Predicting plasticity with soft vibrational modes: From dislocations to glasses

We show that quasilocalized low-frequency modes in the vibrational spectrum can be used to construct soft spots, or regions vulnerable to rearrangement, which serve as a universal tool for the identification of flow defects in solids. We show that soft spots not only encode spatial information, via their location, but also directional information, via directors for particles within each soft spot. Single crystals with isolated dislocations exhibit low-frequency phonon modes that localize at the core, and their polarization pattern predicts the motion of atoms during elementary dislocation glide in two and three dimensions in exquisite detail. Even in polycrystals and disordered solids, we find that the directors associated with particles in soft spots are highly correlated with the direction of particle displacements in rearrangements.

Figure 1

3D split partial dislocations connected by a stacking fault in a fcc lattice composed of $38000$ particles. (a) Polarization field of the first localized mode at dislocation core (green), and particles forming a soft spot from the 500 largest polarization vectors (red). (b) Displacement field resulting from an elementary dislocation slip. Configurations are projected along the [$\overline{1}\overline{1}$2] direction (into plane).

Figure 2

$10000$ particle polycrystal in 2D. (a) Example configuration with grain boundaries (green), soft spots (red circles), and a superposition of the polarization directions of the modes that define the soft spots. (b) Nonaffine displacement map after step shear strain of ${10}^{-3}$.

Figure 3

$10000$ particle amorphous solid in 2D. (a) Example configuration with soft spots (red) and polarization directions from the contributing soft modes. (b) Soft spot map with nonaffine displacements from a step shear strain of ${10}^{-3}$.

Figure 4

(a) Fraction of particles in a soft spot as a function of $d_{\text{min}}^2$ divided by the soft spot covering fraction. Data represent an average over 1000 different configurations subjected to forward-backward simple shear. (b) Trace of product $2\mathrm{tr}(U·V)$ of the total director tensor $U_{ij}={\widehat{u}}_i{\widehat{u}}_j-{\delta }_{ij}/2$ and the displacement tensor $V_{ij}={\widehat{v}}_i{\widehat{v}}_j-{\delta }_{ij}/2$, where $\widehat{u}$ and $\widehat{v}$ are normalized polarization and nonaffine displacement vectors of particles in soft spots as a function of $d_{\min }^2$. Dashed (dotted) lines show for comparison the alignment of plastic displacements with only the lowest-energy (low participation ratio) mode in the glassy system. Configurations with $d_{\text{min}}^2>6{\sigma }^2$ were not considered to remove avalanches.

Figure 5

Difference between saturation value of soft spot-$d_{\min }^2$ overlap probability and soft spot covering fraction vs number of modes $N_m$ included in the soft spot population for the amorphous system.

Figure 6

Ratio between saturation value of soft spot-$d_{\min }^2$ overlap probability and soft spot covering fraction vs number of modes $N_m$ included in the soft spot population and for the amorphous system.