Two clusters of the same sizes dissolve and grow

Figure 6.7: Evolution of two clusters initially of same size of a single sim- ulation at six different times. Image (a) shows the initial configuration of the two clusters. Images (b), (c) and (d) show the clusters growing and dis- solving at three different times (t = 50, 100, 146). Image (e) indicates that one of the clusters is fully dissolved at t = 155. In the last image (f) arms of the dendrite have started to grow. All images were coloured according to coordination number.

The model extends the growth of two clusters of the same size, which is completed through dissolution, single atom hop and surface diffusion mechanisms. The simulation is performed using two spherically symmet-

6.5. SUMMARY 121 ric clusters which originally contained 87 atoms. The initial configuration of the two clusters is shown in Figure 6.7 (a). Figure 6.7 (b), (c), and (d) show that one of the clusters is dissolved and attached to the other clus- ter. The last two images (e) and (f) illustrate that one of the clusters is completely dissolved. Then the remaining crystal will grow and obtain a similar shape as in Figure 6.6 (f).

6.5

Summary

We have constructed a KMC model to simulate the process of a single atom diffusion event in a liquid environment. That is to allow a single atom to hop and to ensure that creating atoms are placed at the edge of the simulation cells so that they can diffuse inward. The single atom hopping rate can be set at any rate, but preferably quite fast to mimic diffusion in the solute. The dendrite shape of the crystal in the current model is similar to our result in Chapter 5. The current model shows the ability to model multiple seeds. We also discussed what happens if the dissolution event is included in the current model. We demonstrated that crystals have a similar crystal shape as we discussed in Chapter 5. The dissolution process in two different size clusters shows that the atoms from the smaller cluster dissolve and appear to join the larger cluster.

Chapter 7

Conclusion and Future Work

7.1

Conclusion

The current study explored this main goal: To develop a multiscale sim-

ulation method for the growth of nanocrystals in solution that couples a KMC description of the crystal relaxation process to solute reaction dif- fusion equations. This goal set out to collect in-depth information for a better understanding of the phase crystal growth using the KMC method, supported by a quantitative analysis of the results. Therefore, it was nec- essary to couple the theory of crystal relaxation with the mathematical understanding of solute diffusion fields.

The kinetic Monte Carlo model was developed to perform realistic sim- ulations over a useful range of growth and dissolution conditions. The emphasis of this work was to establish a model which can serve as a devel- opment platform for other more advanced models. The main motivation behind this work was to advance the current state of knowledge and un- derstanding of crystal growth mechanisms. The thesis was divided into three parts: the Deposition and Growth KMC method, the Continuum- KMC Method, and the Diffusion and Growth KMC method. These phe- nomena were discussed and analyzed by adopting a discrete, atomistic model in the spirit of the KMC simulation which is popular in the growth

7.1. CONCLUSION 123 and dissolution literature. This led to interesting results and valuable in- sights. We will now summarize the findings, and review the main goals of the research, and what we have achieved in terms of the research objec- tives.

In Chapter 4 the interplay of the deposition and surface diffusion rates were demonstrated in the gas phase. The first case, when deposition rate is far greater than hopping rate, indicated that the cluster grows very fast. As the simulation time increased, the shape of the crystal became completely spherical. The physical picture of the crystallite above the roughening temperatures has many visible kinks and steps, indicating that the contin- uous approximation for the curvature might be valid; the mass transfer of solute is via atomic diffusion from kinks or steps from the high curvature regions to the existing kinks or steps of the low curvature regions. Below this temperature, however, large facets do appear in the low curvature re- gions and no kinks or steps are available, thus preventing the diffusing atoms from sticking there. The final case showed the growth of the cluster is not as fast as in the first case, but leads to the same structure. The solu- tion in Chapter 4 has contributed to guide us to be able to respond to the main goal of the research.

The most essential underlying finding in the current study is that the KMC appears to be a promising model for the simulation of dendrite growth on atomistic scales. We adopted an atomistic growth model that uses a KMC technique to track the free boundary. The model allows for both phase change and exchange between liquid and solid atoms on the surface of the crystal and is coupled to a continuum model for heat and diffusion equations at the solid-liquid interface. The present study has demonstrated that the KMC algorithm is useful in contributing to our understanding of solution phase crystal growth, especially of nanocrystal growth. For small length and time scales, this approach provides a simple, effective front tracking with fully resolved atomistic detail. The technique was used to make realistic predictions regarding surface morphology of

crystals.

We have achieved all the research objectives mentioned in Chapter 1. Firstly we showed, that the KMC algorithm is suitable to study the process of solidification in order to explore the evolution and morphology of crys- tal structure. Secondly, the model was then applied to the numerical finite difference method by using an explicit discretization to solve the heat and diffusion equations. The solute concentration (CL) on the liquid site of the

interface was calculated. Finally, the solidification and hopping rates were calculated. The anisotropy is included in the model as a surface diffusion process, and the growth rate of a dendrite is found to increase monotoni- cally with the surface anisotropy. Thus, the branching process (including secondary and nascent tertiary branches) occurs at earlier stages of growth when the value of the surface anisotropy is increased. On the contrary, at low surface anisotropy the crystallite is fully faceted.

In Chapter 5, as expected from the large value of Lewis number, the thickness of the thermal boundary layer is much larger than that of the solutal boundary layer. Even though the Lewis number is large, the initial melt concentration is made low enough that the interface temperature is significantly different from the far-field value. The tip velocity when Lewis number Le = 1 of the Continuum-KMC model is successfully compared with the prediction by Schulze [101] and Tan et al. [139]. The prediction of tip velocity by Karma [162] also agree well with our result when the Le = 1000.

The kinetic Monte Carlo algorithm for the deposition code in Chap- ter 4 was extended to include the process of a solute diffusion in the fluid in Chapter 6. We placed a series of single atoms at the edge of the sim- ulation cells and allowed them to diffuse inward. The uniform rate of a single atom was set quite fast to imitate diffusion in the solute. The final shape of the crystal in the two models, the Continuum-KMC model and the Diffusion and Growth KMC model, is similar.

7.2. RECOMMENDATIONS FOR FUTURE WORK 125