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.

3):333–339, 1957.
[2] F.Y. Génin, W.W. Mullins, and P. Wynblatt. The effect of stress on grain
boundary grooving. Acta Metallurgica et Materialia, 41(12):3541 – 3547,
1993.
[3] W.W. Mullins and P.G. Shewmon. The kinetics of grain boundary grooving
in copper. Acta Metallurgica, 7(3):163 – 170, 1959.
[4] W.W Mullins. The effect of thermal grooving on grain boundary motion.
Acta Metallurgica, 6(6):414 – 427, 1958.
[5] C. Herring. Surface tension as a motivation for sintering, in The Physics
of Powder Metallurgy,edited by Walter E. Kingston. Mcgraw-Hill, New York,
1951.
[6] G Gottstein and LS Shvindlerman. Kinematics of connected grain boundaries
in 2d. Zeitschrift für Metallkunde, 95(4):219–222, 2004.
[7] U Czubayko, VG Sursaeva, G Gottstein, and LS Shvindlerman. Influence
of triple junctions on grain boundary motion. Acta materialia, 46(16):5863–
5871, 1998.
[8] G Gottstein, AH King, and LS Shvindlerman. The effect of triple-junction
drag on grain growth. Acta materialia, 48(2):397–403, 2000.
[9] M. McLean and E.D. Hondros. A study of grain-boundary grooving at the
platinum/alumina interface. Journal of Materials Science, 6(1):19–24, 1971.
67
[10] D. Amram, L. Klinger, N. Gazit, H. Gluska, and E. Rabkin. Grain boundary
grooving in thin films revisited: The role of interface diffusion. Acta
Materialia, 69(0):386 – 396, 2014.
[11] K. Elder, Mark Katakowski, Mikko Haataja, and Martin Grant. Modeling
elasticity in crystal growth. Phys. Rev. Lett., 88:245701, Jun 2002.
[12] K. Elder and Martin Grant. Modeling elastic and plastic deformations in
nonequilibrium processing using phase field crystals. Phys. Rev. E, 70:051605,
Nov 2004.
[13] K. Elder, Nikolas Provatas, Joel Berry, Peter Stefanovic, and Martin Grant.
Phase-field crystal modeling and classical density functional theory of freezing.
Phys. Rev. B, 75:064107, Feb 2007.
[14] J. Swift and P. Hohenberg. Hydrodynamic fluctuations at the convective
instability. Phys. Rev. A, 15:319–328, Jan 1977.
[15] ADJ Haymet and David W Oxtoby. A molecular theory for the solid–liquid
interface. The Journal of Chemical Physics, 74(4):2559–2565, 1981.
[16] Kuo-An Wu and Alain Karma. Phase-field crystal modeling of equilibrium
bcc-liquid interfaces. Phys. Rev. B, 76:184107, Nov 2007.
[17] Kuo-An Wu, Ari Adland, and Alain Karma. Phase-field-crystal model for fcc
ordering. Phys. Rev. E, 81:061601, Jun 2010.
[18] Lev Davidovich Landau and E. M. Lifschitz. Theory of Elasticity. Pergamon
Press, 1970.
[19] Richard Phillips Feynman, Robert B Leighton, and Matthew Sands. The
Feynman Lectures on Physics, volume 2. Basic Books, 2013.
[20] Paul M Chaikin and Tom C Lubensky. Principles of condensed matter physics,
volume 1. Cambridge Univ Press, 2000.
[21] Robert Spatschek and Alain Karma. Amplitude equations for polycrystalline
materials with interaction between composition and stress. Phys. Rev. B,
81:214201, Jun 2010.
68
[22] Gemunu Gunaratne, Qi Ouyang, and Harry Swinney. Pattern formation in
the presence of symmetries. Phys. Rev. E, 50:2802–2820, Oct 1994.
[23] Kuo-An Wu, Alain Karma, Jeffrey Hoyt, and Mark Asta. Ginzburg-landau
theory of crystalline anisotropy for bcc-liquid interfaces. Phys. Rev. B,
73:094101, Mar 2006.
[24] Yashwant Singh. Density-functional theory of freezing and properties of the
ordered phase. Physics Reports, 207(6):351 – 444, 1991.
[25] Hartmut Löwen. Melting, freezing and colloidal suspensions. Physics Reports,
237(5):249 – 324, 1994.
[26] T. V. Ramakrishnan and M. Yussouff. First-principles order-parameter theory
of freezing. Phys. Rev. B, 19:2775–2794, Mar 1979.
[27] M. I. Mendelev, S. Han, D. J. Srolovitz, G. J. Ackland, D. Y. Sun, and
M. Asta. Development of new interatomic potentials appropriate for crystalline
and liquid iron. Philosophical Magazine, 83(35):3977–3994, Dec 2003.
[28] W. Shih, Z. Wang, X. Zeng, and D. Stroud. Ginzburg-landau theory for the
solid-liquid interface of bcc elements. Phys. Rev. A, 35:2611–2618, Mar 1987.
[29] Jesper Mellenthin, Alain Karma, and Mathis Plapp. Phase-field crystal study
of grain-boundary premelting. Physical Review B, 78(18):184110, 2008.
[30] W. Read and W. Shockley. Dislocation models of crystal grain boundaries.
Phys. Rev., 78:275–289, May 1950.
[31] Gunnar Pruessner and A. Sutton. Phase-field model of interfaces in singlecomponent
systems derived from classical density functional theory. Phys.
Rev. B, 77:054101, Feb 2008.
[32] T.A. Roth. The surface and grain boundary energies of iron, cobalt and
nickel. Materials Science and Engineering, 18(2):183 192, Apr 1975.
[33] E. D. Hondros. The influence of phosphorus in dilute solid solution on the
absolute surface and grain boundary energies of iron. Proceedings of the
69
Royal Society of London. Series A, Mathematical and Physical Sciences,
286(1407):pp. 479–498, 1965.
[34] RL Fullman. Interfacial free energy of coherent twin boundaries in copper.
Journal of Applied Physics, 22(4):448–455, 1951.
[35] Arthur W Adamson, Alice Petry Gast, et al. Physical chemistry of surfaces.
Interscience publishers New York, 1967.
[36] Lilin Wang, Xin Lin, Meng Wang, and Weidong Huang. Solid liquid interfacial
energy and its anisotropy measurement from double grain boundary grooves.
Scripta Materialia, 69(1):1 – 4, 2013.
[37] W.R. Tyson and W.A. Miller. Surface free energies of solid metals: Estimation
from liquid surface tension measurements. Surface Science, 62(1):267 – 276,
1977.
[38] John W. Cahn, Yuri Mishin, and Akira Suzuki. Coupling grain boundary
motion to shear deformation. Acta Materialia, 54(19):4953 – 4975, 2006.
[39] S.Y. Yeh, C.C. Chen, and C.W. Lan. Phase field modeling of morphological
instability near grain boundary during directional solidification of a binary
alloy: The hump formation. Journal of Crystal Growth, 324(1):296 – 303,
2011.
[40] Yasushi Shibuta, Kanae Oguchi, and Toshio Suzuki. Large-scale molecular
dynamics study on evolution of grain boundary groove of iron. ISIJ international,
52(12):2205–2209, 2012.
[41] M. Khenner, A. Averbuch, M. Israeli, and M. Nathan. Numerical simulation
of grain-boundary grooving by level set method. Journal of Computational
Physics, 170(2):764 – 784, 2001.