Computation of Five-Dimensional Grain Boundary Energy

doi:10.1007/s40195-015-0242-8

Abstract

Abstract:

A broken-bond type computational method has been developed for the calculation of the five-dimensional grain boundary energy. The model allows quick quantification of the unrelaxed five-dimensionally specified grain boundary energy in arbitrary orientations. It has been validated on some face-centred cubic metals. The stereo projections of grain boundary energy of ∑3, ∑5, ∑7, ∑9, ∑11, ∑17b and ∑31a have been studied. The results of Ni closely resemble experimentally determined grain boundary energy distribution figures, suggesting that the overall anisotropy of grain boundary energy can be reasonably approximated by the present simple model. Owing to the overlooking of relaxation matter, the absolute values of energy calculated in present model are found to be higher than molecular dynamic-based results by a consistent magnitude, which is 1 J/m² for Ni. The coverage of present method forms a bridge between atomistic and meso-scale simulations regarding polycrystalline microstructure.

Key words: Boundaries, Grain boundary energy, Grain boundary structure

Yong-Kun Luo, Rong-Shan Qin. Computation of Five-Dimensional Grain Boundary Energy[J]. Acta Metallurgica Sinica (English Letters), 2015, 28(5): 634-640.

Figures/Tables 4

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Metal	a(10 -10 m)	ϕ b 1 (eV)	ϕ b 2 /ϕ b 2 ϕ b 1 ϕ b 1	ϕ b 3 /ϕ b 3 ϕ b 1 ϕ b 1	ϕ b 4 /ϕ b 4 ϕ b 1 ϕ b 1	∑b 1d	∑b 2d	∑b 3d
Ni	3.524	0.2249	0.434	0.175	0.078
Cu	3.615	0.1794	0.424	0.168	0.073			a^[800] ⁱ
Au	4.079	0.2348	0.301	0.087	0.028			a^[721] ⁱⁱ
Ag	4.086	0.1673	0.359	0.122	0.046	a^[210]	a^[111]	a^[640] ⁱⁱⁱ
Pt	3.924	0.3514	0.318	0.097	0.032			a^[552] ^iv
Pd	3.891	0.2334	0.320	0.098	0.033			a^[633] ^v
Pb	4.951	0.1213	0.323	0.100	0.034

Metal	a(10 -10 m)	ϕ b 1 (eV)	ϕ b 2 /ϕ b 2 ϕ b 1 ϕ b 1	ϕ b 3 /ϕ b 3 ϕ b 1 ϕ b 1	ϕ b 4 /ϕ b 4 ϕ b 1 ϕ b 1	∑b 1d	∑b 2d	∑b 3d
Ni	3.524	0.2249	0.434	0.175	0.078
Cu	3.615	0.1794	0.424	0.168	0.073			a^[800] ⁱ
Au	4.079	0.2348	0.301	0.087	0.028			a^[721] ⁱⁱ
Ag	4.086	0.1673	0.359	0.122	0.046	a^[210]	a^[111]	a^[640] ⁱⁱⁱ
Pt	3.924	0.3514	0.318	0.097	0.032			a^[552] ^iv
Pd	3.891	0.2334	0.320	0.098	0.033			a^[633] ^v
Pb	4.951	0.1213	0.323	0.100	0.034

Coincidence site lattice (CSL) notation	Axis-angle pair notation	Coordinate transformation matrix (YJX)
∑3	?111?60 °	? ? 0.667 -0.333 0.667 0.667 0.667 -0.333 -0.333 0.667 0.667 ? ?
∑5	?100?36.86 °	? ? 1 0 0 0 0.8 -0.6 0 0.6 0.8 ? ?
∑7	?111?38.21 °	? ? 0.857 -0.286 0.429 0.429 0.857 -0.286 -0.286 0.429 0.857 ? ?
∑9	?110?38.94 °	? ? 0.889 0.111 0.444 0.111 0.889 -0.444 -0.444 0.444 0.778 ? ?
∑11	?110?50.47 °	? ? 0.818 0.182 0.545 0.182 0.818 -0.545 -0.545 0.545 0.636 ? ?
∑17b	?221?61.9 °	? ? 0.894 0.029 1 0.818 0.894 -0.577 -0.577 1 0.577 ? ?
∑31a	?111?17.9 °	? ? 0.968 -0.161 0.194 0.194 0.968 -0.161 -0.161 0.194 0.968 ? ?

Coincidence site lattice (CSL) notation	Axis-angle pair notation	Coordinate transformation matrix (YJX)
∑3	?111?60 °	? ? 0.667 -0.333 0.667 0.667 0.667 -0.333 -0.333 0.667 0.667 ? ?
∑5	?100?36.86 °	? ? 1 0 0 0 0.8 -0.6 0 0.6 0.8 ? ?
∑7	?111?38.21 °	? ? 0.857 -0.286 0.429 0.429 0.857 -0.286 -0.286 0.429 0.857 ? ?
∑9	?110?38.94 °	? ? 0.889 0.111 0.444 0.111 0.889 -0.444 -0.444 0.444 0.778 ? ?
∑11	?110?50.47 °	? ? 0.818 0.182 0.545 0.182 0.818 -0.545 -0.545 0.545 0.636 ? ?
∑17b	?221?61.9 °	? ? 0.894 0.029 1 0.818 0.894 -0.577 -0.577 1 0.577 ? ?
∑31a	?111?17.9 °	? ? 0.968 -0.161 0.194 0.194 0.968 -0.161 -0.161 0.194 0.968 ? ?

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