Energy landscape of deformation twinning in bcc and fcc metals

Energy landscapes of $\left(2\overline{1}\overline{1}\right)⟨111⟩$ deformation twinning in bcc Mo and $\left(111\right)⟨11\overline{2}⟩$ deformation twinning in fcc Al and Cu are determined using density functional theory for sliding of layers numbering up to 7. In bcc Mo, the minimum thickness of a metastable twin is two layers, while twin embryos of three and four layers are unstable. Starting from five layers, the Mo twin can grow in a layer-by-layer fashion. The twin boundary formation and migration energies are found to be 607 and $40\mathrm{mJ}∕\mathrm{m}$, respectively, implying that partial dislocations on twin boundaries will have wide cores and high mobilities. The stress to homogeneously nucleate a partial loop on the boundary of a thick twin is determined to be only $1.4\mathrm{GPa}$, indicating that once a deformation twin in Mo reaches a critical thickness, which we estimate to be six layers, it can grow rather easily. Based on simple defect mechanics considerations, we estimate the condition for runaway defect growth requires twin embryo thickness to be tens of layers. Comparing the twinning energy landscape for Mo with those of Al and Cu, we find the former to have a longer ranged interlayer mechanical coupling, which is due to angular bonding and weaker electron screening in the intervening layers. Between Al and Cu, interactions in the former are relatively longer ranged.

Figure 1

(Color online) Schematic illustration of the twinning pathway examined. ${\mathbf{b}}_p$ is partial Burgers vector. See the text for the detail of definitions of the $\lambda$ and $i$ and sliding plane and direction.

Figure 2

Energy landscape of twin migration in Mo. Initially a twin boundary is introduced at the center of the 24-layer slab model, and then the boundary position is shifted 1 layer by rigid sliding of the layers on top. The twin boundary migration energy is estimated at $40\mathrm{mJ}∕{\mathrm{m}}^2$. The twin boundary formation energy, given by the energy difference between the perfect crystal and twinned bicrystal, is estimated to be $607\mathrm{mJ}∕{\mathrm{m}}^2$.

Figure 3

Difference of layer displacements between adjacent layers $\Delta {\mathbf{x}}_i-\Delta {\mathbf{x}}_{i-1}$ near the ideal and relaxed twin boundary.

Figure 4

(Color online) The $\left(2\overline{1}\overline{1}\right)⟨111⟩$ twinning energy pathway of bcc Mo for sliding of layers from 1 to 7 layers in the 24-layer sandwich slab model.

Figure 5

(Color online) The $\left(2\overline{1}\overline{1}\right)⟨111⟩$ twinning energy pathway of bcc Mo up to seven-layer sliding (zoomed). The curve on the right end is Fig. 2 shifted by $2{\gamma }_{\mathrm{TBF}}=1214\mathrm{mJ}∕{\mathrm{m}}^2$.

Figure 6

Difference of shear displacements between adjacent layers $\Delta {\mathbf{x}}_i-\Delta {\mathbf{x}}_{i-1}$ normalized by ${\mathbf{b}}_p$ near the ideal (initial unrelaxed) and relaxed twin embryo configurations. (a) $\lambda =2$, (b) $\lambda =3$, (c) $\lambda =4$, (d) $\lambda =5$. They clearly show that two- and five-layer twin embryos are metastable, while three- and four-layer twin embryos are unstable, which fall back to the two-layer twin embryo configuration spontaneously upon relaxation.

Figure 7

(Color online) (a) Energy landscape of twin boundary migration in Al using the 24-layer model. The twin boundary migration energy is estimated as $97\mathrm{mJ}∕{\mathrm{m}}^2$. (b) The $\left(111\right)⟨11\overline{2}⟩$ twinning energy pathway of fcc Al up to five-layer sliding using the 24-layer model. The curve on the right end of (b) is just (a) shifted by $2{\gamma }_{\mathrm{TBF}}=120\mathrm{mJ}∕{\mathrm{m}}^2$.

Figure 8

(Color online) (a) Energy landscape of twin boundary migration in Cu using the 15-layer model. The twin boundary migration energy is estimated as $168\mathrm{mJ}∕{\mathrm{m}}^2$. (b) The $\left(111\right)⟨11\overline{2}⟩$ twinning energy pathway of fcc Cu up to five-layer sliding using the 15-layer model. The curve on the right end of (b) is just (a) shifted by $2{\gamma }_{\mathrm{TBF}}=42\mathrm{mJ}∕{\mathrm{m}}^2$.

Figure 9

(Color online) Distribution of valence charge density deviation (with and without twin boundary) for Mo, Al, and Cu. $z$ is distance from twin boundary, normalized by the interlayer distance in ${\mathbf{e}}_3$ direction. The charge density is averaged over the plane normal to ${\mathbf{e}}_3$, then normalized by number of valence electrons $N_v$ used in the pseudopotential for each type of atom.