Anisotropic structural predictor in glassy materials

There is growing evidence that relaxation in glassy materials, both spontaneous and externally driven, is mediated by localized soft spots. Recent progress made it possible to identify the soft spots inside glassy structures and to quantify their degree of softness. These softness measures, however, are typically scalars, not taking into account the tensorial, anisotropic nature of soft spots, which implies orientation-dependent coupling to external deformation. Here, we derive from first principles the linear response coupling between the local heat capacity of glasses, previously shown to provide a measure of glassy softness, and external deformation in different directions. We first show that this linear response quantity follows an anomalous, fat-tailed distribution related to the universal ${\omega }^4$ density of states of quasilocalized, nonphononic excitations in glasses. We then construct a structural predictor as the product of the local heat capacity and its linear response to external deformation, and show that it offers an enhanced predictability of plastic rearrangements under deformation in different directions, compared to the purely scalar predictor.

Figure 1

The local heat capacity (LHC) $c_{\alpha }$ [cf. Eq. (1)] for a glass composed of $N=10000$ particles [27]. The magnitude of $c_{\alpha }$ is represented by the thickness of the lines connecting particles and black (red) correspond to positive (negative) values (see Ref. [27] for details about the thresholding procedure employed). Regions with anomalously large $|c_{\alpha }|$, i.e., soft spots, are clearly observed. The right (left) triangles correspond to the first plastic events under positive (negative) simple shear AQS deformation and the up (down) ones to the first plastic events under positive (negative) pure shear AQS deformation.

Figure 2

The probability distribution functions (a) $p(d{c}_{\alpha }/d\gamma )$ and (b) $p(c_{\alpha }d{c}_{\alpha }/d\gamma )$ for both simple (squares) and pure (circles) shear deformation. The curves in each panel are vertically shifted one with respect to the other for visual clarity, while in fact they perfectly overlap, as expected from the initial glass isotropy. The theoretical power-law predictions are marked by the solid lines and the triangles (see text for details).

Figure 3

(a) The same as Fig. 1. (b) $c_{\alpha }d{c}_{\alpha }/d\gamma$ [black (red) corresponds to positive (negative) values, and the line thickness represents the magnitude] for simple shear in the positive direction [the right (left) triangles correspond to the first plastic events under positive (negative) simple shear AQS deformation]. (c) The same as (b), but for pure shear in the positive direction [the up (down) triangles correspond to the first plastic events under positive (negative) pure shear AQS deformation].

Figure 4

Quantifying the predictive power of structural indicators with respect to plastic events under AQS deformation through the function $C\left(\lambda \right)$ (see text for definitions), where $C\left(\lambda \right)=\theta \left(\lambda \right)$ (Heaviside step function) corresponds to perfect predictability and $C\left(\lambda \right)=\lambda$ to no predictability. (a) Results for positive values (negative ones are set to zero) of $c_{\alpha }d{c}_{\alpha }/d\gamma$ (diamonds), for negative values (positive ones are set to zero) of it (squares) and for $|c_{\alpha }|$ (circles) under simple shear AQS deformation in the positive direction. (b) The same as (a), but under simple shear AQS deformation in the negative direction.