Response of jammed packings to thermal fluctuations

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures $T_{cb}$ required to break a single contact in the MS packing for both single- and multiple-eigenmode perturbations of the $T=0$ MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat ${\overline{C}}_V$ and deviations of the average disk positions $\mathrm{\Delta }r$ from their $T=0$ values in the temperature regime $T_{{\overline{C}}_V}<T<T_r$, where $T_r$ is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle $\mathrm{\Delta }{\overline{C}}_V^0/{\overline{C}}_V^0$ relative to the zero-temperature value ${\overline{C}}_V^0$ can grow rapidly above $T_{cb}$; however, the deviation $\mathrm{\Delta }{\overline{C}}_V^0/{\overline{C}}_V^0$ decreases as $N^{-1}$ with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations $\mathrm{\Delta }{r}^{ss}/\mathrm{\Delta }{r}^{ds}$ for single- and double-sided linear and nonlinear spring interactions. We find that $\mathrm{\Delta }{r}^{ss}/\mathrm{\Delta }{r}^{ds}>100$ for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range $T_{cb}<T<T_r$ for model jammed systems.

Figure 1

Examples of isostatic mechanically stable bidisperse disk packings at zero temperature with (a) $N_c=N_c^0=127$ contacts and $N_r=0$ rattler particles and (b) $N_c=N_c^0=115$ contacts and $N_r=6$ rattler particles. The nonrattler (rattler) disks are outlined in black (red). For both (a) and (b), the total number of disks $N=64$, the potential energy per particle $U={10}^{-12}$, and the solid black lines connecting disks centers indicate force-bearing interparticle contacts. (c) The fraction of mechanically stable packings $N_m\left(U\right)/N_{\mathrm{tot}}$ that possess $m=N_c-N_c^0=0$ (circles), 2 (crosses), and 4 (plus signs) excess contacts as a function of $U{N}^4$ for three system sizes $N=32$ (solid lines), 128 (dashed lines), and 256 (dotted lines). (d) The number of excess contacts $m$ normalized by $N$ averaged over 5000 MS packings and plotted versus $U$ for three system sizes $N=32$ (circles), 128 (crosses), and 256 (plus signs). The dashed line has slope 0.25.

Figure 2

Schematic of four important temperature regimes when studying the response of MS packings to thermal fluctuations. For $T<T_{cb}$, the $T=0$ contact network remains intact. In this regime, “form” nonlinearities occur when the interparticle potential cannot be written exactly as a harmonic function of the disk positions. For $T_{cb}<T<T_r$, the $T=0$ contact network changes, both form and contact-breaking nonlinearities occur, and the system remains in the basin of attraction of the original MS packing. For $T_r<T<T_g$, the system can move to the basins of attraction of other MS packings, but the relaxation times are sufficiently long that structural relaxation is not complete. For $T>T_g$, the system is liquid-like with finite structural relaxation times. This article focuses on the (unshaded) temperature regimes that occur for $0<T\le T_r$.

Figure 3

The minimum temperature $T_1(m,m-1)$ required to break a single contact when perturbing an MS packing along one of the eigenmodes of the dynamical matrix averaged over 5000 MS packings, normalized by the potential energy per particle $U$, and plotted as a function of system size $N$. $T_1(m,m-1)$ was obtained by minimizing over all single-mode perturbations. We include results for $U={10}^{-12}$ (circles), ${10}^{-8}$ (crosses), and ${10}^{-4}$ (pluses). The slope of the dashed line is $-2.6$. Rattler disks are removed from the packings prior to performing these calculations.

Figure 4

(a) The minimum temperature $\langle T_n(m,m-1)\rangle$ (normalized by $U$ and averaged over 5000 MS packings) required to break a single contact in response to perturbations that include $n=1,2,...,6$ eigenmodes of the dynamical matrix. $\langle T_n(m,m-1)\rangle$ is obtained by minimizing over all possible $n$-mode combinations of the $2N^{\prime }-2$ eigenmodes for each MS packing at $U={10}^{-12}$ (dashed line), ${10}^{-8}$ (solid line), and ${10}^{-4}$ (dotted line). The horizontal lines give the minimum temperature $\langle T_{\min }/U\rangle$ required to remove the smallest overlap between a pair of contacting disks at each $U$ (averaged over 500 MS packings). The inset shows the scaling of $\langle T_{\min }/U\rangle$ with system size $N$ for the same values of $U$ as in the main panel. The slope of the dashed line is $-2.9$. (b) Difference in the potential energy per particle between MS packings before ($U$) and after ($U^{\prime }$) separating the pair of disks with the smallest interparticle overlap as a function of the angle $\theta$ between the old and new separation vectors between the two disks. (c) and (d) Schematic of the process to measure $T_{\min }/U$. In panel (c), the disk pairs with the smallest overlap are shaded in blue. In panel (d), this pair of disks is shifted so that $r_{ij}={\sigma }_{ij}$. The original positions are indicated by the dashed circles. The new separation vector makes an angle $\theta$ with the old separation vector (as indicated by the dotted lines). After shifting disks $i$ and $j$, potential energy minimization is performed allowing all disks to move except $i$ and $j$. In both panels, the contact networks of the blue-shaded disks are indicated by solid lines.

Figure 5

(a) The fraction of time $f(T,N_{bc})$ that the system (with $N=64$ and $U={10}^{-4}$) possesses a given number of broken contacts $N_{bc}=N_c^0+m-N_c$ at temperature $T$. The color scale from yellow to blue represents decreasing $f$ on a ${log}_{10}$ scale. The horizontal line indicates the rearrangement temperature $T_r$. The solid curve with crosses gives the characteristic temperature $T^{*}\left(N_{bc}\right)<T_r$ for multiple contact breaking for which the fraction $f=0.1$. (b) The characteristic temperature $T^{*}\left(N_{bc}\right)$ for three system sizes, $N=32$ (solid lines), 64 (dashed lines), and 128 (dotted lines), and three values of $U$, ${10}^{-5}$ (circles), ${10}^{-4}$ (crosses), and ${10}^{-3}$ (pluses), for each $N$. The inset shows the same data as in the main panel, but $T^{*}$ is plotted as a function of $N_{bc}^{\gamma }{N}^{\delta }{U}^{\zeta }$, where $\gamma \approx 2.2\pm 0.3$, $\delta \approx -2.2\pm 0.2$, and $\zeta \approx 1.0\pm 0.1$. The slope of the dashed line is 1.

Figure 6

(a) The normalized deviation in the specific heat per particle at constant volume from the value at $T=0$, $\mathrm{\Delta }{\overline{C}}_V/{\overline{C}}_V^0$, at $U={10}^{-5}$ for purely repulsive linear spring interactions as a function of temperature $T$ normalized by $T_{cb}$, where the first contact breaks. The data are obtained from MD simulations at constant energy following equal velocity-amplitude perturbations applied to $50T=0$ MS packings with $N=16$ (circles), 32 (crosses), 64 (pluses), and 128 (stars). The inset shows $\mathrm{\Delta }{\overline{C}}_V/{\overline{C}}_V^0$ versus system size $N$ for 10 values of $T/T_{cb}$ from 1 to ${10}^2$ (from bottom to top). The dotted line has slope $-1$. (b) $\mathrm{\Delta }{\overline{C}}_V/{\overline{C}}_V^0$ as a function of $T/T_{cb}$ for MS packings with $N=32$ disks that interact via purely repulsive linear (circles) and Hertzian spring interactions (crosses) at $U={10}^{-5}$.

Figure 7

(a) The deviation $\mathrm{\Delta }r$ in the disk positions from their $T=0$ values as a function of temperature $T$ for a MS packing with $N=64$ and $U={10}^{-5}$. The data are obtained from constant energy MD simulations with equal velocity-amplitude initial perturbations involving all eigenmodes. We consider both purely repulsive linear spring interactions [circles; $\alpha =2$ in Eq. (1)] and double-sided linear spring interactions [crosses; $\alpha =2$ in Eq. (2)]. The dashed line has slope 1. The three dotted vertical lines indicate (1) the measured temperature $T_{cb}$ at which the first contact breaks, (2) the temperature $T_r$ at which the system transitions to the basin of a new MS packing, and (3) the temperature $T_g$ at which the structural relaxation time (from the self-part of the intermediate scattering function) appears to diverge. (b) The average disk positions at a temperature $T<T_{cb}$ (gray-shaded disks). White solid lines indicate contacts between disks in the backbone. The arrows represent the displacement of the disks relative to their positions at $T=0$, where the length of each arrow is proportional to the logarithm of the displacement of the disk. (c) Same as in (b) except for the average disk positions at a temperature $T_{cb}<T<T_r$. Gray-shaded disks without edges are rattlers, circular outlines with dashed edges indicate the initial positions of rattler disks, and white-dotted lines show contacts that include rattlers. (d) Same as (c) except for the average disk positions at a temperature $T_r<T<T_g$.

Figure 8

The deviation $\mathrm{\Delta }r$ in the disk positions from their $T=0$ values as a function of temperature $T$ for a MS packing with $N=32$, $U={10}^{-5}$, and single- (circles) and double-sided (crosses) Hertzian spring interactions. The three dotted vertical lines indicate $T_{cb}$, $T_r$, and $T_g$ (from left to right).

Figure 9

(a) The ratio $\mathrm{\Delta }{r}^{ss}/\mathrm{\Delta }{r}^{ds}$ between the deviations in positions for single- and double-sided linear spring interactions as a function of temperature normalized by contact-breaking temperature $T/T_{cb}$ for packings with $N=128$ and three values of $U$ $[{10}^{-5}$ (circles), ${10}^{-4}$ (crosses), and ${10}^{-3}$ (pluses)]. Each curve is averaged over 50 packings in the temperature range $1<T/T_{cb}<{10}^4$. The three vertical lines indicate $\langle T_r\rangle$ for these packings, $U={10}^{-5}$ (solid line), ${10}^{-4}$ (dashed line), and ${10}^{-3}$ (dotted line). (b) The ratio of position deviations $\mathrm{\Delta }{r}^{ss}/\mathrm{\Delta }{r}^{ds}$ from single- and double-sided linear (circles) and Hertzian (crosses) spring interactions as a function of $T/T_{cb}$ for MS packings with $N=32$ and $U={10}^{-5}$. Each curve is averaged over 50 packings in the temperature range $1<T/T_{cb}<{10}^4$. The two vertical lines indicate $\langle T_r\rangle$ for MS packings with purely repulsive linear (solid line) and Hertzian spring interactions (dotted line). (c) $\langle \mathrm{\Delta }{r}^{ss}/\mathrm{\Delta }{r}^{ds}\rangle$ averaged over the temperature range $1<T/T_{cb}<{10}^4$ for linear spring interactions as a function of system size $N$ for $U={10}^{-5}$ (circles), ${10}^{-4}$ (crosses), and ${10}^{-3}$ (pluses).

Figure 10

(a) $\langle \mathrm{\Delta }{r}^{ss}\rangle$ and (b) $\langle \mathrm{\Delta }{r}^{ds}\rangle$ averaged over the temperature range $1<T/T_{cb}<{10}^4$ for linear spring interactions as a function of system size $N$ for $U={10}^{-5}$ (circles), ${10}^{-4}$ (crosses), and ${10}^{-3}$ (pluses). Each point is averaged over 50 packings with the same $N$ and $U$.

Figure 11

$\mathrm{\Delta }r$ versus temperature $T$ for an initial MS packing with purely repulsive linear spring interactions, $N=64$, and $U={10}^{-5}$ with the two rattlers kept in the system (triangles) and the two rattlers removed (circles).

Figure 12

The fraction $f_i$ of time that three particular MS packings occur in the temperature range ${10}^{-10}<T<{10}^{-4}$ for systems with $N=64$ and $U={10}^{-5}$. The dashed vertical line indicates the rearrangement temperature $T_r$ for the $T=0$ MS packing. At the lowest temperatures, the system only populates the $T=0$ MS packing (circles). At intermediate temperatures a different MS packing (crosses) becomes most frequent. At the highest temperatures, the system spends all of the time in a third MS packing (pluses).

Figure 13

Structural relaxation time $\tau$ (from the decay of the self-part of the intermediate scattering function) as a function of temperature $T$ for a system with $N=64$ and $U={10}^{-5}$. The dashed line gives $\tau \left(T\right)=Cexp[A{T}_g/(T-T_g)]$, where $C=1.1$, $A=15$, and $T_g=1.3\times {10}^{-4}$. The inset shows the self-part of the intermediate scattering function at $q{\sigma }_S=2\pi$, $F_s(q,t)$, for several temperatures from $T={10}^{-4}$ to ${10}^{-3}$ from top to bottom. The horizontal line indicates $F_s(q,\tau )=e^{-1}$.

Figure 14

(a) The potential energy $U\left(r\right)$ as a function of position $r$ for a quadratic form, $U^q\left(r\right)=A{r}^2/2$ (solid line), and a cubic form, $V^c\left(r\right)=A{r}^2/2+B{r}^3/6$ (dashed line). The vertical dotted line indicates $r=0$. Note that the cubic potential is asymmetric about $r=0$. (b) The absolute value of the average position $|\langle r\rangle |$ versus temperature $T$ for the quadratic (circles) and cubic (crosses) potentials.