Effects of compression on the vibrational modes of marginally jammed solids

Glasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze, in particular, two systems whose density $D\left(\omega \right)$ of vibrational modes of angular frequency $\omega$ display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature and (ii) a system with the same network of contacts, but where the force between any particles in contact (and therefore the total pressure) is set to zero. We account in the two cases for the observed (a) convergence of $D\left(\omega \right)$ toward a nonzero constant as $\omega \to 0$, (b) appearance of a low-frequency cutoff ${\omega }^{*}$, and (c) power-law increase of ${\omega }^{*}$ with compression. Differences between these two systems occur at a lower frequency. The density of states of the modified system displays an abrupt plateau that appears at ${\omega }^{*}$, below which we expect the system to behave as a normal, continuous, elastic body. In the unmodified system, the pressure lowers the frequency of the modes in excess. The requirement of stability despite the destabilizing effect of pressure yields a lower bound on the number of extra contact per particle $\delta z:\delta z⩾p^{1∕2}$, which generalizes the Maxwell criterion for rigidity when pressure is present. This scaling behavior is observed in the simulations. We finally discuss how the cooling procedure can affect the microscopic structure and the density of normal modes.

Figure 1

The density of vibrational states, $D\left(\omega \right)$, vs angular frequency $\omega$ for the simulation of Ref. 8. 1024 spheres interacting with repulsive harmonic potentials were compressed in a periodic cubic box to packing fraction $\varphi$, slightly above the jamming threshold ${\varphi }_c$. Then the energy for arbitrary small displacements was calculated and the dynamical matrix inferred. The curve labeled $a$ is at a relative packing fraction $\varphi -{\varphi }_c=0.1$. Proceeding to the left the curves have relative volume fractions ${10}^{-2},{10}^{-3},{10}^{-4},{10}^{-8}$, respectively.

Figure 2

Illustration of the boundary contact removal process described in the text. 18 particles are confined in a square box of side $L$ periodically which is continued horizontally and vertically. An isostatic packing requires 33 contacts in this two-dimensional system. An arbitrarily drawn vertical line divides the system. A contact is removed wherever the line separates the contact from the center of a particle. 28 small triangles mark the intact contacts; removed contacts are shown by the five white circles.

Figure 3

One soft mode in two dimensions for $N\approx 1000$ particles. Each particle is represented by a dot. The relative displacement of the soft mode is represented by a line segment extending from the dot. The mode was created from a previously prepared isostatic configuration, periodic in both directions, following [8]. 20 contacts along the vertical edges were then removed and the soft modes determined. The mode pictured is an arbitrary linear combination of these modes.

Figure 4

Scaling of ${\omega }^{*}$ with the excess coordination number $\delta z$ in the system with relaxed springs. The line has a slope one.

Figure 5

Log-linear plot of the density of states for a three-dimensional system with $N=1024$ for three values of $\varphi -{\varphi }_c$ in the soft-sphere system (dotted line) and the system where the applied stress term has been removed (solid line).

Figure 6

Sum of the transverse terms (full curve) $ϵ=\frac{1}{2}{\Sigma }_{⟨ij⟩}{\left[{(\delta {\vec{R}}_j-\delta {\vec{R}}_i)}^{\perp }\right]}^2$ and longitudinal terms (dotted curve) $ϵ=\frac{1}{2}{\Sigma }_{⟨ij⟩}{[(\delta {\vec{R}}_j-\delta {\vec{R}}_i)\bullet {\vec{n}}_{ij}]}^2$ for each mode of frequency $\omega$ at the jamming threshold in three dimensions. The longitudinal term is equal to the energy of the modes and vanish quadratically at 0 frequency. The transverse term converges toward a constant different from 0.

Figure 7

Phase diagram of rigidity in terms of the coordination number and pressure. When $p>0$, the line separating the stable and unstable regions is defined by Eq. (20). When the pressure is negative, any connected system can be rigid.

Figure 8

The overlap function $f\left(x\right)$ as defined in Eq. (A1) for different system sizes in three dimensions. Soft modes were created from isostatic configurations as described in Fig. 3.