4716
Langmuir 1997, 13, 4716-4721
Modeling of Charge-Potential Isotherms of Anodic Monolayers Using the Ising Lattice Gas Model U. Retter* and W. Kant Bundesanstalt fu¨ r Materialforschung und -pru¨ fung, D-12489 Berlin, Germany Received February 26, 1997. In Final Form: May 19, 1997X Theoretical charge density-potential isotherms have been derived for anodic monolayers of honeycomb, square, and triangular lattice geometry on the basis of the Ising lattice model. For mercury sulfide monolayers at the mercury/electrolyte interface, the charge-potential isotherms have been measured in the temperature region between 279 and 313 K. The critical temperature of the monolayer was indicated by a sudden change in the isotherm shape. From the critical temperature, the lateral interaction energy of the molecules in the HgS monolayer has been directly calculated. On the basis of the Ising lattice gas model, these isotherms can be theoretically well described with a low-temperature series expansion of the degree of coverage with respect to the activity of the solution phase species. The adsorption energy and number of electrons involved in the electrode reaction obtained are practically independent of temperature.
Introduction Only in a few cases have the charge-potential isotherms of anodic layers been measured, namely for the formation of HgS on Hg1,2 and for the formation of CdS on Cd.3 The two isotherms showed a step-like change of the charge with the potential, indicating a 2D first-order phase transition. The trail failed to model this step isotherm with the Frumkin model.1 Further attemps to model charge-potential isotherms of anodic layers have not been performed with the consequence that such fundamental quantities of the anodic film as the adsorption energy and the lateral interaction energy are not known. Lattice models have been used in molecular modeling of adsorption.4-13 We have shown in several papers,7-13 that the Ising lattice gas model is a valid model for 2D first-order phase transitions, contrary to the Frumkin model. The Ising lattice gas model considers the existence of 2D gas clusters and 2D liquid clusters (the Frumkin model only monomers) and the corresponding possible configurations. For the adsorption of 5-iodocytosine at the mercury/electrolyte interface, the degree of coveragepotential dependence of organic molecules in the presence of a 2D first-order phase transition could be theoretically well described for the first time using the Ising lattice gas model, and from that, the adsorption as well as the lateral interaction energy can be determined.7 The aim of the present paper is, therefore, to model steplike charge-potential isotherms of a selected anodic monolayer (HgS) with the Ising lattice gas model and to obtain the adsorption and lateral interaction energies of the molecules in the condensed film. As an experimental example of an anodic monolayer, mercury sulfide layers * Corresponding author. X Abstract published in Advance ACS Abstracts, July 15, 1997. (1) Peter, L.; Reid, J.; Scharifker, B. J. Electroanal. Chem. 1981, 119, 73. (2) Philipp, R.; Retter, U. Thin Solid Films 1992, 207, 42. (3) Saidman, S.; Vilche, J.; Arvia, A. Electrochim. Acta 1987, 32, 1153. (4) Rangarajan, S. J. Electroanal. Chem. 1977, 82, 93. (5) Trasatti, S. Electrochim. Acta 1983, 28, 1083. (6) Brodsky, A.; Daichin, L. Elektrokhimiya 1989, 25, 435. (7) Retter, U. J. Electroanal. Chem. 1984, 165, 221. (8) Retter, U. J. Electroanal. Chem. 1987, 236, 21. (9) Retter, U.; Vetterl, V.; Jursa, J. J. Electroanal. Chem. 1989, 274, 1. (10) Kharkats, Y.; Retter, U. J. Electroanal. Chem. 1990, 287, 363. (11) Retter, U. J. Electroanal. Chem. 1992, 329, 81. (12) Retter, U. J. Electroanal. Chem. 1993, 349, 41. (13) Retter, U. Electrochim. Acta 1996, 41, 2171.
S0743-7463(97)00213-8 CCC: $14.00
at the mercury/electrolyte interface have been selected because their formation and dissolution had been intensively studied.14-16 Theory In the following it is described in which way the lateral interaction energy and the adsorption energy of molecules in an anodic monolayer can be obtained from the potential dependence of the charge if the monolayer exibits a 2D first-order phase transition. Let us first assume that the critical temperature Tc can be determined experimentally, i.e. that the charge-potential isotherms change their shape from steplike to continous in a narrow temperature region, increasing the temperature. Then the lateral interaction energy AA can be directly obtained from the critical temperature and the critical interaction coefficient aGc according to ref 8.
AA ) -aGckTc/2
(1)
The critical lateral interaction coefficient aGc depends on the kind of the 2D lattice. The values of aGc are for the honeycomb, the square, and the trigonal lattice, respectively10
aGc ) 4 ln(1/(2 - x3)) ) 5.267831...
(2)
aGc ) 4 ln(1/(x2 - 1)) ) 3.525494...
(3)
aGc ) 4 ln(x3) ) 2.197224...
(4)
The quantities x and z are defined as
x ) exp(-aGcTc/(4T))
(5)
z ) Bc
(6)
Here, z is the activity, B is the adsorption coefficient of the molecules in the anodic film, and c is the concentration of the anion in the solution. Let us start from the reaction which describes the formation of an adsorbed anodic monolayer MA from a metal M and a solution phase species A- associated with an oxidation and the desorption of the monolayer combined with a reduction17
M + A- a (MA)ads + e-
(7)
Then, the adsorption rate kads and the desorption rate kdes obey (14) Philipp, R.; Retter, U. Thin Solid Films 1992, 207, 42. (15) Retter, U.; Kant, W. Thin Solid Films 1995, 256, 89. (16) Retter, U.; Kant, W. Thin Solid Films 1995, 265, 101. (17) Noel, M.; Vasu, K. Cyclic Voltammetry and the Frontiers of Electrochemistry; Aspect Publications Ltd.: London, 1990; Chapter 7.
© 1997 American Chemical Society
Charge-Potential Isotherms of Anodic Monolayers
Langmuir, Vol. 13, No. 17, 1997 4717
the Butler-Volmer relations
kads ) kads0 exp(RnFE/RT)
(8)
kdes ) kdes0 exp(-(1 - R)nFE/RT)
(9)
Here, kads0 and kdes0 are the adsorption and desorption rate constants, respectively, R is the charge transfer coefficient, n denotes the number of electrons involved in the electrode reaction, and E is the electrode potential. According to the lattice gas model, the activity z (eq 6) corresponds to the degree of coverage of the gaseous adsorbed phase θg when turning to very low activities. The net rate of adsorption is
dθg/dt ) kadsc - kdesθg
(10)
For adsorption equilibrium, i.e., dθg/dt ) 0, one obtains
(11)
Figure 1. Cyclic voltammogram of Hg in a 10-2 M Na2S + 1 M NaHCO3 solution at 298 K; sweep rate 17 mV s-1.
Comparison of eqs 6 and 11 leads to the following potential dependence of the adsorption coefficient B:
up to j ) 10 in ref 20. In the following, the Aj values are written down up to j ) 3 for the honeycomb lattice
θg ) c(kads0/kdes0) exp(nFE/RT)
B ) (kads0/kdes0) exp(nFE/RT)
(12)
With increasing potential E, a condensed 2D equilibrium lattice phase may form at the potential Etrans, involving that the coverage of the lattice phase attain its stationary value in the monolayer formation. Defining
A1 ) 1;
A2 ) (3/2)x-2 - 2;
A3 ) x-4 - 9x-2 + 19/3
(20)
A3 ) 6x-4 - 16x-2 + 31/3
(21)
for the square lattice
A1 ) 1;
A2 ) 2x-2 - 5/2;
and for the triangular lattice 0
0
Btrans ) (kads /kdes ) exp(nFEtrans /RT)
(13)
K2 ) nF/RT
(14)
(22)
(15)
With increasing hole cluster size, the modeling of the degree of coverage-potential dependence becomes more detailed, since, at the same cluster size, also the corresponding possible configurations are considered. From eqs 15, 17, and 19, the theoretical potential dependence of the degree of coverage results
it follows
B ) Btrans exp(K2(E - Etrans))
The standard free energy of adsorption can be obtained in the following way:7
∆Gtrans ) -RT ln(55.5Btrans)
(16)
Let k be the number of nearest neighbors in the 2D lattice; then the adsorption coefficient Btrans at the potential Etrans obeys the following relation:7
A1 ) 1;
A2 ) 3x-2 - 7/2;
θ)1-
A3 ) 2x-6 + 9x-4 - 30x-2 + 58/3
∑jA x
kj
j
exp(-K2j(E - Etrans))
The degree of coverage θ can experimentally be determined by the charge density q at a particular potential and the maximal possible charge density qm
θ ) q/qm Btransc ) xk
(17)
From eqs 16 and 17
∆Gtrans ) -RT ln(55.5xk/c)
(18)
For the condensed phase (z > xk), the low-temperature series expansion of the degree of coverage θ of the monolayer with respect to the activity Bc gives 12
θ)1-
∑jA x
kj
j
(xk/z)j
(19)
j
(23)
j
(24)
Experimental Section All measurements were performed in the system 0.01 M Na2S + 1 M NaHCO3. As working electrode, a home-made hanging mercury drop electrode was used, and as reference electrode the Hg/HgS electrode (-712 mV with respect to the SCE) was used. All potentials were related to this electrode. The potentiostatic current i-time t transients were measured with a Schlumberger model 1286 electrochemical interface, amplified with a Keithley model 428 current amplifier and recorded with a Hewlett Packard 54600A digital oscilloscope. Other details of the measuring technique have been described in refs 15 and 16. The measuring of the electrode admittance was performed with the LF Impedance Analyzer 4192A of Hewlett Packard.
Here, j is the number of molecules which have to be desorbed to form a corresponding hole cluster in the fully condensed monolayer. The coefficients Aj have been calculated for the honeycomb lattice up to j ) 21 in refs 18 and 19, for the square lattice up to j )15 in refs 18 and 19, and for the triangular lattice
Figure 1 shows the cyclic voltammogram of the system. The formation of the first and second HgS monolayers is indicated by sharp oxidation peaks. Electrocapillary data
(18) Sykes, M.; Essam, J.; Gaunt, D. J. Math. Phys. 1983, 6, 283. (19) Sykes, M.; Gaunt, D.; Mattingly, S.; Essam, J.; Elliot, C. J. Math. Phys. 1973, 14, 1066.
(20) Domb, C. In Phase transitions and Critical Phenomena; Domb, C.; Greene, M., Ed.; Academic Press: New York, 1974; Vol. 3, Chapter 6.
Results and Discussion
4718 Langmuir, Vol. 13, No. 17, 1997
Figure 2. Current-time transient for the formation of the first HgS monolayer after a potential step from -68 mV to +3 mV; T ) 293.4 K.
Figure 3. Enlarged part of the transient in Figure 2 with tDmono the time at which the current just vanishes.
and impedance data have shown21 that the first HgS monolayer is entirely formed by adsorption at some 10 mV more negative than the reversible potential of the Hg/HgS electrode.1 From voltammetric investigations of the system considered here22 above 276 K, the adsorption/desorption of the first HgS monolayer is totally reversible. The anticipated adsorption/desorption reaction is (see eq 7)
Hg + OH- + SH- H (HgS)ads + H2O + 2e- (25) Starting from an initial potential (-68 mV), a potential step has been applied to the electrode which corresponded to the first and second HgS monolayers. As a criterion for an established equilibrium of the monolayers, the attainment of negligible current values in the current i-time t transients for larger times (t ) tDmono) has been used (Figure 2). With increasing potential tDmono decreases and has to be determined separately for every potential. Beginning at a special potential, a second monolayer forms on the first one. However, the formation of the second layer is generally much slower than that of the first layer and is indicated by a reincrease of the current. Figure 3 illustrates this slow reincrease of the current. To determine the overall charge corresponding to the formation (21) Armstrong, R.; Porter, D.; Thirsk, H. J. Electroanal. Chem. 1968, 16, 219. (22) Benucci, C.; Scharifker, B. J. Electroanal. Chem. 1985, 190, 199.
Retter and Kant
Figure 4. Charge-potential dependence of the first (curve 1) and second (curve 2) HgS monolayer at 298 K.
of a monolayer, we have used the dissolution transients (starting from anodic monolayers with the formation time tDmono). This procedure leads to more precise values of the charge, because the dissolution compared to the formation was generally much faster and consequently, the current values obtained were much larger. A precondition for the procedure is that the charges corresponding to formation and dissolution coincide. That has been verified experimentally. Figure 4 represents the charge q-potential E dependence for the first and second HgS monolayers at 298 K. The formation of the first and second monolayers is indicated by a sudden change of the charge in each case. However, the first monolayer shows a ramplike potential dependence of the charge whereas the second monolayer exibits a steplike one. The maximal charge density of the first monolayer is 183 µC cm-2. This value corresponds to the theoretical charge density which would have been obtained if a metacinnabarite monolayer with (010) orientation was oxidized.14 The (010) plane of metacinnabarite corresponds to a square lattice. Indeed, electrolytically formed HgS consists of the black metacinnabarite modification,23 which is of the cubic-facecentered zinc blende type.24 Here, the sulfur atoms are arranged in a cubic closed packing in which half of the tetrahedral sites are filled with mercury atoms and the coordination numbers of sulfur and mercury are both four.25 Figure 5 shows the experimental degree of coverge θ-potential E curves for the temperatures 312.15 and 313.65 K. Although at 312.15 K the θ-E curve exibits a step, the corresponding curve for 313.65 K shows a continous dependence of θ on the potential. Such a marked change of the coverage dependence in a relatively narrow temperature region is typical of the behavior of a monolayer around its critical temperature Tc. Therefore, the mean value of 312.15 and 313.65 K has been selected as the critical temperature Tc ) 312.9 K of the first HgS monolayer. Using eqs 1 and 3, the lateral interaction energy AA of the HgS molecules in the first monolayer can be determined. The result is AA ) -4.59 kJ mol-1. The lateral interaction energy of -4.59 kJ mol-1 corresponds to a strong attractive lateral interaction between the adsorbed HgS molecules. This is the first time that (23) Armstrong, R.; Porter, D.; Thirsk, H. J. Phys. Chem. 1968, 72, 2300. (24) Lundberg Aurivillius, K. Acta Chem. Scand. 1950, 4, 1413. (25) Kleber, W. Kristallographie; VEB Verlag Technik: Berlin, 1956.
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Langmuir, Vol. 13, No. 17, 1997 4719
Figure 5. Degree of coverage-potential dependence of the first HgS monolayer at 312.15 K (curve 1) and 313.65 K (curve 2).
Figure 7. High-frequency capacity and frequency-nomalized conductance vs potential of the first HgS monolayer at 297 K. Sampling times tA: (1) 110 ms; (2) 1100 ms; (3) 2200 ms.
Figure 6. High-frequency capacity-potential dependence and degree of coverage-potential dependence of the first HgS monolayer at 297 K. Sampling times tA: (1) 110 ms; (2) 1100 ms; (3) 2200 ms; (4) (triangles) θ-E data.
Figure 8. Degree of coverage-potential dependence of the first HgS monolayer: (1) (crosses) 280.95 K, θ-E data; (2) (triangles) 308.1 K, θ-E data. Solid lines: theoretical isotherms according to eq 23.
a value of the lateral interaction energy could be obtained for anodic monolayers. Considering the degree of coverage θ-potential E curves (for instance at T ) 297 K in Figure 6) more in detail, the slope of the ramp decreases suddenly at a special potential Etrans of about -18 mV with increasing potential. This finding is supported by measurements of the electrode admittance as a function of the potential. In Figure 6, also the high-frequency capacity CP as a function of the potential E is shown. Figure 7 represents this dependence together with that of the frequency normalized conductance 1/ωRP. The measuring frequency was 25 kHz, and the electrolyte resistance has been mathematically eliminated in the usual way.26 The potential dependences of CP and 1/ωRP have been measured for various sampling times tA after a potential step from -68 mV to the particular measuring potential. As for the θ-E curves, the slopes of the CP-E dependence change suddenly at about -18 mV. Another indication for a sudden change in the monolayer is the marked decrease of 1/ωRP up to negligible values at about -18 mV. The steplike increase of the degree of coverage can be traced back to the formation of a honeycomb lattice. (The maximal theoretical charge density of a honeycomb HgS lattice is 133 µC cm-2 due to its total oxidation.14 The honeycomb lattice (26) Jehring, H.; Retter, U.; Horn, E. J. Electroanal. Chem. 1983, 149, 153.
corresponds to the (110) plane of metacinnabarite.) Only for potentials larger than Etrans does a square lattice exist. For these potential regions, the experimental degree of coverage θ-potential E curves have been modeled in the temperature range mentioned. In the following, the experimantal θ-E curves will be modeled for 11 different temperatures in the range from 279.45 to 308.1 K using the low-temperature series expansion of θ (eq 23) and with jmax ) 15 and the number of nearest neighbors k ) 4. Figures 8-10 show the experimental θ-E curves together with the theoretical ones obtained by nonlinear regression (Marquardt). The modeling can be considered as good, because the standard deviation between theoretical and experimental values was below 2% for all curves. Figure 11 represents the values of n the number of electrons involved in the electrode reaction, obtained by the nonlinear regression analysis as a function of the temperature. In the temperature region considered, the number of electrons involved in the electrode reaction are constant, as expected. The mean value of n is 1.78, somewhat smaller than the expected of n ) 2 for the electrode reaction considered (eq 25). Figure 12 shows the adsorption energies ∆GA determined by using eqs 15-18 with the n values obtained and for (E - Etrans) ) 20 mV. At this potential E of about 2 mV compared to more negative potentials, the monolayer
4720 Langmuir, Vol. 13, No. 17, 1997
Figure 9. Degree of coverage-potential dependence of the first HgS monolayer: (1) (rhombuses) 281.8 K, θ-E data; (2) (circles) 297.5 K, θ-E data. Solid lines : theoretical isotherms according to eq 23.
Retter and Kant
Figure 12. Adsorption energy as a function of temperature for the first HgS monolayer.
Figure 13. Degree of coverage-potential dependence of the first HgS monolayer at 279.45 K. Solid lines: theoretical isotherms according to eq 23 with (1) jmax ) 1 and (2) jmax ) 15. Figure 10. Degree of coverage-potential dependence of the first HgS monolayer: (1) (squares) 283.45 K, θ-E data; (2) (triangles) 293.5 K, θ-E data. Solid lines: theoretical isotherms according to eq 23.
Figure 11. Number of electrons involved in the electrode reaction as a function of temperature for the first HgS monolayer.
has its maximal density. Figure 12 shows a relatively small linear increase of the adsorption energy; i.e., ∆GA changed from -14 to -16 kJ mol-1 in the temperature region considered. The relatively low value of the free energy speaks to a physisorption of the HgS molecules at Hg. These values are in the order of those for organic
molecules adsorbed at the mercury/electrolyte interface.27 Till now no values of the adsorption energy of molecules in anodic films are known. The question arises how an increase of the size of the hole clusters j affects the quality of the modeling using eq 23. To answer that question, a modeling of the potential dependence of θ with jmax ) 1 and jmax ) 15 (lowtemperature series expansion of θ with respect to the activity z) has been performed for 279.45 K. The result can be seen in Figure 13. At 279.45 K, the percentage standard deviation ∆θ is 0.95% for jmax ) 1 and 0.53% for jmax ) 15. Also for 302.4 K, the percentage standard deviation ∆θ is 2.17% for jmax ) 1, with 2.17% markedly higher than that for jmax ) 15 (1.36%). The difference in the modeling quality becomes much more evident, when comparing the values of the number of electrons involved in the electrode reaction obtained by the nonlinear regression analysis. For 279.4 K and jmax ) 1 results, n ) 0.96 ( 0.31; i.e., the percentage standard error is 33%. For the same temperature and jmax ) 15, one obtains 1.8 ( 0.13, i.e. a much lower percentage standard error of 7.2%. Summing up, the low-temperature series expansion of θ with a maximal hole cluster size of 15 represents a much better theoretical description of the degree of coverage-potential curves than that for jmax ) 1. It is known that the low-temperature series expansions of θ converges more weakly with increasing temperature (27) Damaskin, B.; Petrii, O.; Batrakov, V. Adsorption organischer Verbindungen an Elektroden; Akademie-Verlag: Berlin, 1975.
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Langmuir, Vol. 13, No. 17, 1997 4721
Recently, a model of the underpotential deposition of metal layers has been developed based on the lattice gas model.28
Figure 14. Influence of the maximal hole cluster size jmax on the number n of electrons involved in the electrode reaction for T ) 302.4 K.
and T < Tc, and the question arises if jmax ) 15 is sufficiently high at higher temperatures. Therefore, the maximal hole cluster size jmax has been varied between 1 and 15 for 302.4 K, and for each case, the number n of electrons involved in the electrode reaction has been determined. The n - jmax dependence is shown in Figure 14. Once again, jmax ) 1 gives a much lower value of n, which is furthermore undetermined. For jmax ) 2, the n obtained is already a good approximation, and for jmax ) 3, the n value obtained is even equal to that for jmax ) 4-15. However, its standard error of 0.384 exceeds that for jmax ) 15. Considering the jmax values between 6 and 15, the values of n coincide one with another, as do the ∆n values. This result can only be interpreted so that hole clusters of sizes between 1 and 6 already completely determine the degree of coverage of the first HgS monolayer and that the contribution of higher hole clusters can be neglected. Consequently, jmax ) 15 is fully sufficient for the modeling of the degree of coverage θ-potential E dependence using the low-temperature series expansion of θ. Adsorption and lateral interaction energy represent characteristic quantities of an anodic film and determine its stability. The method developed to obtain these quantities from current-time transients enables for the first time a substantive classification of anodic films according their stability. The model derived can also be applied to chargepotential isotherms of metal monolayers supposing that the monolayer and the substrate lattice differ not too much.
Conclusions A special method has been developed to obtain the charge-potential isotherms of anodic monolayers from current-time transients. Such isotherms have been determined for HgS at the mercury/electrolyte interface in the temperature range between 279 and 313 K. A change from the steplike shape of the isotherm to a continuous shape in a narrow temperature region enabled us to obtain the critical temperature of the monolayer from the experiment. On the basis of the Ising lattice gas model, the lateral interaction energy of the molecules in the anodic film can directly be calculated from the critical temperature. For the first time, theoretical degree of coveragepotential isotherms for anodic monolayers of honeycomb, square, and triangular lattice geometry have been derived on the basis of the Ising lattice gas model. Here, the degree of coverage was represented as low-temperature series expansion with respect to the activity. All experimental degree of coverage-potential isotherms exhibit a kink, and so do the high-frequency capacity-potential curves. A square lattice has been assumed only for potentials more positive than the kink potential. In this potential region the isotherms have been modeled with the low-temperature series expansion of the degree of coverage considering hole clusters up to a size of 15. The low-temperature series expansion of the degree of coverage represents a valid model. The number n of electrons involved in the electrode reaction and the adsorption energy obtained are practically independent of temperature. The n values of about 2 correspond exactly to the electrode reaction proposed for the HgS monolayer formation. The adsorption energy has not been known up to now. For one high temperature, the influence of the hole cluster size in the low-temperature series expansion on the value of n has been investigated. For hole cluster sizes larger than 6, the values of n coincided, as did their standard errors; i.e., a maximal hole cluster size of 6 is sufficient for the modeling. The model developed can also be used for the theoretical description of charge-potential isotherms of metal monolayers on electrodes, supposing that the deposit and the substrate lattice differ not too much. Acknowledgment. The authors wish to thank the Deutsche Forschungsgemeinschaft for financial support and Dr. H. Lohse for stimulating discussions. LA970213A (28) Blum, L.; Huckaby, D.; Legault, M. Electrochim. Acta 1996, 41, 2207.
