Three-Dimensional Monte Carlo Simulations of Roughness

Langmuir 1997, 13, 833-841

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Three-Dimensional Monte Carlo Simulations of Roughness Development from Different Mechanisms Applicable to the Dissolution of a Pure Solid A. Hernańdez Creus,† P. Carro,‡ R. C. Salvarezza,‡ and A. J. Arvia*,‡ Instituto de Investigaciones Fisicoquı´micas Teo´ ricas y Aplicadas (INIFTA), Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina, and Departamento de Quı´mica-Fı´sica, Universidad de La Laguna, Tenerife, Spain Received May 6, 1996. In Final Form: November 26, 1996X Three-dimensional (3D) Monte Carlo simulations based upon different models for the dissolution of a pure solid under a surface reaction control are presented. The analysis of the x-y and y-z profiles resulting from Monte Carlo snapshots using the dynamic scaling theory shows that when stochastic noise and lateral advance of the reacting interface are included in the dissolution processes, the roughness exponents are R = 1/3 and β = 1/4, approaching the approximate values predicted by the Kardar, Parisi, and Zhang motion equation in 3D. Otherwise, the addition of surface diffusion relaxation to the model results in R = 1.0 and β = 0.25, for the early stages of the process, and in an unstable surface with R = 1.0 and β > 0.5 for the advanced ones. The progressive hindrance of the interlayer mass transport results in a value of β which increases from 0.25 to 0.45, whereas the value of R remains unchanged. Nonlocal effects stabilizing cavities also result in an unstable surface with β > 1. Results from these models are compared to those obtained from real time STM imaging data on silver and copper electrodissolution in aqueous perchloric acid solution at 298 K.

1. Introduction Solid attack in aggressive environments leads to the loss of material and development of a moving interface. According to the kinetics and mechanism of formation of this interface, a particular morphology of attack can be observed. Then the characteristics of the interface motion constitute by itself a sort of fingerprint of the dominating mechanism involved in the process. The topography of the moving interface results from the noise in the flux of depositing/detaching particles which promotes the development of an irregular profile and a number of physical processes leading to surface smoothing. Therefore, the topography characterization on different scales appears to be a suitable way to investigate the mechanism of the overall dissolution process sustaining the interface motion.1-3 Due to the stochastic nature of the dynamics of solid dissolution, the interface morphology is expected to change with time, in many cases evolving to rough structures.4 Kinetic roughening of the interface occurs when the system is far from the equilibrium and the possibility of a kinetic roughening transition has been investigated by several authors.4 In contrast to the number of models which have been advanced to explain aggregation processes such as vapor deposition,5 molecular beam epitaxis,6 and metal electrodeposition,7 a rather limited number of models have †

Universidad de La Laguna. Instituto de Investigaciones Fisicoquı´micas Teoricas y Aplicadas. X Abstract published in Advance ACS Abstracts, January 15, 1997. ‡

(1) Daccord, G. In The fractal Approach to the Heterogeneous Chemistry; Avnir, D., Ed.; J. Wiley & Sons: New York, 1989; p 183. (2) Evesque, P. In The fractal Approach to the Heterogeneous Chemistry; Avnir, D., Ed.; J. Wiley & Sons: New York, 1989; p 81. (3) Morales, J.; Esparza, P.; Gonza´lez, S.; Va´zquez, L.; Salvarezza, R. C.; Arvia, A. J. Langmuir 1996, 12, 500. (4) Fernandes, M. G.; Latanision, R. M.; Searson P. C. Phys. Rev. B 1993, 47, 11749, and references therein. (5) Meakin, P.; Ramanlal, P.; Sander, M. L.; Ball, R. C. Phys. Rev. A 1986, 34, 509. (6) Family, F. Physica A 1990, 168, 561, and references therein. (7) Meakin, P. In The fractal Approach to the Heterogeneous Chemistry; Avnir, D., Ed.; J. Wiley & Sons: New York, 1989; p 131.

S0743-7463(96)00449-0 CCC: $14.00

been proposed to describe the evolution of the topography of a solid under dissolution8 including the influence of the applied potential.4 Results9 from two-dimensional (2D) simulations of solid dissolution models under surface reaction control have recently shown that this process can be basically considered as a vacancy aggregation process which leads to an interface evolution similar to that expected for models of interface growth. The topographic profiles resulting from each model were analyzed using the dynamic scaling theory6 to determine the characteristic roughness exponents. These exponents, namely, the roughness exponent R, the growth exponent β, and the dynamic exponent z ) R/β, reflected relevant mechanistic aspects of the solid dissolution process. These results can be considered as a first attempt to explore the possibilities of the dynamic scaling theory for interpreting the mechanism of corrosion of a solid. However, to test the validity of the different models for the dissolution of a real solid based upon scanning tunneling microscopy (STM) or atomic force microscopy (AFM) imaging data,10 three-dimensional (3D) Monte Carlo simulations were certainly required. In this work different 3D models for the dissolution of a solid substrate kinetically controlled by a surface reaction are presented. Simulations are based upon Monte Carlo calculations, similar to those described elsewhere.9 Monte Carlo 3D snapshots are analyzed using the dynamic scaling theory applied to the x-z and y-z profiles to evaluate the roughness exponents. Results show that when the topography of the reaction front emerging from the dissolution process involves the contribution of the stochastic noise of detaching particles and lateral contributions at detachment sites, the roughness exponents are close to those which have been assigned to the Kardar, (8) Sieradzki, K. J. Electrochem. Soc. 1983, 140, 2868. Coderman, R. R.; Sieradzki, K. In Fractal Aspects of Materials; Laibowitz, R. B., Mandelbrot, B. B., Passoja, D. E., Eds.; Material Research Society: Pittsburgh, PA, 1985; p 55. Beale, M. I. J.; Benjamin, J. D.; Uren, M. J.; Chew, G.; Cullis, A. J. J. Cryst. Growth 1986, 75, 408; 1984, 73, 622. Louis, E.; Guinea, F. Europhys. Lett. 1987, 3, 871. (9) Hernańdez Creus, A.; Carro, P.; Salvarezza, R. C.; Arvia, A. J. J. Electrochem. Soc. 1995, 142, 11. (10) Salvarezza, R. C.; Va´zquez, L.; Herrasti, P.; Ocoń, P.; Vara, J. M.; Arvia, A. J. Europhys. Lett. 1992, 20, 727.

© 1997 American Chemical Society

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Herna´ ndez Creus et al.

Parisi, and Zhang (KPZ) interface motion model11 for a 3D process, namely, R = 0.38, β = 0.25, and z = 1.5. However, it should be noted that at present the values of R, β, and z for the KPZ 3D-model are not exactly known. On the other hand, the addition of surface diffusion relaxation to the model results in R = 1.0 and β = 0.25, for the early stages of the process, and R = 1.0 and β = 1.0 for the advanced ones, as expected from growth models using a complete surface diffusion equation.12 The progressive hindrance of the interlayer mass transport13 results in an increase in the value of β from 0.25 to 0.45, the value of R remaining unaltered. The inclusion of nonlocal effects stabilizing cavities to the model results in an unstable surface topography with β > 1. The predictions of the 3D models for the dissolution of a solid can be directly compared to the STM and AFM image data on galvanostatic silver and copper electrodissolution in aqueous perchloric acid under surface reaction control. 2. Monte Carlo Simulation 2.1. Basic Aspects. Three-dimensional Monte Carlo simulations for the dissolution of a solid substrate were performed by extending the procedure already described for the two-dimensional (2D) case.10 Briefly, the starting substrate consisted of an initially smooth cubic lattice surface either 20 × 20 × 200, 60 × 60 × 60, 80 × 80 × 26, 90 × 90 × 20, or 100 × 100 × 17 in grid size. The use of different grid sizes allowed us to approach the conditions t f 0 and t f ∞ required to evaluate β and R, respectively. The dissolution process was restricted to the upper face of the cubic lattice. Periodic boundary conditions were included to avoid the preferential dissolution of surface edges. The Monte Carlo time (t) unit was taken as equivalent to the number of movements which were required to remove a monolayer of particles from the substrate. In those models where the reaction is controlled by a surface process the direct proportionality between t and 〈h〉, the average penetration depth of the reaction front referred to the initial upper face cubic lattice plane, was checked. 2.2. Models for Solid Dissolution. From the standpoint of particle dynamics, four models for the dissolution of a solid are considered. For all models the dissolution process is kinetically determined by a surface reaction. In model I (Figure 1a) a particle from the smooth substrate surface is removed at random with a sitedependent detachment probability, Pd(N), given by

Pd(N) ) 6 - N/5

(1)

and a vacancy is simultaneously produced. In eq 1 N denotes the coordination number of the detaching particle at the substrate surface. A more realistic possibility such as an exponential dependence of Pd on N was also considered. As already reported,9 this type of dependence has no significant influence on the results, except that a longer computational time is required. After a particle is removed from the smooth substrate, the dissolution continues around the vacancy in all directions of the cubic lattice with the detachment probability Pd(N). According to this dissolution rule, the lateral advance of the reaction front is possible, and this fact leads to the formation of overhangs. Therefore, this model is a site-dependent version of the ballistic deposition model which is described (11) Kardar, M.; Parisi, G.; Zhang, Y. C. Phys. Rev. Lett. 1986, 56, 889. (12) Family, F. in Fractals in the Natural and Applied Sciences; Novak, M. M., Ed.; North-Holland: Amsterdam, 1994; p 1. (13) Zhang, Z.; Detch, J.; Matiu, H. Phys. Rev. B 1993, 48, 4972.

Figure 1. (a) Scheme for model I. Shaded squares correspond to sites occupied by the aggressive environment. Open squares represent probable dissolution sites. The thick black trace indicates the solid phase/aggressive environment interface. (b) Scheme for roughening relaxation by the surface diffusion of particles (solid squares) left on the surface by the dissolution process corresponding to model II. Arrows indicate possible particle movements in the direction of increasing N. (c) Schemes for the restricted interlayer mass transport used in model III. (1) Displacement of particle A on a terrace and attachment of particle B to a step edge C. The terrace length (l) is smaller than the diffusion length (ld). (2) Displacement of particles and clustering at terraces for l > ld. (3) Scheme showing the physical origin of the energy barrier at steps edges restricting interlayer mass transport. U(x) is the potential energy along the direction x perpendicular to the step edge. (d) Indicative scheme for h and ho in models IVa and IVb. The black square indicates a dissolution site and the full trace corresponds to the moving interface.

by the KPZ interface motion equation, although in this case the aggregation of vacancies rather than that of particles takes place. In model II, particle removal also occurs as in model I, and following this event all remaining particles which are comprised within a domain of a maximum length (ld) from the created vacancy are allowed to diffuse on the surface to a site with a higher N (Figure 1b). Then, the surface diffusion of particles takes place with the same probability in all directions. In model III, the same rules for detachment and surface restructuring used in model II are applied, although in

Roughness Development by Dissolution

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Figure 2. (a) Snapshot obtained from the Monte Carlo simulations of model I. (b) W vs 〈h〉 plot. (c) log W vs log 〈h〉 plot. (d) log W vs log L plot.

model III the terrace-to-terrace displacement caused by the jump of particles involved in the range of ld through a step edge (Figure 1c) is determined by a probability of such a jump, Psc, which is comprised in the range 0 e Psc e 1. Hence, for Psc ) 0, the interlayer surface mass transport is suppresed. In this case, those particles within the domain of ld can only move on the same terrace to reach a site with a higher value of N (Figure 1c,1). Otherwise, when ld is smaller than the terrace length, l, clustering occurs (Figure 1c,2). Conversely, for Psc ) 1, the possibility of interlayer surface mass transport is admitted, and then the features of model II are recovered. Model III accounts for the existence of an energy barrier at step edges (Figure 1c,3).14 Model IV is based upon the same rules described above for model III including the influence of nonlocal effects. During the dissolution of the solid, nonlocal effects may contribute in two different ways, either promoting surface leveling (model IVa) or assisting the development of an unstable interface (model IVb). These situations can be properly described by defining a heigth-dependent particle detachment probability, Pd(N,h), which for model IVa is given by the expression

Pd(N,h) ) Pd(N) P(h/ho)

(2a)

consisting of two probability terms, one which depends on N and another which depends on h and ho, where h and ho are the height of the dissolution site and the height of the highest dissolution site, respectively, measured with (14) Erlich, G. Surf. Sci. 1994, 299/300, 628, and references therein.

respect to the initial basal plane of the substrate. This model simulates the leveling of a surface profile assisted by either an electric or a concentration field acting at protrusions normal to the reaction front. In model IVb the preferred dissolution at cavities due to defects, local acidity, and local salt formation already formed (t > 0), is considered. In this case, for h < ho, the following expression for Pd(N,h) is applied

Pd(N,h) ) Pd(N) Pd((ho - h)/ho)

(2b)

Equations 2a and 2b incorporate nonlocal effects as a factor in the expression of the detachment probability of a surface particle. Correspondingly, the dissolution rate depends on the height at which the particle detachment occurs, i.e., particle detachment is determined not only by the local morphology but also by the entire geometry of the interface. 2.3. Results from Monte Carlo Simulations. Monte Carlo 3D snapshots resulting from the different models described above are shown in Figures 2a, 3a, 4a, and 5a. Accordingly, a considerable difference in the topographic features results from the various dissolution models, these features being reflected in both the x - z and the y - z profiles. The analysis of these profiles applying the dynamic scaling theory provides a way for obtaining kinetic parameters related to the development of roughness during the dissolution of the solid which can be straightforwardly compared to experimental data, allowing us to deduce for each case the likely mechanism of the overall process leading to dissolution and vacancy formation.

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Herna´ ndez Creus et al. Table 1. Values of Scaling Exponents r, β, and z Predicted by the Continuum Growth Equations or Obtained Using Simulations in 3Da model

R

β

z

random deposition Edwards-Wilkinson KPZ Wolf-Villain Lai-Das Sarma

0 0.38b 1 0.66

1/2 0 0.24b 0.24 0.20

2 1.58b 4 3.3

a

From ref 20 and references therein. b Approximate values.

W(L,t) ∝ LR f(x)

(3)

where W(L,t) is defined by

W(L,t) ) [1/N Σ[h(xi) - 〈h〉]2]1/2

(4)

h(xi) is the deposit height measured along the x-direction at the point xi and x ) t/LR/β. Furthermore, f(x) has the following properties: f(x) ) constant for x f ∞, and f(x) ) xβ for x f 0, β and R being the dynamic and static growth exponents, respectively. Note that according to the properties of f(x) for t f 0 eq 3 is reduced to

W ∝ tβ

(5)

The value of the exponent β indicates the time evolution of the interface width of the growing surface. In the case of processes controlled by a surface reaction, t ∝ 〈h〉 so that eq 5 can be also written as

W ∝ 〈h〉β

(5)

On the other hand, for t f ∞ eq 1 is reduced to

W ∝ LR

Figure 3. (a) Snapshot obtained from the Monte Carlo simulations of model II, ld ) 2. (b) W vs 〈h〉 plot. (c) log W vs log 〈h〉 plot.

3. The Dynamic Scaling Theory 3.1. A summary of the Theory. The concept of scaling was introduced in the field to provide a framework for understanding fractal-like topographics of nonequilibrium surfaces.6 The dynamic scaling theory describes the development of a contour on a flat substrate consisting of N points and length L on the x-axis at time t ) 0, and the surface roughness growth in a single direction normal to L (z-axis) increasing in height h without overhangs. The instantaneous surface height can be described by the function h(x,t). Then W(L,t), the instantaneous surface width, can be taken as a measure of the surface roughness. The value of W(L,t) is given by the root mean square roughness of the interface height fluctuations. For an irregular interface the dynamic scaling theory predicts that W scales with t and L as6

(6)

The set of relevant exponents derived from the dynamic scaling theory is assembled in Table 1. The W vs h and W vs L plots shown in this paper to obtain the values of R and β are the average of ten independent runs. 3.2. Interpretation of the Dynamic Scaling Exponents. The value of R is related to the surface texture, and hence to Ds, the fractal surface dimension of the selfaffine surface, by Ds ) 3 - R.6 Thus, for R f 1 (Ds f 2) the surface tends to be Euclidean (ordered), whereas when R f 0 (Ds f 3) the surface exhibits an increasing degree of disorder. Key parameters R and β can be derived from the analysis of surface profiles resulting from adequate imaging procedures. In fact, this is the case of those profiles derived from images obtained by STM which provide high lateral resolution 3D images in real space. Therefore, eqs 3-6 can be extended to data on STM images10 by replacing W by WSTM, the root mean square roughness resulting from STM profiles, and L by Ls, a segment of the STM scan. The values of R and β can be compared to those derived from atomistic and continuum models for interface evolution.6 Nonequilibrium growth models based on an atomistic description such as the Eden model,15 ballistic deposition model,5 and restricted solid-on-solid model,16 result in objects with a nonfractal mass and a self-affine fractal surface. These models can be successfully described by the KPZ continuous equation11 for interface motion, which leads to β = 0.25 and R = 0.4 in 3D growth, and R + (R/β) ) 2 in all dimensions. Otherwise, those (15) Eden, M. In Proceeding of the 4th Berkeley Symposium on Mathematical Statics and Probability; Neyman, F., Ed.; University of California Press: Berkeley, CA, 1961; Vol. 4. (16) Gilmer, G. H.; Bennema, P. J. Appl. Phys. 1992, 43, 1347.


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Figure 4. (a) Snapshot obtained from the Monte Carlo simulations of model III, ld ) 2, Psc ) 0. (b) W vs 〈h〉 plot. (c) log W vs log 〈h〉 plot. (d) log W vs log L plot. (e) β vs Psc plot.

models incorporating surface diffusion lead to either R ) 1.0 and β ) 0.2517,18 or R ) 0.66 and β ) 0.20.19 On the other hand, when nonlocal effects resulting from either an electric or a concentration field are included, the interface growth becomes either unstable or marginally stable, and a value β > 0.5 might be expected.20 4. Dynamic Scaling Data Derived from Monte Carlo Snapshots 4.1. Model I. Monte Carlo 3D snapshots resulting from model I (Figure 2a) show the appearance of a “noisy” (17) Wolf, D. E.; Villain, J. Europhys. Lett. 1990, 13, 389. Villain, J. J. Phys. I 1992, 1, 19. (18) Siegert, M.; Plischke, M. Phys. Rev. Lett. 1994, 73, 1517. (19) Lai, Z. M.; Das Sarma, S. Phys. Rev. Lett. 1991, 66, 2348.

topography similar to that obtained in the formation of a solid phase by a ballistic aggregation.5 Accordingly, the value of W increases initially with 〈h〉 to attain finally a saturation regime (Figure 2b). The log W vs log 〈h〉 plot (Figure 2c) exhibits a short initial region with the slope β ) 0.5, a figure which is due to an uncorrelated random dissolution of the first layer following a Poisson distribution.20 This region, which appears in all the models studied in this work, is irrelevant from the standpoint of the solid surface roughening. Following the initial uncorrelated random dissolution region, a second linear region with the slope β ) 0.23 (20) Baraba´si, A. L.; Stanley, H. E. In Fractal Concepts in Surface Growth; Cambridge University Press: Cambridge, 1995.

838


Figure 5. (a) Snapshot obtained from the Monte Carlo simulations of model IV, ld ) 2, Psc ) 0. (b) W vs t plot.

(Figure 2c) can be observed. Finally, for surfaces in the saturation roughness regime, the log W vs log L plot


approaches a straight line with the slope R ) 0.33 (Figure 2d). The values of R and β derived from model I agree with those values expected from the KPZ equation for interface motion in 3D.11 It can be argued that the asymptotic behavior of R and β would not be reached due to the small grid size used in our 3D simulations, turning the agreement with the approximate KPZ exponents fortuitous. To account for this drawback, we run large scale 2D Monte Carlo simulations based on this model leading to R ) 1/2 and β ) 1/3, as expected for the KPZ interface motion equation in 2D. These results from 2D and 3D Monte Carlo simulations are not surprising as model I is simply the counterpart of the ballistic deposition model for a dissolution process. Therefore, it is reasonable to admit that the values of R and β derived from the 3D Monte Carlo simulations are reliable and describe the asymptotic behavior of the system. 4.2. Model II. The 3D Monte Carlo snapshots resulting from model II considering ld ) 2, lead to the development of cavities with a rather smooth inner surface (Figure 3a). In this model, the W vs 〈h〉 plot, at least in the range of time covered by our simulations, does not reach saturation (Figure 3b) indicating that an unstable interface is produced. The log W vs log 〈h〉 plot exhibits two linear regions yielding either β ) 0.25 or β > 0.512 depending whether the early or advanced stages of dissolution are considered (Figure 3c). Despite the fact that the reaction front is unstable as no saturation is reached, attempts were made to estimate an “effective” value of R from the log W vs log L plot. In this case, it resulted in R f 1.0, irrespective of t. Values of R and β become independent of ld for ld g 1. At the early stages of roughening, the values of R and β related to model II are in the neighborhood of R = 1 and β = 0.25, respectively. Therefore, these figures are consistent with the predictions of those aggregation models incorporating surface diffusion.15,16

Figure 6. (a) Sequential STM images (top view) (85 × 85 nm2) obtained during the electrodissolution of silver in 1 M HClO4 at j ) 30 µA cm-2. The electrodissolution time is indicated in the lower part of each picture. T ) 298 K. (b) WSTM vs t plot at j ) 30 µA cm-2.


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Figure 7. STM images (3D-view) (1500 × 1500 nm2) obtained during the electrodissolution of copper in 1 M HClO4 at j ) 6 µA cm-2: (a) t ) 0; (b) t ) 1500 s; (c) WSTM vs t plot at j ) 6 µA cm-2.

On the other hand, the values R = 1 and β > 0.5, which are observed at advanced stages of roughening, agree with those predicted by solving the complete surface diffusion

equation describing the interface motion.12 It should be noted that at this stage the interface no longer behaves as self-affine, and therefore, the dynamic scaling breaks

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down. Accordingly, exponents R and β are considered as “effective” rather than true scaling exponents. 4.3. Model III. The 3D Monte Carlo snapshots resulting from model III for Psc ) 0 exhibit cavities with rather smooth inner surfaces (Figure 4a). The W vs 〈h〉 plot (Figure 4b) shows the trend to reach a certain saturation which presumably results from the development of a single, deep cavity. The log W vs log 〈h〉 plot for t f 0 leads to a linear region with the slope β ) 0.45 (Figure 4c). For t f ∞, the log W vs log L plot results in a linear relationship extending from log L ) 0.4 to log L ) 1.3 with a value of R f 1 (Figure 4d). For model III no change in the roughness exponents was observed by increasing ld from 1 to 4. Model III with Psc ) 0.2 leads to β ) 0.35 and R ) 1.0, but for Psc > 0.3, the surface roughening behavior resembles that predicted by model II, i.e., β ) 0.25 for the early stages and β ) 1.0 for the advanced stages of dissolution. The β vs Psc plot (Figure 4e) indicates clearly that the incorporation of a small contribution of interlayer mass transport to model III is already sufficient for this model to reproduce the behavior of model II. The dependence shown in Figure 4e resembles that found for aggregation processes with an energy barrier at step edges,13,21 The continuous change from β = 0.25 to β = 1/2 as Psc is increased probably indicates a transient regime when the system goes from a driven surface diffusion to a random evolving interface. As already reported,19 surfaces originated from models incorporating restricted interlayer mass transport would not be strictly self-affine, and therefore, the values of R and β should be also considered as “effective” exponents which, together with the resulting topography, can be compared to experimental data providing thus a test for the validity of the model. 4.4. Model IV. The addition of nonlocal effects (eq 2a) to model III (model IVa) produces a change in neither the surface morphology nor the roughness exponents. In fact, for Psc ) 0 and ld ) 2, model IVa, which includes the leveling effect, leads to R ) 1.0 and β ) 0.40. This means that the interface evolution is dominated by the biased diffusion roughening. On the other hand, when cavities are stabilized (Figure 5a) by nonlocal effects as in model IVb, the dependence of W on t (Figure 5b) implies β > 1. In this case, deep cavities can be related to pit formation in real systems. 5. Monte Carlo Simulations and Experimental Data The topography of the solid resulting from the various models can be compared to that resulting from in-situ STM imaging of single crystal surfaces immersed in aggressive environments and subjected to an applied electric potential, as nanoscopy provides imaging in real time and space. 5.1. The Electrodissolution of Silver in Aqueous Perchloric Acid Solution. Sequential in-situ STM images corresponding to the electrodissolution of a smooth domain of a silver single crystal in aqueous 1 M HClO4 (Figure 6a) under a constant current regime, i.e., at the apparent current density j ) 30 µA cm-2 and 298 K, show the development of large cavities with a rather smooth inner surface. The W vs t plot (Figure 6b) indicates that saturation is reached after a certain dissolution time. As already reported,22 the dynamic scaling analysis applied to the in-situ STM images leads to R ) 0.90 and β ) 0.36. The topography resulting from this system resembles (21) Smilauer, P.; Vvedenski, D. D. Phys. Rev. B 1995, 52, 14263. (22) Vela, M. E.; Andreasen, G.; Salvarezza, R. C.; Hernańdez Creus, A.; Arvia, A. J. Phys. Rev. B 1996, 53, 10217.


closely those topographies predicted by models II or III. However, the value β ) 0.36 indicates that model III with a low value of Psc appears to be the most adequate to represent the electrodissolution of silver at 30 µA cm-2 in the acid solution. From the preceding results it can be concluded that the kinetics of silver electrodissolution proceeds via a surface reaction in which there is an important contribution of the surface diffusion of silver adatoms on terraces. This restriction on the silver adatom interlayer displacement can be attributed to the existence of a Schwoebel barrier at step edges of silver crystal terraces.14 However, although the value β ) 0.36 seems to indicate that the interlayer mass transport is not completely hindered, i.e., a certain number of silver adatoms produced by electrodissolution can move either down or up to either a lower or upper terrace, respectively, to reach high coordination sites. It should be noted that electrochemical kinetic studies of silver dissolution in aqueous HClO4 at low current densities have concluded that the rate controlling step of the electrochemical reaction is the surface diffusion of silver adatoms.23 In this case, a roughening mechanism without pitting dominates the evolution of the silver interface. Nevertheless, when large silver surface domains are imaged, a small number of etch pits at defective sites can also be observed. When domains contain etch pits, a completely different interface evolution is found, as is the case of the copper surface that is discussed further on. 5.2. The Electrodissolution of Copper in Aqueous Perchloric Acid. Sequential STM images of a terrace domain at a copper single crystal, under electrodissolution in aqueous 1 M HClO4, at j ) 6 µA cm-2, show the development of a rough topography with etch pits (Figure 7a).24 In this example, the W vs t plot presents an initial power-like increase with β = 1/3, followed by another power-like increase with β > 1 (Figure 7b). The value β > 1, which is predicted for an unstable reaction front, results from the development of etch pits. Otherwise, in the free-pit regions the interface evolves with W ∝ t1/3, as observed for silver electrodissolution. Accordingly, in these regions model III appears to be valid. On the other hand, when the development of etch pits is considered, model IVb seems to be operative. The origin of this instability is presumably related to the anisotropy in the copper electrodissolution rate at the different crystallographic faces of copper.25 The mechanism of copper electrode reactions involves two consecutive charge transfer steps,26

Cu(hkl) ) Cu(I) + e-

(Ia)

Cu(I) ) Cu(II) + e-

(Ib)

and

and the following equilibrium,

Cu(II) + Cu(s) ) 2Cu(I)

(II)

(23) Despic, A. R.; Bockris, J. O’M. J. Chem. Phys. 1960, 32, 389. Vedenskii, A. V.; Marshakov, I. K. Russ. J. Electrochem. 1995, 31, 234, and references therein. (24) Aziz, S. G.; Vela, M. E.; Andreasen, G.; Salvarezza, R. C.; Arvia, A. J. in preparation. (25) Hamelin H. In Moderns Aspects of Electrochemistry; Conway, B. E., White, R. E., Bockris, J. O’M., Eds.; Plenum Press: New York, 1985; Vol. 16, p 1. Vorotyntsev, M. A. In Moderns Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., White, R. E., Eds.; Plenum Press: New York, 1986; Vol. 17, p 131. (26) Bertocci, U.; Turner, D. R. In Enciclopedia of the Electrochemistry of the Elements; Marcel Dekker: New York, 1974; Vol. 2, p 383.


Unlike reaction Ib, reaction Ia is sensitive to the crystallographic face (hkl). On the other hand, when neither Cu(I) nor Cu(II) species are present in the environment at the beginning of the electrodissolution process, step II is not relevant. According to the copper electrode reaction mechanism in aqueous solutions, there is a change in the ratecontrolling step in going from a low to a high current density range.27 The reaction mechanism in the low current density range implies a rate-determining step resulting from the consortial effect of the charge transfer step (Ib) and surface adatom diffusion, whereas in the high current density range it implies that the surface diffusion of adatoms is extremely fast as compared to the charge transfer step (Ib). The latter becomes then the rate-determining step. Then if one admits that a nonuniform current density distribution exists on the copper surface due to crystallographic defects, local domains of low and high current density could be distinguished, as shown by the STM images. This anisotropy in the reaction rates contributes to the appearance of an unstable interface. A similar line of reasoning can be extended to possible differences in competitive adsorption of solution constituents26 in those regions at defective sites and terraces28 also contributing to an unstable interface motion and stabilizing the electrodissolution at cavities. Therefore, for copper in aqueous perchloric acid when etch pits

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dominate the evolution of the interface, model IVb seems to capture the essential physics of the electrodissolution process. 6. Conclusions The consistency of experimental data with the predictions of rather simple models is promising. Models III and IVb originate unstable interfaces with effective exponents which show a trend for the interface evolution close to that observed in the experimental systems. Despite the fact that these exponents would correspond to transient regimes, from our results it emerges that modeling has to include surface diffusion with restricted interlayer mass transport for the case of silver electrodissolution and nonlocal contributions for copper etch pit formation. Further work in this field is certainly welcome, particularly in the development of more realistic models, larger scale computer simulations of the models, and a much more quantitative set of experimental data than that available at the present time. Acknowledgment. This work was financially supported by the Consejo Nacional de Investigaciones Cientı´ficas y Tećnicas of Argentina (CONICET) and Projects PB94-0592-A and IN94-0553 DGICYT (Spain). R.C.S. thanks the Secretarıá de Estado de Universidades e Investigacioń (Spain) for financial support. LA960449Z

(27) Bockris J. O’M.; Reddy, A. K. K. In Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 2, p 1202.

(28) LaGraff, J. R.; Gewirth, A. A. Surf. Sci. 1995, 326, L461.

Three-Dimensional Monte Carlo Simulations of Roughness

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