固液介面能的不勻向性主宰了在結晶過程中微觀結構的型態，而微觀結構的型態對於材料的特性有關鍵性的影響。因此，固液介面能的不勻向性是一個重要的物理量。相場晶格模型具備了描述材料的結構及行為的能力在原子尺度，自然地可以捕捉到固液介面能、彈力性質等材料特性。因此，我們使用相場晶格模型來研究固液界面能的不勻向性。而相場晶格模型可以進一步推導出振福方程式以及所對應的自由能泛函。我們回顧了過去十年的文獻，發現有許多不同型式的振幅方程式。我們做了統整並歸納出兩個因子來區分這些不同的振幅方程式，分別是旋轉不變性以及固液密度差，而用這兩個因子來探討它們對於固液介面能不勻向性的影響。模擬的結果顯示，旋轉不變性讓固液界面能的不勻向性降低而固液密度差會使固液界面能的不勻向性增強。我們檢視了振幅的型態隨轉角的變化，發現正是振幅型態的變化給予了固液介面能的不勻向性。同時發現，具備旋轉不變性的振幅方程式，可以抓住原子間距增長的現象在靠近固體的地方。而固液密度差使得振幅在跨過介面時更加劇烈，使得固液介面能的不勻向性更強烈。

The anisotropy of solid-liquid interfacial energy dominates the morphology of micro-structures in the solidification process, and material properties deeply depend on the morphology of microstructures. Therefore, the anisotropy of solid-liquid interfacial energy is an important physical parameter that controls properties of materials. On the other hand, because the phase field crystal model can describe the density wave of structure and dynamics at the atomic scale, it reasonably can describe the solid-liquid interfacial energy, elastic properties, etc. Thus, we use the phase field crystal (PFC) model to study the anisotropy of solid-liquid interfacial energy. We derive the amplitude equations and corresponding free energy functional from the PFC model under different assumptions. There are two factors that distinguish various amplitude equations, namely, the rotational invariance and the mean density difference between solid and liquid phases. We study how these two factors affect the anisotropy of solid-liquid interfacial energy. We show that the anisotropy of solid-liquid interfacial energy is weaker while the rotational invariance is present and it is stronger while the mean density difference between solid and liquid is considered. In addition, we analyse the amplitude profiles across the interface and we find that the physical origin of the anisotropy of solid-liquid interfacial energy is due to different amplitude profiles at the different orientation. Furthermore, we find that the rotational invariance gives rise to variation of the atomic spacing which genuinely captures fundamental properties across the solid-liquid interface. Finally, we show that the coupling between the amplitude profiles and the mean density is not negligible, and the mean density difference results in sharp amplitude profiles across the interface, hence a larger anisotropy of solid-liquid interfacial energy.

Contents ii
List of Figures vii
1 Introduction 1
1.1 Significance of anisotropy of interfacial energy in crystal growth . . 1
1.1.1 Morphology and material properties . . . . . . . . . . . . . 3
1.2 Overview of phase field crystal model . . . . . . . . . . . . . . . . . 6
2 Phase Field Crystal Model : Amplitude Equations 9
2.1 Formulation and one-mode approximation . . . . . . . . . . . . . . 9
2.1.1 Liquid phase . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Solid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Coexistence state and phase diagram . . . . . . . . . . . . . . . . . 12
2.3 Full amplitude equations . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Various formulation of amplitude equations . . . . . . . . . . . . . 17
2.4.1 Expansion of amplitudes and AE . . . . . . . . . . . . . . . 18
2.4.2 Amplitude equations with rotational invariance (RI-AE) . . 21
2.4.3 DC-AE, DC-RI-AE and Full-AE . . . . . . . . . . . . . . . 23
2.5 Simulation calculations of interfacial energy . . . . . . . . . . . . . 25
2.5.1 Excess energy and rotation of interface . . . . . . . . . . . . 25
2.5.2 Interfacial energy of AE and RI-AE . . . . . . . . . . . . . . 27
2.5.3 Interfacical energy of DC-AE, DC-RI-AE and Full-AE . . . 28
3 Simulation Results and Discussion 30
3.1 Numerical results for the interfacial energies . . . . . . . . . . . . . 30
3.2 Profile of amplitudes of density wave across the interfacial region . 36
3.2.1 AE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Analysis of amplitude profiles near the liquid . . . . . . . . 40
3.2.3 RI-AE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.4 DC-AE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.5 DC-RI-AE . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.6 Full-AE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Conclusion and Future Work 70
Appendix 71
A. Amplitude Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B. Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Reference 75

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