Thermodynamic derivation and model calculations of the metal underpotential dependence on electron work function differences

J. Electroanul. Chem., 350 (1993) 1-14 Elsevier Sequoia S.A., Lausanne

JEC 02546

Thermodynamic derivation and model calculations of the metal underpotential dependence on electron work function differences E.P.M. Leiva Unidad Docenre de Matemdka, Facultad de Ciencias Quimicas, Universidad National de Cbrdoba, Sucursal16, C.C. 61, 5016 Cbrdoba (Argentina) (Received 26 June 1992; in revised form 14 September 1992)

Abstract The relationship between the work function of the substrate+ adsorbate system and the underpotential shift is derived from a thermodynamic argument. An alternative explanation of the correlation formulated in the literature between the underpotential shift and the work function difference of the metals involved is proposed. A simple model is used to illustrate the meaning of the equations derived. Abrupt adsorption-desorption behaviour is predicted for those electrochemical systems where the work function shows a pronounced minimum as a function of the coverage, provided that this is a first-order contribution to the chemical potential of the system.

INTRODUCTION

Attempts to correlate the underpotential deposition (UPD) of metals with properties of the adsorbate and substrate have a long history. An interesting discussion of UPD phenomena on polycrystalline surfaces was given by Kolb et al. [l], and Kolb [2] has reviewed the properties of these systems. Considering a metal A adsorbed on a different metal S, these authors defined the underpotential shift AE, as the potential difference between the peak potential corresponding to the stripping of bulk atoms of A, and that corresponding to the stripping of the monolayer atoms of A adsorbed on S. They found a linear correlation with slope 0.5 between AE, and the difference between the work functions of the substrate and monolayer materials. The important contribution of Trasatti [3,4], who assumed an ionic bond with total charge transfer from A to S and considered the underpotential value for the deposition of the first A atom, should also be noted. He predicted that this shift should show a linear relationship with the work 0022-0728/93/$06S!il 0 1993 - Elsevier Sequoia S.A. All rights reserved

2

function difference, but with slope of unity. His predictions were in agreement with the experimental results for a number of metal couples. Interest in the theoretical investigation of underpotential phenomena has been renewed in recent times [5-71. Schmickler and colleagues have used the jellium model [5] and improved versions of it [6,7] in order to study the under-potential phenomenon. In the first of these papers [Sl, the jellium model was used to study UPD on polycrystalline substrates and its correlation with the work function. In the later work [6,7], the model was extended to include pseudopotentials and was used to consider single-crystal surfaces, giving the correct trend for the underpotential shift on various surfaces. The purpose of the present work is to derive some simple thermodynamic equations which will throw light on the relationship between the underpotential shift and work function differences. The meaning of the equations derived will be illustrated with some calculations made using a simple jellium model. However, the main conclusions should have a general validity, and calculations using more complex models (like those described in ref. 7) are desirable to test the present ideas. DISCUSSION

We start the present discussion by recalling the work of Kolb et al. [ll who considered the reactions A% + 2 emeta @ Ametal

(la>

A’,,:, + *

(lb)

esibstrate

*

Aad

which correspond to the equilibrium between ions of metal A in solution, denoted A:& and either bulk (eqn. (la)) or monolayer (eqn. (lb)) atoms of metal A, represented by Ameta, and A, respectively. Kolb et al. [l] found that the relationship between the underpotential shift A E,, the chemical potential P,,~~, of the A atoms deposited on bulk A and the chemical potential pad(O) of the A atoms deposited on S to be I-Lmetal

-

,&d(

0)

= Zq,

A E,

(2)

We have retained the notation used by Kolb et al. but have written P,~(@) in order to stress the fact that this quantity is a function of the coverage 8 by atoms A. According to their definition, the chemical potentials ,umeta, and ,uLad(0) are given by &d(@)

dG*+s = dN A

(3a) (3b)

3

where GA+s represents the Gibbs energy of the adsorbate + substrate system and GA is the Gibbs energy of a system formed exclusively by atoms of type A. The change dG,+s in the Gibbs energy when dN, atoms are adsorbed also includes solvent effects. This is because interaction of the solvent with the bare substrate before adsorption is replaced by interaction of the solvent with the adsorbed atom after adsorption. This effect has already been discussed by Schmickler [6,7] and seems to be small; therefore we shall neglect it in this work. In order to relate these quantities to the electronic properties of the systems, we make two further assumptions: the core ions of the adsorbates are considered to have a definite position in space, i.e. no diffusion into the bulk substrate occurs; the valence electrons of the adsorbate are in equilibrium with those of the substrate. From these assumptions, the Gibbs energy GA+s can be written as a function of the number NA+ of ionic cores of the adsorbate (which is linearly related to the coverage by A atoms) and the total number N, of valence electrons in the system: GA+s=~(NA+,

(4)

4)

For arbitrary changes in the quantities dN,+ and dN, respectively, we have

NA+ and N,, which we shall denote

(5) Let us now consider the Gibbs energy change dGA+s for the addition of dNA neutral atoms of type A to the system, as required by eqn. (3a) for the calculation of the chemical potential. Since in the present approximation the valence electrons of the adsorbed atoms are shared with the substrate, we have dN, = z d NA, i.e. the number of electrons added to the system when d NA atoms are adsorbed is the product of this number and the valence z. Furthermore, mass balance requires that dNA+= dNA since the change in the number of adsorbed ionic cores is equal to the change in the number of adsorbed A atoms. Therefore eqn. (5) reduces to dGA+s=

(+)N,-=‘%+ ($),

dNA

or, rearranging, d$s

=*( $s),.+(

Z)&

(7) =&id(@) In order to interpret this equation, we use the definition for the work function of a metal surface given by Monnier et al. [81: @ = dG/dZ

I I=o

(8)

4

where G is the Gibbs energy of the metal and Z is the number of electrons which have been carried off to rest at x = + 03.If we compare eqn. (8) with the first term on the right-hand side of eqn. (71, we arrive at the conclusion that

= -@*+s(@) where -@Jo+, is the work function of the substrate + adsorbate system at the coverage 0. Using eqns. (3a), (7) and (91 we obtain

Pad(@)= -z@A+s(@)+

(10)

i.e. the chemical potential of the adsorbate atoms deposited on S contains two contributions, one corresponding to the work required to bring z electrons from infinity into the system and the other corresponding to the work required to bring the core ions of the adsorbate from infinity onto the substrate. We shall call the latter quantity the adsorption Gibbs energy of the ionic cores. If we perform the same analysis for pmeta, we obtain P metal

(11)

=

where GA is the work function of the metal A. An alternative and more straightforward derivation of eqn. (11) has been proposed by Trasatti [9] and is included in Appendix A. This derivation emphasizes the role of the surface potential of the metal in the determination of its work function. The two derivations are equivalent, in an analogous manner to the different expressions for the work function given by Monnier et al. [8]. Combining eqns. (10) and (11) and substituting in eqn. (2), we obtain the following expression for the underpotential shift: Z?, AE, =r[@A+s(@)

-@A] +

(%)Njz)Ne

(12)

It is clear from eqn. (12) that the underpotential shift is related to the work function of the substrate + adsorbate system at coverage 0 minus the work function of the metal A, and the corresponding changes in Gibbs energy for the adsorption of the ionic cores of A on the substrate and on the metal A. It should be noted that the quantity (aGA/aNA+jNe is also dependent on coverage. In order to interpret the correlation between the underpotential shift and the work function difference proposed by Trasatti [3,4] in terms of the present formulation, let us consider the limit 0* + 0 in eqn. (12). In this limit the work function of the substrate + adsorbate system will be equal to the work function of the substrate, i.e.

lim [@A+S(@)] @+O

= @S

(13)

Then, by rearranging (12), we obtain

(14) From eqn. (14) we see that a linear relationship between the underpotential shift for 0 + 0 and the work function difference should exist if the contribution in square brackets becomes negligible. This may happen under the following conditions. (1) Both terms in the square bracket on the right-hand of (14) are small. (2) The terms in the square brackets compensate each other, with a resulting small contribution. This would mean that the adsorption energies of the cores of A on, metal A and metal S are similar. This is the conclusion that can be drawn from the model formulated by Trasatti [3,4], where the binding of the adsorbate to the substrate surface was assumed to be fully ionic. In such a case, the last term in eqn. (14) vanishes since the image energy involved is independent of the nature of the substrate. Returning to eqn. (14), we can predict that the linear correlation between A,!&(@ + 0) and A@ will break down in those systems where the differences between the adsorption energies of the ionic cores of A are large. In the rest of the paper we illustrate the use of the formula derived here by means of a very simple picture for the metal-vacuum interface: the jellium model. CALCULATIONS USING THE JELLIUM MODEL

In order to calculate adsorption energies (i.e. the second term in eqn. (10)) and work functions for two systems, we shall use a model similar to that employed by Lang [lo] to study the adsorption of alkali atoms on a substrate with a high work function (Fig. 11, which was first applied to the study of UPD by Schmickler and coworkers [5-71. From a qualitative point of view, diagrams of this type have also been used by Trasatti [ll] to illustrate UPD as a particular case of metal-metal contact. The ionic cores of the substrate are represented by a semi-infinite uniform background of average electronic density ii,, and the corresponding ionic charges of the adsorbate are replaced by a homogeneous positive slab of thickness d and density !&. The latter is placed immediately adjacent to the substrate background. Changes in coverage are represented by changes in Eid, so that coverage is defined as 0 = rl&/rlad where IZ,~ is the bulk electronic The distance d is fixed for a the spacing between the 111 or perform calculations for these

(15) density of the adsorbate. given substrate and is equal to the bulk value for 110 body-centred cubic (b.c.c.1 lattice planes. We two surfaces in the hope of finding qualitative

background

-10

0

-5

(0)

x I

5

10

5

10

a.u.

-0.002, -10

(b)

-5

0

x /

a.u.

Fig. 1. Jellium model for the substrate + adsorbate system. The bulk electronic densities are 0.006 a.u. and 0.0021 a.u. for substrate and adsorbate respectively. The width of the adsorbed layer was chosen to represent adsorbates with the following structures: (a) 110 b.c.c. surface; (b) 111 b.c.c. surface.

differences between adsorbates of different compactness. We shall use somewhat arbitrary electronic densities for Es and ?ii,, since we only seek qualitative results. The inhomogeneous electron gas is described by the Hohenberg-Kohn (HK) formalism [12,131 so that the ground state energy of the system can be written as a function of the electron number density n(r):

E[n(r)] =/V(r)n(r) dr+ fj’y:y;:;)dr

dr’+G(n)

where the first term represents the interaction of the electrons with an external potential V(r), the second term is the’ self-interaction energy and G(n) is a universal function of density which accounts for kinetic, exchange and correlation energy.

The correct density profile is that which minimizes the system energy (16). Thus approximate solutions can be found by considering a suitable family of trial functions containing a certain number of variational parameters. These are determined by the condition that E is minimal within the family. TRIAL FUNCTIONS

Trial function for the adatoms

The trial function chosen for the electron density associated with the adatom is

=~:,dp/4[PIZ-Z,,Ielrp(-PIZ-Z,,I)+exp(-BIZ-Z,,l)]

E,,(Z)

(17) where C and /3 are variational parameters and Z,, = d/2. C is a measure of the charge renormalization at the interface. If C = 1, the integral of the charge density (17) will result in Ti,,d, which is minus the surface charge of the positive background corresponding to the adsorbate. C < 1 or C > 1 implies a transfer of negative charge from the adsorbed atoms to the metal or from the metal to the adatoms respectively. Trial function for the metal

The electronic charge density of the metal is represented

%W =

z>o

E, ewaz i 7i,( 1 -A

ear)

2I 0

by

(18)

where cy is a measure of the localization of the atomic electrons. Imposing on (17) and (18) continuity of the electronic density and charge neutrality as boundary conditions, we obtain A=;

l+ I

wz,,d(

C - 1)

% wz,,d( C - 1)

%

I

1

(19

If Nad = 0, the function (19) reduces to the one-parameter trial function used by Smith 1141for calculating the surface properties of jellium. We must stress that the separation of the electron density into adatom and metal charge contributions is made only for calculation purposes. The electrons belonging to the substrate or to the adatoms are certainly physically indistinguishable.

CALCULATION OF CHEMICAL POTENTIALS, WORK FUNCTIONS AND CORE ADSORPTION ENERGIES

Chemical potential According system is

to eqn. (3a), the chemical potential

of the substrate + adsorbate

dG,+s PXl(@) = dN A

This quantity can be calculated in the jellium model by means of the following procedure. (a> The Gibbs energy Gi+s of the substrate + adsorbate system is calculated for a given 0 . (b) 0
(20)

where & is the mean chemical potential of the electrons in the substrate and A4 is the potential difference at the substrate + adsorbate/vacuum interface. With the present choice for the trial functions, the second term on the right-hand side of (20) becomes

The first term on the right-hand side of (21) represents the potential difference across the substrate, while the second term is a measure of the dipole originating from the charge transfer between adsorbate and substrate. For example, if C < 1, we have a transfer of negative charge in the direction of the substrate metal. This would produce a surface dipole which pulls the electrons out of the metal, thus producing a decrease in the work function. However, as pointed out above, a partial charge cannot be assigned directly to the adsorbate atoms since this quantity would depend on the trial function chosen for the variational procedure. An alternative way of calculating the work function is to use eqn. (8): @ = dG/dZ

I x=0

Thus @A+S(@) can be computed by considering a slab with a very small charge density placed at infinity and calculating the energy change which the induced

9

charge produces on the jellium. Since (8) is a very accurate expression, which is rather insensitive to the nature of the variational profile, we use this second method in order to obtain our @A+s(0). Core adsorption energies (c3G,,+,/~N,+),

The second term on the right-hand side of eqn. (10) represents the energy required to bring an adatom core from infinity to the surface. From now on, we shall denote it as G,. As in the case of the work function, there are two alternative methods of calculating this quantity. In the first, G, is obtained through variational parameters as described in Appendix A:

+ dii,~

-+++em(-k%,)+

22, -exp( P

-&Ld I

The second method of obtaining G, consists in using its definition (aG,+,/ a&+>, . As described above for the calculation of the chemical potential, two calculations of the Gibbs energy GA+s are required. In the first the energy of the substrate + adsorbate system is evaluated for a given ?I, and &. In the second step, the charge density of the positive slab EL, is decreased by SEig, and a corresponding positive charge density is located at infinity. The energy is again evaluated and G, is obtained by numerical differentiation. Both methods give very similar results for G,. The use of eqn. (22) is advisable at low coverages, where the parameter C is close to zero and some numerical instabilities arise in the determination of the derivative (aG,+s/aN,+),c. RESULTS OF THE CALCULATIONS

Figure 2 shows calculations of @*+s(O), G, and p,JO) for two different choices of the distance d. As previously stated, d is calculated from d=Fr,

(23)

where the geometrical factor g is equal to 1.4361 for the’110 b.c.c. face and 0.5863 for the 111 b.c.c. face and r,, is the radius of the sphericalized Wigner-Seitz cell (4/37r$ = z/Z where z is the valence). E,, and 7is lay in the range of the alkaline metals and were selected so that the correction to our Gibbs energies due to Madelung energies and pseudopotential effects would be rather small. In fact, Lang and Kohn [15] found that, for relatively low electronic densities, the uniform positive background model gives a reasonably good description of the surface energy of an sp metal as a function of the electronic density.

10 43-

__----

---

B 26 ;

l0

..._...... *e** *..___ .'

-..._

._..s___

--.._ --_

-1 0.0

0.2

0.4

0.6

0.8

IO

4 3-

---_______--------

> T2 B. 2

lO-

-1 (b)

_._._____.-.---._----......__._

0.0

0.2

0.4

06

_-.....__ 0.8

1.0

8

Fig. 2. Negative of the chemical potential - pad(@) (), work function @_s@) (- - -_) and core desorption energy -G+ (..o.*.) as functions of coverage (As = 0.006, R,, = 0.0021): (a) d corresponding to a 110 b.c.c. surface; (b) d corresponding to a 111 b.c.c. surface.

At low coverages, the more compact surfaces show a more rapid decrease of the @A+s(0) with 0. This can be qualitatively understood as follows. For 0 --) 0, the electronic density on the adatoms is small since almost all their electronic charge has been transferred to the metal. Thus the substrate has an excess of negative charge with its centre of gravity in front of the jellium edge, i.e. at position x,,. However, the slab of positive charge density corresponding to the ionic cores of the adsorbate has its centre of gravity located at x = d/2. Therefore the situation is equivalent to the existence of a dipole pointing outwards from the surface with a charge separation Al = d/2 -x0. Since d is greater for the more compact faces, a larger dipole is obtained which favours the extraction of electrons at low coverages. A similar conclusion can be drawn by allowing C to approach zero in eqns. (20) and (21). For the two types of surfaces investigated here, the work function was found to make a first-order contribution to the binding energy.

11

Dependence of the underpotential shift on the coverage All the results described above not only include all the approximations inherent in the jellium model but also ignore all temperature effects; therefore we cannot calculate adsorption isotherms in a rigorous manner. However, there is an interesting fact in the curves of Fig. 2 that deserves attention. Let us recall eqn. (2) for the underpotential shift: AE,(0)

= [pmetal - ~L,d(OWe,

(24)

where we have written AE,(O) in order to emphasize the fact that this equation provides the equilibrium potential for each coverage as a function of the chemical potential of the adsorbate atoms for this value of 0. Thus the curves for -p,JO) in Fig. 2 should give an indication of the underpotential shift as a function of 0, since the quantity timeta, in eqn. (24) is independent of 0. For instance, for a higher -pa,(O), a higher equilibrium potential at that coverage should be obtained. Let us first analyse Fig. 2(b). For these more open faces a monotonic decrease of -pJ0) with increasing 0 is observed. This indicates a smaller underpotential shift for higher 0 according to eqn. (24). Physically speaking, this means that as 0 increases, repulsion between the adsorbed atoms makes the adsorption of more atoms less favourable. This is not an unexpected result. The interesting question arises when we analyse Fig. 2(a). For 0 < 0.12 the behaviour of the -pL,JO) versus 0 curve is similar to that of Fig. 2(b), but at higher coverages -p,JO) increases again. In other words, we obtain two different values of 0 with the same chemical potential for the adsorbed atoms, or, what is equivalent, two different coverages can exist at the same equilibrium potential. From an experimental point of view, this could result in abrupt adsorption-desorption behaviour over a small potential range. This latter fact, which in the simple model used here appears for the more compact faces as a direct consequence of the important contribution of the work function to -p,JO), should be tested using a more elaborate model for UPD, and may take place in those systems where a pronounced minimum is found in the curves of work function versus coverage. Underpotential shift for 0 + 0 Since the jellium model used here is expected to give a rather poor description of the present system in the limit of low coverage, we warn the reader that the following discussion is only an illustration of the type of calculation which should be performed using a more elaborate model (e.g. the model described in ref. 7). The under-potential shift at 0 --) 0 for a given substrate and different adsorbates can be calculated from eqn. (2), by using the jellium model as described above for the computation of the chemical potentials ~~~~~ and ~~~(0 --) 0). AEp is plotted as a function of the bulk work function difference A@ = Gs - Q+,, in Fig. 3. The

12

Ad/eV Fig. 3. Calculated underpotential shift as a function of the work function difference A@ = @s - @,,, for 110 b.c.c. (- - -_) and 111 b.c.c. (. . . . . ) surfaces. (3, = 0.006 ax.). The full line indicates a slope of unity.

important contribution of A@ to AE, in our model produces a plot which is almost linear with slope unity. This can be compared favourably with the correlation between underpotential shift and work function difference found by Trasatti [3,4] for several systems. However, as stated above, this theoretical prediction is not very reliable at low coverages. CONCLUSIONS

We have derived a relationship between the work function of the substrate + adsorbate system and the under-potential shift. By taking the limit of low coverages, we have been able to explain the empirical correlation formulated by Trasatti between the underpotential shift and the work function difference of the two metals, provided that the adsorption energy of the ionic core of the adsorbate on the substrate is compensated by the adsorption energy of that core on a surface of the same nature. We have used a simple model in order to illustrate the calculation procedure which should be performed using a more powerful model than that employed here. In those electrochemical systems where the work function shows a pronounced minimum as a function of coverage, abrupt adsorption-desorption behaviour may occur when the adsorption potential is varied. ACKNOWLEDGEMENTS

This work was supported by CONICET (Consejo National de Investigaciones Cientificas y Tecnicas), Fundacion Antorchas, CONICOR (Consejo de Investigaciones Cientificas de CXrdoba) and Secretaria de Ciencia y Tecnologia de la Universidad National de Cordoba, Argentina. We are indebted to Professor Wolfgang SchmickIer for suggestions concerning the application of the jellium model to underpotential deposition, as well as for useful comments about this phenomenon. Language assistance from P. Falcon is also acknowledged.

13 APPENDIX A

We describe here an alternative derivation of eqn. (11) proposed by Trasatti [93. Since a metal has two constituents [16], metal ion cores and valence electrons, its chemical potential can formally be split into corresponding contributions: &n&l = Zpe -!-PM+ (AI) If the surface potential x of the metallic phase is added and subtracted, we obtain + PM++ ze& p metal =zCLe -.x,x (W from which eqn. (11) follows since (aG,/aN,+jNC =pM++ze,x. This quantity is just the real potential [16] of the ion cores in the metal, which corresponds to the negative of the ion work function. APPENDIX B

We derive here an alternative expression for the energy required to move an atom core from the surface to a point far from the surface. The energy required to take a charge dq from point x to infinity is dG = [& - 4(x)]

dq

(BI) where 4(x) is the electrostatic potential at point X. Since in our model the substrate cores are represented by a constant charge density Ea,, extending over the region 0 > x > d, the energy per unit area required to move a slab of constant charge density An;, from this region is

GA= f[C

- 4(x)] A&,

dx

(Jw

Rearranging this expression gives (B3) so that

C#J(X)can be obtained from the total charge density using the Poisson equation

d24

= -4?Tp(x) dx* and on substituting this in eqn. (B4) we obtain eqn. (14).

W)

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and Electro-

14 3 S. Trasatti, Z. Phys. Chem. NF, 98 (1975) 75. 4 S. Trasatti, in H. Gerischer and C.W. Tobias (Eds.), Advances in Electrochemistry and Elect chemical Engineering, Vol. 10, Wiley, New York, 1977, p. 213. 5 E. Leiva and W. Schmickler, Chem. Phys. L.&t., 160 (1989) 75. 6 W. Schmickler, Chem. Phys., 141 (1990) 95. 7 W. Lehnert and W. Schmickler, 27 (1991) 27. 8 R. Monnier, J.P. Perdew, DC. Langreth and J.W. Wilkins, Phys. Rev. B, 18 (1978) 656. 9 S. Trasatti, Personal communication (1992). 10 N.D. Lang, Phys. Rev. B, 12 (1971) 4234. 11 S. Trasatti, in J.O’M. Bockris, B.E. Conway and E. Yeager (Eds.), Comprehensive Treatise Electrochemistry, Vol. 1, Plenum, New York, 1980, Ch. 2. 12 P. Hohenberg and W. Kohn, Phys. Rev. B, 136 (1964) 864. 13 W. Kohn and L.J. Sham, Phys. Rev. A, 140 (1965) 1133. 14 J.R. Smith, Phys. Rev., 181 (1969) 522. 15 N.D. Lang and W. Kohn, Phys. Rev. B, 1 (1970) 4555. 16 S. Trasatti and R. Parsons, J. Electroanal. Chem., 205 (1986) 359.