Theory of harmonic dissipation in disordered solids

Mechanical spectroscopy, i.e., cyclic deformations at varying frequencies, is used theoretically and numerically to compute dissipation in model glasses. From a normal mode analysis, we show that in the high-frequency terahertz regime where dissipation is harmonic, the quality factor (or loss angle) can be expressed analytically. This expression is validated through nonequilibrium molecular dynamics simulations applied to a model of amorphous silica (${\mathrm{SiO}}_2$). Dissipation is shown to arise from nonaffine relaxations triggered by the applied strain through the excitation of vibrational eigenmodes that act as damped harmonic oscillators. We discuss an asymmetry vector field, which encodes the information about the structural origin of dissipation computed by mechanical spectroscopy. In the particular case of silica, we find that the motion of oxygen atoms, which induce a deformation of the Si-O-Si bonds, is the main contributor to harmonic energy dissipation.

Figure 1

Vibrational density of states (VDOS) of the ${\mathrm{SiO}}_2$ model. The partial VDOS of the oxygen atoms is decomposed on the rocking, stretching, and bending motions of the Si-O-Si bonds. The inset shows a sketch of a Si-O-Si bond with the vectors used to decompose the bond deformation into bending, stretching, and rocking components.

Figure 2

Evolution of the hydrostatic pressure during cycles of isostatic applied strain. The dashed curves show the instantaneous pressure. The solid curve is the average computed over 250 cycles.

Figure 3

Energy dissipation as a function of frequency ($\omega /2\pi$) in amorphous ${\mathrm{SiO}}_2$ modeled with the full nonlinear BKS potential at three different temperatures. The friction of the Langevin thermostat was set to 1 THz. The yellow circles are experimental data obtained by fitting the excitation peak of the x-ray scattering spectra with the DHO model [38].

Figure 4

Energy dissipation as a function of frequency computed with the nonlinear BKS potential, the harmonic approximation of Eq. (4) and the analytical expression of Eq. (12). The same thermostat friction of 1 THz is used in all calculations.

Figure 5

Energy dissipation as a function of frequency computed using the analytical expression for three different thermostat frictions: 0.3, 1, and 3 THz. The inset shows a log-log view of the low-frequency region.

Figure 6

Mean energy dissipation as a function of frequency, calculated with Eq. (12) for 20 different ${\mathrm{SiO}}_2$ glasses, with a thermostat frequency $\gamma =1$ THz. The blue area represents the standard deviation of the energy dissipation computed on the 20 samples.

Figure 7

Square of the mode-dependent nonaffine coefficient, $C_m^2$, computed from Eq. (11), as a function of the mode eigenfrequency.

Figure 8

Asymmetry vector field $\mathrm{\Xi }$ in the case of isostatic deformations, represented in two-dimensional projection for an 8-$\AA$ slab in a ${\mathrm{SiO}}_2$ sample. Black arrows are for Si atoms, red arrows for oxygen atoms. The scaled forces ${\xi }_{j\to i}^{\alpha }$ for the mode of maximum dissipation are shown in gray, with a width proportional to their intensity. The inset shows the stereographic projection of the $\mathrm{\Xi }$ vectors (normalized to unity) of the O atoms in the basis formed by the rocking, stretching, and bending vectors ($V_{\mathrm{Rock}}, V_{\mathrm{Stre}},V_{\mathrm{Bend}}$) of each Si-O-Si bond.

Figure 9

Asymmetry vector field ${\mathrm{\Xi }}_{xy}$ and its stereographic projection for an applied shear strain ${\epsilon }_{xy}$ in the same 8-$\AA$ slab as in Fig. 8, projected on the $xy$ plane of the slab. The scale of the arrows is tripled compared to Fig. 8.