Langmuir 2005, 21, 2235-2241
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Role of Desorption Kinetics in Determining Marangoni Flows Generated by Using Electrochemical Methods and Redox-Active Surfactants Guiyu Bai, Michael D. Graham,* and Nicholas L. Abbott* Department of Chemical and Biological Engineering, University of WisconsinsMadison, Madison, Wisconsin 53706 Received July 13, 2004. In Final Form: November 17, 2004 We report quantitative measurements of Marangoni flows generated at the surfaces of aqueous solutions by using water-soluble redox-active surfactants in combination with electrochemical methods. These measurements are interpreted within the framework of a simple model that is based on lubrication theory and the proposition that the kinetics of the desorption of redox-active surfactants from the surfaces of aqueous solutions plays a central role in determining the strength of the Marangoni flow. The model predicts that the leading edge velocity of the Marangoni flow will decay exponentially with time and that the rate constant for the decay of the velocity can yield an estimate of the surfactant desorption rate constant. Good agreement between theory and experiments was found. By interpreting experimental measurements of electrochemically generated Marangoni flows within the framework of the model, we conclude that the desorption rate constant of the redox-active surfactant Fc(CH2)11-N+(CH3)3Br-, where Fc is ferrocene, is 0.07 s-1. We also conclude that the ionic strength of the aqueous solution has little effect on the desorption rate constant of the ferrocenyl surfactant.
1. Introduction Recent studies have demonstrated that ferrocenecontaining surfactants of the type shown in Figure 1 can be combined with electrochemical methods to form the basis of an experimental system that permits spatial and temporal control over the surface tensions of aqueous solutions.1-4 This capability results from the fact that the oxidation of ferrocene (an electrically neutral moiety) to ferrocenium (a cation) within the surfactant leads to large changes in the surface activity of these molecules. For example, the equilibrium surface tension of an aqueous solution (0.1 M Li2SO4) containing 0.1 mM of the ferrocenyl surfactant (11-ferrocenylundecyl)trimethylammonium bromide (Fc(CH2)11-N+(CH3)3Br-) changes from 49 to 72 mN/m upon oxidation of the surfactant.1 The change in surface tension is reversible. Bennett et al. first reported that application of a reducing electrical potential of -0.3 V (vs saturated calomel electrode (SCE)) to a platinum working electrode immersed through the surface of an aqueous solution of oxidized ferrocenyl surfactant (II2+ in Figure 1) causes the surface of the fluid to move radially outward and away from the working electrode.4 This flow was consistent with the effects of a spatially localized increase in surface pressure near the working electrode caused by reduction of the surface-inactive molecule II2+ to the surface-active molecule II+ at the electrode. The Marangoni flow generated near the electrode was also observed to decay in strength with distance away from the electrode, presumably because surfactant (II+) convected away from the electrode desorbed from the interface * To whom correspondence should be addressed. E-mail:
[email protected] (M.D.G.);
[email protected] (N.L.A.). (1) Gallardo, B. S.; Hwa, M. J.; Abbott, N. L. Langmuir 1995, 11 (11), 4209-4212. (2) Gallardo, B. S.; Metcalfe, K. L.; Abbott, N. L. Langmuir 1996, 12 (17), 4116-4124. (3) Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig, V. S.; Shah, R. R.; Abbott, N. L. Science 1999, 283 (5398), 57-60. (4) Bennett, D. E.; Gallardo, B. S.; Abbott, N. L. J. Am. Chem. Soc. 1996, 118 (27), 6499-6505.
Figure 1. Molecular structure of FTMA (II+) and oxidized FTMA (II2+).
during equilibration with the bulk solution (largely free of II+) distant from the electrodes. In this paper, we report a study that sought to quantify the strengths of Marangoni flows that are driven by electrochemical transformations of redox-active surfactants near electrodes. In addition, we aimed to develop an understanding of the factors that control the Marangoni flows driven by redox-active surfaces by comparing the experimental measurement to a simple model of Marangoni flows based on lubrication theory. Our study was structured around the hypothesis that the kinetics of desorption of the ferrocenyl surfactants plays a major role in determining the rate of decay of the Marangoni flow away from electrodes immersed into solutions of oxidized surfactant. By comparing the quantitative measurements of the Marangoni flows to the model based on lubrication theory, we sought to determine if it was possible to make
10.1021/la048238e CCC: $30.25 © 2005 American Chemical Society Published on Web 02/18/2005
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Figure 2. Schematic illustration of the experimental setup. A reference electrode (not shown) was located in a corner close to the counter electrode.
estimates of the kinetic constant for the desorption of the redox-active surfactant from the surface of the aqueous solution. We note here that there are relatively few past studies of the desorption kinetics of ionic surfactants from aqueous interfaces.5-7 One past study has indirectly exploited Marangoni flows to rapidly and uniformly spread surfactant across a surface and thereby measure the desorption kinetics of an ionic surfactant from a liquidliquid interface.5 In this study, a drop of an aqueous solution of sodium dodecyl sulfate (SDS) was injected into an oil-water interface and the resulting Marangoni flow rapidly spread the SDS over the interface. Following cessation of the flow, the interfacial tension was monitored by using a Wilhelmy plate. In their model, mass transport from the interface was ignored and the LangmuirHinshelwood kinetic expression was used to describe the kinetics of adsorption and desorption of surfactant. Their measurements and analysis led to the conclusion that the rate constant for the desorption of SDS is around 0.02 s-1. Whereas past studies of the kinetics of the desorption of ionic surfactants are few, several reports on the desorption kinetics of nonionic surfactants do exist.6-9 Some of these studies were performed by compressing the interface of aqueous surfactant solutions.8,9 The decrease in surface area leads to overcrowding of surfactant at the interface, and the onset of desorption of surfactant as the interface re-equilibrates with the bulk solution. By using this method, the desorption of the nonionic surfactant C12E8 (octaethylene glycol mono n-dodecyl ether (C12H25(OCH2CH2)8OH)), 1-decanol, and diazinon were studied.8,9 These studies led to the conclusion that, under most experimental conditions investigated, the desorption processes were diffusion controlled. However, under some conditions, like elevated surfactant concentration, evidence of a mixed diffusion-kinetic mechanism was found. The remainder of this paper is structured as follows. First, we report a model for Marangoni flows generated by redox-active surfactants near electrodes. This description of the model is followed by a report of experimental measurements of the strengths of Marangoni flows generated by redox-active surfactants. Finally, a comparison of the experimental measurements and model (5) Shioi, A.; Nagaoka, R.; Sugiura, Y. J. Chem. Eng. Jpn. 2000, 33 (4), 679-683. (6) Chang, C. H.; Franses, E. I. Colloids Surf., A 1995, 100, 1-45. (7) Franses, E. I.; Basaran, O. A.; Chang, C. H. Curr. Opin. Colloid Interface Sci. 1996, 1 (2), 296-303. (8) Tsay, R. Y.; Lin, S. Y.; Lin, L. W.; Chen, S. I. Langmuir 1997, 13 (12), 3191-3197. (9) Lin, S. Y.; Tsay, R. Y.; Hwang, W. B. Colloids Surf., A 1996, 114, 131-141.
predictions is presented. From this comparison, we provide an estimate of the rate constant for the desorption of the ferrocenyl surfactant Fc(CH2)11-N+(CH3)3Br- from the surface of an aqueous solution. 2. Flow Geometry, Theory, and Simulations Figure 2 shows a schematic illustration of the experimental geometry. Initially, the bulk solution contains oxidized ferrocenyl surfactant (II2+). The oxidized surfactant is not surface active at the concentrations of ferrocenyl surfactant used in the experiments reported in this paper (0.1 mM), and thus, it is not present at the surface of the solution. By application of a reducing electrical potential at the working electrode at time t ) 0, reduced ferrocenyl surfactant (II+) is generated. Because the reduced surfactant is surface active at the concentrations generated near the working electrode, the reduced ferrocenyl surfactant exerts a surface pressure at the interface near the working electrode. Because the solution far from the electrode does not contain reduced ferrocenyl surfactant, a gradient in surface pressure extends away from the working electrode and thus drives a Marangoni flow. The experimental observations reported previously4 as well as those reported below reveal that the Marangoni flow weakens with increasing distance from the working electrode. A central goal of the work reported in this paper was to characterize the decay in the Marangoni flow away from the electrodes and to test the hypothesis that the rate of decay is related to the kinetics of desorption of surfactant from the surface. In the section below, we use lubrication theory to describe the concentration profile of the surfactant near the leading edge of the Marangoni flow as well as the rate of decay of the velocity of the Marangoni flow with increasing distance from the working electrode. The model developed below establishes a connection between these two quantities in the limit of low surface coverage. 2.1. Lubrication Theory. Away from the electrodes and side walls of the vessel containing the surfactant solution, the flow under consideration here is expected to be nearly unidirectional, as the thickness of the film of surfactant solution in the vessel does not vary rapidly with position. Such a situation is amenable to treatment with lubrication theory,10 allowing us to develop an analytical relationship between the fluid motion and the surfactant desorption kinetics. The coordinate system and geometry on which the model is based is shown in Figure 3. We assume quasisteady unidirectional flow, so the (10) Deen, W. M. Analysis of Transport Phenomena; Oxford University Press: New York, 1998.
Role of Desorption Kinetics in Marangoni Flows
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Figure 3. Schematic illustration of the leading edge of the Marangoni flow. The horizontal arrows indicate the velocity of the fluid motion.
Navier-Stokes equation reduces to a single equation for the x-component of the velocity, u:
and a “decay” period. In the acceleration period, which is typically the first several seconds, the flow near the working electrode is rapid and inhomogeneous. This heterogeneity is then smoothed quickly by Marangoni effects, after which point we consider the flow to be in the decay period. In this decay period and in the region near the surfactant “front” (i.e., the boundary between reduced and oxidized surfactant), we expect the concentration to be a function only of time and distance from the (moving) front. Thus, we seek a solution Γ(x, t) ) Γ(x - xf(t), t), where xf(t) is the position of the surfactant front. Equation 6 then becomes
∂Γ(x, t) ∂ + ((U - Uf)Γ) ) ra + rd ∂t ∂x
(7)
where µ is the viscosity and p is pressure. The boundary condition at the bottom of the liquid layer, y ) 0, is u ) 0. At the free surface, y ) h(x, t), the shear stress balance requires that
where Uf ) dxf/dt is the velocity of the front. Finally, we assume that the surface velocity, U, near the front is nearly equal to the front velocity, Uf. Equation 5 shows that this simplification is equivalent to assuming that the surface tension near the front varies linearly with position; this same assumption appears again below, so in section 2.3, we validate it with numerical simulations. With this final assumption, we arrive at the following simple expression (in the reference frame moving with the speed Uf):
du dγ ) dy dx
∂Γ ) ra + rd ∂t
0)-
2
∂p du +µ 2 ∂x dy
(1)
(2)
µ
where γ is the surface tension. The surface tension profile will be addressed in the following subsection. Finally, we impose a condition of no net flow
∫0h(x)u dy ) 0
∂ ∂x
(
)
γ ∂γ 3 ∂γ y2 - hy + u(x) ) 2µh ∂x 2 µ ∂x
(4)
At the free surface, y ) h, this reduces to a simple expression for the local free surface velocity, U:
U ≡ u(y ) h) )
h ∂γ 4µ ∂x
(5)
2.2. Desorption Kinetics and Marangoni Flow. In this subsection, we address the issue of the surfactant transport and combine it with the above results to develop an expression for the velocity of the leading edge of the moving surfactant front, Uf, as the surface tension gradient drives fluid away from the electrode. The species conservation equation for surfactant transport on an approximately flat interface is
∂ ∂2Γ ∂Γ + (UΓ) ) D 2 + ra + rd ∂t ∂x ∂x
Many kinetic expressions for surfactant adsorption and desorption have been proposed.6,7 Here, we use a simple Langmuir-Hinshelwood kinetic expression to describe the desorption kinetics:
(3)
This condition sets the local pressure gradient. Solving these equations yields that
(6)
where Γ(x, t) is the surfactant concentration on the surface of the solution, D is the surfactant diffusion coefficient, U is the free surface velocity given by eq 5, and ra and rd are the rates of adsorption and desorption, respectively. In our experiments, the Peclet number (Pe ) Ud/D) is on the order of 104 (using the estimates U ) 0.1 cm/s, d ) 0.2 cm, and D ) 10-6 cm2/s), so in the following, analysis diffusion will be neglected. The dynamics of Marangoni flows in our experiments can be divided into two periods: an “acceleration” period
(8)
(
ra ) KaCs(0, t) 1 -
)
Γ Γ∞
rd ) -KdΓ
(9)
where Γ∞ is the maximum interfacial surfactant concentration, Ka is the adsorption rate constant, Kd is the desorption rate constant, and Cs(0, t) is the sublayer concentration. In the experiments to be modeled in this paper, the solutions are electrochemically preoxidized and the sublayer concentration of reduced ferrocenyl surfactant, Cs(0, t), away from the electrode is negligible. Furthermore, as the front moves, fresh preoxidized solution is swept under the edge of the surfactant front. For these reasons, we expect the sublayer concentration at the front to be very small; therefore, we neglect ra, in which case eq 8 yields that
Γ(x, t) ) Γ0(x)e-Kdt
(10)
Under the assumption that the surfactant concentration varies linearly with position near the leading edge of the Marangoni flow (see section 2.3), we arrive at the expression
Γ(x, t) ) -K0(x - xf(t))e-Kdt
(11)
where -K0 is the slope of the concentration profile. We note here that a simple equation of state for low surfactant surface concentration that is consistent with eq 9 is
(
γ0 - γ ) -Γ∞RT ln 1 -
)
Γ ≈ RTΓ Γ∞
(12)
Finally, by combining eqs 5 and 11 with eq 12, we arrive
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at the prediction that the speed, Uf, of the leading edge of the Marangoni flow is given by
Uf ) K0e-Kdt
RTd 4µ
(13)
The significance of eq 13 lies in its prediction that a measurement of the time dependence of the leading edge velocity of the Marangoni flow can yield an estimate of the desorption rate constant of the surfactant from the interface. This prediction is founded on the assumption that the concentration profile of surfactant near the leading edge is linear with position. In the following section, we justify this assumption with a numerical simulation. We then use eq 13 to estimate the desorption rate constants for ferrocenyl surfactants. 2.3. Numerical Analysis of Surfactant Transport. To justify the assumption that the surfactant concentration profile is linear at the leading edge of the Marangoni flow, a numerical description of surfactant transport in one dimension at the surface of the liquid was developed by combining eqs 6 (with diffusion neglected), 5, and 12. In our numerical simulation, an upwind finite difference scheme and a forward Euler time integration were used.11 The following initial and boundary conditions were imposed: at the left boundary (the working electrode), the surfactant concentration was defined to be a constant, while, in the bulk of the domain, the initial surfactant concentration is zero. The desorption rate constant, Kd, was set to be 0.1 s-1, which is close to the desorption rate constants estimated from our experiments (see below). Figure 4 shows the evolution of the surface coverage, Γ/Γ∞, obtained using two initial conditions that corresponded to different, nonlinear concentration profiles. It reveals that both concentration profiles rapidly (in
