Langmuir 1995,11, 4583-4587
4583
Rigorous Analysis of Low-Temperature Phases in a Model for Underpotential Deposition of Copper on the (111) Surface of Gold in the Presence of Bisulfate Dale A. Huckaby" Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129
Lesser Blum Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931-3343 Received June 8, 1995. I n Final Form: September 11, 1995@ A model for two-component coadsorption is introduced and used to study the structure of the phases which occur at low temperature during the underpotential deposition of copper on the (111)surface of gold in the presence of bisulfate ions. The structure of the voltage dependent ground state configurations is rigorously obtained, and the F'irogov-Sinai theory is used to prove the existence oflow-temperaturephases in the model which are similar in structure to the ground state configurations. The sequence and structure of the low-temperature phases which occur in the model as the voltage is changed are essentially the same as those obtained experimentally at ambient temperatures during the underpotential deposition process.
I. Introduction In the present paper we rigorously obtain the ground state structures and low temperature phases in a model for the underpotential deposition of copper on the (111) surface of gold in the presence of bisulfate. A constant temperature plot of current versus voltage, or voltammogram, of this deposition process exhibits two current spikes.' These spikes are associated with firstorder phase transitions occurring a t the fluid-solid interface during the deposition.2 We previously introduced a statistical mechanical model for underpotential deposition which could be used to generate a model voltammogram of the deposition process.2-8 The model is based on another model for the adsorption of hard ~pheres,~-l* that model being modified so that the spheres are charged and represent ions in the deposition process. The three-dimensional one-componentadsorption model, in which the hard spheres are in contact with a planar wall of sticky sites, has been shown to be mathematically
* To whom correspondence should be addressed. Abstract published in Advance A C S Abstracts, November 1, 1995. (1)Kolb, D. M.; Al Jaaf-Golze, K.; Zei, M. S. Dechema Monographien; VCH: Weinheim, 1986; Vol. 102, p 53. (2) Huckaby, D. A.; Blum, L. J . Electroanal. Chem. 1991,315,255. (3) Blum, L.; Huckaby, D. A. J . Chem. Phys. 1991,94, 6887. (4) Huckaby, D. A.; Blum, L. In X-Ray Methods in Corrosion and Interfacial Electrochemistry; Davenport, A., Gordon, J. G., II., Eds.; Electrochemical Society: Pennington, NJ, 1992; p 139. ( 5 ) Blum, L.; Huckaby, D. A. In Microscopic Models of ElectrodeElectrolyte Interfaces; Halley, J. W., Blum, L., Eds.; Electrochemical Society: Pennington, NJ, 1993; p 232. (6) Blum, L.; Huckaby, D. A. J . Electroanal. Chem. 1994,375, 69. (7) Huckaby, D. A,; Blum, L. In Diffusion Processes: Experiment, Theory, and Simulations; Pqkalski, A., Ed.; Springer-Verlag: Berlin, 1994; p 213. (8) Blum, L.; Legault, M.; Turq, P. J . Electroanal. Chem. 1994,379, 35. (9) Rosinberg, M. L.; Lebowitz, J. L.; Blum, L. J . Stat. Phys. 1986, 44, 153. ( 10)Badiali, J . P.; Blum, L.; Rosinberg, M. L. Chem. Phys. Lett. 1988, 129, 149. (11)Huckaby, D. A.; Blum, L. J . Chem. Phys. 1990,92, 2646. (12)Blum, L.; Huckaby, D. A. In Liquids, Freezing, and Glass Transition;Hansen, J. P., Levesque, D., Zinn-Justin,J., Eds.; Elsevier: Amsterdam, 1991; p 983. (13) Huckaby, D. A.; Blum, L. In Condensed Matter Theories; Blum, L., Malik, F. B., Eds.; Plenum: New York, 1993; Vol. 8, p 637. (14) Huckaby, D. A.; Blum, L. J . Chem. Phys. 1992,97, 5773. @
equivalent to a two-dimensional lattice gas. The fugacity of a particle in the equivalent lattice gas is equal to the product of a stickiness parameter times the hard sphere contact density in the adsorption model, and the Boltzmann factor for a n n-body interaction energy in the equivalent lattice gas is equal to an n-particle contact correlation function in the adsorption model. In the one-component adsorption model for the underpotential deposition process, the hard spheres are charged and represent a single type of ion. The contact density and stickiness parameter are then voltage dependent, as is the fugacity in the equivalent lattice gas. By assuming a reasonable form for this voltage dependence, a model voltammogram can be constructed. In particular, for a constant rate of change ofthe voltage, the current intensity in the model voltammogram is proportional to the derivative with respect to voltage of the adsorption isotherm of the equivalent lattice gas. Although the underpotential deposition of copper on (111)gold in the presence ofbisulfate is a two-component system which involves voltage dependent coadsorption of the copper and bisulfate, we were able in earlier work to model this process with the one-component hard sphere adsorption model by focusing on the adsorption of only one type of ion, the effect of the other type of ion on the process being approximated as resulting in a voltage dependent change in the adsorbate surface. In our first treatment of the system, we focused on the deposition of the copper ions, the effect of the bisulfate being to alter the available lattice of adsorption sites for copper d e p o ~ i t i o n .Because ~ ~ ~ of packing considerations, a t high voltage the bisulfate ions adsorb on one triangular sublattice of the triangular lattice of adsorption sites, leaving a honeycomb lattice of sites available for copper deposition. As the potential is made less positive, the copper ions are deposited on the honeycomb lattice, resulting in a first-order phase transition and two-thirds coverage of the (111)gold surface. As the potential becomes less positive, copper replaces the adsorbed bisulfate, undergoing a first-order phase transition and completing the copper monolayer. The resulting model voltammogram contains two current spikes with the same relative areas as the spikes in the experimental voltammogram.
0743-746319512411-4583$09.00/0 0 1995 American Chemical Society
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4584 Langmuir, Vol. 11, No. 11, 1995
Some of the structure of the experimental voltammogram, however, is not reproduced by the above approximation. In particular, the first current spike is preceded by a "foot" which does not appear in the model voltammogram. Indeed, experiments indicate that the bisulfate begins to desorb at a potential above that a t which copper deposits.15J6 The copper and bisulfate then coadsorb onto the gold surface near the first current spike. We envisioned that the voltammogram foot is a consequence of this rather complicated coadsorption process. In a second treatment we used the one-component adsorption model and the exact isotherm of the hard hexagon lattice gas to treat the bisulfate adsorption in this region, the effect of the copper coadsorption being treated in a Langmuir fashion, coupled to the bisulfate adsorption as though it changed the surface composition and hence the reference potential for bisulfate adsorpti~n.~ The ? ~model voltammogram resulting from this approximation qualitatively reproduced the foot on the first current spike. Neither of the above two treatments, one focusing on copper adsorption and the other on bisulfate adsorption, is able to accurately model the voltammogram ofthe entire deposition process. Since the copper and bisulfate coadsorption is highly coupled, it seems appropriate to use a true two-component statistical mechanical model in order to accurately model this underpotential deposition process. Two-componentlattice gases have been used previously to model two-component competitive adsorption on the triangular l a t t i ~ e , ' ~ -including '~ a recent calculation for the present system.20 In addition to rather extensive analyses of the ground states, Monte Carlo and transfer matrix techniques were used to study the various phases a t finite temperatures. For a model of two-component adsorption on the square lattice, which included both hard core and finite repulsions, the existence of ordered phases a t low temperatures was rigorously proved using reflection positivity combined with the Peierls argumenLz1 A two-component model for underpotential deposition is developed in the present paper. In section I1 the onecomponent hard sphere adsorption model is generalized to treat the case of two types of charged hard sphere adsorbates. In section I11 the two-component adsorption model is used to develop a two-component model for underpotential deposition of copper on (111)gold in the presence ofbisulfate. For a reasonable range of interaction parameters, the voltage dependent ground state structures are determined for the system. The Pirogov-Sinai theory22is used to prove the existence of low temperature phases which are similar to the ground state structures but which contain defects. For the phases with symmetryrelated, multiple ground state structures, it is proved that a phase transition occurs as the temperature is raised. In future work we plan to develop a cluster approximation and calculate a n adsorption isotherm and a corresponding model voltammogram for the two-component (15) Borges, G. L.; Kanazawa, K. K.; Gordon, J. G., 11.; Ashley, K.; Richer, J. J . Electroanal. Chem. 1994,364, 281. (16)Shi, Z.; Lipkowski, J. J . Electroanal. Chem. 1994,365,303. (17) Rikvold, P. A,; Collins, J. B.; Hansen, G. D.; Gunton, J. D. Surf Sci. 1988,203, 500. (18)Collins, J.B.;Rikvold, P. A.; Gawlinski, E. T. Phys.Reu. B 1988, 38, 6741. (19) Rikvold, P.A. Electrochim. Acta 1991,36, 1689. (20)Zhang, J.;Rikvold,P. A.; Sung,Y.-E.;Wieckowski,A. In Computer Simulation Studies in Condensed-MatterPhysics; Landau, D. P., Mon, K. K., Schiittler, H. B., Eds.; Springer-Verlag: Berlin, in press. (21)Huckaby, D. A.; Kowalski, J. M. J.Chem. Phys. 1984,80,2163. (22) Pirogov, S.;Sinai, Ya. G. Theor. Math. Phys. 1976,25, 1185; 1976,26,39.
model. The phase transitions which occur a t constant temperature as the voltage is changed can then be modeled.
11. A Model for the Coadsorption of Two Types of H a r d Spheres In this section a model for the adsorption of one type of hard sphereg-14 is generalized to treat the case of coadsorption of two types of hard spheres. The general structure of some of the results derived in this section has been previously anticipatednZ3 Consider N1 spheres of diameter u1 and Nz spheres of diameter a2 in a volume V a t temperature T. The hard sphere gas is in contact with a planar wall a t z = 0 which contains a lattice A of sticky sites. The partition function for the system is
where p = l/kT and Ni
N?
is the Hamiltonian. HOis the Hamiltonian for the smooth wall problem, and UlSand UzSare sticky potentials defined by
where j = 1or 2. Here Aj is a lattice a t the contact plane for adsorption of spheres of type j , located a t z = aj2. If eqs 2 and 3 are inserted into eq 1and integrated, the partition function can then be written as
where Zo is the partition function for the smooth wall problem and 0
Pn,,n, ( r 1 , 1 , ~ ~ ~ ~ r 1 , n 1 ~ r 2 , 1 ~ ~ = ~~~r2,nz)
Letting pl0 = pl,oo(r1,1=(x,y,aJ2))be the contact density of spheres of type 1 and pz0 = p0,lo( r 2 , 1 = (xy,od2))be the contact density of spheres of type 2, then
where (23)Blum, L.Adu. Chem. Phys. 1990,78, 171.
Rigorous Analysis of Low-Temperature Phases
Langmuir, Vol. 11, No. 11, 1995 4585
+
is a n (nl n2)-body contact correlation function. The ,...,rz,,,Jis defined potential ofmean force U(rl,l,...,rl,nl,rz,l by
If the sum over labeled spheres in eq 6 is changed to a sum over lattice sites of A1 and Az, then
E = ZlZ0 =
c ~~
{tl,J2,J
where tJ,Lis the occupation number of site i of Aj (j = 1or 2). The two-component adsorption model is thus equivalent to a two-component lattice gas which has a many-body interaction energy U({tl,i,tZ,i})and chemical potentials p1 and pz for the two components given by
The inclusion of only painvise or triplet interactions in the interaction energy corresponds to a n appropriate superposition approximation for the corresponding correlation functions.14 The fraction of sites of Aj which are occupied by spheres of type j (j = 1 o r 2) is given by
(11) 111. The Low Temperature Phases in a Two-ComponentModel for Underpotential Deposition If the two-component adsorption model discussed in section I1 is generalized to the case in which the hard spheres are charged ions, the product of the stickiness parameter times the contact density then depends on the applied electrical potential q.2-8 We shall assume this dependence has the form5
process, potential dependent coadsorption of C and S occurs on a triangular lattice A of adsorption sites on the gold(ll1) surface. We shall assume the C ions are discharged upon adsorption, but the S ions retain their charge. In the present model we assume the following types of first-neighbor interactions: two S ions are excluded from occupying neighboring adsorption sites, a pair of neighboring adsorbed C's interact with a n energy ECC < 0 after adsorption, and a n adsorbed C interacts with a neighboring adsorbed S with an energy ECS < 0. As we shall presently show, these first-neighbor interactions are sufficient to rigorously obtain the structure of low temperature phases in the model which have the same structures as the potential dependent phases which occur during the experimental underpotential deposition process. However, in order to obtain the first-order phase transitions seen experimentally as the potential is changed a t constant temperature during a voltammogram plot, it is necessary to introduce some longer-ranged interactions,2s20as was done in the original HB model.2 Since we do not herein consider the of any transitions, such long-ranged interactions will not need to be considered in the present treatment. The Hamiltonian for an allowed configuration E in the present model can be written as
H(5) = Cse(q - vs)ns(5)+ Cce(lC)- lyc)nc(O + ncS(6)~cS+ ncc(5kcc (16) where, in configuration E , ns({) is the number of S ions, nc(5)is the number of C's, nc&) is the number of firstneighbor CS pairs, and ncc(E) is the number of firstneighbor CC pairs. Assuming periodic boundary conditions for A, the Hamiltonian can be written as the sum of Hamiltonians, H,(4),restricted to elementary triangles, t,of A. That is (17)
For convenience,H , will be written as HrSt,where the three vertices o f t are occupied by species r, s, and t. Equation 17 will be satisfied if we let
Hooo = 0
(12) where 5, is the electrosorption valency of a n ion of type j and q, is a reference potential which we shall herein assume is concentration dependent but temperature independent. The reference potential is thus given as
(13)
1 3 Hccc = $ce(w - Wc) + Z'cc
and the chemical potential of a n ion of type j is given by Because of the assumed form of Ajpjo(W) given by eq 12, the grand canonical partition function, given by eq 9, can be written as
1
where H(E)is the temperature independent Hamiltonian for a configuration E. We now focus on the specific case of the underpotential deposition of copper, C, on the (111)surface ofgold in the presence ofbisulfate, S.1-8J5J6!24325 During the deposition (24) Hachiya, T.; Honbo, H.; Itaya, K. J. Electroanal. Chem. 1991, 315, 275.
Hscc = @W
1
+ $ce(W
- Wc) + ECS
1
+p
c (18) If there is a configuration 5~ for which, in a domain D of the parameter space (ECC,ECS,~S,~C,SS,SC), the restricted Hamiltonian for every triangle t c A has the value - ws)
(25)Zelany, P.; Rice, L. M.; Wiekowski, A. Surf. Sci. 1991,256,253.
/"\
c-c then f;G is a ground state configuration in D and the restricted Hamiltonian constitutes a n m-potential in D.26 (This potential equals H,(f;)on triangles z c A and equals zero on other subsets of A,) Each ground state configuration in D is then composed of triangles which have restricted Hamiltonians equal to HO. The proof of the existence of low temperature phases in the model which are similar in structure to the ground states proceeds by first establishing a certain probabalistic stability of the ground states.26 This requires that a criterion known as the Peierls condition be satisfied, which will now be described with emphasis on the present model. If in a configuration f; in the present model a triangle z has a restricted configuration f;* which differs from every ground state configuration restricted to z, it will be said to be an "excited state" triangle. If the value of the restricted Hamiltonian for every excited state triangle in every configuration exceeds the value of the Hamiltonian restricted to z in every ground state configuration f ; ~then ,
H(E) - H(E,)
2
aB
(20)
where B is the number of excited state triangles in f; (the union of which is equivalent to the boundary26of f;), and where
a = m i n H r -19 >o 5*
'
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4586 Langmuir, Vol. 11, No. 11, 1995
(21)
If a > 0 in a domain D in which there are a finite number of ground states, Peierls' condition is said to be satisfied in D. Although it is straightforward to show directly that Peierls' condition is satisfied for such domains in the present model, Holsztynski and S l a ~ n proved y ~ ~ the more general result that Peierls' condition is satisfied if the restricted Hamiltonian is a finite-ranged m-potential with a finite number of ground states. Pirbgov and Sinai proved that if a model satisfies Peierls' condition, the model exhibits multiple equilibrium states which are small perturbations of the ground states.22,26 Moreover, if the ground state configurations are related by symmetry, then these multiple equilibrium states exist so long as the temperature is sufficiently low. Since there is only one equilibrium state in the present model a t sufficiently high temperatures,2s then a phase transition occurs as the temperature is raised if there is more than one ground state configuration. For the present model, if we assume [C > 0, [S < 0, E cc < 0, and ECS < 0, it is straightforward to show that eqs 18 and 19 yield
At sufficiently low 7/1, HCCCalone equals W , the single ground state configuration corresponds to a full monolayer coverage of C atoms, and a low temperature phase exists which, except for some defects, has the same structure as the ground state configuration. At sufficiently high potential, Hsoo alone equals Ho. There are then three ground state configurations, each corresponding to a 4 3 x 4 3 structure in which S ions occupy all the sites of one triangular sublattice of A, the remaining honeycomb (26) Slawny, J.Inphase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic: New York, 1987; Vol 11, p 127. (27) Holsztynski, W.; Slawny, J. Commun. Math. Phys. 1978, 61, 177. (28) Dobrushin, R. L.Funct. Anal. Appl. 1968,2, 302.
I JlC1 A 0-0
0.07V
0.22v
/"\
0-0
0.40V
Figure 1. In each voltage range given above, a ground state configurationrestricted to any triangle z has the configuration illustrated, and the low temperature phases have essentially these same structures. Here we have set vs= 0.40 V and v c = 0.12 V and have assumed HSCCis minimal in the range between the spikes of the experimental voltammogram, 0.07 V 5 V 5 0.22 V. The resulting interaction energies are ECC = -0.017 eV and csc = -0.088 eV.
sublattice being vacant. The multiple equilibrium states a t low temperatures have the same basic structure, except for some defects, as the three ground states, and a phase transition occurs as the temperature is raised. Depending upon the values of 7/1c,7/Is,E C C , and ECS, there may be ranges of 7/1 in which either HOOO or HSCCalone equals HO. IfHooo alone equals HO,then the single ground state configuration is the vacant lattice, and the low temperature phase is a nearly vacant lattice. IfHscc alone equals HO, there are three ground state configurations, each corresponding to a 4 3 x 4 3 structure in which S ions are on each site of one triangular sublattice of A, the remaining honeycomb sublattice being completely occupied by C atoms, giving a two-thirds monolayer coverage of C atoms. As is known, these four types of ground states have the same structures as the phases which occur in the experimental system as the voltage is the vacant phase possibly occurring for a range of voltage at low temperatures. In particular, the experimental voltammogram indicates that for an intermediate range of 7/1 a n ordered phase occurs in which, except for some defects, one triangular sublattice is occupied by bisulfate ions, the remaining honeycomb sublattice being occupied by copper atoms.2J6 In the model system at low temperatures, this type of phase occurs if Hscc alone is minimal. If we assume HSCCequals HO in the range V I 5 v 5 ~ 2 and we assume, quite reasonably, that ~2 < 7/1s,then eqs 18 and 19 yield interaction energies ECC and ECS in the form ECC
ECS
= 5[CC(3VC e - V I - 27/92)+ CS(lv1 - 7/92)]
- -%C 18 c ( 3 v c + ly,
+ 5s(3vs - 27/92 - 7 / 4 1
- 4742)
(23) If, in addition, 2 5 ~> -[s, which is quite reasonable to assume, then eqs 18 and 19 yield that Hccc alone is minimal if 7/1 < 7/11, HSCCalone is minimal if ~1 < 7/1 < 7/12, HOOO alone is minimal if 7/12 < q < 7/1s,and HSOOalone is minimal if 7/1 > 7/1s. Thus, if 2 5 ~> -5s > 0, ccc < 0, E C S < 0, and 7/1s> 7/12 > wl, then the sequence of phases which occurs in the model as is varied is consistent with those of the experimental system.2,6J5J6z24 We shall assume further that HSCCalone equals W for potentials 7/1 between the twovoltammogram peaks,lZ2i.e., we shall assume = 0.07 V and ~2 = 0.22 V. Since bisulfate ions strongly desorb for 7/1 < 0.40 V,7J5J6we shall set 7/Is = 0.40 V. Equation 23 indicates that the values of ECC and ccs depend linearly on 7/1c,which is unknown. If we let 5s = -1 and 5~ = +2, eq 23 indicates ECC < 0 requires that 7/1c< 0.145 V. If, for example, we set y c = 0.12 V, eq 23 yields ECC = -0.017 eV and ECS = -0.088 eV, which are physically reasonable in magnitude. The sequence of phases which occurs in the model as 7/1 is varied for this choice of parameters is illustrated in Figure 1.
,
Langmuir, Vol. 11, No. 11, 1995 4587
Rigorous Analysis of Low-Temperature Phases (Note that suitable values of the parameters also result from eq 23 if we let 5s = -2, i.e., if sulfate rather than bisulfate is the adsorbed species.) In another paper we plan to calculate a model voltammogram using a cluster variation approximation of the voltage dependent coadsorption. We will include longerranged interactions in that approximation in order to accurately model the experimental voltammogram which indicates the occurrence of two first-order phase transitions. A generalization of the HB model,2 containing longer-ranged and three-body interactions, has been recently studied using Monte Carlo techniques.20 These additional interactions result in several other types of
ground states being exhibited by the model. The resulting model voltammogram could be fitted quite well to the experimental voltammogram of the deposition.
Acknowledgment. L.B. was supported by the Office of Naval Research and by the EPSCoR program EHR91-08775, and D.H. was supported by the Robert A. Welch Foundation grant P-0446. D.H. wishes to express his appreciation for the hospitality he received during his visit to the University of Puerto Rico. LA9504523
