Near-contact structure and low-frequency vibrations of randomly pinned jammed solids: marginal stability versus the geometric pinning length

When a randomly chosen fraction $f$ of particles in a hard-sphere glass is frozen in place, translational invariance is broken and the counting of mechanical constraints changes. Brito, Parisi and Zamponi (Soft Matter 9, 8540, 2013) showed that if particles are pinned while the system is still at equilibrium, the resulting packings satisfy a generalized isostaticity condition $z_J=2d(1-f)$, remain marginally stable throughout the glass phase, and their low-frequency vibrational spectrum is essentially unaffected by pinning, in sharp contrast with crystals; pinning particles only at the jamming point instead yields hyperstatic packings (excess coordination $\delta z\approx 2df$) whose soft modes are shifted to higher frequencies. Two questions raised in that work remain open. First, the near-contact power-law growth $Z(\delta)\sim\delta^{\alpha}$ ($\alpha\approx 0.6$ in two dimensions) of the integrated pair correlation, traditionally connected to isostaticity and marginal stability, persists even in hyperstatic pinned packings where those mechanisms are absent, and no theoretical account explains this. Second, pinning introduces a geometric length $l_f\sim f^{-1/d}$ (the mean inter-pin distance) in addition to the marginality length $l^{*}\sim 1/\delta z$; naively the softest modes should be stabilized once $l^{*}>l_f$, but this is not observed, and whether a 'strong-pinning' regime in which $l_f$ controls the vibrations exists at all remains undetermined. The problem is to establish what mechanism — constraint counting and marginal stability, or quenched pinning geometry — sets the near-contact structure and the low-frequency spectrum of near-jammed pinned amorphous solids, and how robust the answer is to the operational definition of the contact network. Both central questions remain open in the literature.

Background

Jamming of hard spheres. Athermal hard spheres form a disordered rigid ('jammed') packing when compressed beyond a critical packing fraction. At the jamming point of frictionless spheres in $d$ dimensions, the average number of contacts per particle takes the minimal value required for mechanical stability, $z=2d$: this is the isostaticity condition, obtained from Maxwell's counting of constraints (contacts) against degrees of freedom (particle translations). Packings with more contacts than this bound are called hyperstatic; particles carrying too few contacts to bear any force ('rattlers') are conventionally removed from the counting. Near and above jamming, many properties are governed by the excess coordination $\delta z=z-z_{\mathrm{iso}}$: the vibrational density of states $D(\omega)$ develops a plateau of anomalous low-frequency modes with onset frequency $\omega^{*}\propto\delta z$, and this onset is tied to a length scale $l^{*}\sim 1/\delta z$ below which continuum elasticity breaks down (Wyart, Silbert, Nagel and Witten, Phys. Rev. E 72, 051306 (2005)). In hard-sphere systems below jamming, where the pressure $p$ is finite and configurations are sampled by event-driven molecular dynamics (ballistic free flight between successive elastic collisions), marginal stability takes the form of a lower bound $\delta z\ge A\,p^{-1/2}$ on the excess coordination, with $A$ a numerical constant; the bound is saturated by marginally stable glasses. In hard spheres the microscopic frequency scale is set by the pressure, so the same onset relation is quoted below in the dimensionless form $\omega^{*}/p\sim\delta z$.

Near-contact structure. The pair correlation of jammed packings is singular near contact: counting pairs with normalized gap $\delta_{ij}=(2r_{ij}-\sigma_i-\sigma_j)/(\sigma_i+\sigma_j)$ below $\delta$, the integrated pair correlation $Z(\delta)$ first plateaus at the contact number and then grows as a power law, $Z(\delta)\sim\delta^{\alpha}$ with $\alpha\approx 0.5$-$0.6$ (Donev, Torquato and Stillinger, Phys. Rev. E 71, 011105 (2005)). For isostatic packings this divergence is constrained by marginal stability: Wyart (Phys. Rev. Lett. 109, 125502 (2012)) showed that stability against contact-network rearrangements requires $\gamma\ge 1/(2+\theta)$, relating the divergence exponent $\gamma$ of $g(r)\sim(r-\sigma)^{-\gamma}$ to the small-force exponent $\theta$ of the contact force distribution $P(f)\sim f^{\theta}$. The mean-field theory of the hard-sphere glass, obtained with the replica method of disordered-systems statistical mechanics (averaging over coupled copies of the system) in its full replica-symmetry-breaking scheme, fixes $\gamma\approx 0.41$ and $\theta\approx 0.42$ (Charbonneau, Kurchan, Parisi, Urbani and Zamponi, Nat. Commun. 5, 3725 (2014)), and these scalings are universal across dimensions $d=3$-$10$ and preparation protocols for isostatic jammed packings (Charbonneau, Corwin, Parisi and Zamponi, Phys. Rev. Lett. 109, 205501 (2012); Phys. Rev. Lett. 114, 125504 (2015)). Within this framework the near-contact power law is regarded as a consequence of marginal stability at isostaticity.

Random pinning. Freezing a randomly chosen fraction $f$ of particles at their current positions breaks translational invariance explicitly. This is an established tool in the glass-transition literature (Cammarota and Biroli, Proc. Natl. Acad. Sci. USA 109, 8850 (2012), who distinguish 'weak' and 'strong' pinning regimes according to whether the mean distance between pinned particles, $l_f\sim f^{-1/d}$, exceeds or falls below the relevant cooperative length). For jammed solids, pinned particles remove degrees of freedom but keep their contacts, so the isostatic counting generalizes to $z_J=2d(1-f)$ when particles are pinned in the equilibrium liquid and the system is then jammed (Brito, Parisi and Zamponi, Soft Matter 9, 8540 (2013)). Pinning only at the jamming point instead keeps the contact number of the unpinned system while removing degrees of freedom, producing hyperstatic packings with $\delta z\approx 2df$.

The source results. Brito, Parisi and Zamponi (2013) simulated two-dimensional bidisperse hard disks (a 50:50 mixture of disks with diameter ratio 1.4; $N=4096$; event-driven Lubachevsky-Stillinger compression, in which particle diameters are inflated at a fixed rate during the dynamics until the system jams; contacts defined dynamically as pairs colliding within a window $\Delta t=1000N$ collisions, with an effective potential $V_{\mathrm{eff}}=-k_BT\log h_{ij}$ built from time-averaged gaps) and found: (i) for packings pinned at equilibrium, the jamming density is only slightly reduced, generalized isostaticity holds, marginal stability $\delta z\sim p^{-1/2}$ persists throughout the glass phase, and the vibrational spectrum is essentially unaffected by pinning fractions of several percent, in sharp contrast to crystals, whose entire spectrum shifts upwards as soon as any particle is pinned; (ii) for hyperstatic packings the excess coordination $\delta z\approx 2df$ stabilizes the soft modes, the characteristic frequency saturating at $\omega^{*}/p\sim 2df$; (iii) the near-contact growth $Z(\delta)\sim\delta^{\alpha}$ with $\alpha\approx 0.6$ is unaffected by pinning for both protocols. Two observations did not fit the existing marginal-stability framework and were left as open questions, as stated in the paper's Discussion: the persistence of the near-contact power law in hyperstatic packings, and the absence of any regime in which the pinning length $l_f$ appears to control the vibrations.

Problem Statement

Determine what controls the structural and vibrational signatures of the jamming transition of amorphous solids in the presence of random pinning. The three questions below are jointly posed and probe this single objective from different observables - whether constraint counting and marginal stability (the excess coordination $\delta z$ and the marginality length $l^{*}\sim 1/\delta z$) or the quenched pinning geometry (the mean inter-pin distance $l_f\sim f^{-1/d}$) governs near-jammed pinned solids.

1. Near-contact power law versus isostaticity. The power-law growth $Z(\delta)\sim\delta^{\alpha}$ of the integrated pair correlation beyond the contact plateau persists in randomly pinned packings even when they are hyperstatic ($\delta z\approx 2df\gg A\,p^{-1/2}$ in the hard-sphere convention), i.e. when both the generalized isostatic condition and mechanical marginality are violated - although the standard theoretical arguments tie this power law to marginal stability at isostaticity. Provide a theoretical account of the origin of this power law in pinned systems: does the exponent remain fixed by the marginal-stability mechanism (and if so, how does that argument survive the excess of constraints), or is it set by more generic caging or geometric statistics of the amorphous glass, with a definite prediction for the dependence of $\alpha$ on the pinning fraction $f$ and on $\delta z$? The account must be consistent with the unpinned limits (measured $\alpha\approx 0.5$-$0.6$ in two and three dimensions; the Wyart bound $\gamma\ge 1/(2+\theta)$ and mean-field $\gamma\approx 0.41$ for isostatic packings) and with the observed pinning-independence of $Z(\delta)$ for both the isostatic and the hyperstatic preparation protocols.

2. Marginality length versus pinning length. Pinning introduces the geometric length $l_f\sim f^{-1/d}$ alongside the marginality length $l^{*}\sim 1/\delta z$. Naively, once $l^{*}>l_f$ the softest (anomalous) modes should be stabilized by the pinned geometry; existing hard-sphere data show no such effect for isostatic-protocol packings at the largest pinning fractions and highest pressures explored, suggesting instead that the modes adapt their spatial structure to the pins while keeping their characteristic frequency ($\omega^{*}/p\sim\delta z$). Determine whether a genuine 'strong-pinning' regime exists - at larger $f$, closer to jamming (larger $l^{*}$), in other spatial dimensions, or under other protocols - in which $l_f$ rather than $\delta z$ controls the low-frequency spectrum (for example a stabilization or quantization of the anomalous modes, a low-frequency cutoff of the plateau, or a qualitative change of the modes' spatial structure), or establish that the soft modes always adapt so that the distance to isostaticity remains the only relevant parameter. A resolution should identify which observable discriminates between the two scenarios and measure or derive it.

3. Robustness of the operational definitions (auxiliary to questions 1 and 2). The original measurements rest on a dynamical definition of the contact network - pairs colliding within an averaging window $\Delta t=1000N$ collisions, with an effective interaction potential $V_{\mathrm{eff}}=-k_BT\log h_{ij}$ - in event-driven hard-sphere simulations, and on the time and length scales accessible there. Establish whether the above conclusions, in particular the pinning-independence of $\omega^{*}$ for isostatic-protocol packings and the measured $\alpha$, are independent of this operational choice and of longer-time relaxations.

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