is sufficiently high that activities must be used in fitting the surface tension data. The activities are related to the bulk ion concentrations, ci,∞, through the relationship
ai,∞ ) γ (ci,∞
(25)
where the activity coefficient γ ( was calculated through the semiempirical formula25
log γ( ) -
A|z+z-|xI 1 + BRixI
+ bI
(26)
z+ and z- are the valences of the cations and anions, I ) 0.5∑izi2((ci,∞/co)) is the total ionic strength of the solution, co is the standard state concentration (1 mol dm-3 to conform with the conventions of Robinson and Stokes, see below), Ri is the sum of the ionic radii (though in practice a fitting parameter), and A, B, and b are parameters whose values have been tabulated by Robinson and Stokes25b (see Appendix 7.1 therein). For example, for NaCl at 298 K, A ) 0.5115, BRi ) 1.32, and b ) 0.055. (22) Battal, T.; Shearman, G. C.; Valkovska, D.; Bain, C. D.; Darton, R. C.; Eastoe, J. Langmuir 2003, 19, 1244. (23) Eastoe, J.; Rankin, A.; Wat, R.; Bain, C. D.; Styrkas, D.; Penfold, J. Langmuir 2003, 19, 7734. (24) Simister, E. A.; Thomas, R. K.; Penfold, J.; Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A. J. Phys. Chem. 1992, 96, 1383. (25) (a) Debye, P.; Huckel, E. Phys. Z. 1923, 24, 185. (b) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth: London, 1959.
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Figure 3. Surface expansion rate, θ, as a function of surfactant concentration for (a) C16TAB (0), C14TAB (3), and di-CF4 (b); (b) C12TAB + 0.1 M NaBr (b), C14TAB + 0.1 M NaBr (4), and C16TAB + 0.1 M NaBr (9).
Figure 4. Surface expansion rate, θ, as a function of concentration of NaBr for C12TAB at concentration c1,∞ ) 0.6 mM.
4. Comparison of Experimental and Calculated Surface Excesses for Ionic Surfactants below the cmc To compare the theoretical calculations with the experimental data, we need first to measure the surface expansion rate, θ , and the dynamic surface excess, Γ1, as a function of surfactant and salt concentrations, c1,∞ and c3,∞. The surface expansion rate at the center of the OFC was determined from the gradient of the surface velocity, υr(r), at r ) 0, measured by laser Doppler velocimetry.12 Experimental values of θ are shown in Figure 3a for C14TAB, C16TAB, and di-CF4 in the absence of salt, in Figure 3b for the CnTAB (n ) 12, 14, 16) with added 0.1 M NaBr, and in Figure 4 for C12TAB as a function of concentration of NaBr for a fixed value of c1,∞ of 0.6 mM. These graphs clearly show the large surface accelerations that can occur in the presence of surfactant in the solution. As discussed above, these surface expansion rates are a necessary input into the mathematical model for calculating the surface excess, Γ1. The dynamic surface excess at the center of the OFC, Γ1(r ) 0), was deduced for the CTAB family of surfactants from ellipsometric measurements.12 The coefficient of ellipticity was related to the surface excess through calibration curves constructed from independent neutron reflection and surface tensiometry experiments.22 For diCF4, ellipsometric measurements cannot easily be used to determine Γ since the refractive index contrast between
the surfactant and water is too small. Consequently, for this surfactant we have only drawn comparisons with concentrations at which neutron reflection experiments were performed.23 Detailed explanations of the experimental procedure for measuring Γ have been published previously.12,14,22,23 The experimental values of Γ1(r ) 0) are shown in Figures 5 and 6. Theoretical values for Γ1(r ) 0) were calculated from the adsorption isotherms in Table 1 and the surface expansion data in Figures 3 and 4 with no adjustable parameters. The diffusion coefficients used in these calculations are listed in Table 2. Smooth curves were drawn through the theoretical values and are shown as lines in Figures 5 and 6. Very good agreement between theory and experiment is achieved for all the surfactants over the whole range of concentration. The biggest systematic discrepancy was observed for C12TAB at high salt concentrations (Figure 6), but even here the maximum deviation is about 6% of the surface excess, which lies within the experimental error in the value of Γ∞ determined from the adsorption isotherm. We note, that in our previous analysis of the adsorption kinetics of C16TAB in the OFC,15 we assumed a simpler Langmuir isotherm and concluded that adsorption was not diffusion-controlled at low surface concentrations. The good agreement with a diffusion-controlled model for the improved isotherm employed here demonstrates the importance of reliable adsorption isotherms for the interpretation of kinetic data. We return now to justifying the neglect of the term 2δ2γk in eq 16. For the co-ion, γ3 ) 0 and the net flux automatically vanishes. For the counterion, the concentration and concentration gradients are (except at z ) zλ) much higher in the EDL than in the diffusion layer for any appreciable amount of surfactant adsorption. Consequently, the deviation in the counterion distribution necessary to create a net flux of 2δ2γ2 is small. If the two opposing fluxes of the surface-active ion, due to diffusion and migration, on the right-hand side of eq 16, are large compared to the net flux, 2δ2γ1, then the whole of the electrical double layer will be close to thermodynamic equilibrium and the Γk are determined from ck,λ by the equilibrium adsorption isotherm, as discussed above. To verify this point, we calculated the ratio of the diffusive and convective fluxes, [dk(dcjk/dzj)]/2δ2γ), for the range of surfactant concentrations studied for C16TAB without added salt. (The presence of salt lowers the surface potential and hence reduces the magnitude of any electrostatic barrier to adsorption.) Representative curves
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Valkovska et al.
Figure 5. Surface excess in the center of the cylinder as a function of surfactant concentration: (a) C16TAB (2), C14TAB (b), and C12TAB + 0.1 M NaBr (O); (b) di-CF4. The lines represent theoretical predictions.
Figure 6. Influence of the salt on the adsorption of surfaceactive ions, Γ1(0), at the center of the OFC for 0.6 mM C12TAB: (b) experimental values; (-) theoretical calculations. Table 2. Diffusion Coefficients in Water at 298 K for Surfactants D Is Calculated from the Model of Wilke and Chang,26 for Ions D Is Taken from Ref 27 surfactant system
D (10-10 m2 s-1)
Deff (10-10 m2 s-1)
C16TAB C14TAB C12TAB di-CF4 Na Br
4.5 4.3 4.6 4.4 13.3 20.8
7.4 7.1 7.5 6.6
at two surfactant concentrations (c1,∞ ) 0.46 and 0.92 mM) are plotted in Figure 7. For both concentrations, the convective flux is much smaller than the diffusion and migration fluxes within the whole EDL. The ratio of diffusive to convective fluxes is lowest at the surface, but even here the convective flux is an order of magnitude lower than the diffusive flux for the bulk concentration, c1,∞ ) 0.46 mM, where the surface potential is close to its maximum value (and hence where there is the greatest chance for an electrostatic barrier to provide a kinetic limitation to surfactant adsorption). The calculations in Figure 7 show that the potential distribution within the double layer does not create an electrostatic barrier to adsorption under the conditions studied in this paper. Therefore, the assumption of diffusion-controlled adsorption holds for all the surfactants studied here. The reason for the absence of an electrostatic
Figure 7. Comparison of the diffusive and convective fluxes in the electric double layer. The ratio [dk(dcjk/dzj)]/2δ2γ is calculated for two concentrations of C16TAB without salt: c1,∞ ) 0.46 mM (solid lines) and c1,∞ ) 0.92 mM (dash-dot lines).
barrier is that counterion binding limits the rise in the surface potential with surface excess of the surfactant. As the surface potential starts to build up, so does the subsurface concentration of counterions, which leads to the adsorption of the counterions to the monolayer and a suppression of the growth of net surface charge. 5. Micellar Solutions: Diffusion with Infinitely Fast Micellar Breakdown When the bulk concentration of surfactant is above the critical micelle concentration, both micelles and monomers are present in solution. The mass-transport equations for micelles and monomers are coupled by a source term that takes account of the interconversion of micelles and monomers. According to the model of Aniansson and Wall,28 the demicellization reaction occurs in two steps: a fast process, on the time scale of microseconds, in which single molecules are exchanged between the micelle and monomers in solution, and a slow one, on the time scale (26) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264. (27) Atkins, P. W. Physical Chemistry, 7th ed.; OUP: Oxford, 2002; Table 24.8. (28) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1974, 78, 1024; J. Phys. Chem. 1975, 79, 857.
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of milliseconds and longer, in which the micelles totally disintegrate into monomers. In this section, we analyze the effect of micelles on the steady-state surface excess at the expanding surface of the OFC in the limiting case where micelle breakdown is very fast. On the time scale provided by the OFC, θ -1 ∼ 0.1-1 s, this assumption is likely to apply for many simple surfactant systems. Furthermore, we will treat the cmc as a sharply determined transition point so that below the cmc no micelles exist and above the cmc the monomer concentration is constant. With these assumptions, equilibrium between micelles and monomers is established instantaneously and diffusion of the surfactant is governed solely by the micelles when the local concentration is higher than the cmc and solely by monomers below the cmc. As in section 2, we will restrict ourselves to the case of local equilibrium between the surface and subsurface (no kinetic barriers). To simplify the notation, we will not include transport through the double layer explicitly but will treat the adsorption of ionic surfactants in the same way as a nonionic surfactant by setting the concentration at the edge of the diffusion layer (cλ) to be the subsurface concentration, cs. This simplification does not reduce the generality of the model so long as δ , 1 and the appropriate adsorption isotherm is used. The mean concentration of micelles in the bulk, cm∞, is given by
cm∞ )
ct - ccmc m
(27)
where ct is the total bulk concentration and m is the mean number of monomers within a micelle (the aggregation number). The concentration of monomer in the bulk is ccmc. For infinitely fast micellar breakdown, the diffusion equations for monomers and micelles in the center of the OFC take the form
dc d2c D 2 + zθ ) 0 dz dz dcm )0 Dm 2 + zθ dz dz
|
dc )D h dz
(x ) (x ) (x ) (x ) erf z
θ 2D
erf h
θ 2D
c - cs ) (ccmc - cs)
[
erfc z
θ 2Dm
erfc h
θ 2Dm
(31a)
]
(31b)
The fluxes of monomers and micelles at z ) h must match according to eq 29, so from (31) we have
ct - ccmc ) ccmc - cs
x
( ) (x ) (x ) ( ) θh2 2D
exp -
D ‚ Dm erf h
θ 2D
erfc h
‚
θ 2Dm
θh2 exp 2Dm
(32)
The flux of the monomers to the surface at z ) 0 is h
(29)
1. At z ) 0, (30a)
|
(30d)
The concentration profiles in the diffusion boundary layer adjacent to the surface are shown schematically in Figure 8. The solutions of the diffusion equations (28) together with the boundary conditions (30) are
(28b)
The boundary conditions to be used in solving the diffusion equations are as follows.
c ) cs
cm ) cm∞
cm ) cm∞ 1 -
where D is the diffusion coefficient of monomers, Dm the diffusion coefficient of micelles, c the concentration of monomers, and cm the concentration of micelles. At a distance h from the surface the total concentration passes through the cmc. This plane marks the transition from micellar diffusion to monomer diffusion. At z ) h, the monomer and micelle profiles have a finite slope and continuity requires that their diffusion fluxes are equal:
|
3. At z ) ∞,
(28a)
d2cm
dcm mDm dz
Figure 8. Schematic diagram of the concentration profiles of monomers and micelles in the diffusion boundary layer of an expanding surface.
dc θΓ ) D dz 0
(30b)
c ) ccmc; cm ) 0
(30c)
2. At z ) h,
x [ ( x )]
dc θΓ ) D |0 ) (ccmc - cs) dz
2θD erf h π
θ 2D
-1
(33)
Equations 32 and 33 are implicit relations, from which we can calculate h and Γ if we know the surfactant adsorption isotherm, Γ(cs). The error function in eq 33 has values between zero and unity, so the presence of micelles enhances the flux of surfactant above what it would be for a solution of monomers at the cmc. On the other hand, since micelles diffuse more slowly than monomers on account of their greater size, the adsorption flux is reduced below the value that would be observed if all the surfactant were present
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Figure 9. Surface excess in the center of the OFC, Γ(0), as a function of surfactant concentration for (a) C14TAB + 0.1 M NaBr and (b) C16TAB + 0.1 M NaBr. Solid lines are theoretical predictions for a model that assumes infinitely fast micellar breakdown.
as monomers. If we define a pseudo-coefficient D*(ct) by (compare eq 21)
Γ ) (ct - cs)
x2D* πθ
(34)
then (from (33) and (34))
D* ) D
[(
1+
) ( x )]
ct - ccmc erf h ccmc - cs
θ 2D
-2
(35)
When ct e ccmc, D* ) D and (from eq 32) when ct . ccmc, D* f Dm, as one would intuitively expect. 6. Comparison of Experimental and Calculated Surface Excesses for Ionic Surfactants above the cmc Ellipsometric measurements for C14TAB and C16TAB in the OFC in the presence of 0.1 M NaBr show that the surface excess is below its saturation value (and hence that cs < ccmc) even at bulk concentrations up to 10 × cmc. This conclusion is supported by the observation of Marangoni effects (Figure 3b), which would not be observed if the surface were saturated. To apply the model of section 5, we have employed the Frumkin adsorption:
Kcs )
Γ 2βΓ exp Γ∞ - Γ kT
(
)
(36)
For C14TAB + 0.1 M NaBr, the best-fit parameters of the isotherm are Γ∞ ) 3.5 × 10-6 mol m-2, K ) 263 m3 mol-1, and β ) 0; for C16TAB + 0.1 M NaBr, they are Γ∞ ) 3.5 × 10-6 mol m-2, K ) 6.20 × 10-3 m3 mol-1, and 2βΓ∞/kT ) -1.8. In excess electrolyte, the simpler Frumkin isotherm provides an equally good fit to the more complicated Stern-van der Waals isotherm used in section 4 since the high ionic strength screens the electrostatic interactions effectively. The effective diffusion coefficient of monomers of C14TAB and C16TAB in the presence of 0.1 M NaBr are simply the self-diffusion coefficients of the surface-active ions (see Table 2) as explained in section 2. We estimate the diffusion coefficient of the micelles, Dm ) 1.0 × 10-10 m2s-1, from the simple relationship, Dm ) D/3xm, and using a value for the mean aggregation number of C16TAB, m ) 100. (Lianos and Zana29 reported the value m ) 89 for the salt-free case, but m is likely to (29) Lianos, P.; Zana, R. J. Colloid Interface Sci. 1981, 84, 100.
increase in the presence of added electrolyte due to screening of the lower electrostatic repulsion between the charged headgroups. The value of Dm is insensitive to the exact value of m.) In Figure 9, we plot the experimental values of the dynamic surface excess, deduced from ellipsometric measurements as described above, together with values of Γ1(r ) 0) calculated from the model in section 5 and the surface expansion rates in Figure 3b. The arrows on the graphs show the value of the cmc. For C14TAB there is excellent agreement between theory and experiment. The small deviations almost certainly arise from the NR measurements that are used to generate a calibration curve for the coefficient of ellipticity against Γ. For C16TAB the experimental values of Γ1 are lower than the theoretical predictions. The deviation is too large to be explained by experimental errors in the determination of the surface expansion rate and surface excess. The limit of fast micellar breakdown gives the highest possible values for the surface excess; the discrepancy between theory and experiment for C16TAB points to a breakdown of the assumption that the rate of micellar breakdown is fast on the time scale of the OFC. To confirm that micellar breakdown kinetics, rather than some other form of kinetic barrier, is responsible for the deviations from diffusion-controlled adsorption will require independent measurements of micelle breakdown rates and the development of a (numerical) adsorption model for finite rates of micelle breakdown. Conclusion We have developed a theoretical model for adsorption of ionic surfactants at an expanding liquid surface under steady-state conditions. The model is simplified by the fact that the EDL is much thinner than the diffusion layer, which enables us to separate the diffusion problem into two sets of equations that can be solved numerically. In this model, we have assumed that the flux of surfactant to the surface does not significantly perturb the ion distribution in the electrostatic double layer from its profile at local equilibrium. The assumption is supported both by the good agreement between experiment and theory and by model calculations showing that the convective flux is much lower than the diffusion and migration fluxes in EDL. The agreement between theory and experiment below the cmc for all the surfactant systems investigated shows that the adsorption of CnTAB (n ) 12, 14, 16) and di-CF4 is diffusion-controlled on the time scale of the overflowing cylinder (0.1-1 s).
Adsorption of Ionic Surfactants
We stress the importance of having accurate equilibrium isotherms for modeling adsorption kinetics. The isotherm we employ accounts for both lateral interactions and counterion binding. Counterion binding serves to limit the rise in the surface potential with surface excess and hence to constrain the height of the electrostatic barrier to adsorption. An electrostatic barrier is unlikely to affect the kinetics of adsorption except at appreciably higher strain rates, O(102 s-1), than those operating in an OFC. For surfactant concentrations above the cmc, we have developed an adsorption model for the limiting case of fast micellar breakdown. This model yields excellent agreement with experimental data for surfactant systems C14TAB + 0.1 M NaBr, but overpredicts the surface excess for C16TAB + 0.1 M NaBr. To obtain quantitative agreement for the latter system, a more general model incorporating a finite micellar breakdown rate is required. Such a model is currently under development at Oxford. Note Upon review, two useful observations were made regarding the equilibrium isotherms presented in Table 1. First, the values of R ) Γ-1 ∞ , which are interpreted in a van der Waals model as the excluded area per molecule,
Langmuir, Vol. 20, No. 11, 2004 4445
give areas per molecule for the CTABs of 37-40 Å2, which are very close to the cross-sectional area of 38 Å2 of the trimethylammonium headgroup. Second, the equilibrium constant K1 can be written as20 K1 ) Rδ exp(E/kT) where δ is the length of the molecule and E is the adsorption energy. From this equation, we can derive values of E/kT ) 11.2, 14.3, and 15.9 for C12TAB, C14TAB, and C16TAB, respectively. According to Traube’s Rule,30 E/kT varies by ln(3) for each additional CH2 group in the molecule. We observe that E/kT increases by 4.3 ln(3) as the chain length is increased from 12 to 16 carbons, in very good agreement with Traube’s Rule. Acknowledgment. We thank Dr. D. Styrkas, for obtaining the surface expansion rate data for di-CF4, Dr. R. K. Thomas for supplying surface tension data for C14TAB, R. Campbell and M. Sekine for obtaining ellipsometric data on C14TAB + NaBr, and Dr. C. Breward and M. Weiss for helpful discussions. This work was supported by the EPSRC under Grants GR/M83797 and GR/M83780. LA035739B (30) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990; p 96.
