Extracting the properties of quasilocalized modes in computer glasses: Long-range continuum fields, contour integrals, and boundary effects

Low-frequency nonphononic modes and plastic rearrangements in glasses are spatially quasilocalized, i.e., they feature a disorder-induced short-range core and known long-range decaying elastic fields. Extracting the unknown short-range core properties, potentially accessible in computer glasses, is of prime importance. Here we consider a class of contour integrals, performed over the known long-range fields, which are especially designed for extracting the core properties. We first show that, in computer glasses of typical sizes used in current studies, the long-range fields of quasilocalized modes experience boundary effects related to the simulation box shape and the widely employed periodic boundary conditions. In particular, image interactions mediated by the box shape and the periodic boundary conditions induce the fields' rotation and orientation-dependent suppression of their long-range decay. We then develop a continuum theory that quantitatively predicts these finite-size boundary effects and support it by extensive computer simulations. The theory accounts for the finite-size boundary effects and at the same time allows the extraction of the short-range core properties, such as their typical strain ratios and orientation. The theory is extensively validated in both two and three dimensions. Overall, our results offer a useful tool for extracting the intrinsic core properties of nonphononic modes and plastic rearrangements in computer glasses.

Figure 1

An example of a quasilocalized mode in a two-dimensional computer glass. Shown is $ln\left(\right|\mathit{u}\left(\mathit{r}\right)\left|\right)$, the logarithm of the displacement field $\mathit{u}\left(\mathit{r}\right)$ of a nonlinear quasilocalized mode (see Appendix pp1 for details about the computer glass model and Appendix pp2 about nonlinear modes). $\mathit{r}$ is the position vector relative to the center of the mode (white arrow, represented by the polar coordinates $(r,\theta )$) and hotter (colder) colors correspond to larger (smaller) displacements. The mode exhibits intense displacements at its core (marked by a dashed circle) of linear size $a$, which are accompanied by a long-range decaying field. The mode also exhibits azimuthal quadrupolar structure [16, 19] oriented at an angle ${\varphi }^{*}$ relative to a fixed Cartesian coordinate system ($x,y$), which is aligned with the simulation box (bottom left corner).

Figure 2

${\overline{I}}_2\left(r\right)=\sqrt{{\left[I_2^{\left(1\right)}\left(r\right)\right]}^2+{\left[I_2^{\left(2\right)}\left(r\right)\right]}^2}$ vs $r/L$, cf. Eqs. (5b)–(5c), for three different nonlinear quasilocalized modes $\mathit{u}\left(\mathit{r}\right)$ identified in a 2D computer glass with $L=345a_0$ (see Appendix pp1 for additional information about the computer glass model, and Appendix pp2 for information about how the nonlinear modes were identified and calculated). The presented results are discussed in detail in the text.

Figure 3

(a) ${\varphi }_{\text{I}}\left(r\right)\equiv \frac{1}{2}arctan\left[I_2^{\left(2\right)}\left(r\right)/I_2^{\left(1\right)}\left(r\right)\right]$, where $I_2^{\left(1\right)}\left(r\right)$ and $I_2^{\left(2\right)}\left(r\right)$ are defined in Eqs. (5b)–(5c), vs ${\varphi }^{*}$ (obtained using the microscopic measure of Sec. 5b) for many nonlinear quasiocalized modes and $r=0.2L,0.4L,0.5L$ (discrete symbols, see legend). The results indicate the orientation-dependent fields' rotation—see text for additional details, explanations, and discussion—as explicitly demonstrated in the inset. The inset presents a nonlinear quasilocalized mode with ${\varphi }^{*}=9.7^{\circ }$ (cf. Fig. 1), which corresponds to the vertical line in the main panel. The white line shows ${\varphi }_{\text{I}}\left(r\right)$, rotated by ${90}^{\circ }$ relative to ${\varphi }^{*}$ for visual clarity. The dashed circles correspond to $r=0.2L,0.4L,0.5L$, i.e., to the three intersections of the vertical line with the data points in the main panel (marked by thick circles). The dashed lines in the main panel are the theoretical predictions of the image interaction theory of Sec. 4; see text for details. (b) $r{\langle \mathit{u}\left(\mathit{r}\right)\rangle }_{\theta }$ (solid lines) averaged over many atomistic quasilocalized modes with ${\varphi }^{*}=0^{\circ }\pm 1^{\circ },22.5^{\circ }\pm 1^{\circ },{45}^{\circ }\pm 1^{\circ }$ (see legend), as a function of $r/L$ (${\langle \mathit{u}\left(\mathit{r}\right)\rangle }_{\theta }\equiv \sqrt{{\int }_0^{2\pi }{\left|\mathit{u}\left(\mathit{r}\right)\right|}^{2}d\theta }$). The theoretical predictions of the image interaction theory of Sec. 4 are superimposed (dashed lines). Both the atomistic and theoretical curves are normalized by their maximal value. See text for additional details, explanations, and discussion.

Figure 4

The continuum quantities ${\varphi }_{\text{I}}$ (left $y$ axis) and $({\phi }_{\text{I}},{\psi }_{\text{I}})$ (right $y$ axis), see Sec. 5a and Fig. 5 for details, vs ${\varphi }^{*}$ (lower $x$ axis) and $({\phi }^{*},{\psi }^{*}$) (upper $x$ axis), see Sec. 5b for details. The black dashed line corresponds to perfect agreement between the two approaches.

Figure 5

(a) The contour integrals of Eqs. (5a)–(5c) for a 2D mode [the same one previously shown in Fig. 3] are shown on the left panel. On the right panel, the amplitude of the atomistic mode is shown under “Atomistic mode” and the mode obtained by the extracted core properties is shown under “Continuum theory” (see text for additional details). The core properties are extracted at a distance from the core's center where the largest contour integral [here ${\overline{I}}_2\left(r\right)$] attains its maximum, which is marked by a vertical dashed line. The $R$-squared correlation coefficient of the atomistic and continuum fields, for $r>0.15L$, is $R^2=0.98$. (b)–(c) The contour integrals of Eqs. (7a)–(7b) and (8) for two 3D modes are shown on the left panels. The vertical dashed lines, as in (a), mark the distance from the core's center where the largest contour integral attains its maximum. The corresponding $R$-squared correlation coefficients for $r>0.15L$ are $R^2=0.92$ for (b) and $R^2=0.93$ for (c). On the right panel, surfaces of constant value of the magnitude of each field are shown. The principal directions extracted from the analysis are superimposed (they are identical in both the “Atomistic mode” and the “Continuum theory” columns). In the continuum theory modes, $2^8$ Fourier modes per spatial dimension have been used in all panels.