Heat transport in model jammed solids

We calculate numerically the normal modes of vibrations in three-dimensional jammed packings of soft spheres as a function of the packing fraction and obtain the energy diffusivity, a spectral measure of transport that controls sound propagation and thermal conductivity. The crossover frequency between weak and strong phonon scattering is controlled by the coordination and shifts to zero as the system is decompressed toward the critical packing fraction at which rigidity is lost. We present a scaling analysis that relates the packing fraction dependence of the crossover frequency to the anomalous scaling of the shear modulus with compression. Below the crossover, the diffusivity displays a power-law divergence with inverse frequency consistent with Rayleigh law, which suggests that the vibrational modes are primarily transverse waves, weakly scattered by disorder. Above it, a large number of modes appear whose diffusivity plateaus at a nearly constant value before dropping to zero above the localization frequency. The thermal conductivity of a marginally jammed solid just above the rigidity threshold is calculated and related to the one measured experimentally at room temperature for most glasses.

Figure 1

(Color online) Plots of diffusivity, measured in units of $\sigma \sqrt{ϵ/M}$ versus frequency, measured in units of $\sqrt{ϵ/{\sigma }^{2}M}$ for an unstressed packing of 2000 particles with harmonic interactions at packing fractions $\Delta \varphi =0.3$ (green circles), 0.1 (red squares), 0.05 (blue upside triangles), 0.02 (purple downside triangles), and 0.01 (yellow crosses). The black dots indicate the crossover frequency ${\omega }_d$ at each $\Delta \varphi$, while the dashed lines show a power law of ${\omega }^{-4}$, expected for weakly scattered plane waves. The plateau diffusivity is $d_0\approx 0.35$. The inset shows the packing fraction scaling of ${\omega }_d$.

Figure 2

(Color online) Plots of diffusivity vs frequency for an unstressed packing of 2000 particles interacting via a Hertzian potential. The packing fractions are approximately $\Delta \varphi =0.3$ (green circle), 0.1 (red squares), 0.05 (blue triangles), and 0.02 (yellow crosses). (a) $d\left(\omega \right)$ in units of $\sigma \sqrt{\frac{ϵ}{M}}$) vs $\omega$ (in units of $\sqrt{\frac{ϵ}{{\sigma }^{2}M}}$), (b) scaled diffusivity, $d\left(\omega \right)/\Delta {\varphi }^{1/4}$, vs scaled frequency, $\omega /\Delta {\varphi }^{1/4}$, showing data collapse in the plateau region. The inset shows the packing fraction scaling of the crossover frequency ${\omega }_d$.

Figure 3

(Color online) Plot of the transport crossover versus the boson peak at different $\Delta \varphi$ in jammed packings of 2000 particles with harmonic (black open circles) and Hertzian (red open triangles) repulsions. The ratio is near unity and is nearly constant with $\Delta \varphi$.

Figure 4

(Color online) Transverse mode structure factor $f_T\left(q,\omega \right)$ for $N=2000$ and $\Delta \varphi =0.1$ at $\omega =0.23$ (black circles) below the transport crossover at ${\omega }_d=0.36$ and at $\omega =0.74$ (red crosses) well above the crossover. (b)–(d) Phonon dispersion relations for (b) $\Delta \varphi =0.1$, (c) 0.05, and (d) 0.01, respectively. The data are at discrete values of $q$ because of the finite system size. The horizontal (blue) dotted line in each panel (b)–(d) marks ${\omega }_d$, while the solid (red) line marks the transverse sound speed that varies with compression as given by Eq. (10). As a comparison, the dashed (green) line is the same in each panel (independent of compression) and has a slope proportional to the speed of longitudinal sound. Note that the solid line gives the dispersion for frequencies below ${\omega }_d$.

Figure 5

(Color online) Diffusivity just above the jamming transition at $\Delta \varphi ={10}^{-6}$. (a) Contour plot of the heat-flux matrix elements ${\left|\Sigma \left(\omega ,{\omega }^{\prime }\right)\right|}^2$ plotted versus $\omega$ and ${\omega }^{\prime }$ at $N=2000$. (b) Scaling plot showing collapse of ${\left|\Sigma \left(\omega ,{\omega }^{\prime }\right)\right|}^2$ at $N=2000$ (black solid), 1000 (red dashed), and 500 (blue dotted) with scale factors $s^2$ and $w$. (c) Scale factors $s^2$ (red symbols), and $w$ (black symbols) versus $\omega$. We find $s^2\propto {\omega }^2$ (red dashed line) and $w\propto \omega$ (black dotted line) except at high $\omega$. (d) Scatter plot of $d\left(\omega \right)$ with $N=2000$ and delta function broadening $\eta =0.002$. Solid line shows the predicted $d\left(\omega \right)$ for the infinite system.

Figure 6

(Color online) Diffusivity for $\Delta \varphi =0.1$ in stressed (black circles) and unstressed (red crosses) packings composed of 1000 particles with harmonic repulsions.

Figure 7

(Color online) Density of states $D\left(\omega \right)$ vs $\omega /{\omega }^{\ast }$ for $\Delta \varphi =0.0001$ (black dotted), 0.01 (yellow dashed-dot-dot), 0.05 (blue dot-dashed), 0.1 (red dashed) and 0.3 (green solid), for a stressed system with $N=2000$ with harmonic repulsions. Note that ${\omega }^{\ast }$ scales according to Eq. (12). This scaling produces a good collapse of the vibrational spectrum at low frequency.