Particle pinning as a method to manipulate marginal stability

TL;DR

This work shows that random pinning of particles provides a tunable handle on marginal stability in jammed packings. By comparing two protocols (CP and PC), the authors demonstrate that ω^* remains governed by the isostatic distance via δz, while CP induces hyperstaticity and shifts vibrations below ω^*, and PC preserves marginal stability with the non-Debye scaling intact. The study reveals that plateau modes stay extended under pinning, whereas pinning can induce localization in the quasi-localized regime, and it uncovers novel 2D non-Debye behavior under PC. Overall, pinning offers a practical method to control low-frequency vibrational properties and stability, with potential implications for transport and plastic rearrangements in amorphous solids.

Abstract

We study the critical behavior of low-frequency vibrations of packings with pinned particles near the jamming point. Soft modes form a plateau in the density of states and its frequency is controlled by the contact number as the ordinary jamming transition. The spatial structure of these modes is not largely affected by pins. Below the plateau, the non-Debye scaling predicted by mean-field theories and quasi-localized modes breaks down depending on the pinning procedures. We comprehensively explain these behaviors by the impact of pinning operations on the marginal stability of the packings.

Figures (12)

• Figure 1: Stability phase diagram of jammed harmonic spheres and the summary of this study. The diagram is divided into stable and unstable phases by the marginal stability line $\delta z=Cp^{1/2}$. Ordinary jammed packings (the circle at the center of the figure) are located on this line and thus they are marginally stable. In previous studies, unstressed analysis has been used to stabilize the packing by moving it away from this line. In this study, we show that particle pinning can stabilize the packing in another direction from unstressed analysis and that the distance from the marginal stability line can be manipulatively controlled.
• Figure 2: The excess contact number versus pressure in 2D packings generated by the CP protocol. Inset: $\delta z$ vs $p$ for packings with $N=1000$ generated by the PC protocol. The legends are the same as the main panel. The error bars are estimated by the bootstrap method and the dashed lines represent $\delta z \sim p^{1/2}$.
• Figure 3: The vibrational density of states of $p=\numrange{e-6}{e-2}$ with (a) $c=0.01$ by CP protocol, (b) $c=0.10$ by CP protocol, (c) $c=0.01$ by PC protocol, and (d) $c=0.10$ by PC protocol. 2D packings with $N=1000$ are used for this analysis. (e) The characteristic frequency $\omega^*$ of the onset of the plateau. The dashed line represents $\omega^* \sim \delta z$.
• Figure 4: The visualization of plateau modes. (a) Plateau modes in unpinned ($c=0.00$), pinned ($c=0.01$, CP), and pinned ($c=0.01$, PC) packings are shown (from left to right). (b) Plateau modes at $c=0.00, 0.01, 0.10$ with PC protocol are shown (from left to right). Packings ($N=1000$) at $p=e-4$ are used for both panels and frequencies of each mode are $\omega=4.5e-2$ close to $\omega^*$. The particles in magenta represent pinned particles.
• Figure 5: Spatial mode correlation $C(r)$ of the plateau modes illustrated in Fig. \ref{['fig:visualize']} (b). Packings at $p=e-4$ generated through the PC protocol is used in this figure.