Teaching the Growth, Ripening, and Agglomeration of Nanostructures

Article pubs.acs.org/jchemeduc

Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments Jan Philipp Meyburg and Detlef Diesing* Faculty of Chemistry, University of Duisburg−Essen, 45117 Essen, Germany S Supporting Information *

ABSTRACT: This article describes the implementation and application of a metal deposition and surface diffusion Monte Carlo simulation in a physical chemistry lab course. Here the self-diffusion of Ag atoms on a Ag(111) surface is modeled and compared to published experimental results. Both the thin-film homoepitaxial growth during adatom deposition onto a single-crystal surface and the agglomeration of grown nanostructures due to Ostwald ripening during a heating experiment are simulated. In addition, a Monte Carlo classroom experiment is presented that simulates the random walk of a single Ag atom on the Ag(111) plane. These two Monte Carlo simulations allow for the teaching of random walk, surface diffusion, island formation, and agglomeration. The universal Arrhenius law is evaluated for thermally activated processes. KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Computational Chemistry, Crystals/Crystallography, Surface Science, Thermodynamics, Statistical Mechanics, Nanotechnology

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INTRODUCTION

SIMULATION The hexagonal symmetry of a (111) surface of a face-centered cubic (fcc) crystal must be transformed into a rectangular matrix form in order to be processable for a computer. Here a selected parallelogram of the hexagonal lattice symmetry is transformed into a rectangular or square matrix M: {1, ..., imax} × {1, ..., jmax}. This is accomplished by shifting each row i of the i parallelogram by − 2 in the j direction so that a rectangular (or square form if imax = jmax) results (see Figure 1). In the code, imax is defined as the number of rows (NR) and jmax is defined as the number of columns (NC). For proper output, the calculated rectangular matrices are transformed back to parallelograms, and the distance between the rows i is corrected by a factor of 3 regarding the height of an equilateral triangle. Among the 2 many approaches for making a hexagonal structure processable for a computer, this transformation turned out to be easily understandable and therefore suitable for teaching. As Ag atoms prefer to be adsorbed on fcc sites, these are the only allowed sites for adatom adsorption in our model. In the output files (encapsulated postscript), the next two atom layers beneath the fcc adsorption layer are colored gray (with layer B in dark gray and layer C in light gray), and the occupied adsorption sites (representing the adsorbate layer or the first fcc layer above the crystal substrate) are colored yellow (see Figure 2). Therewith, all three layers (A, B, and C) of the unit cell are drawn. The hollow sites where white space is visible are fcc sites, and the hollow sites where the dark-gray layer B atoms are visible are hcp sites. If the surface size exceeds an adjustable limit (termed the upper limit for substrate display in the code), only the edge atoms of layer C are drawn. The area that is

1

As homoepitaxial growth during adatom deposition and agglomeration of grown nanostructures2 are of great interest in chemistry, various approaches have been developed to model these processes.3,4 One useful tool is the Monte Carlo (MC) method,5−7 which can also be taught in undergraduate courses. This work illustrates the power of the MC method by a comprehensible and instructive code (file growth_ripen.c in the Supporting Information). Therewith, students can be trained in this modeling method and learn to compare modeling results with experimental data. Furthermore, they learn how to influence a chemical system on the nanoscale by varying thermodynamic and kinetic parameters. In addition, the students can study the limitations of Monte Carlo simulation methods.8 While the Monte Carlo method was introduced for simple classroom experiments 40 years ago9,10 and various surface diffusion processes were simulated using Monte Carlo algorithms principally on mainframe computers in the following decades,3,4,11−15 the computing performance and graphical display options of commercial personal computers available nowadays enable a user-friendly and instructive implementation of simulations of the growth and ripening of nanostructures in chemical education. As diffusion and growth processes on sufficiently large surface sectors (e.g., 250 × 250 substrate atoms with periodic boundary conditions) can be calculated in a few minutes using common workstations, this lab course experiment can be carried out economically not only in laboratories but also in classrooms. Here the epitaxial growth and ripening of Ag structures on Ag(111) crystal surfaces are in focus. Nonetheless, the code presented here can be used to simulate various similar systems by modifying the code parameters, as written in the manual for the code. © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: December 23, 2016 Revised: June 6, 2017

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DOI: 10.1021/acs.jchemed.6b01008 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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Figure 1. (left) Real-space parallelogram of a (111) surface of a face-centered cubic crystal. (right) Transformed computable matrix form of the same (111) fcc surface structure. Each adsorbed atom (yellow) is surrounded by six nearest-neighbor fcc sites (light gray), six next-nearest-neighbor fcc sites (green), and six next-next-nearest neighbor fcc sites (blue). The possible diffusion directions are numbered (1−6).

The pre-exponential frequency factor ν0 can be interpreted as the number of diffusion attempts per unit of time or, more figuratively, how often the atom tries to hop from the initial site to the final site per second. In this experiment, the frequency factor is set to 1011 s−1. A microscopic understanding of this factor is often motivated by comparing the above-mentioned attempt frequency with a vibration frequency at the surface or a frustrated translational mode. The activation energy E*a is equal for all possible diffusion processesand depends only on a general activation barrier E0 for surface diffusionif there is no interaction between the adsorbed atoms. However, in the case of self-diffusion and epitaxial growth, there are attractive pair interactions between adsorbed atoms that influence the possibilities of the processes. Diffusion toward adjacent adatoms is more likely, and diffusion away from adatom clusters is less likely. Therefore, for each adatom, interacting neighbor atoms are listed. As drawn in Figure 1, the six nearestneighbor (N) positions (light gray), the six next-nearestneighbor (NN) positions (green), and the six next-next-nearest neighbor (NNN) positions (blue) are taken into account. The diffusion activation energy Ea for an adatom is modified by the number of neighbors. For instance, NN is the number of nearest neighbors, and EN is the interaction energy for the nearest neighbors. Therewith, the activation barrier for each diffusion process is calculated as

Figure 2. An output file supplemented with the fcc crystal structure beneath the adsorption layer.

enclosed by these edge atoms is filled completely by a parallelogram that is drawn between the centers of those edge atoms. Therewith, the file size is reduced and the modeled substrate still exhibits the atomistically correct dimensions. For each adsorbed Ag atom, there are six possible diffusion pathways toward the six nearest neighbor fcc sites, if the final sites are unoccupied. In the case of diffusion, there are thus a maximum of six processes per adatom from which to choose within a Monte Carlo step. The rate (i.e., how often each diffusion process is attempted) can be evaluated by an Arrhenius calculus according to

ν = ν0e

−Ea*/ kBT

Ea = E0 + NN·E N + NNN·E NN + NNNN·E NNN

(2)

Accordingly, this approach increases the activation barrier for accumulated adatoms. Furthermore, it is essential to influence the diffusion toward islands for adatoms, as drawn in Figure 3.

(1)

Figure 3. (center) Diffusion of an adsorbed atom toward an island (green pathway) and away from the island (red pathway). (left) Activation barriers for an actual jump toward an island, Ea,1 (gray), and a virtual jump back from this island, Ea,2 (blue). (right) Reduced activation barrier (green) for a jump toward the island. B



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#5-adatom preferably diffuses along those A-step edges via the two unblocked adjacent hcp sites. In the further course of the simulation, several mobile #5-adatoms perform edge running and thus coarsen the triangular island structure. As a result, entire islands can also diffuse on the surface.15,17−19 In addition, dendritic structures with typically three main branches (one pointing in the [1̅1̅2] direction) result during growth simulations for low temperatures.3,4,11−15 Despite the fact that dimer diffusion may occur at temperatures higher than 220 K, this code simulates only single-atom hops. Therefore, temperature values in the range of 60−210 K are meaningful parameters for this simulation. Nonetheless, dimer diffusion and also the migration of islands can be observed with this code as a sequence of consecutive MC processes. In order to teach not only surface diffusion but also epitaxial growth, the code is supplemented with the option of adsorption processes. The rate of adsorption correlates to a deposition rate that is an adjustable parameter and is set according to experimental deposition rates in the range of 10−3 ML s−1 (monolayer per second). The code tries to adsorb an adatom on a randomly chosen surface site when an adsorption processes has to be performed. When the chosen surface site is already occupied, the code chooses another random surface site that is empty. The growth of a second adsorbate layer is not possible with this code. The diffusion rates for all possible diffusion pathways of all adatoms are listed in a table (called the table of adsorbate data in the code). If Nads is the number of adatoms on the surface, this table has n = Nads × 6 entries. The n sum of all of the diffusion rates in the table, ∑k = 1 νk , is used to calculate a normalized sum of diffusion rates:

Here the diffusion of an adsorbed atom toward an already existing island (green pathway) is distinguished from an alternative path further away from that island (red pathway). In order to favor the green pathway over the red pathway, the activation barrier Ea is modified. For this purpose, the difference between the activation barrier for the actual jump, Ea,1, and the activation barrier for a virtual jump back, Ea,2, is calculated as ΔEa,1,2 = Ea,1 − Ea,2

(3)

Thus, this difference is negative in the case of a jump toward an island and positive in the case of a jump away from that island. Now this difference is used to modify Ea as follows:

Ea* = Ea +

ΔEa,1,2 a

(4)

Here a fraction of the barrier difference ΔEa,1,2 is used to reduce or increase the actual activation barrier. The factor a (termed the barrier deformation factor in the code) is an adjustable parameter. Therewith, all rates ν are calculated using eq 1. In addition, the diffusion pathways to unoccupied fcc sites differ depending on the nearest-neighbor constellations. A single adatom diffuses via hexagonal close-packed (hcp) sites as intermediate states toward the adjacent fcc sites (blue pathways in Figure 4). If one of these hcp intermediate states is blocked

n

∑ νk ≙ 1

(5)

k=1

The code generates a random value between 0 and 1. The random determination of a diffusion process to be performed is done in a way originally suggested by Fisher and Yates20 and later on improved by Durstenfeld.21 A randomly selected entry from the table of adsorbate data is picked up and swapped with the last entry of this list. Then again one entry is selected and is swapped now with the penultimate (so far unmodified) entry. Meanwhile, the swapped diffusion rates (normalized to the sum of diffusion rates) are summed. As soon as this sum exceeds the generated random value, the diffusion process that caused that exceedance is executed (see Figure 5). Therewith, higher diffusion rates are rather selected than lower rates because the higher rates are more likely to cause the limit value to be exceeded. The shuffled list that results is updated regarding the executed diffusion process and is then used as the table of adsorbate data for the following MC step.

Figure 4. Diffusion via hcp sites (blue pathways) and via on-top sites (purple pathways) for possible adatom positions #1−5.

but the final fcc site is unoccupied, the adatom can still diffuse via an on-top site toward the final position (purple pathways in Figure 4). In the case of diffusion via on-top sites, the activation barrier Ea is multiplied by a factor b = 1.375 and is therefore increased, as such diffusion processes are energetically less favorable.3,4,11,12,14 As drawn in Figure 4 the #1-adatom is able to move freely via the three adjacent hcp sites (blue pathways) to all six nearest-neighbor fcc sites. The number of possible pathways for the #2-adatom is reduced to five because one fcc site is blocked by a nearest-neighbor adatom. As one adjacent hcp site is also blocked, one fcc site can be reached only via an unfavorable on-top pathway (purple). For all of the #3adatoms, only one adjacent hcp site is accessible. Thus, there are two pathways away from the cluster via this hcp site, and two pathways via on-top sites are possible. The #4-adatoms (called A-step edge atoms) can diffuse only via one on-top site toward the two possible nearest-neighbor fcc sites. Thus, the diffusion activation barriers for these processes are higher, and therefore, the processes are less probable. As a consequence, for higher temperatures thermodynamically stable triangular constellations with three A-step edges are expected.3,16 The

Time Axis

The elapsed time during one Monte Carlo stepthe time increment Δt with each stepis given as the reciprocal of the sum of all diffusion rates and the adsorption rate for all empty surface sites that are calculated within the Monte Carlo step:5 Δt =

1 n ∑k = 1 νk

+ νads

(6)

in which the adsorption rate is given by νads = (NC·NR − Nads) ·νdep

(7)

where νdep is the deposition rate in monolayers per second. C



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Figure 5. Partial sum of the swapped diffusion rates normalized to the sum of all diffusion rates plotted against the count of the swapped diffusion rates. The diffusion process (green circle) that makes the partial sum strike the limit set by the selected random value is executed.

Figure 6. Epitaxial growth at constant T and increasing θ (period I), heating at constant θ (period II), and Ostwald ripening at constant T and constant θ (period III).

Therefore, the time increment for one MC step decreases with increasing matrix size, as one MC step on a larger surface represents a more minor change. Depending on the course of time and according to the set deposition rate parameter, additional atoms are placed randomly on the surface. In case of a set deposition rate of νdep = 5 × 10−3 ML s−1, a coverage of θ = 0.02 is reached after 4 s: tθ= 0.02 =

0.02 ML = 4s 5 × 10−3 ML s−1

Here surface diffusion events are executed solely. As a result of the temperature increase, the dendritic island shapes transform to more round structures, and larger islands grow at the expense of smaller ones (see Figures 8 and 9). This phenomenon is the basis of Ostwald ripening. The students can observe these shape transformations and the diffusion of smaller islands toward bigger islands in order to merge with them. Thereafter (period III), the surface temperature is held constant for another selectable time interval (duration of ripening at constant temperature). Here the Ostwald ripening of the grown adsorbate structures at constant temperature can be observed as a function of time for different initial conditions (see Figures 10 and 11). Again, the number of islands is reduced as the system tends to form one single island that is rather triangular and exhibits more stable A-step edges. The shapes and sizes of the islands and their evolution over periods I, II, and III should be discussed and compared with experimental results obtained by scanning tunneling microscopy.11,14,23 With the preset parameters (see Table 1), the published STM results can be matched. Nevertheless, it should be addressed that the analyzed structures in the literature grew up to different coverages or with different deposition rates. The influence of these parameter variations should be discussed by the students. In addition, the students should plot the number of islands n normalized to the number of adsorption sites N against 1 in T order to investigate the influence of the set temperature on the homoepitaxial growth and the Ostwald ripening (see Figure 12). Regarding the size of the calculated surface (here 250 × 250 sites), the code may run into a regime where the surface shape (here a parallelogram) and the periodic boundary conditions have effects on the island structures. An increase in the number of adsorption sites N (e.g., to 500 × 500 sites) could somewhat compensate for the effects of the boundary conditions (see the purple curve in Figure 12). Hence, it is common to publish simulation results that show only the inner part of a much bigger calculated surface and neglect the boundaries in order to avoid those effects. The students should become aware of those limitations of the simulation and reflect on the periodic boundary conditions (Why is there only one island growing at 120 K and will there

(8)

Students should look up further details on Monte Carlo experiments in the literature.22

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EXPERIMENTS After a brief introduction to the topic of surface diffusion and KMC simulations, each student sets up two KMC experiments on a desktop computer: one experiment in order to simulate the epitaxial growth and ripening of Ag atoms on a Ag(111) surface and another in order to simulate a random walk over the fcc surface. The growth and ripening code and the random walk code as well as a manual on how to compile the codes and execute the compiled programs on classroom computers are provided in the Supporting Information. Epitaxial Growth and Ostwald Ripening

Starting with the adsorption of a single Ag atom on the Ag(111) surface, first the epitaxial growth is simulated (period I; see Figure 6). Here surface diffusion and adsorption events occur. The code should be implemented for at least five different surface temperatures (temperature during growth), for instance, 60, 75, 90, 105, and 120 K, as shown in Figure 6. According to the selected deposition rate and the final coverage θ, this regime of epitaxial growth has a defined duration (here 4 s; see eq 8). At 120 K, only one island is grown, which exhibits a typical shape with three branches that are separated by 120°, with one branch pointing in the [1̅1̅2] direction. More islands grow at lower temperatures. Preferably also dendritic structures with three branches grow at 105 and 90 K (see Figure 7). The students can observe the influence of the set temperature on the island shapes and the number of islands grown during period I. Subsequently (period II), the surface is heated to a selected temperature (temperature during ripening; here 210 K) during a selectable time interval (duration of heating process; here 10 s). D



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Figure 7. Epitaxially grown structures on 250 × 250 sites at the end of period I (t = 4 s, θ = 0.02) at (from left to right) 60, 75, 90, 105, and 120 K.

Figure 8. Shape transformation of the grown structures shown in Figure 7 during heating within period II (t = 10 s, θ = 0.02).

Figure 9. Shape transformation of the grown structures shown in Figures 7 and 8 at the end of period II (t = 14 s, θ = 0.02, T = 210 K).

Figure 10. Ostwald ripening of the grown structures shown in Figures 7, 8, and 9 in period III (t = 24 s, θ = 0.02, T = 210 K).

Figure 11. Further Ostwald ripening of the grown structures shown in Figures 7, 8, 9, and 10 in period III (t = 34 s, θ = 0.02, T = 210 K).

Table 1. KMC Code Parametersa parameter

value

meaning

E0 EN ENN ENNN ν0 a b

0.1 eV 0.025 eV 0.005 eV 0.00125 eV 1011 s−1 1.25 1.375

diffusion activation energy nearest-neighbor interaction energy next-nearest-neighbor interaction energy next-next-nearest-neighbor interaction energy pre-exponential factor barrier deformation factor (see Figure 3) via on-top diffusion factor (see Figure 4)

a

These parameters were tuned in order to match values from the literature.3,4,14

be only one island for all initial temperatures at t = ∞?). Also, the influence of different adsorption rates and coverages on the number of islands and the island shapes can be investigated by the students in additional experiment runs by adjusting the deposition rate νdep, the pre-exponential frequency factor ν0, and the final coverage θ.

Figure 12. Number of islands n normalized to the number of sites N plotted against 1/T at θ = 0.02 for two different surface sizes.

Random Walk

neighbors are added, the code executes one out of six possible diffusion processes with each Monte Carlo step. All of the diffusion directions are always equiprobable. The trajectory of the randomly diffusing adatom is marked. If an fcc site has been occupied once by an adatom it is colored black. Frequently

In order to teach the basics of surface diffusion using the meansquare displacement, a second code that simulates the random walk of a single adatom on the surface is presented here. The adatom is initially placed at the center of an fcc surface displayed as a parallelogram (see Figure 13). As no interacting E



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Figure 13. Diffusion trajectory of a single adatom on 200 × 200 sites. The blue circle is the smallest centered circle that encloses the whole trajectory.

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CONCLUSION The set of computer experiments presented here is suitable for the purpose of teaching the fundamentals of surface diffusion, crystal structures, epitaxial crystal growth, island formation, Ostwald ripening, and also random walk. Furthermore, students gain an understanding of stochastic simulations and the kinetic Monte Carlo (KMC) method in particular. The results calculated by using the parameter values listed in Table 1 agree with already published experiments and calculations for the Ag/Ag(111) system.11,13−15,23 Also, the diffusion of islands at elevated temperatures due to edge running is observable.12,18,19 In addition and depending on the parameter settings, this code can also be used to simulate hetero- and selfdiffusion on other fcc transition-metal surfaces. For instance, concave equilateral hexagon structures (which are observable at temperatures above 200 K for Pd on Pd(111)25) can be obtained. Since the first implementation of these experiments in physical chemistry courses three years ago, evaluation data could be collected by interviewing the students and the teaching staff. As for today it can be concluded that the computer experiments are highly appreciated by the students. In the students’ curriculum of physical chemistry, the presented computer experiments are the only simulation tasks. It turned out that the quality of the graphical output files increases the interest in and the understanding of the simulated processes. Over 80 students have already passed the physical chemistry course and performed these simulation experiments. The students reported that their knowledge in statistical chemistry and surface dynamics significantly improved. The students understood the connection of the mean-square displacement to the diffusivity and were delighted by observing the adatom perform a random walk on the surface. Several students raised the question whether periodic boundary conditions are misleading in the end and became aware that for very long simulation times there is always only one island left on the calculated surface. Finally, it should be mentioned that one can discuss the factors a and b in eq 4 with the students also in context with the detailed balance condition. A comparative study (a = 1.0 and a

visited sites are colored according to a color scheme ranging from dark red representing rarely visited sites to yellow representing the most frequently visited sites. For a single adatom, the diffusivity can be derived from the Einstein equation for the mean-square displacement ⟨Δx2⟩ along two coordinates: ⟨Δx 2⟩ = 2dDΔt

(9)

where for two coordinates the dimensionality is d = 2.6The diffusivity in turn is given by D = D0e−E0 / kBT

(10)

where E0 is the unaffected activation energy for diffusion without any neighbor interaction. As the site-to-site hopping distance λ on the hexagonal surface arrangement of an fcc(111) surface, for instance Ag(111), is λ=

l 2

(11)

and the surface lattice constants l of transition metals are in the range of 2−5 Å (lAg = 4.079 Å),24 λ is on the order of 10−8 cm. From the spatial extent of the diffusing adatom, ⟨Δx2⟩, and the elapsed time Δt, the diffusivity D is calculated for various surface temperatures T. For this purpose, the radius r of the smallest centered circle that encloses the whole trajectory (blue circle in Figure 13) is used to calculate the square displacement as r 2 = Δx 2

(12)

Therewith, the activation barrier E0 can be determined using an Arrhenius plot analysis. By running the code 30 times, five times at each selected temperature (here 60, 90, 120, 150, 180, and 210 K), an activation barrier that differs slightly from the set parameter value E0 and depends on the set pre-exponential frequency factor ν0 and a reasonable diffusion coefficient at infinite temperature, D0, are calculated. A model analysis of the random walk experiment is provided in the Supporting Information. F



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(12) Brune, H. In Metal Clusters at Surfaces: Structure, Quantum Properties, Physical Chemistry; Meiwes-Broer, K.-H., Ed.; Springer: Berlin, 2000; Chapter 3, pp 67−105. (13) Ovesson, S.; Bogicevic, A.; Lundqvist, B. I. Origin of Compact Triangular Islands in Metal-on-Metal Growth. Phys. Rev. Lett. 1999, 83 (13), 2608−2611. (14) Cox, E.; Li, M.; Chung, P.-W.; Ghosh, C.; Rahman, T. S.; Jenks, C. J.; Evans, J. W.; Thiel, P. A. Temperature Dependence of Island Growth Shapes During Submonolayer Deposition of Ag on Ag(111). Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 71 (11), 115414. (15) Nandipati, G.; Shim, Y.; Amar, J. G.; Karim, A.; Kara, A.; Rahman, T. S.; Trushin, O. Parallel Kinetic Monte Carlo Simulations of Ag(111) Island Coarsening Using a Large Database. J. Phys.: Condens. Matter 2009, 21 (8), 084214. (16) Giesen, M.; Steimer, C.; Ibach, H. What Does One Learn from Equilibrium Shapes of Two-dimensional Islands on Surfaces? Surf. Sci. 2001, 471 (1−3), 80−100. (17) Voter, A. F. Classically Exact Overlayer Dynamics: Diffusion of Rhodium Clusters on Rh(100). Phys. Rev. B: Condens. Matter Mater. Phys. 1986, 34 (10), 6819−6829. (18) Wang, S.; Ehrlich, G. Structure, Stability, and Surface Diffusion of Clusters: Irx on Ir(111). Surf. Sci. 1990, 239 (3), 301−332. (19) Hamilton, J.; Daw, M.; Foiles, S. Dislocation Mechanism for Island Diffusion on fcc (111) Surfaces. Phys. Rev. Lett. 1995, 74 (14), 2760−2763. (20) Fisher, R. A.; Yates, F. Statistical Tables for Biological, Agricultural and Medical Research; Oliver and Boyd: London, 1948. (21) Durstenfeld, R. Algorithm 235: Random Permutation. Commun. ACM 1964, 7 (7), 420. (22) Newman, M. E. J.; Barkema, G. T. Monte Carlo Methods in Statistical Physics; Oxford University Press: New York, 1999. (23) Morgenstern, K.; Rosenfeld, G.; Comsa, G. Local Correlation During Ostwald Ripening of Two-dimensional Islands on Ag(111). Surf. Sci. 1999, 441 (2−3), 289−300. (24) Davey, W. P. Precision Measurements of the Lattice Constants of Twelve Common Metals. Phys. Rev. 1925, 25 (6), 753−761. (25) Steltenpohl, A.; Memmel, N. Energetic and Entropic Contributions to Surface Diffusion and Epitaxial Growth. Phys. Rev. Lett. 2000, 84 (8), 1728−1731.

> 1.0) revealed that the growth of nanostructures is only weakly influenced by a, in contrast to the ripening and agglomeration regime. A detailed study of these effects is beyond the scope of this article, but the C code in the Supporting Information enables such detailed studies. The main objective that is met here is to present a very comprehensible and instructive code that is tunable for specific purposes and scientifically correct but still accessible and understandable for chemistry students.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.6b01008. Manual for compiling and using the C codes (PDF) Model analysis of the random walk experiment (PDF) C code for the growth and ripening experiment (ZIP) C code for the random walk experiment (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Detlef Diesing: 0000-0002-5587-2557 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Thomas Michely, Eckart Hasselbrink, and Eckhard Spohr for helpful discussions. The University of Duisburg-Esssen assisted in paying the open access fee.

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REFERENCES

(1) Bauer, E. Phänomenologische Theorie der Kristallabscheidung an Oberflächen. I. Z. Kristallogr. 1958, 110, 372−394. (2) Ostwald, W. Ü ber die vermeintliche Isomerie des roten und gelben Quecksilberoxyds und die Oberflächenspannung fester Körper. Z. Phys. Chem. 1900, 34U, 495−503. (3) Michely, T.; Krug, J. Islands, Mounds and Atoms: Patterns and Processes in Crystal Growth Far from Equilibrium; Springer: Berlin, 2004. (4) Antczak, G.; Ehrlich, G. Surface Diffusion: Metals, Metal Atoms, and Clusters; Cambridge University Press: Cambridge, U.K., 2010. (5) Maksym, P. A. Fast Monte Carlo Simulation of MBE Growth. Semicond. Sci. Technol. 1988, 3 (6), 594−596. (6) Gomer, R. Diffusion of Adsorbates on Metal Surfaces. Rep. Prog. Phys. 1990, 53 (7), 917−1002. (7) Lombardo, S. J.; Bell, A. T. A Review of Theoretical Models of Adsorption, Diffusion, Desorption, and Reaction of Gases on Metal Surfaces. Surf. Sci. Rep. 1991, 13 (1−2), 3−72. (8) Soltzberg, L. J.; Waber, F. G. Caution in Using Monte Carlo Kinetics Modeling. J. Chem. Educ. 1974, 51 (9), 576. (9) Foglio Para, A.; Lazzarini, E. Some Simple Classroom Experiments on the Monte Carlo Method. J. Chem. Educ. 1974, 51 (51), 336−342. (10) Moebs, W. D.; Haglund, E. A. A Simple Monte Carlo Method for Teaching Chemical Kinetics. J. Chem. Educ. 1976, 53 (8), 506− 342. (11) Brune, H.; Röder, H.; Bromann, K.; Kern, K.; Jacobsen, J.; Stoltze, P.; Jacobsen, K.; Nørskov, J. Anisotropic Corner Diffusion as Origin for Dendritic Growth on Hexagonal Substrates. Surf. Sci. 1996, 349 (1), L115−L122. G


Teaching the Growth, Ripening, and Agglomeration of Nanostructures

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