A Molecular-Thermodynamic Model for Gibbs Monolayers Formed

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Langmuir 1999, 15, 722-730

A Molecular-Thermodynamic Model for Gibbs Monolayers Formed from Redox-Active Surfactants at the Surfaces of Aqueous Solutions: Redox-Induced Changes in Surface Tension Nihal Aydogan, Benedict S. Gallardo, and Nicholas L. Abbott* Department of Chemical Engineering, University of Wisconsin, Madison, Wisconsin 53706 Received June 18, 1998. In Final Form: November 16, 1998 We report the development of a molecular-thermodynamic model for Gibbs monolayers formed from the redox-active surfactant (11-ferrocenylundecyl)trimethylammonium bromide (II+), or oxidized II+ (II2+), at the surfaces of aqueous solutions. This model provides an account of past experimental measurements (Gallardo, B. S.; Metcalfe, K. L.; Abbott, N. L. Langmuir 1996, 12, 4116-4124) which demonstrated electrochemical oxidation of II+ to II2+ to lead to large and reversible changes in the excess surface concentrations and surface tensions of aqueous solutions of this redox-active surfactant. The results of the model lead us to conclude that II+ assumes a looped conformation at the surfaces of aqueous solutions. This looped conformation lowers the surface tensions of aqueous solutions of II+ to ∼49 mN/m at a limiting surface area of 85 Å2/molecule (in 0.1 M Li2SO4). The underlying cause of the reduction in surface tension is not an electrostatic contribution to the surface pressure (as is the case with classical ionic surfactants) but rather an entropic contribution due to the constrained (looped) configuration of the surfactant at the surface of the solution (chain packing). At concentrations around the critical micelle concentration (CMC) of II+ (0.1 mM), oxidation of II+ to II2+ results in the desorption of surfactant from the surface of the solution and an increase in surface tension from 49 to 72 mN/m. The process of desorption is driven by an oxidation-induced decrease in the hydrophobic driving force for self-association of the surfactants as well as an electrostatic repulsion between adsorbed surfactants. In contrast, at concentrations of II+ that substantially exceed its CMC, oxidation of II+ to II2+ drives the disruption of micelles to monomers in the bulk solution, thus increasing the chemical potential and excess surface concentration of surfactant: the oxidation-induced increase in excess surface concentration of surfactant leads to a decrease in surface tension. These results, when combined, provide principles for the design of redox-active surfactants.

1. Introduction We recently reported electrochemical control of the oxidation state of the ferrocenyl moiety within the watersoluble, redox-active surfactant (11-ferrocenylundecyl)trimethylammonium bromide (Figure 1, II+) to lead to large and reversible changes in the surface tensions of aqueous solutions.1,2 For example, oxidation of 0.1 mM II+ to II2+ (in aqueous 0.1 M Li2SO4) caused the surface tension of the solution to increase from 49 to 72 mN/m (Figure 2). The increase in surface tension was reversed by reduction of II2+ back to II+. This capability is a useful one because it makes possible the “tuning” of the surface tensions of aqueous solutions through application of an external potential1,2 and permits electrochemical control of a variety of surfactant-induced phenomena including convection at the surface of aqueous solutions driven by gradients in surface tension3 as well as the wetting and spreading of aqueous solutions on surfaces.4 While these past studies have demonstrated the usefulness of redoxactive surfactants for active control of the interfacial properties of aqueous solutions, the balance of molecularlevel forces responsible for the redox-induced change in surface activity has not been reported. In fact, as described * To whom correspondence should be addressed. E-mail: abbott@ engr.wisc.edu. (1) Gallardo, B. S.; Hwa, M. J.; Abbott, N. L. Langmuir 1995, 11, 4209. (2) Gallardo, B. S.; Metcalfe, K. L.; Abbott, N. L. Langmuir 1996, 12, 4116. (3) Bennett, D. E.; Gallardo, B. S.; Abbott, N. L. J. Am. Chem. Soc. 1996, 118, 6495. (4) Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig V. S.; Shah, R. R.; Abbott, N. L. Science 1999, 283, 57.

Figure 1. Molecular structures of (11-ferrocenylundecyl)trimethylammonium bromide (II+) and oxidized II+ (II2+).

below, several features of the surface tension plots shown in Figure 2 were unexpected and have been hitherto unexplained. The goal of this paper is to present a molecular-thermodynamic model for Gibbs monolayers formed from II+ and II2+ that is capable of describing the essential features of the surface tension plot shown in

10.1021/la9807208 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/09/1999

Model for Gibbs Monolayers

Figure 2. Equilibrium surface tensions of aqueous solutions (0.1 M Li2SO4, pH 2, 25 °C) of ferrocenyl surfactants: experimental measurements1,2 for II+ (triangles) and II2+ (circles); values calculated using the model reported in this paper for II+ (dashed line) and II2+ (solid line).

Figure 2. We use this model to elucidate the balance of forces that controls the activity of these redox-active surfactants at the surfaces of aqueous solutions. In short, the model reported in this paper is used to address three principal questions that result from an inspection of the surface tension plots for II+ and II2+ shown in Figure 2. Question 1: What is the molecular-level mechanism by which II+ reduces the surface tensions of aqueous solutions to 49 mN/m? Application of the Gibbs adsorption equation (with swamping electrolyte) to the plot of surface tensions reported in Figure 2 reveals the surface tensions of solutions of II+ to be reduced to 49 mN/m at a surface area per molecule of II+ of 85 ( 5 Å2.1 In contrast, when dissolved in 0.1 M Li2SO4, the ionic surfactant dodecyltrimethylammonium bromide (DTAB) reduces the surface tension of an aqueous solution to only 71 mN/m at an area per molecule of DTAB of 85 ( 5 Å2.1 That is, the principal contribution to the surface pressure of II+ appears to differ substantially from that of DTAB (it is not electrostatic). In this paper, we seek to understand the dominant molecular-level contributions to the surface pressure of Gibbs monolayers formed from II+. Question 2: What oxidation-induced change in the balance of forces drives the desorption of II2+ from the surface of a solution containing a bulk surfactant concentration of 0.1 mM? Through the development of the model reported in this paper, we aimed to distinguish between three factors plausibly responsible for the desorption of II2+ from the surface of an aqueous solution containing 0.1 mM of II2+. First, oxidation of II+ to II2+ doubles the electrostatic charge on each surfactant molecule adsorbed to the surface of the solution: we hypothesized that the resulting electrostatic repulsion between molecules of II2+ could drive its desorption from the surface of the solution. Second, upon oxidation of II+ to II2+, the hydrophobic driving force5 for adsorption of surfactant to the surface of the solution may be reduced: whereas ferrocene partitions into the oil phase of a twophase system of water and oil, the ferrocenium ion partitions into the water phase.6,7 Third, because the (5) We use the term “hydrophobic driving force” in this paper to refer to contributions to the standard free energy of transfer of II+ or II2+ from dilute aqueous solution to a Gibbs monolayer that do not reflect electrostatic interactions between surfactant molecules and do not reflect changes in the configurational degrees of freedom (chain packing). (6) Georges, G.; Desmettre, S. Electrochim. Acta 1984, 29, 521. (7) Ohsawa, Y.; Aoyagui, S. J. Electroanal. Chem. 1982, 136, 353.

Langmuir, Vol. 15, No. 3, 1999 723

oxidized ferrocene group (ferrocenium cation) is likely hydrated by the aqueous phase, we hypothesized that the conformations accessible to the oxidized surfactant (II2+) might be constrained as compared to the reduced state of the surfactant (II+): the loss of configurational degrees of freedom could, we believed, lead to an entropically driven desorption of surfactant from the surface of the solution. Question 3: What oxidation-induced change in the balance of forces drives the adsorption of II2+ to the surfaces of aqueous solutions containing bulk surfactant concentrations of 10 mM or greater? Whereas the limiting area occupied by II+ at the surface of these aqueous solutions was estimated to be 85 ( 5 Å2/molecule (see above), the results in Figure 2 indicate that the area occupied by each molecule of II2+ is less than 65 ( 5 Å2/ molecule at concentrations of surfactant greater than 10 mM.8 This result is surprising because it suggests that oxidation of II+ to II2+, which results in an increase in the electrostatic repulsion between the molecules of surfactant (see Question 2 above), can drive adsorption of surfactant to the surfaces of solutions containing concentrations of surfactant greater than 10 mM. Through the development of the model reported in this paper, we sought to understand the cause of the oxidation-induced adsorption of the surfactant at high concentrations (>10 mM). Below we first outline our model for Gibbs monolayers formed from II+ and II2+. We demonstrate this model to be capable of capturing the essential features of the surface tension plots shown in Figure 2. We then use the model to answer the three questions posed above. Because the model we report is based on molecular-thermodynamic models of aqueous solutions of surfactants reported in the past, we refer the interested reader to these past papers for the details of our approach.9-12 Below we focus on theoretical developments that were necessary to adapt those models9-12 to the specific case of II+ and II2+. To our knowledge, a molecular-thermodynamic model of a redoxactive surfactant has not been reported in the past. 2. Theory 2.1. General Considerations. We evaluate the surface tensions of aqueous solutions of II+ and II2+ as the work done to expand the area of the surface of an aqueous solution10

() ∂g

∂a

)γ

(1)

T,P

where g is the Gibbs free energy per surfactant molecule within the surfactant layer adsorbed to the surface of the solution (evaluated using an isolated surfactant molecule in bulk solution without translation mobility as the reference state), a is the surface area occupied, on average, by each molecule within the adsorbed layer of surfactant, T is temperature, and P is pressure. To evaluate the surface tension of an aqueous solution containing a given bulk concentration of surfactant, the partial derivative in eq 1 must be performed at the value of a dictated by the diffusional equilibrium between the surfactant in the (8) In ref 2, we reported the smallest value of the surface area per molecule of II2+ to be 75 Å2/molecule. We now believe a more accurate estimate of the limiting surface area to be 65 ( 5 Å2/molecule. (9) (a) Zoeller, N. J.; Blankschtein, D. Ind. Eng. Chem. Res. 1995, 34, 4150. (b) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618 and references therein. (10) Ericksson, J. C.; Ljunggren, S. Colloid Surf. 1989, 38, 179. (11) Brasher, L. L.; Herrington, K. L.; Kaler, E. W. Langmuir 1995, 11, 4267. (12) Tanford, C. The Hydrophobic Effect, 2nd ed.; John Wiley and Sons: New York, 1989.

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adsorbed surface layer and bulk solution,

µbulk ) µsurface

(2)

where µbulk and µsurface are the chemical potentials of the surfactants in the bulk solution and at the surface of the solution, respectively. Thus, to evaluate the surface tension of aqueous solutions of II+ and II2+, we require models for both µbulk and µsurface. 2.2. Model for Chemical Potential of II+ or II2+ within a Gibbs Monolayer. We have modeled the chemical potentials of II+ and II2+ adsorbed at the surfaces of aqueous solutions by assuming these surfactants to form monolayers.1,2 The chemical potential is evaluated (see eq 11 below) from a model of the Gibbs free energy of the monolayer. The molecular-level contributions to the Gibbs free energy that we have included in our model are based on past models that have successfully described the properties of surfactant monolayers,9,10 bilayers (including vesicles),11 and micelles9,12 formed from classical ionic and nonionic surfactants. We consider the following contributions to the Gibbs free energy of a monolayer formed from II+ or II2+

g ) gel + gconf + ghyd + gcont + ghg

(3)

where gel is a contribution arising from electrostatic interactions between charged species, gconf is a contribution arising from the restricted number of conformations accessible to the surfactants confined within a monolayer, ghyd is a hydrophobic contribution arising from the removal of aliphatic chains and ferrocene from the bulk aqueous environment, gcont is the so-called contact free energy contribution (see below), and ghg is a contribution that is specific to the headgroup of the surfactant (also discussed below). Below we briefly sketch our evaluation of each of these contributions to the Gibbs free energy of a monolayer formed from II+ or II2+, with an emphasis on how our evaluation (caused by the presence of ferrocene attached to the end of the chain) differs from past descriptions of Gibbs monolayers of surfactants.9,10 Electrostatic Contribution. The electrostatic contribution to the free energy of the Gibbs monolayer is evaluated as the work performed to bring isolated charges dispersed in bulk aqueous electrolyte (0.1 M Li2SO4) to the surface of the solution at a density of charge corresponding to the excess charge density of the Gibbs monolayer. We evaluate this electrostatic contribution as10

gel ) eψ0 + achargeγel

(4)

where e is the charge of an electron, ψ0 is the surface potential, γel is the electrostatic contribution to the surface tension (see below), and acharge is the area occupied, on average, by each charge. Inspection of eq 4 reveals that gel can be considered to be composed of two parts: (i) the work performed to bring a charged headgroup to the surface with a potential of ψ0, and (ii) the work recovered by expanding the surface by an area a charge ) e/σ, where σ is the surface charge density, at constant surface pressure (to maintain constant surface potential). The electrostatic contribution to the surface tension (γel) is evaluated as13

γel )

∂ ∂ (g ) ) (a ∂a el ∂a

∫0σψ dσ) ) -∫0ψσ dψ

(5)

Evaluation of eq 5 requires knowledge of the surface charge (13) Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1983, 87, 5025.

density as a function of the potential of the surface. This relationship can be obtained from the solution of the Poisson-Boltzmann equation (see Appendix A). For the case of a solution containing Li2SO4, the relationship can be expressed as

σ2 ) 20kT[Li2SO4]{2e-eψ0/kT + e2eψ0/kT - 3}

(6)

where [Li2SO4] is the concentration of Li2SO4 in bulk solution,  is dielectric constant, 0 is permittivity of free space, k is the Boltzmann constant, and T is temperature. Equation 6, when combined with eq 5, permits the numerical evaluation of the electrostatic contribution to the surface tension of an aqueous solution of II+. A straightforward extension of the above development provides the electrostatic contribution to the free energy of Gibbs monolayers formed from II2+. The electrostatic free energy per molecule of II2+ can be calculated from eq 4 as 2gel (acharge ) aII2+/2, where aII2+ is the area per molecule of II2+). Configurational Contribution. The configurational degrees of freedom of surfactants adsorbed at interfaces have been treated by Szleifer et al.,14 Andelman et al.,15 and others.11,16-18 The transfer of a hydrocarbon chain from bulk alkane19 to a monolayer at the surface of water results in a change in the configurational free energy of the chains because one or both ends of the hydrocarbon chains are tethered to the aqueous subphase and because the conformations of the chains are restricted by the presence of neighboring molecules within the adsorbed monolayer. Whereas past studies have treated the packing of aliphatic chains within monolayers,16-18 we require the development of a model to describe the packing of alkylferrocene chains within a monolayer. This situation differs from those treated in the past for two reasons. First, the volume of ferrocene (150 Å3) is three times larger than that of a methyl group (∼50 Å3). Second, because the solubility of ferrocene in water is greater than that of alkanes in water (for alkanes of size comparable to that of ferrocene), we hypothesized that an effective attraction (relative to hydrocarbon) may exist between ferrocene and the aqueous subphase of the Gibbs monolayer. When evaluating the configurational contribution to the free energy of the monolayer, therefore, we weighted the probabilities of conformations that placed the ferrocene within 3 Å of the aqueous subphase by an energy Efc. As discussed below, the value of Efc was determined by fitting the limiting area predicted by the model for II+ to the experimental value (85 ( 5 Å2). Details of the evaluation of the configurational free energy of II+ packed within a monolayer at the surface of water are described in Appendix B. Hydrophobic Contribution. We have estimated the hydrophobic contribution to the free energy of the surface phase by using two sets of experimental measurements. First, from past measurements of the solubility of alkanes in water, it is known that the free energy of transfer of (14) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1990, 94, 5081. (15) Andelman, D.; Brochard, F.; Joanny, J.-F J. Chem. Phys. 1987, 86, 3673. (16) Gruen, D. W. R. J. Colloid Interface Sci. 1981, 84, 281. (17) Gruen, D. W. R.; Lacey, E. H. B. Surfactants in Solutions; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1. (18) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 7830. (19) The process of transfer of surfactant tails between the reference state of the molecules (bulk aqueous environment) and the Gibbs monolayer involves the use of an intermediate state corresponding to bulk hydrocarbon.9 The configurational free energy describes the transfer of tails from bulk hydrocarbon to the confined environment of the interface.

Model for Gibbs Monolayers

Langmuir, Vol. 15, No. 3, 1999 725

a methylene group from water to alkane is -1.5 kT.21 This value has been used in past models of the thermodynamic properties of surfactant systems.9-12 Second, from electrochemical measurements of the partitioning of ferrocene and ferrocenium cations between water and micelles in solution, the standard free energy of transfer of Fc and Fc+ from bulk water to micelles can be estimated.6,22 The surfactant solutions used in these past studies contained water, Fc or Fc+, and cetyltrimethylammonium bromide (CTAB). The location of the ferrocene within the micelles formed from CTAB is unknown, although it is likely that ferrocene is associated with the interfacial region of the micelle (in part, because of its bulkiness) and not solubilized in the hydrocarbon interior (as we conclude for monolayers formed from II+ at the surface of the solution). We assume the change in free energy upon transfer of ferrocene from bulk water to the micelle to be the same as the change in free energy upon transfer of ferrocene from bulk water to the Gibbs monolayer formed from II+. Details of the evaluation of partition coefficients for Fc and Fc+ from measurements of the half-wave potentials are presented in Appendix C. Using the methods described in Appendix C, the value of the partition coefficient for ferrocene (KR, see Appendix C for definition) in a dilute micellar solution of CTAB can be estimated as 0.0337 for a CTAB concentration of 0.055 M.22 Partition coefficients estimated using the methods described above can be translated to standard free energies of transfer by using the equation22

(

∆G° ) -kT ln

55.5KR

(C - CMC)

)

and bulk hydrocarbon-air (ghc/air) interface: gcont ) a(γhc/w + γhc/air), where γhc/w and γhc/air are interfacial tensions between hydrocarbon-water and hydrocarbon-air.25 This description is only approximately valid (see below) for condensed surface phases and is not valid for gaseous surface phases with isolated chains. While the use of a macroscopic surface tension may seem inconsistent with a molecular-level description, this approach has been quite successful in describing a range of properties of surfactantbased systems9 and is, therefore, used here.26 Headgroup Contribution. The headgroup contribution accounts for several less well-understood effects associated with the presence of a polar headgroup at the surface of an aqueous solution. First, the presence of the headgroup of the surfactant reduces the area of contact between the hydrocarbon-rich region of the monolayer and the underlying aqueous subphase.10 Second, specific interactions and volume correlation effects exist, which are not taken into account when calculating electrostatic free energy by assuming a smeared surface charge density.9,11 Because no quantitative theories exist to describe these effects, and because the details present in these terms do not provide the answers to the three questions posed in the Introduction, here we simply use the value of 1.5 kT for this term as was used by Eriksson et al.10 to describe the surface tensions of aqueous solutions of cationic (alkylammonium) surfactants. 2.3. Chemical Potential of Bulk Solution. At concentrations below the CMC of II+, we describe the concentration dependence of the chemical potential of II+ as that of a species within an ideal solution

µbulk ) µ0 + kT ln X

(7)

We estimate the standard free energy of transfer of ferrocene from a bulk aqueous phase to a (cationic) micellar environment to be -9.5 kT. For comparison, values of KR for ferrocene in micellar solutions of CTAB have also been estimated from measurements of rates of chemical oxidation: this second method leads to an estimate of KR of 0.09 ( 0.005 for a CTAB concentration of 0.02 M and a standard free energy of transfer of -9.2 kT.23 The standard free energy of transfer calculated for Fc+ using the same approach includes an electrostatic contribution due to the interaction between the charged headgroups of CTAB and Fc+. To prevent double counting of the electrostatic contribution in our model of the Gibbs monolayer of II2+, we subtract the electrostatic free energy of transfer of Fc+ to the CTAB micelles from the free energy of transfer ∆G° obtained experimentally.24 The resulting hydrophobic free energy of transfer for Fc+ is calculated as -6.8 kT. Contact Free Energy. As described by others,10 when a condensed phase of surfactant is present at the surface of water, interfaces are created between the hydrocarbonrich region of the monolayer and the contacting liquid (aqueous) and vapor (air) phases. This term, called gcont, is estimated as the free energy penalty associated with the creation of a bulk hydrocarbon-water interface (ghc/w) (20) Gallardo, B. S. Ph.D. Thesis, University of California, Davis 1997. (21) Abraham, M. H. J. Chem. Soc., Faraday Trans. 1 1984, 80, 153. (22) Myers, S. A.; Mackay, R. A.; Brajter-Toth, A. Anal. Chem. 1993, 65, 3447. (23) Calvaruso, G.; Cavasino, F. P.; Sbriziolo, C.; Liveri, L. T. J. Colloid Interface Sci. 1994, 164, 35. (24) The electrostatic contribution to the standard free energy of transfer of Fc+ from bulk aqueous solution to the micellar environment was calculated using the solution of the Poisson-Boltzmann equation for an isolated spherical micelle of radius R.9 We calculated this value to be 2.2 kT for micelles formed from II+ in 0.1 M Li2SO4.

(8)

where X is the mole fraction of surfactant in the bulk solution containing surfactant, electrolyte, and water. For the dilute concentrations of surfactant (and electrolyte concentrations) considered in this paper, this approximation is likely to be a good one.9,11 In this paper, we do not attempt to calculate the CMC of II+. Instead, we simply use values of the CMC obtained from experimental measurements of surface tension and light scattering. In our past studies, we have confirmed that the break in the surface tension plot in Figure 2 corresponds to the CMC.1 For X > Xcmc, we describe the chemical potential as +

+

II µII bulk ) µ0 + kT ln Xcmc ) const

(9)

We have also confirmed (by light scattering studies) that aqueous solutions of II2+ do not contain micelles in the concentration range studied in Figure 2. Thus we describe the chemical potential of II2+ as 2+

2+

II + kT ln XII2+ µII bulk ) µ0

(10)

over the range of concentrations considered in this paper. 2.4. Evaluation of Surface Tension. For a fixed reference state (i.e., an isolated surfactant molecule in aqueous solution), the Gibbs free energy of a monolayer of surfactant is the sum of all the contributions in eq 3. The requirement of diffusional equilibrium between the (25) The surface and interfacial tensions used in our calculations are given by (in mJ/m2)11

γhc/w ) 50.0 - 0.094(T - 305) + 0.0011(T - 305)2

γhc/air ) 20.0 - 0.01(T - 293.15)

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Figure 3. Calculated values (circles) of the area occupied by II+ at the surface of an aqueous solution (0.1 M Li2SO4) with a surface tension of 49 mN/m as a function of the interaction energy (Efc) between ferrocene and the aqueous subphase. The solid line is drawn to guide the eye.

Figure 4. Calculated values of the configurational free energy (triangles) and average position of ferrocene (circles) within a Gibbs monolayer formed from II+ at the surface of an aqueous solution (0.1 M Li2SO4) as a function of the interaction energy (Efc) between ferrocene and the aqueous subphase. The solid lines are drawn to guide the eye.

surface and bulk of the solution leads to the equation

( ) ∂Ng ∂N

) kT ln X

(11)

A,T,P

where A is the total surface area and N is number of surfactant molecules at the interface. At constant T and P, the left-side of eq 11 can be expressed as

( ) ( ) ( )( ) ∂Ng ∂N

)

A

∂Ng ∂N

+

a

∂Ng ∂a

N

∂a ∂N

(12)

A

Equation 12, when combined with eq 11, leads to

kT ln X ) g - aγ

(13)

For a specified concentration of surfactant in bulk solution, we used eq 13 to solve for the surface area per surfactant molecule that satisfies the requirement of diffusional equilibrium between surface and bulk. The surface tension of the corresponding solution can be evaluated as the following summation:

γ ) γel + γcont + γconf

(14)

3. Results 3.1. Limiting Area Occupied by II+. The model presented above, which permits calculation of the surface tension of aqueous solutions of II+, involves one unknown parameter, the value of Efc used in the evaluation of the configurational contribution to the Gibbs free energy of the monolayer. Here we consider the effect of the magnitude of Efc on the value of Amin predicted by the model of II+. Recall that the experimental value of Amin, which is measured at a surface tension of 49 mN/m, is estimated from Figure 2 to be 85 Å2/molecule.1 Figure 3 shows values of the surface area per molecule (Amin) calculated at a surface tension of 49 mN/m and as a function of increasing Efc. First, we point out that with Efc ) 0, the calculated value of Amin is 40 Å2/molecule. Inspection of Figure 4 shows that the average position of ferrocene with Efc ) 0 is within the outer region (air-side) of the monolayer. Because the molecular volume of ferrocene is large and because free volume exists at the outer region of the monolayer, the molecule assumes an extended conformation in which the bulky ferrocene group is placed near the outer surface of the monolayer. The

Figure 5. Calculated values of the surface tension of an aqueous solution (0.1 M Li2SO4) of II+ as a function of the area occupied by each molecule of II+ and with Efc ) 0.

value of Amin calculated for Efc ) 0 is, however, lower than that observed experimentally (85 Å2/molecule), thus suggesting that the experimental situation differs substantially from Efc ) 0. To emphasize this point, Figure 5 shows surface tensions calculated as a function of the area per surfactant molecule for the case of Efc ) 0. With Efc ) 0, the surface tension at an area per molecule of 85 Å2/molecule is indistinguishable from that of the electrolyte without added surfactant. As mentioned above, we hypothesized that energetic interactions between the water and ferrocene likely weight those conformations that place the ferrocene near the water. This hypothesis is supported by the results of our past experimental measurements of the limiting areas of ferrocenyl surfactants with 8 and 11 methylene units:2 the limiting area per molecule of II+ (11 methylenes) was greater than its homologue with eight methylenes, a result that is consistent with conformations of these surfactants that locate the ferrocene near the interface.2,15 Figure 3 shows values of Amin calculated as a function of increasing strength of attraction between ferrocene and water (increasing Efc). These results indicate that Efc must be greater than ∼30 kJ/mol so as to obtain agreement between experimental and calculated values of the limiting area (85 Å2/molecule). Inspection of Figure 4 shows the ensemble average position of ferrocene (evaluated using the probability distribution function given in eq B3 of

Model for Gibbs Monolayers

Figure 6. Proposed (typical) conformations assumed by II+ and II2+ at the surfaces of aqueous solutions.

Appendix B) to be near the aqueous subphase and the surfactant to adopt a “looped” conformation for values of Efc that are greater than ∼30 kJ/mol. The entropic penalty associated with the formation of the looped conformation is approximately 4 kT. In the remainder of our calculations, we use a value of Efc of 30 kJ/mol to weight conformations of the ferrocenyl surfactants within monolayers.27 That is, we assume that II+ takes a looped conformation at the surface of water (Figure 6A). Because we also expected II2+ to assume a looped conformation at the surface of the solution (since the Born energy of the ferrocenium ion will keep it in contact with the water, as is observed with ionic bolaform amphiphiles28), we evaluated the configurational contribution to the Gibbs free energy of monolayers of II2+ by requiring that all conformation of II2+ place the ferrocenium ion close to the surface of the solution (Figure 6B). This constraint on the position of the ferrocenium ion was imposed by using Efc ) 100 kJ/mol. 3.2. Surface Tensions Calculated for Aqueous Solutions of II+ and II2+. Figure 2 shows surface tensions calculated for aqueous solutions of II+ and II2+ using the model presented in this paper. Comparison of the calculated and experimental surface tensions in Figure 2 shows good qualitative agreement. While the model reported in this paper is a simple one that contains many approximations (as described above), it does surprisingly well in capturing the essential features of the experimental measurements. For example, at concentrations around 0.1 mM, oxidation of II+ to II2+ leads to an increase in the surface tensions of the aqueous solutions from 49 to ∼72 mN/m, as is observed experimentally. Second, at high concentrations, oxidation of II+ to II2+ leads to a decrease in the surface tensions of aqueous solutions and, as discussed below, to an increase in the extent of adsorption of surfactant to the surfaces of the solutions. In short, the qualitative agreement between the model and experimental measurements shown in Figure 2 leads us to conclude that we have correctly identified the dominant contributions to the free energy of Gibbs monolayers formed from II+ and II2+. (26) The use of the macroscopic interfacial tension is justified, in part, by evaluation of the configurational free energy term. This term “corrects” the macroscopic interfacial tension for the effects of confinement of chains within the monolayer (see discussion in refs 9 and 10). (27) The use of values of Efc greater than 30 kJ/molecule does not substantially change the calculated contribution of the configurational term to the Gibbs free energy of the monolayer. (28) Menger, F. M.; Wrenn, S. J. Phys. Chem. 1974, 78, 1387.

Langmuir, Vol. 15, No. 3, 1999 727

Figure 7. Calculated values of the electrostatic (γel, circles) and configurational (γconf, diamonds) contributions to the surface tension of aqueous solutions (0.1 M Li2SO4) of the redox active surfactant II+. The solid lines are drawn to guide the eye.

4. Discussion In the sections that follow, we use the model described above to address the three questions posed in the Introduction of this paper. 4.1. Answer to Question 1. The first question posed in the Introduction addressed the unknown balance of intermolecular forces controlling the surface tensions of aqueous solutions of II+. In particular, we aimed to understand the mechanism by which a Gibbs monolayer formed from II+ can reduce the surface tension to a value of 49 mN/m at a limiting area per molecule of 85 Å2. As was pointed out in the Introduction, surface tensions of aqueous solutions of II+ are somewhat surprising because DTAB, when occupying a surface area of 85 Å2/molecule, reduces the surface tension of an aqueous solution to only ∼71 mN/m. Although our model of II+ includes five contributions to the Gibbs free energy of the monolayer (see eq 3), only three of them (i.e., electrostatic, configurational, and contact terms) depend on the area occupied by II+ at the surface of the solution and thus contribute directly to the lowering of the surface tension of the solution. Furthermore, only two of these terms (electrostatic and configurational) make an area-dependent contribution to the surface tension. Figure 7 shows the electrostatic and configurational contributions to the surface tension of aqueous solutions of II+ (in 0.1 M Li2SO4) as a function of the area per molecule. At an area per molecule corresponding to the limiting value for II+ (85 Å2/molecule), it is apparent from Figure 7 that the principal contribution to the surface pressure (lowering of surface tension) is configurational (17 mN/m) and not electrostatic (4 mN/m) in origin. The dominant contribution of the configurational term arises from the looped conformation assumed by II+. The looped conformation is, therefore, the principal reason that II+ lowers the surface tension of aqueous solutions by 20 mN/m at values of the surface excess for which classical ionic surfactants such as DTAB do not significantly reduce surface tension.29 Finally we point out that Figure 7 shows the configurational contribution to the surface tension to be strongly dependent on the area per molecule. Furthermore, for areas greater than 115 Å2/molecule, the configurational contribution to the surface tension is opposite in sign to the electrostatic one and it acts to increase the surface tension of the solution. 4.2. Answer to Question 2. The model presented in this paper can also be used to answer the second question (29) Armstrong, X.; Quinn, R. K.; Van der Borgh, N. E. Anal. Chem. 1974, 46, 1759.

728 Langmuir, Vol. 15, No. 3, 1999

Aydogan et al.

Figure 9. Calculated values of the concentration-dependent part of the chemical potential [(µ - µo)/kT] of II+ (solid line) and II2+ (dashed line) dissolved within an aqueous solution. The CMC of II+ is 0.1 mM whereas II2+ does not aggregate over the range of concentrations shown. The arrow shows the increase in [(µ - µo)/kT] that accompanies oxidation of a 5 mM solution of II+ to II2+ and the disassembly of micellar species in solution to monomers.

Figure 8. Calculated values of the configurational (triangles), electrostatic (squares), hydrophobic (diamonds), and contact (circles) contributions to the free energy of formation of Gibbs monolayers at the surfaces of aqueous solutions (0.1 M Li2SO4) of (A) II+ and (B) II2+. Panel C shows the change in the free energies upon oxidation of II+ to II2+ (at constant area per molecule). The lines are drawn to guide the eye.

posed in the Introduction. The second question addressed the origin of the oxidation-induced desorption of surfactant from the surfaces of solutions containing 0.1 mM surfactant (and thus the origin of the increase in surface tension from 49 to 72 mN/m). The adsorption and desorption of II+ or II2+ from the surface of the solution is controlled by the diffusional equilibrium between the surface and the bulk, and thus the Gibbs free energies of these phases. The electrostatic, configurational, hydrophobic, and contact contributions to the free energies of monolayers formed from either II+ or II2+ are presented in Figure 8. Because the area per molecule of II+ at its CMC is 85 Å2, we use Figure 8 to determine the change in free energy of the surface phase upon oxidation at an area per molecule of 85 Å2. First, oxidation of II+ to II2+ leads to little change in the configurational term because in both the reduced and oxidized states the molecules assume looped conformations (and there is no change in the contact term at the same area per molecule). Second, oxidation of II+ to II2+ leads to an increase in the electrostatic contribution to the free energy of ∼3 kT. Moreover, oxidation affects the hydrophobic driving force for the adsorption of surfactants molecules to the interface. Whereas the hydrophobic driving force of the aliphatic chains does not change, oxidation of ferrocene to ferrocenium reduces the contri-

bution of this moiety to the Gibbs free energy by ∼3 kT. We conclude, therefore, that oxidation-induced desorption of the redox-active surfactant is caused by both a reduction in the hydrophobic driving force for adsorption and an increase in the electrostatic contribution to the free energy of the monolayer. The configurational term does not play a significant role in causing oxidation-induced desorption of II+ from the surface of the solution. 4.3. Answer to Question 3. Finally, we use the model presented in this paper to provide an explanation of the oxidation-induced adsorption of surfactant to the surface of a 10 mM solution of the redox-active surfactant. This experimental observation is, at first sight, a surprising one because the Gibbs free energy of the monolayer increases upon oxidation (at the same area per molecule) and thus one might predict desorption of the surfactant from the surface of the solution (as is observed at concentrations of 0.1 mM). An explanation of the oxidation-induced adsorption of surfactant at high surfactant concentrations is found in the oxidation-induced change in the chemical potential of the surfactant in the bulk solution. As noted above, aqueous solutions of II+ possess a CMC of 0.1 mM, and thus at bulk concentrations of II+ greater than 0.1 mM, the chemical potential of II+ changes little with concentration (Figure 9). Indeed, as pointed out in past papers,9-12 for the range of concentrations of surfactant considered here, the chemical potential of II+ at concentrations greater than the CMC is well-described as a constant (equal to µo + kT ln Xcmc). In contrast to aqueous solutions of II+, however, we have found no experimental evidence for the existence of a CMC for II2+ at concentrations less than 30 mM. That is, oxidation of aqueous solutions containing 0.1 to 30 mM II+ leads to the disassembly of micelles of II+ to singly dispersed molecules of II2+. Accompanying the disruption of micelles is an increase in the concentration dependent part of the chemical potential ((µ - µo)/kT) of the dissolved surfactant (Figure 9). It is this increase in the value of (µ - µo)/kT for the surfactant in bulk solution that leads to the oxidation-induced adsorption of surfactant and lowering of surface tension. The effect of the oxidation-induced change in bulk chemical potential can be seen in Figure 10. At a value of (µ µo)/kT of 0.58 (corresponding to a bulk concentration of II+ of 0.1 mM or greater), the surface area occupied by II+

Model for Gibbs Monolayers

Langmuir, Vol. 15, No. 3, 1999 729

Appendix A We found the surface potentials of Gibbs monolayers of II+ and II2+ formed in 1:2 electrolytes (0.1 M Li2SO4) to be sufficiently high (>25.4 mV) that linearization of the PB equation was not justified. Therefore, we used the nonlinear form of the PB equation to evaluate the surface charge density

∇2ψ ) -

Figure 10. Calculated values of the surface area occupied by II+ (solid line) and II2+ (dashed line) as a function of the concentration-dependent part of the chemical potential of these species [(µ - µo)/kT]. The arrow shows the decrease in the specific surface area that accompanies oxidation of a 5 mM solution of II+ to II2+ (same process as depicted by the arrow in Figure 9), thus corresponding to an oxidation-induced adsorption of surfactant to the surface of the solution.

is 84 Å2/molecule. In contrast, values of (µ - µo)/kT for II2+ can greatly exceed 0.58 because II2+ does not form micelles. For example, the value of (µ - µo)/kT for a 5 mM solution of II+ is 4.5 and the corresponding area per molecule is 67 Å2/molecule. Thus, upon oxidation of a 5 mM solution of II+ to II2+, the value of (µ - µo)/kT increases from 0.58 to 4.5 and the area per molecule at the surface of the solution is calculated to decrease from 84 to 67 Å2/molecule. The latter prediction of the model (oxidation-induced adsorption) is qualitatively consistent with the experimental measurements shown in Figure 2. 5. Conclusions This paper reports a molecular-thermodynamic model capable of describing redox-induced changes to the surface tensions of water-soluble, ferrocenyl surfactants. The principal conclusions of the work reported in this paper are twofold. First, the results reported in this paper establish the importance of configurational contributions to the free energy of Gibbs monolayers formed from II+; in contrast to classical ionic surfactants such as DTAB and CTAB, the configurational contribution is the dominant contribution to the surface pressure of II+. Second, the results described in this paper explain why it is that II+, upon oxidation to II2+, desorbs from the surfaces of aqueous solutions at low concentrations (concentrations less than the CMC) and adsorbs to the surfaces of solutions at high concentrations (concentrations greater than the CMC). The explanation is based on redox-induced changes to the bulk chemical potential of the surfactant. At high concentrations, an oxidation-induced disruption of micelles leads to an increase in the chemical potential of surfactant and thereby drives adsorption of surfactant onto the surfaces of the solution. At low concentrations, in contrast, micelles do not exist in bulk solution in either oxidation state and thus oxidation is accompanied by desorption of surfactant from the surfaces of the solution, a consequence of mutual interactions between surfactant molecules (e.g., electrostatic repulsion) hosted within the Gibbs monolayer. Acknowledgment. This work was supported in part by the Camille and Henry Dreyfus Foundation (New Faculty Award), the David and Lucile Packard Foundation, the donors of the Petroleum Research Fund, and the National Science Foundation (CTS-9410147 and CTS9502263).

F(r) 0

)-

( )

e

∑ zinio exp  i

zieψ kT

(A1)

where zi is the valence of charge i, nio is the number density of species i in bulk solution, F is the total charge density, r is the distance from interface,  is the dielectric constant, k is the Boltzmann constant, and T is the temperature. By applying the condition of electroneutrality to the entire system (surface plus bulk solution), the surface charge density can be expressed as

σ2 ) 20kT[

∑Fi

surface

∑Fi

bulk

]

(A2)

where the quantity in the brackets represented the excess charge within the diffuse part of the electrical double layer near the interface. Although the bulk solution contains Li2SO4 as well as the redox-active surfactant (II+ or II2+), we neglect the excess surface charge that arises from the presence of the Li2SO4.2 Under this assumption, eq A1 and eq A2 can be combined to yield eq 6. Appendix B We adapted the rotational isomeric state model for an aliphatic chain to describe the configurational contribution to the free energy of alkylferrocene chains within a Gibbs monolayer. This contribution was evaluated via the canonical partition function14

Z)

Rjk exp ∑i ∏ jk

( ) -Ei kT

(B1)

where the summation is performed over all i bonding configurations (gauche+, gauche-, or trans) of the aliphatic chain, and the product is performed over all methylene groups j positioned in slab k of the monolayer. The parameters ajk are statistical weights that are selfconsistently determined (see ref 20) so as to ensure that the density of hydrocarbon (including ferrocene) within the monolayer does not exceed the bulk density of hydrocarbon. The energy of a particular configuration i of the chain is determined by the number of gauche bonds within the chain, and for configurations that place the ferrocene group within 3 Å of the aqueous subphase, by an energy Efc

Ei ) Ebonding - Efc

(B2)

where Ebonding is given by ngauche × 2.094 kJ/mol. Although those configurations that place ferrocene near the water are weighted with an energy Efc, we do not use this energy when evaluating the internal energy of the chain (see eq B5 below). This contribution is included in our estimate of the hydrophobic contribution to the free energy of the monolayer. Following the self-consistent determination of the values of ajk, all ensemble average properties are evaluated using the probabilities Pi of each conformation

730 Langmuir, Vol. 15, No. 3, 1999

Pi )

1

∏Rjk exp Z jk

( ) -Ei kT

Aydogan et al.

(B3)

and the chain entropy S is calculated as

S ) -k

∑i Pi ln Pi

(B4)

The free energy of the chain is then calculated as 0 ) 〈U〉 - TS gmonolayer

(B5)

where 〈U〉 is the ensemble averaged bonding energy (equal to 〈Ebonding〉). The free energy of the chain in bulk liquid 0 ) is calculated by assuming the chain to experi(gfreechain ence no packing constraints and to have conformational probabilities determined solely by the bonding energies of the chain. The change in free energy upon transfer of the chains from the bulk liquid to the monolayer is evaluated as 0 0 - gfreechain ∆gpacking ) gmonolayer

(B6)

Appendix C The partition coefficients for ferrocene (Fc) and ferrocenium (Fc+) within the micellar solutions can be defined, respectively, as

KR )

KO )

( ) [Fcw]

[Fcm]

( ) [Fc+ w]

[Fc+ m]

where [Fcw] and [Fcm] are the concentrations of Fc within the bulk aqueous and micellar regions, of the solution, + respectively, and [Fc+ w] and [Fcm] are analogous quantities defined for Fc+. Note that all concentrations in eqs C1 and C2 are based on the total volume of the system. As described below, the values of KR and KO can be estimated from measurements of half-wave potentials of aqueous solutions and micellar solutions containing Fc or Fc+.6,22 First, the formal potential can be estimated from measurements of the half-wave potential E1/2 of a micellar solution containing Fc/Fc+ 22

E1/2 ) E0M +

( )

DR RT ln nF DO

1/2

(C3)

where E0M is the formal potential of the microheterogeneous system (a function of the partition coefficients) and DR/DO is the ratio of the diffusion coefficients of Fc and Fc+, respectively. The last term on the right side of eq C3 can be estimated from a controlled potential electrolysis of an aqueous surfactant solution containing ferrocene (Levich equation).22 Values for KO and KR can then be estimated by performing measurements of half-wave potentials and diffusion coefficients as a function of the composition of micellar solutions containing ferrocene and by interpreting these measurement using eq C4.

(C1)

E0M ) E0aq +

(C2) LA9807208

(

)

KO(1 + KR) RT ln nF KR(1 + KO)

(C4)