$\mathrm{\Lambda }$-Invariant and Topological Pathways to Influence the Strength of Submicron Crystals

In small volumes, sample dimensions are known to strongly influence mechanical behavior: especially strength and crystal plasticity. This correlation fades away at the so-called “mesoscale,” loosely defined at several micrometers in both experiments and simulations. However, this picture depends on the “entanglement” of the initial defect configuration. In this Letter, we study the effect of dislocation topology through the use of a novel observable for dislocation ensembles (the $\mathrm{\Lambda }$ invariant) that depends only on mutual dislocation linking: It is built on the natural vortex character of dislocations, and it has a continuum-discrete correspondence that may assist multiscale modeling descriptions. We investigate arbitrarily complex initial dislocation microstructures in sub-micron-sized pillars using three-dimensional discrete dislocation dynamics simulations for finite volumes. We demonstrate how to engineer nanoscale dislocation ensembles that are independent from sample dimensions, either by biased-random dislocation loop deposition or by sequential mechanical loads of compression and torsion.

Figure 1

Dislocation loops and topology. (a) Two dislocation loops on different slip systems can form a topological “link” with linking number 2, 4, or 6, depending on activation of additional latent hardening mechanisms such as double cross slip. (b) The linking number can be directly calculated through a dislocation line double integral: the Gauss integral (see text). (c) The shear strength ${\tau }_Y$ of a submicron finite volume is typically viewed in the ${\tau }_Y-\rho$ space [30], introducing the elusive concept of the dislocation mesoscale defined at a critical ${\rho }_c$. Hereby, we propose that a topological measure might provide a clear understanding of why a transition takes place, from low to high dislocation densities.

Figure 2

Statistics of two loops in a $0.5\mu \mathrm{m}$ pillar: (a) Initial conditions of (upper panel) two prismatic dislocation loops 1 and 2, with randomly selected Burgers vectors but finite $L_{12}$; and (lower panel) two prismatic dislocation loops 1 and 2, with randomly selected Burgers vectors but zero $L_{12}$. (b) Probability of forming sessile dislocation junctions through two randomly placed prismatic loops with or without linking.

Figure 3

Depositing and testing entanglement for various discrete dislocation densities: uniaxial compression of submicron volumes. Initial sessile junction density is always zero. (a) Random deposition and uniaxial compression of prismatic dislocation loops in a finite mesh of $R=0.25\mu \mathrm{m}$ in (i) an unbiased or (ii) $\mathrm{\Lambda }$-biased manner (shown compression to 0.01% strain for ${\rho }_0=2\times {10}^{13}/{\mathrm{m}}^2$, where various ${\rho }_0$ choices are studied). (b) Nanopillar yield stress (solid symbols) and sessile dislocation density (open symbols) at 0.4% strain as function of initial ${\mathrm{\Lambda }}_0$. Every point corresponds to a uniaxial compression simulation of distinct initial conditions at a target ${\rho }_0$: (${\rho }_0:\{\blacksquare :{10}^{13}/{\mathrm{m}}^2\}$, $\{▸:2\times {10}^{13}/{\mathrm{m}}^2\}$, $\{◂:4\times {10}^{13}/{\mathrm{m}}^2\}$, $\{▴:8\times {10}^{13}/{\mathrm{m}}^2\}$, $\{▾:{10}^{14}/{\mathrm{m}}^2\}$). In Figs. 3 and 3, excerpts of responses of three sample initial conditions are shown in detail to demonstrate the major effect heavily entangled have on mechanical response. Loading stress ${\sigma }_{\mathrm{zz}}$ as a function of applied axial $zz$ strain $\epsilon$ is shown in Fig. 3 with solid symbols. The $\mathrm{\Lambda }$ configurational value as a function of applied axial $zz$ strain $\epsilon$ is shown in Fig. 3 with open symbols. Analogously, in Fig. 3, the total (open) and sessile (closed) dislocation densities are shown. The sessile dislocation density increases roughly linearly with ${\mathrm{\Lambda }}_0$ while yield stress quickly saturates. ${\rho }_0$ values are given for the configurations used in Figs. 3 and 3 ($▴:8\times {10}^{12}/{\mathrm{m}}^2$, $▸:{10}^{13}/{\mathrm{m}}^2$, $◂:2\times {10}^{13}/{\mathrm{m}}^2$).

Figure 4

Uniaxial compression size effects and $\mathrm{\Lambda }$: Four realizations at each size with initially random dislocation density constrained so that (a) $\mathrm{\Lambda }<2$ and (b) $\mathrm{\Lambda }>3$. Each color corresponds to a sample size with diameters $D=0.5$, 1, 2, 4, and $8\mu \mathrm{m}$. Color darkness increases with $D$. Insets show engineering yield stress (defined at 0.02% plastic strain), with power law lines $\sim D^{-0.5}$ (left) and $D^{-0.2}$ (right). Also, an initial configuration is shown for $D=1\mu \mathrm{m}$ for small and large $\mathrm{\Lambda }$ (see also SM [36]).

Figure 5

Engineering entanglement by controlling prior torsion strains: Uniaxial compression of submicron volumes. (a) A protocol is followed where a dislocation loop configuration at density ${\rho }_0$ is prepared, torsion is applied on top or bottom pillar surfaces up to prechosen torsion angle ${\theta }_0$, and then uniaxial compression is applied towards yield. Configurations at each step of the process are shown for a particular configuration with ${\rho }_0=3\times {10}^{13}/{\mathrm{m}}^2$. (b) Yield stress (left $y$ scale-filled symbols) and sessile dislocation density (right $y$ scale-open symbols) at the yield point vs ${\mathrm{\Lambda }}_0$ for a variety of initial conditions and torsions ${\rho }_0\in (5\times {10}^{12},2\times {10}^{14})$ and ${\theta }_0\in \{{10}^{-5},0.1\}$. In Figs. 5 and 5, three characteristic cases $\{{\rho }_0,{\theta }_0\}\to (▸:[{10}^{13}/{\mathrm{m}}^2,0.1],◂:[2\times {10}^{13}/{\mathrm{m}}^2,3\times {10}^{-3}],▴:[2\times {10}^{13}/{\mathrm{m}}^2,3\times {10}^{-2}])$ are shown for (c) torsion $T$ (filled symbols) and $\mathrm{\Lambda }$ (open symbols) vs torsion angle $\theta$ and (d) subsequent loading stress ${\sigma }_{\mathrm{zz}}$ (filled symbols) and $\mathrm{\Lambda }$ (open symbols) vs axial strain $\epsilon$.