J. Phys. Chem. 1996, 100, 11175-11183
11175
Second-Order Surface Phase Transitions in Electrode Kinetics A. A. Kornyshev,*,† A. M. Kuznetsov,†,‡ U. Stimming,† and J. Ulstrup§ Institut fu¨ r EnergieVerfahrenstechnik, Forschungszentrum Ju¨ lich GmbH (KFA), D-52425 Ju¨ lich, Germany, A. N. Frumkin Institute of Electrochemistry of the Russian Academy of Sciences, 117071 Moscow, Russia, and Chemistry Department A, The Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark ReceiVed: October 19, 1995; In Final Form: March 22, 1996X
A theory, based on the Landau-Ginzburg formalism, shows that remarkable singularities in currents of electrode charge transfer reactions occur near the critical potentials where second-order surface phase transitions take place. This happens if the fluctuations of the order parameter, which describes the transition, are exponentially coupled to the reaction system via fluctuating tunneling or activation barriers. Coupling depending on the sign of the order parameter is considered; the corresponding singularities in Arrhenius and Tafel plots for electrode currents are predicted, and the perspectives for their experimental investigation are discussed.
Introduction Surface phase transitions at the electrochemical interface, where the electrode potential can be varied in a broad interval, are a subject of current interest.1,2 This refers to transitions in ionic or molecular adsorbed layers and surface reconstruction. The effect of first-order transitions on electrode kinetics was studied both experimentally and theoretically (for a review see refs 3 and 4). The widely used model of two parallel currents, j ) j1θ + (1 - θ)j2, rests on the assumption that the current is composed of two contributions: the current through the adsorbate-free (j1) and adsorbate-covered (j2) sections of the surface, where θ is the coverage. If the adsorbate dynamics is much faster than the reaction rate, the problem is solved by the combination of two issues: (i) the equilibrium adsorption isotherm, θ(c,U), which relates the coverage θ to the concentration c of these particles in the bulk of the electrolyte, and the electrode potential U; (ii) the current-voltage plots for the partial currents, j1 and j2, taken from the measurements on adsorbate-free and on fully covered surfaces. Sharp changes of θ with the variation of U, T, and c will result in abrupt changes in the current. The problem becomes more intriguing for the structural transition in the adlayer induced by temperature or potential variation at unchanged coverage. The related changes in the symmetry of the adlayer (be it the discrete symmetry of the 2D-submonolayer films of adatoms, relative to the symmetry of the substrate surface, or the orientational symmetry in the self-assembled monolayers) are described by the order parameter, the phenomenological quantity, the physical meaning of which depends on the microscopic picture of the system (see below). If the coupling of the reaction rate with the order parameter affects the tunneling or/and activation barriers, the local instantaneous values of the current exponentially depend on the local momentary value of the order parameter. In the calculation of the mean current these exponentials must be averaged. This would make the results very different from those obtained under the assumption that the reaction rates depend only on the mean value of the order parameter. The latter could * Corresponding author. † Institut fu ¨ r Energieverfahrenstechnik. ‡ A.N. Frumkin Institute of Electrochemistry. § The Technical University of Denmark. X Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(95)03096-6 CCC: $12.00
be true only if the fluctuation times in the adsorbate layer were much shorter than the characteristic times of motion along the reaction coordinates and electron tunneling through the layer. This is, obviously, not the case for electron and proton transfer reactions. However, even for reactions where the rate-determining step is not electron or proton transfer across the interface but relatively slow transport of the reactant to the electrode, this assumption will inevitably break down at the critical point due to the critical slowing down of the relaxation dynamics in the adsorption layer. The theories operating with the averaged order parameter will then no longer be valid close to the critical point. A number of papers have been devoted to the relationship between the reaction rates and surface phase transitions on magnetic catalysts.5-11 All the reactions considered were adsorption limited, and the motion along the reaction coordinate was the motion of the adsorbate perpendicular to the electrode surface. The order parameter was the substrate surface magnetization, and its coupling with the reacting molecule was caused by the interaction of the magnetic moment of the molecule with the magnetization of the substrate. Borman et al.,9 Seifert and Dietrich,10 and Seifert and Wagner11 calculated the rate constant in the framework of Kramers’ theory.12 The energy characteristics of the reaction were assumed to depend on the aVerage Value of the order parameter; the friction coefficient was calculated via the time correlation function of the order parameter of the substrate, with no “feedback” effect of the spin of the reactant on the dynamics of the spin of the substrate. The scaling relationships10,11 or the Landau-Ginzburg theory9 was used to calculate the reaction rate as a function of temperature. Earlier, Hentschel and Procaccia8 used the transition state theory for the local rate constant with its consequent averaging over the spin configurations of the substrate, described within the framework of the lattice Ising model. The feedback effect was taken into account by the calculation of the partition function. However, in the averaging of the exponential in the transition probability the latter was linearized in the interaction between the reactant and substrate spins (weak coupling limit), which could not give rise to exponentially large effects. All the reports, except for ref 11, studied the case with no external magnetic field. Having a depressive effect on the second-order phase transitions, the latter was shown to remove singularities in the Arrhenius plots for the reaction rates.11 In the present communication, we consider the interplay © 1996 American Chemical Society
11176 J. Phys. Chem., Vol. 100, No. 26, 1996 between surface phase transitions and interfacial charge trasfer reaction rates, with a focus on electrified interfaces. Subject to the assumption that the coupling with the order parameter of the substrate enters either via the tunneling barrier for the electron (or proton, in the case of proton transfer reactions) or via the activation energy, or both, the averaging over the equilibrium distribution for the order parameter will refer to the whole exponential rate constant. The averaging is undertaken within the framework of the continuum Landau-Ginzburg functional and fluctuations treated within the Gaussian approximation above and below the critical temperature Tc.13 We consider systems in an external (electric) field, which can be varied by variation of the electrode potential. We shall thus try to describe not only the Arrhenius plots but also Tafel (current-voltage) plots for the electrode current. We shall mainly address the electrochemical charge transfer reactions across adsorbate layers, but the results may hold broader perspectives subject to other interpretations of the order parameter and the nature of the transition. We shall show that the averaging of the “nonexpanded” rate constants results in effects which are absent upon the substitution of the mean value of the order parameter into the exponentials. With no claims on a precise description of the currents at the critical points or correct values of critical exponents, we shall rather consider new qualitative effects in Arrhenius and Tafel plots, to be conceivably studied in electrochemical experiments. Indications relating to such effects in the literature are discussed. The Landau-Ginzburg form of the free energy functional is a simple but, of course, not the general one in the context of surface phase transitions.14,15 The meaning of the order parameter is straightforward only in the case of molecular adsorbate orientations, where the Landau-Ginzburg form (eq 1) should, probably, include the main qualitative effects. Preferential orientation of molecules at the electrode is taken into account by the linear term in the Hamiltonian, i.e. the “external field” term. At the potential of zero charge this field is usually nonzero due to residual preferential orientation, and the system of molecular adsorbates cannot show any temperature-driven orientational second-order transitions (the “external field” removes the degeneracy over the orientation sign). Due to the electrostatic interaction of molecular dipoles with the electric field of the electrode, the tendency for residual orientation will be removed at a certain “compensating” potential. (Such a picture implies the linear dependence of the “external field” on electrode potential.) This will make second-order phase transitions possible. The resulting singularities in electrode currents are the main subject of this work. A similar form can be used to describe the decomposition of the binary mixtures18 at the surface. The meaning of the order parameter here is the difference in surface coverage of the two species. The role of the “external field” is played by the difference in surface chemical potentials of the two species. The variation of the electrode potential can change this difference: when it vanishes, second-order transition becomes possible. This case may be typical of a nonlinear dependence of the “external field” on electrode potential. For example, when one of the species is an organic molecule, its free energy in the adsorbed state rises quadratically with the electrode potential due to the Frumkin electrostatic desorption mechanism.16 With suitable extensions one can apply the Landau-Ginzburg formalism to the simplified variants of order-disorder deconstruction of the missing row 1 × 2 structure in Au(110).18 However, already the deconstruction of x2 × x2 structures is not given by this form (it requires a vectorial order parameter and a tensorial form of the Hamiltonian18). Temperature-
Kornyshev et al. induced order-disorder transitions at unchanged coverage in adsorbate overlayers commensurate with the substrate19 may possibly be described by the Landau-Ginzburg Hamiltonian too. However, commensurate-incommensurate transitions (a widely occurring phenomenon in adlayers on single-crystal substrates) require more complicated Hamiltonians.18 The same applies to transitions between incommensurate structures or their melting.19 Roughening transitions need proper statistics of steps and domain walls.2,19 This can be reduced to continuum Hamiltonian models, but of more complicated type, such as the sine-Gordon model, describing the excitations in the lattice of solitons.15,20 We intend to exploit some of these more sophisticated Hamiltonians in future case studies of the effect of structural transitions (in adsorbate overlayers and surface reconstriction) on electrode reaction rates. As a start, however, we limit the analysis to the simplest consequences of the Landau-Ginzburg model. This will impose obvious limitations on the discussion of the experimentally observed effects. Phenomenological Theory Basic Equations. Landau Hamiltonian. We assume that the degree of freedom at the interface that may undergo secondorder transition and that may affect the current across the interface can be described by a scalar order parameter s, which depends on the lateral coordinate R. The free energy functional of this order parameter will be given the Landau-Ginzburg form:13
F[s(R)] ) F0 +
1 ∫dR dR′ Φ(R - R′) s(R) s(R′) + 2 C ∫dR[s(R)]4 - ∫dRhs(R) (1) 2
where the kernel of the quadratic term consists of the local and nonlocal parts,
Φ(R - R′) ) A(T) δ(R - R′) + χ(R - R′)
(2)
with A(T) > 0 at T > Tc and A(T) < 0 at T < Tc. In the context of an electrified interface, the “external field” h reflects the combined effect on the order parameter of the electric field and of the specific interaction with the electrode surface. Its function in the Hamiltonian is, however, independent of its physical nature. At h ) 0, the solution of the Euler equation gives nonzero 〈s(R)〉 at T < Tc degenerate in sign (ordered state), which vanishes at T > Tc (disordered state). A nonzero value of h fixes the sign of s, “killing” the second-order transition.15 The small h case will be most relevant for the electrode current, because peculiarities in Arrhenius plots (ln j versus 1/T) are expected near Tc only for small h and in “Tafel plots” (ln j versus h) for h near 0 at T ) Tc. The partition function is given by the functional integral
Z ) ∫D[s(R)]e-F[s(R)]/kBT
(3)
Coupling with Electrode Current. We adopt the adiabatic approximation which suggests that the charge transfer kinetics follows instantaneously the fluctuations of the order parameter. The coupling with these fluctuations will be given by
j(R) ) j0eas(R)
(4)
Here a is the coupling constant and j0 the current independent of s. This type of coupling corresponds, for example, to the case when the electron tunnel barrier depends on the sign of the order parameter. For the case of oriented molecular dipoles
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J. Phys. Chem., Vol. 100, No. 26, 1996 11177
the sign of a determines the orientation favorable for tunneling. For the case of the mixture of two types of molecules the sign of a reflects the difference in tunneling through the adsorbed species of a given type. We call this type of coupling Vectorial in constrast to scalar coupling, given by j(R) ) j0eb[s(R)]2. In the latter case, the current is influenced by the mere existence of order, and the sign of the coupling constant b determines the “affinity” or “phobicity” to the order. This type of coupling is considered in our following report. The mean current is calculated as
j ≡ 〈j(R)〉 ) j0Z1 ∫D[s(R)]eas(R)e-F[s(R)]/kBT
(5)
For the Landau-Ginzburg form of F[S(R)] this functional integral has no exact close form expression. The approximate way to perform the integration is the renormalization group method, which works right at the critical point, or the, so-called, Gaussian approximation, which is good at the wings around the critical point.13 The Gaussian approximation exploits the expansion of the Hamiltonian up to the accuracy of the secondorder terms in the order parameter around the ground state solutions which are different for temperatures below and above the critical one. It gives the simplest, reference solution, which however breaks down right at the critical point. We first consider the simplest case of T > Tc with no imposed external field. Then we study the case of T < Tc, where the complexity of solution is the same with or without a field. Finally we extend the results to the case of T > Tc (indeed, for nonzero h we get a universal expression for the current, while separate consideration below and above Tc is in fact needed only for h ) 0). Gaussian Approximation at T > Tc without External Field. Here we neglect the fourth-order term in the Landau Hamiltonian (1), so that
F[s(R)] ≈ F0 +
1 ∫dR dR′ Φ(R - R′) s(R) s(R′) (6) 2
{
j ) j0Z-1∫∏d2sq exp aA-1/2∑sqeiqR q
B
(7)
(where A is the surface area) and for the nonlocal kernel χ,
χ(R - R′) ) ∑χ˜ (q)eiq(R-R′)
we obtain
{
j ) j0Z-1∫∏duq dνq exp aA-1/2∑(uq + iνq) q
}
Φ(q)(uq2 + νq2) ∑ 2k T q B
(13)
Integration over νq gives unity (which could not be anything different since j must be real). Integration over uq finally gives j ) j0 exp{(a2kBT/2A)∑q[Φ(q)]-1}. In the limit of a large surface area, A f ∞, we may replace the summation by integration in which we must introduce an upper cutoff qc ≈ 2π/l, where l is greater than or equal to intermolecular distances along the interface. Finally,
{
j ) j0 exp
}
a2kBT qc dq q ∫ 4π 0 Φ(q)
(14)
Gaussian Approximation at T < Tc with External Field. Below Tc one must keep the fourth-order term
F[s(R)] ≈ F0 +
1 2
∑q [A(T) + χ˜ (q)]|sq|2 + ∑sqsq′sq′′s-q-q′-q′′ - hs0
(15)
The minimum of F corresponds to a uniform solution, s(R) ) js, i.e. jsq ) δq0xA js, where js is determined by the equation
js[A(T) + 2Csj2] ) h
(16)
For small h
where
∫ dR e
q
1
(8)
q
(12)
[d2sq ≡ duq dνq, sq ) uq + iνq (uq ) u-q, νq ) -ν-q)],
2
q
}
Rewriting the integral in terms of the real and imaginary components of the variable s
C
s(R) ) A-1/2∑sqeiqR, q ) {qx,qy}
χ˜ (q) ≡ A
Φ(q)|sq|2 ∑ 2k T q 1
Using Fourier expansion for s(R),
-1
q
-iqR
χ(R), χ˜ (0) ) 0
js ≈ (9)
x
-
A(T) h 2C 4A(T)
(17)
Expansion of F near js, up to the terms of second-order, reads
we may write
F[s(R)] ≈ F0 +
1 2
∑Φ(q)|sq|2
(10)
F[s(R)] ≈ F(sj) +
q
1 2
[4Csj2 + h/sj](s0 - js0)2 +
1
Φ h (q)|sq|2 ∑ 2 q*0 (18)
where
Φ(q) ) A(T) + χ˜ (q)
(11)
with Φ(0) ) A(T) Mean Current. The substitution of eqs 9 and 10 into eq 5 gives
where
Φ h (q) ) [4Csj2 + (h/sj) + χ(q)] Substitution of eq 18 into eq 5 gives
(19)
11178 J. Phys. Chem., Vol. 100, No. 26, 1996
{
Kornyshev et al.
j ) j0Z-1easj∫∏d2sq exp aA-1/2{(s0 - js0) + ∑ sq} q
1 2kBT
q*0
} | |}
h (q)|sq|2} {[4Csj2 + h/sj](s0 - js0)2 + ∑ φ q*0
{
) j0Z-1easj∫∏d2sq exp aA-1/2∑sq q
q
1
Φ h (q) sq 2 ∑ 2k T q B
After the integration we get
{
}
a2kBT qc dq q j ) j0e exp ∫ h (q) 4π 0 Φ asj
(20)
Thus, the difference of the result for the mean current from the case of T > Tc is in the appearance of the factor easj and the replacement of Φ(q) by Φ h (q). Gaussian Approximation at T > Tc with Nonzero Magnetic Field. It is easy to show that eq 20 is valid for this case, as well. In other words, the results for h * 0 are the same for T > Tc and T < Tc; the difference is only in the value of js, determined by eq 16. Expressions for the Mean Current To use the formulas obtained, an explicit form of χ(q) must be invoked. Gradient expansion of the free energy (a standard approximation in the theory of critical phenomena) gives
χ˜ (q) ) gq2
(21)
Another standard approximation will be used for A(T) which is the Landau expansion near Tc,
A(T) = A0
expression for the mean current, valid below and above Tc,
T - Tc Tc
{ }
j ) j0
qc g A0
a2kBT/16πg
{
}
Tc |T - Tc|
a2kBT/16πg
{θ(T - Tc) + 2-a kBT/16πgθ(Tc - T)} (23) 2
This result is inaccurate right at Tc, where the Gaussian approximation breaks down. It reproduces, however, a general qualitative tendency: an asymmetric peak in the current around Tc. In units of Tc, i.e. if we introduce a reduced temperature t ≡ T/Tc, eq 23 takes a compact form,
j ) pβt|t - 1|-βt{θ(t - 1) + θ(1 - t)2-βt} j0
(23′)
where we have introduced the definitions
p≡
{
(22)
Mean Current at h ) 0. Substitution of eqs 21 and 22 into eqs 14 and 20 gives 2
Figure 1. Spike-addition to the Arrhenius plot (the case of zero external field), given by eq 23′ for some typical values of the parameters p and β: (a) β ) 1, p ) 4 (1), 20 (2); (b) p ) 20, β ) 1 (1), 2 (2).
qc2g a2kBTc , β≡ A0 16πg
(24)
The “Arrhenius plot”, log(j/j0) versus 1/t, is shown in Figure 1. The value of β scales as a2. The stronger the coupling to the order parameter, the greater the effect (Figure 1b); the nature of the dependence on p is less clear. Mean Current at h * 0. For h * 0 there is formally a unified
A0 j ) easj p 2 j0 4Csh + h/sh
}
βt
(25)
where js is the solution of eq 16. However, instead of trying to solve the cubic equation (16) for js as a function of h, one may benefit from the fact that eqs 25 and 16 determine a parametric dependence of j on h, with js playing the role of the running variable. A similar trick may be used to plot the temperature dependence of the current at a given h: for the running variable js one finds the temperature from eq 16 and inserts it into eq 25. With the help of these equations, two types of curves have been calculated. Figure 2 accumulates the Arrhenius curves at finite values of h (which remove the divergence of the current at t ) 1). Plots of log(j/j0) versus h, related to electrode potential, are shown in Figure 3 for a few typical temperature values. When T * Tc, the current does not diverge at any h. For T ) Tc divergence occurs only at h ) 0 (cf. Figure 1). The families of curves shown in Figures 2 and 3 formally illustrate a few typical situations with regard to the relative values of parameters. To take a step further and plot the current dependence on the overpotential, we must specify the type of electrode process, the meaning of the order parameter and its coupling to the reaction rate, and the relationship between h and the overpotential. Examples The results derived above rest on the Landau-Ginzburg Hamiltonian and “vectorial” coupling of the tunnel barrier to
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J. Phys. Chem., Vol. 100, No. 26, 1996 11179
Figure 2. Arrhenius plots for small external fields, calculated by eqs 25, 16, and 29. E/kBT ) 23, p ) 2.25, 2C/A0 ) 0.1. (a) Negative field (h/A0 ) 0.001); (b) positive field (h/A0 ) -0.001). Curves correspond to the variation of the coupling constant a (and thereby β (∼a2): (s) a ) -5.46, β ) 2.25; (‚‚‚) a ) -3.64, β ) 1; (- - -) a ) β ) 0. Graphs: upper, a < 0; lower, a > 0. (c) Formal variation of the field at unchanged activation energy for a “cathodic process” (a ) -3.64), β ) 1. Curves: h ) -0.002 (s), -0.004 (‚‚‚), -0.008 (- - -).
11180 J. Phys. Chem., Vol. 100, No. 26, 1996
Kornyshev et al. by the electrode potential. At the compensating value of the potential the effective value of h thus vanishes, and a secondorder phase transition therefore still occurs at T ) Tc. Assume that h is related to the overpotential η as
h ) h0 - γη
(26)
The sign of γ is defined by the molecular interpretation of the model. When molecules on an uncharged electrode have a preferential orientation by their negatively charged end (oxygen) toward the electrode, then the increase of the positive overvoltage will stabilize this orientation, which means γ < 0. For this sign of γ, the negative overvoltage will tend to turn molecules around, compensating the value of h0 and making h negative; the value
η* ) h0/γ
(27)
defines the compensating overvoltage. On the other hand the electrode potential also affects j0. For this dependence we adopt the Tafel law
j0 ) j* exp(Reη/kBT)
Figure 3. Tafel curves for a cathodic process, calculated via eqs 25, 16, and 28. R ) -0.5, a < 0, γ < 0 (γ ) -1), h0/kBTc ) 2, for temperatures near Tc: T/Tc ) 1.01 (s), 0.99 (‚‚‚). Variation of coupling constants: (a) a ) β ) 0; (b) a ) -1, β ) 1; (c) a ) -2, β ) 4.
the order parameter. The Gaussian approximation is used for the treatment of fluctuations. For systems that fulfill these symmetry constraints they provide a simple but rather general basis for the description of reaction currents at the wings on both sides of the critical point. We illustrate this by some examples where such notions as the “order parameter”, the “external field”, and the coupling to the electrode current take a definite meaning. We then discuss a few experimentally studied cases, reported in literature. Model Constructions. Orientational Model. Consider a monolayer of dipolar molecular adsorbates on the electrode surface. Orientation of the molecules is characterized by the order parameter s(R): positive values will correspond to the orientation of molecules by its negatively charged end (say oxygen) toward the electrode. Let such orientation be preferential. This will be reflected in the Landau-Ginzburg Hamiltonian by an external field, h0, which polarizes the layer. In the absence of such a field the layer could undergo an orientational order-disorder transition, being ordered below Tc and disordered above Tc. The preferential orientation (external field), however, destroyes transition, but the fluctuations of the order parameter become stronger, the closer the system is to Tc and the smaller the orienting field, h0. Fluctuations will be very large and diverge if the spontaneous field is exactly compensated
(28)
where R > 0 for anodic processes (e.g. oxygen evolution) and R < 0 for cathodic processes (e.g. hydrogen evolution). In both cases we assume that the normal overvoltage region, and |R| ≈ 0.5 prevail. The sign of the coupling constant is also determined by the molecular view of the interface and the direction of the electrode process. If the electron transfer from the electrode to the reactant (cathodic reaction) is impeded by the molecular orientation with the negative end toward the electrode, then a < 0 for the cathodic process. If electron transfer from the reactant to the electrode (anodic reaction) is facilitated by this preferential orientation of the molecules, then a > 0. Situations when electron tunneling from the electrode may be slightly facilitated by a tightly packed layer of oxygen-toelectrode-oriented water molecules are not excluded, however, since this organization contributes an additional potential drop for the electron. The sign of a cannot thus be predefined even in this particular model. In rather general terms, it can be determined by different effects on the structure of the resonance integral and its dependence on the orientation of the molecules. Special conditions can be envisaged, such as the contribution to the current of surface states, quenching of these states by molecular adsorption, and the effect of reorientation on quenching. The sign of a should be referred to detailed quantumchemical information about the electronic structure of adsorbed molecules. For illustrative purposes, we tentatively assume a < 0 for anodic reactions and a > 0 for cathodic reactions. Structural Models. In the case of atomic or ionic adsorbates (e.g. underpotential deposition of monolayers of metal ions on foreign metal substrates or adsorption of anions) the meaning of the order parameter is not associated with “orientation”. Instead, the order parameter characterizes the presence of a certain symmetry which is absent when the order parameter is zero. For zero external field the state is degenerate in the sign of the order parameter, but a finite positive h stabilizes the positive value of the order parameter. Applying the overpotential, we may, in principle, encounter both positive and negative γ. For example when γ < 0, a positive overpotential stabilizes the symmetry characterized by the positive order parameter, but when γ > 0, this state becomes less favorable. The sign of a will be determined by the difference between the tunnel barriers of the two states of the surface.
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J. Phys. Chem., Vol. 100, No. 26, 1996 11181
Figure 4. Tafel curves for a cathodic process, calculated via eqs 25, 16, and 28. R ) -0.5, a < 0 with γ > 0 (γ ) 1), h0/kBTc ) 2, for temperatures near Tc: T/Tc ) 1.01 (s), 0.99 (‚‚‚). γ ) -1. Variation of coupling constants: (a) a ) β ) 0; (b) a ) -1, β ) 1; (c) a ) -2, β ) 4.
Illustrations. A complete classification of the current-voltage and current-temperature curves in the space of all the dimensionless parameters which appear in the resulting expressions is not warranted. We show, however, a few representative effects of the coupling between the reaction rate and the interfacial subsystem, which undergoes phase transitions in the form of basic Arrhenius (Figures 1 and 2) and current-voltage (Figure 3-6) plots. The main purpose of these figures is to demonstrate the role of variation of the coupling constant a. Figures 1 and 2 collect Arrhenius plots with the activation law adopted for the exchange current:
j0 ) j00 exp(-E/kBT)
(29)
Figures 3-5 show Tafel plots for anodic and cathodic processes for different temperatures around Tc and different coupling constants. The main observation from these figures is that the peak in the current around Tc and near the compensating values of the potential is a stable feature of the theory. A general (but rather obvious) conclusion is that the larger the effective external field, the smaller and smoother the singularity on the Arrhenius plots. Similary, the singularity in the Tafel plot becomes weaker, the
Figure 5. Tafel curves for an anodic process, calculated via eqs 25, 16, and 28. R ) 0.5, a > 0, γ < 0, h0/kBTc ) 2, for temperatures near Tc: T/Tc ) 1.01 (s), 0.99 (‚‚‚). γ ) -1. Variation of coupling constants: (a) a ) β ) 0; (b) a ) -1, β ) 1; (c) a ) -2, β ) 4.
farther we are from the critical temperature. Of course, the weaker the coupling constant a, the less pronounced are the singularities in the Arrhenius and Tafel plots. The interplay of this singularity with the background slopes determined by the activation energy or the transfer coefficient R, or the type of coupling of the order parameter with the external field, must be studied separately in each case. One aspect is worth being noted. The coupling of the order parameter to the reaction rate not only contributes the singularity in the current near the critical point but also notably affects the wings around this singularity. For instance if the negative value of the order parameter is stabilized by a negative overpotential, but a is positive, then below Tc this state, unfavorable for the current, will be stabilized. The current will then be lower than in the case of a reaction not coupled with the surface order parameter (see lower graph of Figure 2a). Since a positive a is, presumably, less typical for anodic processes (see, however, counterarguments in the preceding section), this case would be
11182 J. Phys. Chem., Vol. 100, No. 26, 1996
Figure 6. Tafel curves for an anodic process, calculated via eqs 25, 16, and 28. R ) 0.5, a > 0 with γ > 0 (see discussion in text), h0/kBTc ) 2, for temperatures near Tc: T/Tc ) 1.01 (s), 0.99 (‚‚‚). γ ) 1. Variation of coupling constants: (a) a ) β ) 0; (b) a ) -1, β ) 1; (c) a ) -2, β ) 4.
rare and the Arrhenius plots will look rather like the upper graph in Figure 2a. Experimentally Investigated Systems. We are not aware of experimentally studied cases where surface orientational order-disorder transitions are solidly detected and their influence on the charge transfer rate unambiguously registered. Structural potential-induced order-disorder transitions in the UPD (underpotential deposition) monolayers and their electrocatalytic effects were reviewed in ref 21. With the reservations expressed in the Introduction, and with a view on qualitative conclusions, we may attempt to map such transitions on the Landau-Ginzburg formalism, avoiding direct microscopic interpretation of the order parameter (which would be particularly crude for transitions involving incommensurate phases, but may be acceptable for the case of commensurate order-disorder transitions. The electrode potential would primarily affect the adionsubstrate interaction. Larger negative potentials will increase
Kornyshev et al. the interaction of UPD cations with the substrate and thereby may be coupled to the field h in the Landau-Ginzburg Hamiltonian, stabilizing the commensurate order. Note, however, that larger negative potentials may also increase the coverage, and thereby the inter-adcation interactions, which makes the commensurate structures less favorable. Thus, the dependence of h(η) may be nonmonotonous, but somehow, there could be a particular value of η(T) where the effective value of h vanishes and the continuous transition will take place. The coupling to the reaction rate depends on the particular reaction. For instance, the UPD-deposited Pb on Au(111) has a catalytic effect on the H2O2 reduction. The strength of the effect varies with the structure of the adlayer. Maximum effect appears when the surface adions are spaced in a way which facilitates the dissociative adsorption of O2 (e.g. by bridge adsorption complexes which induce stretching and rupture of the O-O bond22). The shift of Pb adions from such favorable arrangements may drastically diminish the catalytic effect. This may happen with different structural transitions, including simple order-disorder transitions. Sharp peaks in the current-voltage plots have been observed near a hypothetical change of the superlattice structure (see Figure 11 of Ref 21). A few other prospective systems of this kind, with catalytic or inhibiting activity, were also discussed in ref 21. Note, however, that all the examples of this kind, mentioned in ref 21, refer to reactions which are not simple outer-sphere redox processes, but more complex, seemingly adsorption-controlled reactions. The mapping of these processes on the formalism developed above would most likely require modifications in which the order parameter fluctuations are also coupled with the activation energy. Investigations of simple redox reactions on electrodes with well-defined structural transitions in adsorbate superlattices would be extremely interesting in search of singularities in the current-voltage or Arrhenius plots near the “critical” potentials or temperatures. Such electrodes could be the Cu layers on Au(111),23 CO on Rh(111)24 and Pt(111),25 Pb on Ag(100),26 sulfates,27 Cl,Br28 and Tl29 adlayers on Au(111), and anions adsorbed on Pt(111) from sulfuric acid solutions.30 One other important implication is held by the clear observations of a singularity in the Arrhenius plot for the rate of the electrochemical hydrogen evolution or the Fe2+/Fe3+ reaction in HClO4‚5.5H2O near the bulk melting point of the electrolyte (∼228 K).31 The singularity, however, does not appear as a peak, but rather like a smeared kink. Across the kink there is virtually no change of the activation energy, but the preexponential factor rises jumpwise (∼5 times for the Fe2+/Fe3+ reaction and some 10 times for the hydrogen evolution) upon freezing. The height of a kink is practically not affected by overvoltage. Whether the electron tunnel barrier decreases upon the phase transition at the surface (e.g. due to a more favorable position of the donor-acceptor sites) or simply the effective environmental vibration frequency increases remains an open question. The phase state of the bulk is not the most crucial factor in these electrode reactions. However, the bulk phase transition is reflected in the state of the interface,32 and this change seemingly increases the preexponential factor in the transition probability. Bulk melting is a first-order transition. Nevertheless, at the surface the order parameter may change abruptly or continuously33 (“second-order-like”) subject to criteria, related to the boundary condition.33 The absence of any peak at that point indicates that the fluctuations of the order parameter are not important, and it is likely that we face an abrupt first-order transformation at the surface, which induces a jumpwise variation of the electrode current.
Second-Order Surface Phase Transitions Conclusion and Outlook If the orientation of molecules affects the current passing across a monolayer, say, via the tunneling factor, fluctuations near the critical point will give rise to a current peak, because the tunneling will be stimulated by the most favorable fluctuation. As a result, we would see spikes in the Arrhenius and Tafel plots (with a jump in the preexponential factor across the compensating potential). Such a clear picture would, however, take place when the electrode potential affects, primarily, the molecular reorientations, but has only minor influence on other system parameters (e.g. molecular adsorption energy, as in the case of organic inhibitors16). The lateral structure of the layer and the coverage may then be affected by potential, and the resulting behavior will be much more complicated. In situations not related with reorientations but generally with the symmetry of the interfacial layers, it is likely that the effects can be mapped on a similar theoretical scheme, and the effects discussed above could be observed. We have discussed only a few experimental examples which resemble the effects discussed. Availability of these systems and the simple formalism might urge a more systematic search of singularities in the current, induced by surface phase transitions. Broader varieties of examples could also become an impetus for a more sophisticated theory, which goes beyond the Gaussian approximation. Note that similar singularities in the tunneling current might be encountered in in situ STM, if the tip does not perturb the structure of the surface. The natural concluding question would be whether the putative spikes in the electrode current can be exploited to accelerate electrode processes. In other words, whether the adsorbed layers near the critical points of surface phase transitions can be brought to exhibit systematic pronounced catalytic effects on electrode kinetics.21,34 This perspective may hold also in broader terms, such as conceivable charge transfer enhancement across biological membranes and macromolecules induced by structural transitions.35,36 If so, coupling between the structural phase transitions and charge transfer kinetics could prove crucially important not only for electrocatalysis and longrange electron tunneling but also for comprehensive bioenergetic cell functioning. Acknowledgment. Thanks are due to Udo Seifert and Alexei Ioselevich for useful and stimulating discussions. This work was made possible due to the grant of the Volkswagen Stiftung for the cooperation between the Institut fu¨r Energieverfahrenstechnik, Forschungszentrum Ju¨lich GmbH, and the A. N. Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Moscow. A.M.K. and J.U. acknowledge the support from the EU INTAS program. J.U. also thanks the Danish National Research Council for financial support. References and Notes (1) Buess-Herman, C. In Adsorption of Molecules at Metal Electrodes; Lipkowski, J., Ross, P. N., Eds.; VCH: New York, 1992, p 77.
J. Phys. Chem., Vol. 100, No. 26, 1996 11183 (2) Kornyshev, A. A.; Vilfan, I. Electrochim. Acta 1995, 40, 109. (3) Damaskin, B. B.; Petrii, O. A. Introduction to Electrochemical Kinetics; Vysshaya Shkola: Moscow, 1975 (in Russian). (4) Lipkowski, J. In Modern Aspects of Electrochemistry; Conway, B. E., Bockris, J. O’M., White, R. E., Eds.; Plenum: New York, 1992; Vol. 23. (5) Suhl, H. In Physical Basis for Heterogeneous Catalysis; Drauglis, E., Jaffee, R. I., Eds.; Plenum: New York, 1975; p 427. (6) Lagos, R. E.; Suhl, H. Surf. Sci. 1978, 78, L801. (7) Galleani d.Agliano, E.; Huberman, B. H. Solid State Commun. 1979, 32, 1255. (8) Henschel, H. G. E.; Procaccia, I. J. Chem. Phys. 1982, 77, 5234. (9) Borman, V. D.; Pivovarov, A. N.; Troyan, V. I. JETP Lett. 1984, 39, 556. (10) Seifert, U.; Dietrich, S. Europhys. Lett. 1987, 3, 593. (11) Seifert, U.; Wagner, H. Phys. ReV. Lett. 1989, 63, 287. (12) Kramers, H. A. Physica 1940, 7, 284. (13) Ma, S.-K. Modern Theory of Critical Phenomena; Benjamin: London, 1977. (14) Phase Transitions and Adsorbate Restructuring at Metal Surfaces/ The Chemical Physics of Solid Surfaces; King, D. A., Woodruff, D. P., Eds.; Elsevier: Amsterdam, 1994. (15) Lyuksyutov, I.; Naumovets, A. G.; Pokrovsky, V. Two-Dimensional Crystals; Academic Press: Boston, 1992. (16) Damaskin, B. B.; Petrii, O. A.; Batrakov, V. V. Adsorption of Organic Compounds on Electrodes; Plenum: New York, 1971. (17) Bak, P.; Paszuski, M. In ref 16, p 1. (18) Naumovets, A. G. In ref 16, p 163. (19) Bernasconi, M.; Tosatti, E. Surf. Sci. Rep. 1993, 17, 363. (20) Nijs, M.den In ref 16, p 115. (21) Ju¨ttner, K. Electrochim. Acta. 1986, 31, 917. (22) Griffiths, J. S. Proc. R. Soc. A 1956, 235, 73. (23) For review see: D.Abrun˜a, Electrochemical Interfaces, Modern Techniques for in Situ Interface Characterization; VCH: Weinheim, 1991. Blum, L.; Hackaby, D. H. In Microscopic Models of Electrode-Electrolyte Interfaces; Halley, J. W., Blum, L., Eds.; The Electrochem. Soc. Inc.: Pennington, 1993; Vol. 93-5, 232. (24) Yau, S.-L.; Gao, X.; Chang, S.-C.; Schardt, B. C.; Weaver, M. J. J. Am. Chem. Soc. 1991, 113, 6049. (25) Yau, S.-L. Ph.D. Thesis, Purdue University, 1990. (26) Schmidt, U.; Vinzelberg, S.; Repphun, G.; Staikov, G.; Lorenz, W. J. Electrochemically Deposited Thin Films; The Electrochem. Soc. Inc., Proceedings; Penington, in press. (27) Edens, G. J.; Gao, X.; Weaver, M. J. Electroanal. Chem. 1994, 375, 350. Gao, X.; Edens, G. J.; Weaver, M. J. Electroanal. Chem. 1994, 376, 21. (28) Magnussen, O. M.; Ocko, B. M.; Adzic, R. R.; Wang, J. X. Phys. ReV. B 1995, 51, 5510. (29) Adzic, R. R.; Ocko, B. M.; Wang, J. Electrochim. Acta 1995, 40, 83. Polewska, W.; Wang, J. X.; Ocko, B. M.; Adzic, R. R. J. Electroanal. Chem. 1994, 376, 35. Wang, J.; Adzic, R. R.; Ocko, B. M. J. Phys. Chem. 1994, 98, 7182. (30) Funtikov, A. M.; Linke, U.; Stimming, U.; Vogel, R. Surf. Sci. 1995, 324, L343. (31) Stimming, U. In Electrochemical Surface Science; Soriaga, M., Ed.; ACS Symposium Series 378; American Chemical Society: Washington, DC, 1988; p 275. (32) Binder, K. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1983; Vol. 8, p 1. Dietrich, S. Ibid. 1988; Vol. 12, p 1. (33) Lipowski, R. Phys. ReV. Lett. 1982, 49, 1575. (34) Note that not only the current peaks but also mean square current fluctuations are large near the critical point. Current fluctuations can be calculated in the same manner and will be considered elsewhere. (35) Mouritzen, O.; Jorgensen, K. BioEssays 1992, 14, 129. (36) Kuznetsov, A. M.; Ulstrup, J. In Protein Structure: Molecular and Electronic ReactiVity; Austin, B., Bukhs, E., Chance, B., DeVault, D., Dutton, P. L., Frauenfelder, H., Goldanskii, V. I., Eds.; Springer: New York, 1987; p 309.
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