Formation of hexagonal close packing at a grain boundary in gold by the dissociation of a dense array of crystal lattice dislocations

D. L. Medlin; J. C. Hamilton

doi:10.1007/s10853-009-3488-6

Journal of Materials Science

July 2009, Volume 44, Issue 13, pp 3608–3617 | Cite as

Formation of hexagonal close packing at a grain boundary in gold by the dissociation of a dense array of crystal lattice dislocations

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D. L. Medlin
J. C. Hamilton

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First Online: 01 July 2009

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Abstract

We analyze a thin (~1 nm) hexagonal-close-packed (HCP) intergranular layer at a 29° 〈110〉 tilt grain boundary in gold. Our analysis, which is based on HRTEM observations and atomistic calculations, shows that this boundary consists of a dense array of 60° 1/2〈110〉 crystal lattice dislocations that are distributed one to every two {111} planes. These dislocations dissociate into paired Shockley partial dislocations, creating a stacking fault on every other plane and thereby producing the …abab…, or HCP, stacking sequence. This distribution of dislocations is consistent both with the measured intergranular misorientation and with the calculated rigid-body translation along the tilt axis. By establishing the interfacial dislocation arrangement, we also show how the HCP layer at the 29° boundary observed here is geometrically related to that found previously at the 80.6° Σ = 43 〈110〉 boundary. This result helps to link dislocation-based descriptions for boundary structures between the high- and low-angle misorientation regimes.

Keywords

Partial Dislocation Dense Array Tilt Boundary Embed Atom Method Tilt Axis

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Notes

Acknowledgements

Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy, National Nuclear Security Administration, under Contract DE-AC04-94AL85000. This work was supported in part by the DOE Office of Basic Energy Science, Division of Materials Sciences. The authors acknowledge helpful comments from J. Sugar, S. Foiles, and the anonymous reviewer.

Appendix A: Analysis of a step at the {111}/{131} interface

In the main text, we have analyzed the defect content of the interface with reference to one of the two adjacent single crystals. We chose this approach because it directly addresses the origin of the stacking fault pattern and its relationship to boundaries at other misorientations. However, because the boundary in this paper is vicinal to {111}/{131}, it is also instructive to analyze the interface relative to a reference frame in which these planes are aligned. This complementary analysis approach, which we discuss below, provides further insight concerning the nature of the interface at the facet length scale.

For this analysis, we adopt a coherent reference frame following an approach used previously for boundaries in FCC metals vicinal to {111}/{121} [22, 23]. The dichromatic pattern representing this strained reference state is shown in Fig. 9. Here, $ (\bar{1}3\bar{1}) $ and $ (\bar{1}1\bar{1}) $ are rotated into alignment (θ = 29.496°) and periodic lengths parallel to the interface in the upper (λ) and lower (μ) crystals (1/2[323]_λ and [121]_μ) are strained into coherency. The matrix defining this coherent reference state, P_coh, is given by:

Fig. 9
Dichromatic pattern for the coherently strained reference state. $ (\bar{1}3\bar{1}) $ and $ (\bar{1}1\bar{1}) $ are rotated into alignment (θ = 29.496°) and periodic lengths parallel to the interface in the upper (λ) and lower (μ) crystals (1/2[323]_λ and [121]_μ) are strained into coherency. The analyzed disconnection can be envisioned as the difference between two lattice translation vectors, $ {\mathbf{t}}_{\lambda } = [101] $ and $ {\mathbf{t}}_{\mu } = \tfrac{1}{2}[211] $. The difference in step heights (2 $ (\bar{1}3\bar{1}) $ planes λ and 1 $ (\bar{1}1\bar{1}) $ plane in μ) gives a component of Burgers vector that is perpendicular to the $ \left( {\bar{1}3\bar{1}} \right)_{\lambda } //\left( {\bar{1}1\bar{1}} \right)_{\mu } $ terrace

$$ {\mathbf{P}}_{\text{coh}} = {\mathbf{P}}_{\text{rel}} {\mathbf{U}}_{\mu } {\mathbf{AU}}_{\mu }^{ - 1} $$

(4)

P_rel relates the coordinate frames of the two relaxed and unstrained crystals, A describes the uniaxial compression of μ to bring it into coherency with λ (in unit orthogonal coordinates), and U_μ converts from unit orthogonal coordinates to μ crystal coordinates:

$$ {\mathbf{P}}_{\text{rel}} = \frac{1}{{2\sqrt {33} }}\left( {\begin{array}{*{20}c} {5 + \sqrt {33} } & 4 & {5 - \sqrt {33} } \\ { - 4} & {10} & { - 4} \\ {5 - \sqrt {33} } & 4 & {5 + \sqrt {33} } \\ \end{array} } \right) $$

$$ {\mathbf{A}} = \left( {\begin{array}{*{20}c} {\frac{1}{2}\sqrt {\frac{11}{3}} } & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right); $$

(5)

$$ {\mathbf{U}}_{\mu } = \frac{1}{\sqrt 6 }\left( {\begin{array}{*{20}c} 1 & { - \sqrt 2 } & { - \sqrt 3 } \\ 2 & {\sqrt 2 } & 0 \\ 1 & { - \sqrt 2 } & {\sqrt 3 } \\ \end{array} } \right) $$

As noted in the results sections, the observed section of boundary consists of two terraces that are separated by a step-like defect (or “disconnection” in the terminology of Hirth [24], which emphasizes the dual roles of step and dislocation content of such defects). By constructing a circuit around the disconnection, and then mapping this circuit into the coherent reference frame, the topological properties of the defect can be established. Figure 10 shows the intensity peak positions extracted from the HRTEM image of the dissociated boundary. This is the same data as in Fig. 3 of the paper, but here the data are rotated so that $ (\bar{1}3\bar{1}) $ and $ (\bar{1}1\bar{1}) $ are horizontal to better highlight the step and to allow comparison with Fig. 9.

Fig. 10
Circuit analysis of the observed interfacial disconnection. The data are the same as in Fig. 3, but is rotated here by 90°

Denoting the circuits in the upper and lower crystals as C_λ and C_μ, respectively, the Burgers vector associated with this disconnection is given by:

$$ {\mathbf{b}} = - \left( {C_{\lambda } + {\mathbf{P}}_{\text{coh}} C_{\mu } } \right) $$

(6)

Resolving b into edge components parallel and normal to the terrace gives: b_|| = 0.3020a and b_n = −0.0256a. The defect also possesses a screw component of b_screw = ±0.3535a. The step heights in the upper and lower crystals are −2d₁₃₁ = −0.603a and −d₁₁₁ = −0.577a, respectively. As illustrated in Fig. 9, this defect can be envisioned as the difference between two lattice translation vectors, $ {\mathbf{t}}_{\lambda } = [101] $ and $ {\mathbf{t}}_{\mu } = \tfrac{1}{2}[211] $ (note: $ {\mathbf{b}} = {\mathbf{t}}_{\lambda } - {\mathbf{P}}_{\text{coh}} {\mathbf{t}}_{\mu } $).

A periodic array of such disconnections would rotate both the boundary inclination and crystal misorientation from $ (\bar{1}3\bar{1})_{\lambda } //(\bar{1}1\bar{1})_{\mu } $ due to the step and dislocation content of the defects. Pond et al. have derived the relationships for the interface inclination and misorientation when disconnections are spaced at the separation required to fully accommodate the interfacial coherency strain [25]. Using the expressions in Appendix B of reference [25], we compute that an array of disconnections of the type observed here would accommodate the coherency strain at the $ (\bar{1}3\bar{1})_{\lambda } //(\bar{1}1\bar{1})_{\mu } $ interface when distributed at a separation of L = 7.166a (an equal distribution of positive and negative screw components would be required to cancel any twist). At this spacing, the normal component of the Burgers vector of the disconnections adds 0.206° to the crystal misorientation, bringing the total misorientation to 27.702°. Moreover, the step components of the disconnections rotate the macroscopic boundary inclination from $ (\bar{1}3\bar{1})_{\lambda } //(\bar{1}1\bar{1})_{\mu } $ by 4.83° in λ and 4.62° in μ (both in the clockwise direction). This misorientation and macroscopic boundary inclination are equivalent to that computed in section 1 of the discussion using the Frank-Bilby equation for a Burgers vector density of one 60° 1/2〈110〉 dislocation for every two (111) planes intersected by the interface. This analysis, then, gives a microscopic picture of the relationship between the macroscopic geometric parameters of the interface and its specific atomic-scale defects.

References

1.
Krakow W, Smith DA (1987) Ultramicroscopy 22:47CrossRefGoogle Scholar
2.
Merkle KL (1990) Colloq Phys C1 51:251Google Scholar
3.
Ernst F, Finnis MW, Hofmann D, Muschik T, Schönberger U, Wolf U, Methfessel M (1992) Phys Rev Lett 66:991Google Scholar
4.
Wolf U, Ernst F, Muschik T, Finnis MW, Fischmeister HF (1992) Philos Mag A 66:991CrossRefGoogle Scholar
5.
Merkle KL (1994) J Phys Chem Solids 55:991CrossRefGoogle Scholar
6.
Medlin DL, Campbell GH, Carter CB (1998) Acta Mater 46:5135CrossRefGoogle Scholar
7.
Radetic T, Lançon F, Dahmen U (2002) Phys Rev Lett 89:85502CrossRefGoogle Scholar
8.
Brown JA, Mishin Y (2007) Phys Rev B 76:134118CrossRefGoogle Scholar
9.
Tschopp MA, Tucker GJ, McDowell DL (2007) Acta Mater 55:3959CrossRefGoogle Scholar
10.
Rittner JD, Seidman DN (1996) Phys Rev B 54:6999CrossRefGoogle Scholar
11.
Medlin DL, Foiles SM, Cohen D (2001) Acta Mater 49:3689CrossRefGoogle Scholar
12.
Lucadamo G, Medlin DL (2003) Science 300:1272CrossRefGoogle Scholar
13.
Jenkins ML (1972) Philos Mag 36:747CrossRefGoogle Scholar
14.
Balk TJ, Hemker KJ (2001) Philos Mag A 81:1507CrossRefGoogle Scholar
15.
Thompson N (1953) Proc Phys Soc Sect B 66B:481CrossRefGoogle Scholar
16.
Hirth JP, Lothe J (1992) Theory of dislocations. Krieger Publishing Company, Malabar, FLGoogle Scholar
17.
Read WT (1953) Dislocations in crystals. McGraw-Hill, New York, p 17Google Scholar
18.
Sutton AP, Balluffi RW (1995) Interfaces in crystalline materials. Clarendon Press, Oxford, p 70Google Scholar
19.
Daw MS, Baskes MI (1983) Phys Rev Lett 50:1285CrossRefGoogle Scholar
20.
Daw MS, Baskes MI (1984) Phys Rev B 29:6443CrossRefGoogle Scholar
21.
Dash S, Brown N (1963) Acta Metall 35:1067CrossRefGoogle Scholar
22.
Medlin DL, Cohen D, Pond RC (2003) Philos Mag Lett 83(4):223CrossRefGoogle Scholar
23.
Pond RC, Medlin DL, Serra A (2006) Philos Mag 86(29–30):4667CrossRefGoogle Scholar
24.
Hirth JP (1994) J Phys Chem Solids 55:985CrossRefGoogle Scholar
25.
Pond RC, Celotto S, Hirth JP (2003) Acta Mater 51:5385CrossRefGoogle Scholar

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D. L. Medlin
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J. C. Hamilton
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1.Sandia National LaboratoriesLivermoreUSA

Formation of hexagonal close packing at a grain boundary in gold by the dissociation of a dense array of crystal lattice dislocations

Abstract

Keywords

Notes

Acknowledgements

References

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