^{1}and Kevin M. Rosso

^{2,a)}

### Abstract

The solid-on-solid kinetic Monte Carlo model of Lasaga and Blum [Geochim. Cosmochim. Acta50, 2363 (1986)] for dislocation-controlled etch-pit growth has been extended to the growth of etch pits under the control of multiple dislocations and point defects. This required the development of algorithms that are times faster than primitive kinetic Monte Carlo models for surfaces with areas in the range of lattice sites. Simulations with multiple line defects indicate that the surface morphology coarsens with increasing time and that the coarsening is more pronounced for large bond-breaking activation energies. For small bond breaking activation energiesdissolution enhanced by line defects perpendicular to the dissolving surface results in pits with steep sides terminated by deep narrow hollow tubes (nanopipes). Larger bond breaking activation energies lead to shallow pits without deep nanopipes, and if the bond breaking activation energy is large enough, step flow is the primary dissolution mechanism, and pit formation is suppressed. Simplified models that neglect the far field strain energy density but include either a rapidly dissolving core or an initially empty core lead to results that are qualitatively similar to those obtained using models that include the effects of the far field stress and strain. Simulations with a regular array of line defects show that microscopic random thermal fluctuations play an important role in the coarsening process.

We would like to than Jens Lothe, Center for the Physics of Geological Processes at the University of Oslo, for valuable discussions. This work was made possible by a grant from the U.S. Department of Energy, Office of Basic Energy Sciences, Geosciences Program. Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract No. DE-AC06-76RLO 1830.

INTRODUCTION

Linear elasticity and elastic energy density

Point defects

Line defects

KINETIC MONTE CARLO SIMULATIONS

Point defects

Screw dislocations

EDGE DISLOCATIONS

Simplified models for line defect controlled pit growth

DISCUSSION

### Key Topics

- Dissolution
- 113.0
- Activation energies
- 80.0
- Dislocations
- 39.0
- Point defects
- 28.0
- Monte Carlo methods
- 26.0

## Figures

Results of a simulation of dissolution with a point defect density of . On the left-hand side (a), the bond-breaking activation energy is 10, while on the right-hand side (b), the bond-breaking activation energy is 4. The gray scale indicates the height of the surface (a darker shade indicates a lower height), which is shown after an average of 20 layers have been removed. A similar gray scale is used to indicate the surface height in all of the simulations. The surface area is .

Results of a simulation of dissolution with a point defect density of . On the left-hand side (a), the bond-breaking activation energy is 10, while on the right-hand side (b), the bond-breaking activation energy is 4. The gray scale indicates the height of the surface (a darker shade indicates a lower height), which is shown after an average of 20 layers have been removed. A similar gray scale is used to indicate the surface height in all of the simulations. The surface area is .

Results from a simulation carried out using the simplified no strain/empty core model for dissolution of a solid containing point defects. The point defect density is . On the left-hand side (a), the bond-breaking activation energy is 10, while on the right-hand side (b), the bond-breaking activation energy is 4. The volume of the defect core, which dissolves instantaneously on exposure, was 15 (in units of ). The gray scale indicated the height of the surface, which is shown after an average of 20 layers have been removed. The surface area is .

Results from a simulation carried out using the simplified no strain/empty core model for dissolution of a solid containing point defects. The point defect density is . On the left-hand side (a), the bond-breaking activation energy is 10, while on the right-hand side (b), the bond-breaking activation energy is 4. The volume of the defect core, which dissolves instantaneously on exposure, was 15 (in units of ). The gray scale indicated the height of the surface, which is shown after an average of 20 layers have been removed. The surface area is .

Isolated pits centered on screw dislocations. The pits were generated with a shear modulus of , a Burgers vector with a magnitude of unity and bond-breaking activation energies of , 5.0, 3.5, and 3.0 in (a), (b), (c), and (d). The simulations were carried out using lattices of sites, and the average height was reduced by 50, 50, 500, and 500. The simulations were stopped when the pit growth (approximately) reached the edges of the lattice, and the pit depths (the difference between the maximum and minimum heights shown in the areas) at this stage were 6, 55, 524, and 1171. The apparently darker gray shade along the diagonal corners in (b)–(d) is an optical illusion, and this can be confirmed by covering three of the four quadrants with white paper.

Isolated pits centered on screw dislocations. The pits were generated with a shear modulus of , a Burgers vector with a magnitude of unity and bond-breaking activation energies of , 5.0, 3.5, and 3.0 in (a), (b), (c), and (d). The simulations were carried out using lattices of sites, and the average height was reduced by 50, 50, 500, and 500. The simulations were stopped when the pit growth (approximately) reached the edges of the lattice, and the pit depths (the difference between the maximum and minimum heights shown in the areas) at this stage were 6, 55, 524, and 1171. The apparently darker gray shade along the diagonal corners in (b)–(d) is an optical illusion, and this can be confirmed by covering three of the four quadrants with white paper.

Four stages in a simulation of dissolution of a surface with 100 randomly distributed dislocations. This simulation was performed using Young’s modulus of , a Poisson ratio of , a Burgers vector with a value of unity, , and a bond-breaking activation energy of ). A sharp cutoff in the total strain energy density corresponding to a cutoff distance of for an isolated screw dislocation was used. (a)–(d) show the surface after the average height of the surface has been reduced by 40, 320, 1280, and 5120.

Four stages in a simulation of dissolution of a surface with 100 randomly distributed dislocations. This simulation was performed using Young’s modulus of , a Poisson ratio of , a Burgers vector with a value of unity, , and a bond-breaking activation energy of ). A sharp cutoff in the total strain energy density corresponding to a cutoff distance of for an isolated screw dislocation was used. (a)–(d) show the surface after the average height of the surface has been reduced by 40, 320, 1280, and 5120.

Pitted surfaces obtained using a symmetric array of screw dislocations. The stress/strain fields were calculated using the parameters , , and , with a sharp cutoff in the total strain energy density corresponding to a cutoff distance of for an isolated screw dislocation. (a) shows a simulation with a bond-breaking activation energy of after the average surface height had been reduced by . For (b)–(d) , ; , , and , .

Pitted surfaces obtained using a symmetric array of screw dislocations. The stress/strain fields were calculated using the parameters , , and , with a sharp cutoff in the total strain energy density corresponding to a cutoff distance of for an isolated screw dislocation. (a) shows a simulation with a bond-breaking activation energy of after the average surface height had been reduced by . For (b)–(d) , ; , , and , .

A dissolving surface generated by a simulation similar to that used to generate Fig. 5(b), but with a surface of area of instead of . The line defect density was the same in both simulations, and all other model parameters were also the same. Because Figs. 5(b) and 6 are displayed on the same scale, they can be directly compared.

A dissolving surface generated by a simulation similar to that used to generate Fig. 5(b), but with a surface of area of instead of . The line defect density was the same in both simulations, and all other model parameters were also the same. Because Figs. 5(b) and 6 are displayed on the same scale, they can be directly compared.

Etch-pit patterns from simulations carried out with a variable number of screw dislocations with Burgers vectors of magnitude . Young’s modulus of and a Poisson ration of were used in this simulation. A sharp strain energy density cutoff corresponding to the single dislocation strain energy density at a radius of 1.1 was used, and the bond-breaking activation energy was . In all cases, the etch-pit pattern is shown after the average surface height was reduced by 500. (a) shows the surface obtained with two (well separated) dislocations, but the surface contains only one pit. (b), (c), (d), (e), and (f) show the surface morphologies generated using 8, 32, 128, 512, and 2048 screw dislocations.

Etch-pit patterns from simulations carried out with a variable number of screw dislocations with Burgers vectors of magnitude . Young’s modulus of and a Poisson ration of were used in this simulation. A sharp strain energy density cutoff corresponding to the single dislocation strain energy density at a radius of 1.1 was used, and the bond-breaking activation energy was . In all cases, the etch-pit pattern is shown after the average surface height was reduced by 500. (a) shows the surface obtained with two (well separated) dislocations, but the surface contains only one pit. (b), (c), (d), (e), and (f) show the surface morphologies generated using 8, 32, 128, 512, and 2048 screw dislocations.

Simulation of the growth of a single pit controlled by an edge dislocation. Young’s modulus of 850, a Poisson ratio of , a Burgers vector of magnitude , and a strain energy density cutoff length of were used in these simulations. In (a), a bond-breaking activation energy of was used and in (b), the bond-breaking activation energy was . Both parts show pits formed after an average surface height has been reduced by 250. In an infinite mineral surface, the pit shown would have grown far beyond the area shown here.

Simulation of the growth of a single pit controlled by an edge dislocation. Young’s modulus of 850, a Poisson ratio of , a Burgers vector of magnitude , and a strain energy density cutoff length of were used in these simulations. In (a), a bond-breaking activation energy of was used and in (b), the bond-breaking activation energy was . Both parts show pits formed after an average surface height has been reduced by 250. In an infinite mineral surface, the pit shown would have grown far beyond the area shown here.

Simulation of multiple etch-pit growth controlled by 100 randomly placed edge dislocations intersecting a surface. Young’s modulus of , a Poisson ration of , and a strain energy cutoff of 3.0 were used in the simulation. The magnitude of Burger’s vector was , and equal numbers of edge dislocations with Burgers vectors of , , , and were used. The bond-breaking activation energy was set to . (a), (b), (c), and (d) show the surface height field after the average height was reduced by 10, 20, 40, and 80. Several small, shallow pits, are located inside the large pits in (d) (below the level of the gray scale cutoff). Propagating step waves can be seen in several of the figures; they are particularly clear in (d).

Simulation of multiple etch-pit growth controlled by 100 randomly placed edge dislocations intersecting a surface. Young’s modulus of , a Poisson ration of , and a strain energy cutoff of 3.0 were used in the simulation. The magnitude of Burger’s vector was , and equal numbers of edge dislocations with Burgers vectors of , , , and were used. The bond-breaking activation energy was set to . (a), (b), (c), and (d) show the surface height field after the average height was reduced by 10, 20, 40, and 80. Several small, shallow pits, are located inside the large pits in (d) (below the level of the gray scale cutoff). Propagating step waves can be seen in several of the figures; they are particularly clear in (d).

Simulation of multiple etch-pit growth controlled by 100 randomly placed edge dislocations intersecting a surface. Young’s modulus of , a Poisson ration of , and a strain energy cutoff of 3.0 were used in the simulation. The magnitude of Burger’s vector was , and equal numbers of edge dislocations with Burgers vectors of , , , and were used. The bond-breaking activation energy was set to . (a), (b), (c), and (d) show the surface height field after the average height was reduced by 80, 640, 2560, and 10240. In (d), the gray scale covers a height range of 1200. Most of the dissolved cores extend to depths greater than 50 000 below the maximum surface height.

Simulation of multiple etch-pit growth controlled by 100 randomly placed edge dislocations intersecting a surface. Young’s modulus of , a Poisson ration of , and a strain energy cutoff of 3.0 were used in the simulation. The magnitude of Burger’s vector was , and equal numbers of edge dislocations with Burgers vectors of , , , and were used. The bond-breaking activation energy was set to . (a), (b), (c), and (d) show the surface height field after the average height was reduced by 80, 640, 2560, and 10240. In (d), the gray scale covers a height range of 1200. Most of the dissolved cores extend to depths greater than 50 000 below the maximum surface height.

Simulation of the growth of etch pits using the no strain/empty core model. In all parts the pit formed after the average height had been reduced by 250 is shown. In an infinite mineral surface, the pit shown in (a) and (c) would have grown far beyond the area shown here. (a) and (c) were obtained with a bond-breaking activation energy of , and in (b) and (d). In (a) and (b) the empty cores have a size of , while in (c) and (d), the empty cores have a size of . The pit in (b) has a depth of 75, while the pit in (d) has a depth of 200.

Simulation of the growth of etch pits using the no strain/empty core model. In all parts the pit formed after the average height had been reduced by 250 is shown. In an infinite mineral surface, the pit shown in (a) and (c) would have grown far beyond the area shown here. (a) and (c) were obtained with a bond-breaking activation energy of , and in (b) and (d). In (a) and (b) the empty cores have a size of , while in (c) and (d), the empty cores have a size of . The pit in (b) has a depth of 75, while the pit in (d) has a depth of 200.

Simulation of the growth of pits nucleated by 100 empty cores, each with an area of , in an area of . (a), (b), (c), and (d) show the surface after the average surface height has been reduced by 80, 320, 1280, and 2560. The bond-breaking activation energy is . Like many simulations with large bond-breaking activation energies, step waves can be clearly seen on almost flat regions of the surfaces.

Simulation of the growth of pits nucleated by 100 empty cores, each with an area of , in an area of . (a), (b), (c), and (d) show the surface after the average surface height has been reduced by 80, 320, 1280, and 2560. The bond-breaking activation energy is . Like many simulations with large bond-breaking activation energies, step waves can be clearly seen on almost flat regions of the surfaces.

Simulation of the growth of pits nucleated by 100 empty cores, each with an area of , in an area of . (a), (b), (c), and (d) show the surface after the average surface height has been reduced by 80, 640, 2560, and 10240. The bond-breaking activation energy is .

Simulation of the growth of pits nucleated by 100 empty cores, each with an area of , in an area of . (a), (b), (c), and (d) show the surface after the average surface height has been reduced by 80, 640, 2560, and 10240. The bond-breaking activation energy is .

Surface morphologies obtained from simulations with a regular array of empty cores. (a) and (b) show the etch-pit patterns for an array of 64 empty cores after the surface height was reduced by 320 and 10 240, while (c) and (d) show the etch-pit patterns for an array of 256 empty cores after the surface height has been reduced by 320 and 5120. In both simulations, a bond-breaking activation energy of was used and the empty cores consisted of vertical columns with a cross section.

Surface morphologies obtained from simulations with a regular array of empty cores. (a) and (b) show the etch-pit patterns for an array of 64 empty cores after the surface height was reduced by 320 and 10 240, while (c) and (d) show the etch-pit patterns for an array of 256 empty cores after the surface height has been reduced by 320 and 5120. In both simulations, a bond-breaking activation energy of was used and the empty cores consisted of vertical columns with a cross section.

Surfaces obtained from four no-strain/fast core dissolution model simulations in which the microscopic dissolution rates were larger by a factor of in 64 symmetrically placed columns in a surface. Each vertical column had a cross section of (nine sites). A bond-breaking activation energy of was used in all of the simulations. (a) and (b) show the surface morphology obtained from a simulation with and after the average surface height has been reduced by 640. (c) (, ), (d) (, ), (e) (, ), and (f) (, ) show the surface morphologies obtained using larger values of , which leads to pit formation. However, the total range of surface heights (the “thickness” of the surface) soon reaches an essentially constant value. The simulations indicate that the pits slowly grow and disappear as the dissolution process progresses, but the dynamics of this process becomes increasingly slower as the dissolution rate ratio increases.

Surfaces obtained from four no-strain/fast core dissolution model simulations in which the microscopic dissolution rates were larger by a factor of in 64 symmetrically placed columns in a surface. Each vertical column had a cross section of (nine sites). A bond-breaking activation energy of was used in all of the simulations. (a) and (b) show the surface morphology obtained from a simulation with and after the average surface height has been reduced by 640. (c) (, ), (d) (, ), (e) (, ), and (f) (, ) show the surface morphologies obtained using larger values of , which leads to pit formation. However, the total range of surface heights (the “thickness” of the surface) soon reaches an essentially constant value. The simulations indicate that the pits slowly grow and disappear as the dissolution process progresses, but the dynamics of this process becomes increasingly slower as the dissolution rate ratio increases.

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