晶界（Grain boundary, GB）槽的動力行為在奈米尺度的長晶中具有重要作用。之前的古典理論假設的GB為均勻邊界，細節的結構被忽略。這種假設顯然需要進行個案某些修改，特別是低角度的GB。
相場晶格模型（PFC）的優勢是在於此模型可以描述材料在原子尺度下的行為。我們研究了位錯如何影響晶界槽的成形，此外，我們使用來自PFC模型得出定量描述應力，位錯之間的相互作用與固-液之間夾角的關係。同時發現一些新的現象，如晶格的轉動與缺陷的移動，這提供了另一種方法去控制晶格的成形。

Dynamics of grain boundary (GB) grooving plays an important role in microstructural evolution. Classical theory on GB grooving assumes the solid-solid interface as a homogeneous boundary where details of GB structures are ignored. This assumption clearly requires certain modifications for cases such as low angle GB. The advantage of phase field crystal (PFC) method is its capability to describe materials
with atomic resolutions. We investigated how dislocations influence dihedral angle in low angle GB. Furthermore, we used amplitude equations derived from PFC model to describe quantitatively the interplay between stresses, dislocations,
and the dihedral angle. While GB grooves, interesting phenomena such as grain rotation and dislocation translation were observed, which provide an alternative way to control grain growth at the nanoscale.

Contents ii
List of Figures v
1 Introduction 1
1.1 Grain boundary grooving . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Phase field crystal model . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Methodology of PFC model 5
2.1 One mode approximation . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Elastic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Young’s modulus and Poisson ratio . . . . . . . . . . . . . . 8
2.2.3 Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.4 Shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Examination of elastic constant by simulation . . . . . . . . . . . . 11
2.3.1 Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Poisson ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Coexistence of phases . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Amplitude expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 From density functional theory to phase field crystal . . . . . . . . 19
2.7 Periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . 25
2.7.1 The PFC model . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7.2 Amplitude equations . . . . . . . . . . . . . . . . . . . . . . 26
i
3 Interface 28
3.1 Grain boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Strain in amplitude equation and the PFC model . . . . . . 32
3.2 Solid-liquid interface . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Solid-liquid interface in amplitude equation . . . . . . . . . 39
3.2.2 Isotropic approximation . . . . . . . . . . . . . . . . . . . . 41
3.3 Dihedral angle and Herring’s relation . . . . . . . . . . . . . . . . . 42
4 Simulation of grain boundary grooving and result 44
4.1 Configuration of simulation . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Dihedral angle measurement . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Dihedral angle and grooving depth versus time . . . . . . . . . . . . 48
4.4 Dihedral angle versus misorientation . . . . . . . . . . . . . . . . . 51
4.5 Modeling of evaporation . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Effects of strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 Soft boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Conclusion and Discussion 64
Appendix 66
A. Burgers vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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