Formation of Stable Schwarz Crystals in Polycrystalline Copper at the Grain Size Limit

A prototype Schwarz crystal (SC) structure of dividing–space minimal grain boundaries (GBs) constrained by coherent twin boundaries (CTBs) was recently discovered in extremely fine-grained polycrystalline Cu. In this Letter, constraining effects of $3D$ CTB network on the formation and thermostability of SC are addressed via atomistic simulations. GB migration and evolution of CTB network trigger formation of SC diamond. CTB constraints are critical to generate GBs of zero mean curvature underlying vanishing capillary pressure, and to counterbalance the elastic driving forces of lattice. GB motion can be suppressed at temperatures close to the melting point with GB aperture down to 3 nm.

Figure 1

(a) Schwarz $D$-TPMS based on exact mathematical solution [2]. The unit cell contains 6 primitive patches around [111] axis, each enclosed by a tetrahedron such as “$ADgq$” or “$Agqi$”. The aperture of a $D-SC$ is defined as $D_S=\sqrt{2}/2a$. (b) Schwarz crystal (SC) of type–I $D$-surface morphology ($D$–SC I, $a\sim 13\mathrm{nm}$) observed in MD simulations of Cu. GB atoms colored in gray are recolored in blue if locating within a surface layer of 1 nm thick according to Eq. (1). Atoms at CTB sites are colored in orange. Atoms in fcc lattice sites are filtered out. (c) The two MD-obtained thermal profiles (atomic volume vs temperature) indicate that the transformations from twin-limited Kelvin crystal (KC, denoted as ‘KC0’ in the Supplemental Material [3]) occur at 1172 K to $D$–SC I or at 1250 K to $D$–SC II well below the $T_E$ (1358 K) of Cu. (d) The same KC may transform into a $D$–SC II ($a\sim 13\mathrm{nm}$) with a different partition of $D$ surface [Eq. (1) with a spatial translation of $\pi /2$], depending on intergranular interactions and thermal activations.

Figure 2

Transformations from KC to $D$–SC corresponding to Fig. 1. (A) KC to $D$–SC I via ‘longitudinal’ constriction of GBs. (a)–(f) Snapshots of MD configurations at different temperatures. Thermal noises are eliminated from MD configurations via energy minimizations (indicated by $Q$ as “quenched” state). Frames of initial Kelvin cells are drawn in (b), (d) and (f) to reveal the structural evolution, where atoms in fcc lattice sites colored in blue in (a), (c) and (e) are removed. (B) KC to $D$–SC II via ‘transverse’ constriction of GBs. (g)–(l) Quenched states of MD configurations at different temperatures. To reveal GB activities, the [111] foils ($\sim 1.5\mathrm{nm}$ thick) are shown in (h), (j) and (i). For KC, both O1 and O2 locate in the [111] orientation but sit apart from each other at a distance of $a\sqrt{3}/2$ as explicitly shown in Fig. S1 of the Supplemental Material [3].

Figure 3

(a) $TPMS$ GBs of a twin-free primitive Schwarz crystal ($P$–SC, constructed using two Kelvin cells of equal size) in a fully relaxed MD configuration at 0 K. More than 80% of GB atoms (colored in blue) are within the surface layer of $\sim 1.5\mathrm{nm}$ thick according to Eq. (2). (b) MD-obtained thermal profile of atomic volume as a function of temperature. Upon heating, the $P$-SC remains stable up to 685 K ($0.5T_E$) at which opposite GBs shrink toward each other and annihilate abruptly. (c) ADFs calculated for GB atoms and lattice atoms in $P$-SC and $D$-SC. Similar GB structures and certain degree of disorders typical for general GBs are well expected for both SCs. (d) The [110] cross section view of the $P$-SC at 573 K. Atoms in red mark the trace of $P$ surface according to Eq. (2). (e) The [110] cross section view of a $D$-SC at 1000 K. Atoms in red mark the trace of the $D$ surface according to Eq. (1). CTB constraining effects are schematically represented by arrows underlying counterbalanced and hence vanishing GB tension forces for GB motion.

Figure 4

(a) Size dependence of $D$-SC thermostability, $T_S/T_E$ vs $D_S$. The temperature range of KC–to–SC transformation is marked by the dashed line and error bars. Here, S0, S1, S2, S3, and S4 stand for D-SC structures obtained upon transformations from KCs of various supercell sizes, i.e., $a\cong 13\mathrm{nm}$ (KC0), 6.5 nm (KC1), 5.7 nm (KC2), 4.3 nm (KC3), and 3.3 nm (KC4), see additional details in the Supplemental Material [3]. (b) MD snapshot of a $D$–SC I at 473 K, containing 8 ($2\times 2\times 2$) unit cells of $a\sim 6.5\mathrm{nm}$ and $D_S\sim 4.6\mathrm{nm}$. (c) A HRTEM image of an as-prepared sample of Schwarz crystal Cu, from which a $D$–SC morphology can be deduced ($D_S\sim 5\mathrm{nm}$). Also similar to MD observations, the structure is thermally stable up to nearly 1350 K according to experimental measurements [1].