Langmuir 2004, 20, 869-874
869
The Electrocapillary Effect at an Electrode Modified with an Insoluble Redox-Active Self-Assembled Monolayer Willem H. Mulder* Department of Chemistry, University of the West Indies, Mona Campus, Kingston 7, Jamaica
Juan Jose´ Calvente and Rafael Andreu Departamento de Quı´mica Fı´sica, Facultad de Quı´mica, Universidad de Sevilla, 41012 Sevilla, Spain Received July 26, 2003. In Final Form: November 18, 2003 The interface between an electrolyte solution and a metal electrode coated with an oxidatively adsorbed, redox-active monolayer of long-chain thiols has been examined from a thermodynamic point of view. The electrode potential is assumed to vary within the region where no reductive desorption of the thiol occurs, so that the interface may formally be regarded as ideally polarizable. The analysis leads to an expression describing the potential dependence of interfacial tension in terms of the charge density on the metal, salt concentration, dielectric properties of the organic film, and the redox properties of the active terminal groups, which vary with the (average) distance from the electrode surface. This result generalizes the classical Lippmann equation to modified electrodes of the type considered.
1. Introduction Alkanethiols have long been known to form strong, covalent bonds with a number of metals (notably copper, gold, and also mercury) after oxidation of the thiol group.1 A metal electrode immersed in a solution of alkanethiol (A-SH) becomes covered with a self-assembled monolayer (SAM), either in a spontaneous, electroless fashion (accompanied by, e.g., hydrogen evolution) or oxidatively, upon application of a suitable potential. In either case, the adlayer is firmly attached to the metal, via the sulfur anchors. Only in the case of mercury is there sometimes a tendency to form soluble dimers, held together by mercury atoms if the alkyl chains are short. This (essentially corrosive) process can be suppressed by using long-chain thiols which, by virtue of their increased hydrophobicity, tend to stabilize the SAM. Of particular interest, from the point of view of applications (such as biosensing2 or the development of molecular electronic devices3) are alkanethiols with incorporated redox centers, which render them redox active. Model systems that have been studied rather extensively over the past decade are SAMs containing ferrocenylcarbonyloxy (FcCO2)- or ferrocenyl (Fc)-terminated alkanethiols,4 and mixed monolayers of ferroceneterminated thiols and n-alkanethiols.5 These studies are normally limited to solid (gold) electrodes, with voltammetry being the technique of preference to characterize their electrochemical properties.6 In the present contribution we emphasize the fact that the use of mercury or, in the case of solid electrodes, * Corresponding author. Fax: +1 876 977 1835. E-mail:
[email protected]. (1) Ulman, A. An Introduction to Ultrathin Organic Films: From Langmuir-Blodgett to Self-Assembly; Academic Press: San Diego, 1991; Chapter 3. (2) Gooding, J. J.; Hibbert, D. B. Trends Anal. Chem. 1999, 18, 525. (3) (a) Fox, M. A. Acc. Chem. Res. 1999, 32, 201. (b) Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M. Science 1999, 286, 1550. (4) Chidsey, C. E. D.; Bertozzi, C. B.; Putvinski, T. M.; Mujsce, A. M. J. Am. Chem. Soc. 1990, 112, 4301. (5) Rowe, G. K.; Creager, S. E. Langmuir 1991, 7, 2307. (6) Finklea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19.
contact-angle measurements provides us with an opportunity to test theoretical models of the structure of SAMs in yet another way, via the determination of the potential dependence of interfacial tension (electrocapillarity phenomenon). Insoluble, redox-active organic thiols present an extreme example of reactant adsorption, where the oxidized and reduced moieties remain tethered to the metal surface, such that varying the potential will only give rise to a change in the Ox/Red surface concentration ratio and to a reorganization of the diffuse double layer, not to a sustained current flow in the external circuit. Thus, the interface may be regarded, for all means and purposes, as ideally polarizable, even though electrons can cross the metal/SAM boundary. During the past few years, there has been a surge in research activity focusing on the electrowetting properties of polymer-coated metals, with potentially important implications for, e.g., the control of liquid flow through microchannel networks.7,8 Therefore, it would be worthwhile to revisit the classical thermodynamic approach to electrocapillary phenomena at polarized electrified interfaces and see how it can be extended to include the case of redox-active modified electrodes. 2. Description of the Model An alkanethiol film of thickness δ (approximate chain length) and area A is sandwiched between an electrode (onto which it is chemisorbed via covalently bonded sulfur atoms) and an aqueous solution of a 1:1 electrolyte (see Figure 1). A fraction of the alkane chains terminate in redox groups that can exchange electrons with the metal across the film according to the scheme
OzO + ne - a RzR
(1)
The dielectric constants of film and solvent are 1 and , (7) 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. (8) Prins, M. W. J.; Welters, W. J. J.; Weekamp, J. W. Science 2001, 291, 277.
10.1021/la035364u CCC: $27.50 © 2004 American Chemical Society Published on Web 12/24/2003
870
Langmuir, Vol. 20, No. 3, 2004
Figure 1. Schematic of the interfacial region, showing metal, SAM plus redox sites, and diffuse layer. Also indicated is the potential distribution in the case of a positive surface potential.
respectively (typically, 1 ∼ 3 and ) 78.5). Next to the electroactive groups, the SAM normally contains inert alkanethiols that serve as diluents, such as to render the surface layer more akin to an oil-like, isotropic dielectric slab, impermeable to water or ions. Although it is well known that the water layer adjacent to an organic film has a significantly different structure (and hence dielectric constant) from bulk water, this aspect will not be considered in the model. Moreover, discreteness-of-charge effects, associated with the immobilized redox centers,9 will not be accounted for in the present theory, but we shall instead limit ourselves to a mean-field description of the interfacial potential distribution, following the original work of Smith and White.10 This model applies to monolayers prepared ex situ, whose electrochemical properties are probed over a potential range where the monolayer retains its structural integrity, and its thermodynamic analysis is the main goal of this paper. However, a related experimental situation, corresponding to the free exchange of thiols between the solution and the adsorbed state, is also briefly discussed in the Appendix. 3. Thermodynamics of Insoluble Redox-Active SAMs The system is defined as a simple two-electrode cell with a working electrode that is covered with a thiol monolayer and a reference electrode that is reversible with respect to either the cation (cell potential E+) or the anion (E-) of the electrolyte. Alkanethiols are assumed to be insoluble. The Gibbs energy of this cell will be denoted by G. An infinitesimal reversible change in the state of this system, under conditions of constant temperature T and ambient pressure p, and total numbers of water molecules, cations and anions, henceforth indicated by labels j (e.g., Nj), can then be brought about in several ways, giving rise to as many terms in the attendant change in Gibbs energy, (dG)T,p,{Nj}, each term representing an infinitesimal amount of reversible work. (i) Expansion of surface area of the working electrode by dA gives a contribution γdA, where γ is the interfacial tension. (9) (a) Fawcett, W. R. J. Electroanal. Chem. 1994, 378, 117. (b) Andreu, R.; Fawcett, W. R. J. Phys. Chem. 1994, 98, 12753. (10) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398.
Mulder et al.
(ii) Insertion of alkanethiols in the monolayer would alter G by increments of either (µ jR + µ j e)dNR or (µ j O + (n + 1)µ j e) dNO, depending on whether the redox group becomes oxidized upon chemisorption (The thiols are taken to be fully reduced prior to contact with the metal.) The µ j O,R represent the electrochemical potentials of the oxidized and reduced forms while grafted onto the electrode, µ j e, that of an electron inside the metal. In either case, one electron is donated to the electrode as the thiol moiety gets oxidized, hence the terms µ j edNO,R. If the R species subsequently undergoes oxidation, it transfers a further n electrons to the metal, which accounts for the term nµ j edNO in the second case. (iii) Finally, since the interface is ideally polarizable in the potential range under study, an external power source can perform electrical work on the cell, by pumping charge dQ′ onto the working electrode, to an amount E(dQ′, where E( is the terminal voltage imposed by the source and traversed by the charge dQ′. To preserve electroneutrality and equilibrium, this charging process will inevitably be accompanied by spontaneous adsorption or desorption of ions inside the diffuse layer, as well as by a change in the equilibrium O/R ratio inside the SAM (such that NT ) NO + NR remains constant, though). It is important to emphasize that Q′ is that part of the charge on the electrode surface that reached there via an external circuit and differs from the total charge Q that is actually present by a contribution that derives from the (multiple) oxidation of alkanethiols. Thus
Q ) Q′ - e(NR + (n + 1)NO)
(2)
Putting it all together, the following expression for the change in the Gibbs energy of the cell, resulting from the processes just described, is obtained
jO - µ j R + nµ j e) dNO + (dG)T,p,{Nj} ) γ dA + (µ (µ jR + µ j e) dNT + E( dQ′ (3) The electrochemical potentials of the solution constituents, µ j j, are all fixed at their respective bulk values; i.e., the solution acts as a reservoir of water molecules and ions. Since the reference electrode is in equilibrium with the solution throughout, no contribution due to the redox processes occurring at its surface appears in eq 3. Subject to the additional constraints A ) constant, NT ) constant (i.e., the cell is a closed system), and Q′ ) constant (which is tantamount to having an open circuit), eq 3 simplifies to
jO - µ j R + nµ j e) dNO (dG)T,p,A,{Nj},NT,Q′ ) (µ
(4)
For G to attain a minimum, the following condition of internal (redox) equilibrium, must obviously be fulfilled
j O + nµ je µ jR ) µ
(5)
which, not surprisingly, is essentially a statement of the Nernst equation. j R can be written The electrochemical potentials µ j O and µ more explicitly by using the configurational part of the contribution to G due to the monolayer which, based on standard reasoning, is given by
-TSmix ) kT[ΓO ln(ΓO/ΓT) + ΓR ln(ΓR/ΓT)]A
(6)
with Smix the configurational entropy of mixing, ΓO,R )
Electrocapillary Effect at Monolayers
Langmuir, Vol. 20, No. 3, 2004 871
NO,R/A defines the surface concentration of O or R, and ΓT ) ΓO + ΓR is the total surface concentration of redox sites. The electrostatic part of µ j O,R is equal to zO,Reφ′, with φ′ the micropotential at the position of an O or R species. For simplicity, we shall henceforth neglect the discretenessof-charge effects9 that should normally be accounted for in the presence of immobilized charges in the vicinity of a phase boundary and approximate φ′ simply by the meanfield potential in the plane through the redox centers. Also, possible effects due to ion-pairing11a or spatial distribution of redox centers11b will not be taken into consideration. The electrochemical potentials are then given by the expression
µ j O,R ) µO,R* + kT ln(ΓO,R/ΓT) + zO,Reφ′
()
ΓO kT ln ne ΓR
(8)
defining the “standard potential” φ 0 as
φ0 ) (µO* + nµe - µR*)/ne
(9)
Next, we proceed by further exploiting the fundamental thermodynamic relation, eq 3, making use of the important fact that it may be readily integrated if all intensive variables are held constant, to give
j O + (n + 1)µ j e)NO + G ) G0 + γA + (µ j e)NR + E(Q′ (10) (µ jR + µ where G0 represents the collective contribution to G due to the electrodes and cell solution, which is here regarded as constant. Taking the differential of G, while maintaining T, p, Nj, and µ j j at fixed values, and comparing the result with eq 3 then yields (after dividing through by A) the GibbsDuhem relation
j O + (n + 1)µ j e) + -(dγ)T,p,{µj j} ) ΓO d(µ ΓR d(µ jR + µ j e) + q′ dφM (11) where dµ j e ) -e dφM and q′ ) Q′/A, which is related to the actual charge density q on the metal via
q ) q′ - e(ΓT + nΓO)
zRΓR) dφ′ + q dφM (13) The factor of dφ′ is recognized as the equilibrium charge density, henceforth denoted by q1, in the plane through the redox centers. The first two terms on the right-hand side may be combined into a single differential, dΠ*, which is equal to a change in the hydrostatic surface pressure inside a neutral organic layer (at constant temperature). This is a consequence of the conventional Gibbs-Duhem relation, as applied to the adlayer. Thus, the generalized Lippmann equation for the modified electrochemical interface assumes the more concise form
(7)
where the µ* values represent the nonelectrical parts of the µ j values for a film consisting entirely of either O or R and can be regarded as being functions of temperature and local (surface) pressure inside the monolayer. Substitution of eq 7 and µ j e ) µe - eφM, where φM denotes the potential difference between metal and bulk solution, into eq 5 then leads to the following familiar form of the Nernst equation for this particular system10
φM ) φ0 + φ′ +
-(dγ)T,p,{µj j} ) ΓO dµO* + ΓR dµR* + e(zOΓO +
(12)
where the second term equals the charge density due to electrons donated by the chemisorbed (oxidized) alkanethiols (compare eq 2). Also, use has been made of the fact that, under the stated conditions, dE( ) dφM. If eqs 7 and 12 are substituted into eq 11, it is seen that the “mixing” terms cancel, and what remains can be rearranged to (11) (a) Andreu, R.; Calvente, J. J.; Fawcett, W. R.; Molero, M. J. Phys. Chem. B 1997, 101, 2884. (b) Calvente, J. J.; Andreu, R.; Molero, M.; Lo´pez-Pe´rez, G.; Domı´nguez, M. J. Phys. Chem. B 2001, 105, 9557.
-(dγ)T,p,{µj j} ) dΠ* + q1 dφ′ + q dφM
(14)
The Nernst equation (5) (or (8)) and the electrocapillary equation (14) are the two most significant general results that a classical thermodynamic treatment of this type of system has to offer. It is not difficult to show that they remain unaffected by the inclusion of inert alkanethiol coadsorbates. It is important to note that eq 14 has, in fact, a range of applicability that extends well beyond redox-terminated thiols, as it can be shown in a straightforward manner that this result remains valid also in the case of ionizable (acidic/basic) end groups.12 Furthermore, it can be demonstrated that, as far as the form of eq 14 is concerned, the restriction to insoluble monolayers is not essential. The case of soluble thiols is dealt with in the Appendix, where it is shown that, under certain conditions that have to be imposed in order to guarantee ideal polarizability of the metal/solution interface, the same breakdown of dγ in terms of three contributions is retrieved. Here, we will proceed with the case of an insoluble monolayer. Clearly, to be able to make predictions based on eq 14 that can be tested by experiment, more detailed modeling of the electrical and mechanical properties of the various parts of the interface (SAM and diffuse layer) will be necessary, as discussed in the next section. 4. Model Calculations As mentioned in section 2, the organic layer is represented as a continuous, isotropic dielectric of relative permittivity 1. In solution, where a 1:1 salt is present at concentration c, the diffuse part of the double layer will be described in terms of the Gouy-Chapman theory. First of all, the general differential expression, eq 14, will be integrated to yield the total surface pressure Π, since dΠ ) d(γ0 - γ) ) -dγ, where γ0 is, somewhat arbitrarily, chosen as the surface tension in a reference state with q ) q1 ) 0 (i.e., at the potential of zero charge (pzc) for the nonionized monolayer). Thus
Π ) Π* +
∫0φ -φ′ q d(φ - φ′) + ∫0φ′ (q + q1) dφ′ M
(15)
Here, the term q dφ′ has first been added and then again subtracted in order to separate the contributions to Π due to the SAM (the first integral) and the diffuse layer (the second integral). Inside the organic layer, which is henceforth assumed to be insoluble and incompressible, with constant thickness (12) Smith, C. P.; White, H. S. Langmuir 1993, 9, 1.
872
Langmuir, Vol. 20, No. 3, 2004
Mulder et al.
δ, a uniform field is present of magnitude |φM - φ′|/δ. Introducing a scaled potential via ψ ≡ eφ/kT, the surface charge density on the metal is related to this field according to Gauss’ law
q)
10kT M (ψ - ψ′) eδ
(16)
The first integral then becomes
Πel )
∫0
φM-φ′
10 kT 2 M q d(φ - φ′) ) (ψ - ψ′)2 2δ e
( )
(17)
and is equal to the electrostatic part of the lateral pressure inside the SAM, the so-called Maxwell pressure. The hydrostatic contribution is of course Π*, the value of which follows from a consideration of mechanical equilibrium. As noted before, the SAM may be treated as a uniform, isotropic dielectric spacer, sandwiched between two media carrying opposite charge densities, q and -q. If the latter are modeled as condenser plates, which pull at each other with a force per unit area ∆p ) (1/2)q|φM - φ′|/δ (remembering that each charged plate generates a field equal to half the field inside the condenser), an excess pressure ∆p has to build up inside the film, to keep the charges separated. The effect on the lateral pressure Π* will be dependent upon such factors as chain length and mobility, presence of (short-chain) diluents, and whether the metal is solid or liquid. If the monolayer is rigid and grafted on a solid support, it is likely that Π* is close to zero (as in the case of polymer coatings on metals). At mercury electrodes, the film may be expected to more closely resemble an isotropic fluid. In this limit, Π* is obtained by integrating the hydrostatic (i.e., isotropic) excess pressure profile across the layer, which simply amounts to multiplying ∆p by δ, to yield the corresponding contribution to Π
Π* ) ∆pδ )
10 kT 2 M (ψ - ψ′)2 2δ e
( )
(18)
that is, Π* equals Πel in the case of a mechanically isotropic SAM. In an actual monolayer, Π* will be intermediate between the two limits corresponding to perfect solid or fluid behavior, and more detailed statistical mechanical treatments of the transverse pressure profiles in grafted “brushes” would need to be invoked.13 The third contribution to Π is recognized as the surface pressure exerted by the diffuse part of the double layer, Πdiff, and can be readily obtained by using the GouyChapman expression for the electroneutrality condition14
q + q1 )
20kT ψ′ κ sinh e 2
( )
(19)
where κ ) (2e2c/0kT)1/2 is the inverse Debye length. Hence
Πdiff )
ψ′ cosh( ) - 1) ∫0φ′ (q + q1) dφ′ ) 8ckT κ ( 2
(20)
From a detailed study of the electrical and mechanical properties of double layers,15 it follows that Πdiff is divided (13) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1990, 94, 5081. (14) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; Wiley: New York, 1997; Chapter 5. (15) Mulder, W. H.; Bharaj, B. S. J. Electroanal. Chem. 1997, 421, 59.
equally between hydrostatic (“osmotic”) and Maxwell pressure, exactly as in the case of the fluid organic film (and for the same reason). We now arrive at the following, dimensionless, form of the net surface pressure
δ e 2 ψ′ 4κδ Π) cosh - 1 + λ(ψM - ψ′)2 (21) 10 kT 1 2
( )
( ( ) )
where λ ) 1/2 for a perfect “solid” monolayer and λ ) 1 for a perfect fluid film. Ultimately, we want to express Π as a function of the (scaled) surface potential ψM. To this end, we first need to eliminate q1, in favor of ψM and ψ′. This is done by using the Nernst equation (8), which allows ΓO and ΓR to be written as10
ΓO ) ΓT/[1 + exp(-n(ψM - ψ0 - ψ′))]
(22)
ΓR ) ΓT/[1 + exp(n(ψM - ψ0 - ψ′))]
(23)
and
(with ΓT ) constant, since we are considering an insoluble monolayer) and therefore, with zR ) zO - n, the equilibrium charge density on the redox plane becomes
q1 ) e(zOΓO + zRΓR) )
[
eΓT zO -
]
n (24) 1 + exp(n(ψM - ψ0 - ψ′))
Combining eqs 16, 19, and 24 gives rise to the following relationship
[
]
e2ΓTδ n z ) ψ - ψ′ + 10kT O 1 + exp(n(ψM - ψ0 - ψ′)) M
()
ψ′ 2κδ sinh (25) 1 2 from which ψ′ can be solved (numerically) as function of ψM, which must then be substituted into eq 21. In Figure 2 are shown some trends in the appearance of “electrocapillary curves”, actually plots of -Π vs φM, obtained using eqs 21 and 25, and typical values for the various system parameters, as indicated. The value 1/2 has been chosen for λ in each case, since most experiments have been performed using solid electrodes. It is clear from Figure 2c that the contribution of the diffuse layer is relatively small over a wide range of electrolyte concentrations. In the absence of the redox couple, the curves are essentially parabolic in shape, as also observed in several electrowetting experiments using inert alkanethiols.16 In the presence of oxidizable groups, the most salient feature of the theoretical electrocapillary curves is the rather abrupt departure from the parabolic shape beyond the standard potential φ 0 (see Figure 2a), where a transition takes place to a second parabolic branch. As Figure 2d shows, at sufficiently negative φ 0, two electrocapillary maxima (ecm) are predicted to occur. This is of course indicative of the fact that there is now no simple relationship between the ecm and the pzc (in whatever way one would choose to define it for this system), as there exists in the case of an “ordinary” double layer, described by the Lippmann equation. This is evident from (16) Sondag-Huethorst, J. A. M.; Fokkink, L. G. J. J. Electroanal. Chem. 1994, 367, 49.
Electrocapillary Effect at Monolayers
Figure 2. Dependence of surface pressure on the following: (a) surface concentration of redox groups, ΓT (1) 0, (2) 5 × 10-11, (3) 10-10, (4) 2 × 10-10, (5) 5 × 10-10 mol cm-2; (b) monolayer thickness δ (1) 4, (2) 2, (3) 1, (4) 0.5 nm; (c) concentration of 1:1 electrolyte c (1) 10-3, (2) 10-2, (3) 10-1, (4) 1 mol dm-3; (d) standard potential of the redox couple φ 0, (1) -0.5, (2) 0, (3) 0.5, (4) 1 V. Default parameter values are as follows: zO ) 1, zR ) 0, c ) 0.1 mol dm-3, δ ) 1 nm, ΓT ) 2 10-10 mol cm-2, φ 0 ) 0.5 V, 1 ) 3, ) 78.5, T ) 298 K, λ ) 1/2.
inspection of eq 14, which also precludes any simple relationship with another important experimental quantity, the differential double layer capacity.11,17 5. Discussion and Conclusions In this contribution, we have developed the thermodynamics of electrodes, modified with a redox-active selfassembled monolayer, with particular reference to the aspect of electrocapillarity. This work was motivated both by the intrinsic physicochemical interest of the phenomenon and by the attention that is increasingly being lavished upon these types of modified electrodes, with a view to applications that utilize the potential-controlled wettability of their surfaces (“electrowetting”).18 These coatings have low relative permittivity compared to the aqueous solution and are to some extent impermeable to ions. In the preceding sections, the theory of electrocapillarity has been developed for adlayers, here assumed to be made up of alkanethiols, a fraction of which contain redox-active centers at both ends, one being the sulfur atom, through which each thiol binds to the metal, while the other could be any redox species. It should be pointed out, however, that the validity of the main results of this paper, eq 5 (or 8), essentially the Nernst equation, and eq 14, which is a generalized version of the Lippmann equation, would not be affected if the redox centers were embedded in the thiol layer. As a matter of fact, eq 14 applies more generally to situations where fixed charges are located at some distance from the electrode surface, irrespective of the mechanism by means of which these charges are generated. Here, we (17) Andreu, R.; Calvente, J. J.; Fawcett, W. R.; Molero, M. Langmuir 1997, 13, 1797. (18) Quillet, C.; Berge, B. Curr. Opin. Colloid Interface Sci. 2001, 6, 34.
Langmuir, Vol. 20, No. 3, 2004 873
only discuss oxidation by the metal, but the potential dependence of the surface pressure due to an organothiol layer with acidic end groups, say, could be determined in a way analogous to the model calculation presented in section 4, with the charge density q1 now governed by a local acid/base equilibrium and eqs 24 and 25 modified accordingly.12 Section 4 also deals with the issue of the hydrostatic, or kinetic, contribution (Π*) made by the thiol chains to the lateral pressure. For the rigid layers adsorbed onto solid electrodes, normally employed in the experimental studies reported so far, it is safe to assume that this contribution is practically equal to zero, because of the elastic, anisotropic response of the film to the electrostatic squeeze. This means that the layer only exerts a purely electrostatic transverse (Maxwell) pressure, i.e., minus the Maxwell stress, which completely overwhelms the contribution due to the diffuse part of the double layer.16 For SAMs that behave more isotropically, much like fluids, a situation that may be approached in the case of thiols chemisorbed on mercury, it has been argued here that the kinetic contribution may actually be equal to the Maxwell pressure. The applicability of one model or the other (or, more likely, something between these two extremes) would have to be decided by experiment, or using a more sophisticated (statistical mechanical) description of brushlike monolayers that are subjected to perpendicular compression. A further point of interest is that eq 15, with Π* ) 0, can also be applied to cases in which an electrode is coated with an insulating polymer film that either has ions specifically adsorbed onto it or acidic/basic groups covalently attached to it (redox-active groups are clearly not an option here in view of the high resistivity of the film). Experimentally, the potential dependence of the interfacial tension of a modified (solid) electrode in contact with an electrolyte solution is typically obtained by means of wetting angle (θ) measurements on a sessile drop of the electrolyte solution, which has one metal wire dipped in it, acting as a reference electrode, so as to allow control of the potential drop across the metal/solution interface.19 As an alternative, the Wilhelmy plate method has also been employed.16,20,21 Detailed analysis of the problem of charge distribution near the line of contact of the three phases (air, solution, and film-coated metal)22 has established that one can safely apply the usual Young-Dupre´ equation which, in this case, may be formulated as
cos θ ) cos θpzc + Π/γLV
(26)
where γLV is the (constant) surface tension of the solution. Some extra care should be taken in analyzing experimental results however, since these are known to be plagued by hysteresis effects that often show up as large differences between advancing and receding contact angles as the potential is shifted back and forth.23 This problem seems to be less severe with polymer coatings, though. Comparison of predictions based on eqs 21, 25, and 26 with experiments by Sondag-Huethorst and Fokkink21 and (19) Abbott, N. L.; Whitesides, G. M. Langmuir 1994, 10, 1493. (20) Sondag-Huethorst, J. A. M.; Fokkink, L. G. J. Langmuir 1992, 8, 2560. (21) Sondag-Huethorst, J. A. M.; Fokkink, L. G. J. Langmuir 1994, 10, 4380. (22) Kang, K. H. Langmuir 2002, 18, 10318. (23) Gorman, C. B.; Biebuyck, H. A.; Whitesides, G. M. Langmuir 1995, 11, 2242.
874
Langmuir, Vol. 20, No. 3, 2004
Mulder et al.
it that this time the total number of chemisorbed thiols, NT ) NO + NR, is variable, its value depending on electrode potential φM and electrochemical potential of the (fully reduced) thiols in solution, µ j t. With this prerequisite, the thermodynamic treatment proceeds along similar lines as in the case of an insoluble adlayer, so we will only highlight the differences with the theory expounded in section 3. First of all, the total number of thiols (regardless of state of oxidation) is given as Nt ) NT + Nt,bulk ) constant. A small, reversible, isothermal/isobaric change of state of the cell is now accompanied by a change in Gibbs energy equal to
(dG)T,p,{Nj},Nt ) γ dA + (µ j O + (n + 1)µ je - µ j t) dNO + (µ jR + µ je - µ j t) dNR + E( dQ′ (A1)
Figure 3. Theoretical estimates of equilibrium wetting angle variation upon oxidation of a ferrocene-terminated alkanethiol monolayer: (1) δ ) 1.6 nm, c ) 1 mol dm-3, θpzc ) 74° (conditions analogous to those in ref 21); (2) δ ) 2.0 nm, c ) 0.1 mol dm-3, θpzc ) 71° (conditions analogous to those in ref 19). Other parameter values are as follows: 1 ) 3, ) 78.5, φ 0 ) 1 V, ΓT ) 4 × 10-10 mol cm-2, T ) 298 K, λ ) 1/2. (Note: The wetting angle θ is calculated using eq 26, with γLV ) 72 mN m-1. SondagHuethorst and Fokkink21 obtained ∆θa ) 26° from the advancing-angle measurements, and ∆θr ) 6° from the receding-angle measurements, while Abbott and Whitesides19 obtained values of 28° and 15°, respectively.)
by Abbott and Whitesides,19 shows reasonable agreement of their results with theory (see Figure 3). The numerical results presented in section 4 suggest a strong influence of the redox couple on the potential dependence of interfacial tension, which sometimes manifests itself as a very sharp decrease in γ close to the standard potential of the couple, or what could be described as a shoulder on the electrocapillary curve (depending on the available potential window). The basic shape of the curve is a parabola, which signals the relative insignificance of the diffuse layer contribution, regardless of electrolyte concentration. This is probably only true as long as the SAM remains impermeable to ions and water molecules, a situation that is unlikely to persist at high field strengths. Depending on standard potential, two local maxima (separated by an electrocapillary minimum!) may appear on the curve, an effect that has so far not been reported. One could easily envision even more unusual behavior in the case of mixed monolayers, containing more than one redox couple, and/or dissociable end groups.24 Appendix: Derivation of Equation 14 for a Soluble Monolayer In this case, it is necessary to assume that the thiol moiety is more prone to becoming oxidized than the reduced form of the redox group at the other end of the molecule, i.e., the standard potential of the O/R couple is taken to be (much) higher than that of the thiol moiety. This is to ensure that oxidation of the R species will only occur from thiols that are already anchored to the metal so that no thiols with oxidized end groups are present in solution, which would have rendered the working electrode reversible with respect to the O/R couple. Thus, with this precaution, the electrode is again ideally polarizable, be (24) Hickman, J. J.; Ofer, D.; Laibinis, P. E.; Whitesides, G. M.; Wrighton, M. S. Science 1991, 252, 688.
which replaces eq 3. The condition for internal equilibrium then becomes (dG)T,p,A,{Nj},Nt,Q′ ) 0. Since NO and NR are now independent, this implies that
je ) µ jR + µ je ) µ jt µ j O + (n + 1)µ
(A2)
where the first equality leads again to eq 5 (Nernst equation), while the second expresses the condition for adsorption equilibrium. Noting that for a sufficiently large cell volume the µ jj and µ j t are essentially constant, eq A1 may be rearranged, after substitution of eq A2 and application of a Legendre transformation, to the exact differential
d(G - E(Q′)T,p,{Nj},Nt ) γ dA - Q′ dφM
(A3)
which, upon cross-differentiation, gives rise to25
-(dγ)T,p,{ µj j},µj t ) q′ dφM
(A4)
This is of course simply eq 11, subject to the conditions (A2). Substituting eq 12 for q′, and recognizing that eqs A2, for µ j t constant, produce the equalities ΓO(dµ j O - (n + 1)e dφM) ) 0 and ΓR(dµ j R - e dφM) ) 0, allows eq A4 to be written in the alternative form
j O + ΓR dµ j R + q dφM (A5) -(dγ)T,p,{µj j},µj t ) ΓO dµ Elimination of dµ j O and dµ j R by means of eq 7 shows that the right-hand side of eq A5 is equivalent to that of eq 13, and hence a breakdown of dγ into three contributions results just as in eq 14, as one would expect. The issue regarding the precise potential dependence of Π* (see section 4) is closely related to the fact that a theoretical expression for the potential dependence of q′ is as yet lacking, which prevents us from performing the integration of q′ dφM (i.e., the right-hand side of eq A4), which would have produced the function Π(φM) directly, at least for the case of a soluble monolayer (where ΓT depends on φM). Acknowledgment. This work was supported by the Spanish DGICYT under Grant BQU2002-02603. LA035364U (25) Mohilner, D. M. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1966; Vol. 1.
