Direct Observation of Single Ostwald Ripening Processes by

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Direct Observation of Single Ostwald Ripening Processes by Molecular Dynamics Simulation Thomas Kraska† Physical Chemistry, UniVersity of Cologne, Luxemburger Strasse 116, D-50939 Ko¨ln, Germany ReceiVed: July 17, 2008; ReVised Manuscript ReceiVed: August 13, 2008

Ostwald ripening is an important growth process in many scientific disciplines ranging from material science, geology, biophysics, and product formulation. Here ripening of argon clusters in a vapor phase is observed directly in constant energy molecular dynamics simulations serving as a model system for large-time scale ripening processes. Starting from an initial metastable equilibrium between the vapor phase and two clusters Ostwald ripening is initiated by the addition of kinetic energy. This mimics local thermal fluctuations in a larger system. It appears that there is not necessarily a close encounter of two clusters before ripening sets in. Also no static density bridge between two ripening clusters is observed. The onset of ripening is rather related to the different evaporation dynamics of clusters of different size. It can start at the moment energy is added or with some delay, depending on the difference in cluster size and dynamics. 1. Introduction The decay of a metastable phase passes through several different types of growth. At the very beginning homogeneous nucleation takes place. Small clusters are formed by density or concentration fluctuations which continue to grow once they pass an activation barrier. This activation barrier originates from the competition of bulk and surface properties of the just forming cluster. Stable clusters can continue to grow by surface growth. Once a significant amount of clusters is formed coarsening takes place. Clusters can collide and merge to a larger cluster by coalescence. A collision-free type of coarsening is Ostwald ripening. Ostwald ripening is a near-equilibrium process of aging, redistribution, or coarsening of matter in various areas. During Ostwald ripening large clusters or particles grow at the expense of small ones that eventually evaporate or dissolve completely. The driving force, for both coalescence and ripening, is the lowering of the surface energy which is in turn related to the surface area. The difference in vapor pressure or solubility of clusters with different sizes and surface curvatures leads to the evaporation or dissolution of small clusters and growth of larger ones. The background of ripening has been investigated by Ostwald using mercury oxide particles.1 He found in extensive experiments that fine milled yellow mercury oxide particles have a higher solubility in the order of 7% than more coarse red particles. His work is the first quantitative proof of the influence of surface energy on the equilibrium. Later Hulett confirmed Ostwald’s results with other substances.2 Hulett also discussed the well-known experience that freshly precipitated solids may pass through a regular filter while after some time they do not which is a consequence of ripening. To be able to separate the solid directly after precipitation one uses hot solutions. High temperature enhances ripening as later observed for many other systems. One important area for ripening is material sciences. It is for example observed for metal clusters or islands on metal surfaces.3,4 The ripening process takes place either by diffusion of atoms on the surface or by vacancies. Ostwald ripening has also been used to prepare hollow TiO2 spheres.5 Under hydro† E-mail: [email protected].

thermal conditions for reaction times from 2 to 100 h the rise in crystallinity has been observed which indicates ripening. For sintering of silicon nitride ceramics an anisotropic dynamic Ostwald ripening process has been proposed to be initiated by very rapid heating of the system.6 Recently it has been reported that ripening can be enhanced by dislocations. A flux of silicon atoms from smaller to larger precipitates in aluminum grains by so-called pipe diffusion has been observed.7 This in situ investigation shows single ripening processes for solid state systems with growth curves similar to those of liquid state ripening investigated here. Ostwald ripening is also expected in geological systems like, for example, the coarsening of garnet, minerals used as gemstones, where the narrow crystal size distribution can only be explained by ripening.8 Based on an analysis of the particle size distribution is has been concluded that ripening is a significant growth type in zircon overgrowth formation during anatexis, which is the partial or complete melting of rocks in the earth’s crust.9 In most cases the volume of the zircon overgrowth is too high to originate from existing saturated melt, hence ripening is the most likely process. In protein crystallization and recrystallization Ostwald ripening is involved too. By seeding one can, for example, grow large crystals from a large amount of small crystals obtained from high supersaturation precipitation.10 The small crystals shrink while the larger seed crystal grows being visible by a depletion zone of small particles surrounding the seed crystal after some time. Furthermore, Ostwald ripening has been reported in the evolution of phase separated domains in lipid bilayer membranes.11 It was found that ripening is especially important in immobile systems while mobile systems more likely form larger domains by merging after collision. The growth rate due to merging or coalescence is significantly higher than that of ripening, however, the authors found that both processes are coupled for mobile systems. Finally, Ostwald ripening is important for the formulation and stability of ice cream. Recrystallization by ripening has been investigated for ice cream stored in bulk containers between -15 and -5 °C12 leading to an undesired texture of the ice

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Figure 1. (a) Temperature of the complete system. After equilibration at 94 K the temperature is set to 130 K for 1 ps. After that short period of time the thermostat is switched off and the temperature evolves in the NVE ensemble. (b) Number of atoms in the big and the small cluster and the sum of both. (c) Number of monomers smoothed by a moving average. (d) Temperatures of the two clusters in the system smoothed by a moving average. The vertical line at 33 ns marks the onset of ripening. (e) Kinetic, potential, and total energy of the system.

cream. It has been found that the ripening increases with the temperature of the system. Furthermore, the rate of ripening is higher if temperature fluctuations of (1 °C are imposed on the ice cream. A first model for the influence of Ostwald ripening on the particle size distribution has been proposed by Lifshitz and Slyozhov13 and Wagner.14 Several other models and analysis have been proposed which all yield the particle size distributions caused by ripening as result.15-18 In this work here the focus is not on the properties of the size distributions of large ensembles of particles but rather on the analysis of single ripening processes of two clusters. So far there are few direct investigations of single ripening processes by simulation methods. Ripening has been observed in phase field simulations, which is a continuum method suitable for the simulation of growth.19 In that work ripening in two-dimensional alloys is investigated. Typical Ostwald ripening is observed as well as a combination of ripening and coalescence if two droplets are brought close enough together or when the interface thickness of the clusters is increased. In the two latter cases a density bridge between the droplets exists. This is not the case for the

typical ripening observed when the clusters are in sufficient distance. In an interesting recent work on the complete simulation of phase separation including nucleation and different types of growth it is claimed that Ostwald ripening in a system of Lennard-Jones atoms is observed, however, the information provided, especially the growth curves, suggest a coalescence process.20 While molecular dynamics simulation (MD) is a suitable method to investigate the dynamics of growth processes including ripening, in case of solid systems the time frame of ripening is clearly beyond the current ability of MD simulations.4 In order to be able to analyze ripening in detail by MD one has to choose a system that evolves in the time frame of up to hundreds of nanoseconds. Therefore, the ripening of liquid argon clusters in the vapor phase serves here as a model system for Ostwald ripening. Due to the vapor pressure and the fast diffusion of argon atoms in the vapor phase it is possible to observe ripening in the time frame of MD simulations. Single Ostwald ripening processes are directly observed in LennardJones argon systems.

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Figure 2. Snapshots of the simulation at three different points in time. The big cluster is marked by a large (red) circle, the small cluster by a smaller (blue) circle. At 80 ns only one cluster is left. It is marked by two half-circles because the cluster just leaves on one side and reenters on the other side by periodic boundary conditions.

Figure 3. (a) Distance of the two clusters from center to center. The vertical line at 33 ns marks the onset of ripening. (b) Density profiles of the two clusters averaged over 0.1 at 40 ns.

2. Simulation Method

3. Analysis of Ripening Processes

Molecular dynamics simulations are performed in the NVE ensemble, i.e. the number of argon atoms, the volume, and the total energy of the system are conserved. Programs used and further developed here go back to the code of Wonczak.21 The only exception to the NVE ensemble is the addition of kinetic energy for very short period of time (1 ps) at the beginning of the simulation as described below. All simulation systems contain 8000 argon atoms interacting by the Lennard-Jones potential with the parameters
/kB ) 120 K and σ ) 0.3405 nm and a 5σ cutoff. The length of the cubic box is 29.5 nm (diagonal 51.1 nm). The overall density is 0.517 mol/dm3 or 20.67 kg/m3. The clusters are detected by the Stillinger criterion22 with a reasonable distance for large clusters of 1.5σ. Periodic boundary conditions and minimum image conventions are applied. The initial configurations are obtained from nucleation simulations in the NVE ensemble.23 In some cases systems with two large clusters are obtained in metastable equilibrium with the surrounding vapor phase. A problem of the direct simulation of Ostwald ripening is the interference with coalescence. In several cases the clusters collide and merge before the onset of Ostwald ripening or before the smaller cluster completely vanished by ripening. In order to alter the development of the system after the equilibration simulation and to speed up ripening, kinetic energy is added to the system. This is accomplished by setting the system temperature for 1 ps to an elevated value by velocity scaling. No influence of the thermostat type on the simulation results is expected. The reason is the very short duration of the thermostat application of 0.001 ns compared to typical simulations durations of 10 to 80 ns. Due to the continuing evaporation and condensation of atoms at the clusters any influence of this initial period disappears. In addition special temperature effects are confirmed by calculating the configurational temperature.

Within this work several ripening processes are investigated. A first successful example for Ostwald ripening is shown in Figure 1. After thermal equilibration of a system obtained from particle formation simulation23 the temperature is set from about 94 to 130 K for 1 ps followed by a simulation in the NVE ensemble. The kinetic temperature of the complete system (Figure 1a) goes back to the initial temperature of approximately 94 K in about 7 ns. During the following period of time the temperature exhibits fluctuations on the order of (3 K related to the small cluster size but remains constant on average. The development of the size of the two clusters is shown in Figure 1b. In the first 8 ns both clusters evaporate partially because of the addition of kinetic energy. The plot of the system energy (Figure 1e) shows that at the beginning the kinetic energy is increased. Accordingly the total energy rises. The addition of kinetic energy increases the temperature leading to partial evaporation of the clusters and hence rising potential energy. The total energy remains constant after the first picosecond over the complete period of simulation time of 80 ns of which only 10 ns is shown for better visibility of the initial period of the simulation. Changes in the kinetic and potential energy compensate as usual in an NVE ensemble. Hence, the required heat for vaporization leads to a decreasing temperature of the complete system (Figure 1a). At about 33 ns one can recognize that the size of the large cluster increases while that of the small cluster shrinks until it vanishes at about 73 ns. A numerical analysis of the cluster size data plotted in Figure 1b shows that the sum of the number of atoms in both clusters decreases during Ostwald ripening while it is more or less constant before the onset of ripening; it only slightly decreases as discussed below. This means that there is a slight overall evaporation during ripening, also visible in the slightly rising number of vapor phase monomer (Figure 1c). It can be explained from an energetic

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Figure 4. Density profiles along the connection line of the clusters at different points in time corresponding to Figures 1-3. The dashed horizontal line represents the equilibrium vapor density at the given temperature. For all calculations the distance between the centers of the two clusters is divided into 20 intervals. The density in each interval is averaged over 20 configurations: (a) long time before the onset of ripening, 20.481 to 20.541 ns, to dj ) 13.7 nm; (b) short time before, 32.541 to 32.601 ns, dj ) 17.8 nm; (c) short time after, 33.841 to 33.881 ns, dj ) 17.6 nm; (d) long time after, 39.901 to 39.961 ns, dj ) 12.8 nm.

Figure 5. (a) Temperature of the complete system. After equilibration at 94 K, the temperature is set to 120 K for 1 ps. The simulation is then continued in the NVE ensemble. Inset: system temperature during coalescence. The horizontal lines mark the average temperature before and after coalescence, which is indicated by the vertical line. (b) Number of atoms in the two clusters.

point of view: the single final large cluster has a smaller surface and hence a lower surface energy than the sum of the two clusters before the onset of ripening. This energy difference of the system before and after ripening slightly heats up the system leading to evaporation. Therefore the number of atoms in both clusters decreases slightly during ripening in a constant energy system. In order to understand the origin of the onset of ripening the temperature of the two clusters is plotted in Figure 1d. Before the onset of ripening the temperature of both clusters is more or less the same as that of the total system. Hence, the system is thermally equilibrated. The temperatures of the two clusters start to deviate a few nanoseconds after the ripening sets in. This deviation of the cluster temperatures is verified by calculating the configurational temperature. The small cluster

evaporates and hence its temperature decreases. This suggests that a temperature difference is not the origin of ripening in this system, but rather the result of the ripening process. The temperature jump at 0 ns leads to a strong evaporation followed by a weak evaporation after 7 ns. The origin of the onset of Ostwald ripening at about t ) 33 ns might be related to the fact that the system still evaporates to a small extent in the time frame from 7 to 33 ns. Within this period of time the number of atoms in the big cluster N1 is almost constant (Figure 1b). The slope of the linear regression of the N1-data as function of time is -9.982 × 10-3 ns-1 and hence the large cluster only loses 0.2 atoms on average. The number of atoms in the small cluster N2 has a slope of -0.3877 ns-1 which gives a loss of about 10 atoms in the same period of time. Apparently the

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Figure 6. (a) Temperature of the complete system after a temperature jump to 110 K for 1 ps at 0 ns. At 10 ns the simulation is forked. One simulation is continued for another 10 ns without further temperature jump. The second system is again heated up to 110 K for 1 ps. (b) Cluster sizes corresponding to part a. The second temperature jump at 10 ns leads to a faster evaporation of both clusters and acceleration of ripening compared to the simulation without a second temperature jump.

evaporation caused by the temperature jump affects the small cluster stronger than the large cluster, which eventually leads to the onset of ripening. In Figure 2 some snapshots of the simulation system at different points in time are shown. One can see at 0 and 40 ns that there are two distinct clusters in sufficient distance so they cannot coalesce. The structure of the clusters is that of a typical Lennard-Jones liquid with wide peaks in the pair correlation function at r/σ somewhat lager than 1, 2, and 3. At 80 ns only one cluster is left, located at a top boundary of the box connected to its other part at the bottom of the simulation box by periodic boundary conditions. Another possible explanation for the onset of Ostwald ripening is that the clusters approach closely without coalescence. The distance of the centers of the two clusters is plotted in Figure 3a. The movement of the clusters is random Brownian motion due to the collision of the cluster with vapor phase atoms and the continuous condensation and evaporation of atoms leading to many small impulses in various directions. There is indeed a decrease in the distance at 33 ns but the clusters have been even closer before the onset of ripening. There appears to be no special close encounter of the two clusters at the moment ripening starts. Figure 3b shows the density profiles of the two clusters averaged over 0.1 at 40 ns simulation time. The density profiles are calculated for each cluster separately starting from their respective centers. They are combined in Figure 3b by shifting the cluster center of the small cluster to the distance of 13.2 nm. From this plot there also appears to be no special density bridge in the space between the two clusters. It should be kept in mind however that the density distribution plotted in Figure 3b is normalized the usual way by the volume of the incremental shell volume. This leads to a lowering of the density of a possible cluster in significant distance to the center of the first cluster. A possible density bridge between two clusters is therefore suppressed. In order to get a reliable density profile between the two clusters a new analysis method is implemented. The centers of both clusters are calculated and at points along the connection line between the clusters the density is calculated locally. Typically 20 points along the connection line are chosen and the radius of the spheres around the points is one tenth of the cluster distance. Variations of these parameters over a reasonable range do not change the topology of the curves. The resulting density profiles are plotted in Figure 4. In these diagrams, showing the density domain of the vapor phase only, one can recognize that the density in the space between the clusters is in the order of the equilibrium vapor phase density of argon marked by the dashed horizontal line. Apparently there

are fluctuations of the density in the vapor phase region between the clusters. Figure 4 shows distributions at four different points in time and for different distances between the clusters. Fluctuations are present for large distances (Figure 4, parts b and d) and short distance (Figure 4c). In Figure 4b an example for a density wave is visible with a minimum at about 11.5 nm close to the small cluster (right) becoming a maximum. Such waves appear occasionally but they appear to be a transient phenomenon. If the two clusters are close together the density profile does not exhibit pronounced density maxima (Figure 4c). The density between the two clusters rather fluctuates longitudinally around the vapor phase density. A possible reason is that the wavelength of the transversal fluctuation appears to be in the order of the distance between the two clusters. It is known that there is a depletion zone with a density minimum near the surface of a cluster surrounded by a vapor phase in equilibrium.24 In combination with the evaporation and condensation at the surfaces of the two clustersseven without ripeningssuch fluctuations can appear. Comparison of the density profiles long before the onset of ripening (Figure 4a), shortly before (Figure 4b), shortly after (Figure 4c), and long after (Figure 4d) does not exhibit any difference in this respect. Hence, there is no static density bridge present and density fluctuations always exist with and without ripening. The fluctuation is in the order of (50% of the equilibrium vapor density but it is orders of magnitude lower than the liquid phase density. Therefore density fluctuations are basically not visible if the profiles are scaled to the complete density range of the system including the liquid phase. In Figure 5, a simulation similar to that in Figure 1 is shown. The difference is that the temperature is kept at 120 K for 1 ps after t ) 0 ns. The temperature of the complete system evolves similar to that shown in Figure 1. It reaches approximately 94 K in less than 10 ns (Figure 5a). In this case, however, there is no time lag; Ostwald ripening sets in immediately after this temperature is reached (Figure 5b). In contrast to the simulation shown in Figure 1 ripening is slower, i.e. the slopes are shallower than in Figure 1d after 33 ns. Later, at about 52 ns, the two clusters collide and ripening is bypassed by coalescence. The sum of the atoms in both clusters N1 + N2 again slightly decreases during ripening as in the case discussed above. After coalescence the sum N1 + N2 decreases more rapidly. This is caused by the temperature rise of the coalescence process shown at higher resolution in the inset in Figure 5a. This temperature increase is related to the loss of surface energy by coalescence. Since coalescence is faster than ripening the temperature rises

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faster and leads to faster evaporation. Comparison of the two examples in Figure 1 and Figure 5 shows that ripening and coalescence lead to the same result in a constant energy ensemble in case of fluid clusters. Obviously the difference lies in the velocity of the two processes. A third example is shown in Figure 6. At the beginning of the simulation the temperature is set to 110 K for 1 ps. After about 5 ns the temperature is back to the original value of approximately 95 K (Figure 6a). Due to the addition of kinetic energy both clusters shrink but at some point the big cluster starts to grow again due to ripening. Ripening is terminated at about 18 ns by coalescence. In a second branch of the simulation the temperature is again set to 110 K for 1 ps at t ) 10 ns. The temperature is back to 95 K at about 17 ns (Figure 6a). Both clusters evaporate after kinetic energy is added a second time. The small cluster eventually vanishes without coalescence while the big cluster exhibits a second minimum in size at about 17.5 ns and then slightly grows again. Apparently such heating cycle enhances the ripening dynamics in the system while avoiding coalescence in this case.

potentially avoid coalescence. The elevated temperature causes evaporation of smaller particles while larger ones remain more stable. This is important in cases where recrystallization is desired to yield well ordered particles rather than agglomerates with many defects. In case of the stability of ice cream, it has been found experimentally that the ripening gets faster with higher temperature of the system12 and that an initial temperature cycle between -30 and -15 °C leads to ripening.25 In this work here it is shown that the addition of thermal energy for a short period of time can lead to the onset of ripening even after the system temperature has gone down to the original temperature for some period of time. If the analogy between ripening of solid phases and that of droplets investigated here holds, the temperature cycle may be regarded as activation for the onset of ripening.

4. Conclusions

(1) Ostwald, W Z. Phys. Chem. 1900, 34, 495–503. (2) Hulett, G. A. Z. Phys. Chem. 1901, 37, 385–406. (3) Witterlin, J.; Schuster, R.; Coulman, D. J.; Ertl, G.; Behm, R. J. J. Vac. Sci. Tech. B 1991, 9, 902–908. (4) Hannon, J. B.; Klu¨nker, C.; Giesen, M.; Ibach, H.; Bartelt, N. C.; Hamilton, J. C. Phys. ReV. Lett. 1997, 79, 2506–2509. (5) Yang, H. G.; Zeng, H. C. J. Phys. Chem. B 2004, 108, 3492–3495. (6) Shen, Z.; Zhao, Z.; Peng, H.; Nygren, M Nature 2002, 417, 266– 269. (7) Legros, M.; Dehm, G.; Arzt, E.; Balk, T. J. Science 2008, 319, 1646–1649. (8) Miyazaki, K. Contrib. Mineral. Petrol. 1991, 108, 118–128. (9) Nemchin, A. A.; Giannini, L. M.; Bodorkos, S.; Oliver, N. H. S. Geochim. Cosmochim. Acta 2001, 65, 2771–2788. (10) Bergfors, T. J. Struct. Biol. 2003, 142, 66–76. (11) Frolov, V. A. J.; Chizmadzhev, Y. A.; Cohen, F. S.; Zimmerberg, J Biophys. J. 2006, 91, 189–205. (12) Donhowe, D. P.; Hartel, R. W Int. Dairy J. 1996, 6, 1209–1221. (13) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35–50. (14) Wagner, C. Z. Elektrochem. 1961, 65, 581–591. (15) Binder, K. Phys. ReV. B 1977, 15, 4425–4447. (16) Marqusee, J. A.; Ross, J. J. Chem. Phys. 1983, 79, 373–378. (17) Bhakta, A.; Ruckenstein, E. J. Chem. Phys. 1995, 103, 7120–7135. (18) Madras, G.; McCoy, B. J. J. Chem. Phys. 2001, 125, 6699–6706. (19) Warren, J. A.; Murray, B. T. Modelling Simul. Mater. Sci. Eng. 1996, 4, 215–229. (20) Suh, D.; Yoon, W.; Shibahara, M.; Jung, S. J. Chem. Phys. 2008, 128, 154523. (21) Wonczak, S. Ph.D. Thesis, University of Cologne, 2001. (22) Stillinger, F. H. J. Chem. Phys. 1963, 38, 1486–1494. (23) Kraska, T. J. Chem. Phys. 2006, 124, 054507. (24) Vrabec, J.; Kedia, G. K.; Fuchs, G.; Hasse, H. Mol. Phys. 2006, 104, 1509–1527. (25) Flores, A. A.; Goff, H. D. J. Dairy Sci. 1999, 82, 1408–1415.

Ostwald ripening is most relevant for coarsening of immobile particles while coalescence is more likely for mobile ones. However, while Ostwald ripening of solid particles is beyond the current ability of MD simulations it is possible to simulate it for liquid particles. The examples for Ostwald ripening reported here can be summarized as follows: • Ripening can start with some delay after a heating-cooling cycle due to evaporation/melting of clusters initiated by the heating-cooling cycle. This is caused by slow evaporation of the clusters after the addition of energy. • Coalescence is much faster than ripening and leads to the same end result in case of liquids. In case of solids of course morphological differences are possible. • Repeated heating cycles accelerate ripening. This may help to bypass coalescence. These results are consisted with experimental investigations of ice cream stability,12,25 ceramics sintering,6 pipe diffusion,7 and of geological systems.9 A distinct density bridge between the two ripening cluster is not observed in agreement with phase field simulations.19 Density fluctuations in the vapor phase between the clusters are found with and without ripening. Depending on the distance between the clusters they may be actual transversal waves for big distances or nearly horizontal for short distances. One should bear in mind that big distance in this case means 15 to 20 nm and even larger distances as well as larger simulation systems may affect the fluctuations. Heating cycles can be used to further enhance ripening and

Acknowledgment. The author gratefully acknowledges discussions on the manuscript with N. Lu¨mmen and S. Wonczak. References and Notes

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