[Paper Review] Vibrations in jammed solids: Beyond linear response

Thibault Bertrand, Carl F. Schreck+2 more|arXiv (Cornell University)|Jul 1, 2013

Granular flow and fluidized beds3 citations

TL;DR

This paper proposes a phase diagram for jammed particulate solids under purely repulsive, frictionless contact forces, distinguishing between iso-coordinated solids (ICS) with harmonic vibrational response and hypo-coordinated solids (HCS) with strongly nonharmonic response. The key finding is that vibrational response in HCS is nonharmonic and method-dependent, challenging linear response assumptions in jammed matter near jamming onset.

ABSTRACT

We propose a `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Δϕ=ϕ- ϕ_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Δϕ> Δϕ_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{ m iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies $ω_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $Δϕ> Δϕ^*_-(T_0)$, $Δϕ< Δϕ^*_+(T_0)$, and $Δϕ> Δϕ_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> \approx z_{ m iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {\it i.e} the density of vibrational modes $D(ω)$ is not a collection of sharp peaks at $ω_k^d$, and its precise form depends on the measurement method. For $Δϕ> Δϕ_{cb}(T_0)$ and $Δϕ< Δϕ^*_{-}(T_0)$, the system behaves as a hard-particle liquid.

Motivation & Objective

• To understand the vibrational response of jammed particulate systems under purely repulsive forces, especially near jamming onset.
• To identify and characterize distinct classes of mechanical and vibrational behavior in these systems as functions of kinetic energy $T_0$ and packing fraction deviation $\Delta\phi$.
• To challenge the assumption of linear response in jammed matter by directly measuring vibrational modes instead of inferring them.
• To clarify the role of contact network fluctuations in determining vibrational spectra, particularly in the absence of particle rearrangements.

Proposed method

• Numerical simulations of $N$ bidisperse, frictionless disks with a purely repulsive pairwise potential $V(r_{ij}) = \frac{\epsilon}{2}(1 - r_{ij}/\sigma_{ij})^2 \Theta(1 - r_{ij}/\sigma_{ij})$.
• Measurement of vibrational response via Fourier transforms of velocity autocorrelation functions to obtain $D(\omega^v)$.
• Comparison of $D(\omega^v)$ with $D(\omega^s)$ from displacement correlation matrices and $D(\omega^d)$ from the dynamical matrix at $T_0=0$.
• Identification of three regimes: iso-coordinated solids (ICS), hypo-coordinated solids (HCS), and hard-particle liquids (HPL) based on $\langle z\rangle/z_{\rm iso}$ and $D(\omega)$.
• Use of scaling collapse to analyze the frequency dependence of the mode softening, fitting $\omega_m^s \sim \alpha \sqrt{T_0} (\Delta\phi)^\nu$ with $\nu \approx 0.25$ above jamming.
• Estimation of thermal shift in jamming point $\Delta\phi_s(T_0)$ via effective diameter scaling, yielding $\Delta\phi_s \sim \sqrt{T_0}$.

Experimental results

Research questions

• RQ1 How does the vibrational response of jammed solids deviate from harmonic behavior as a function of $T_0$ and $\Delta\phi$?
• RQ2 What determines the transition between harmonic and nonharmonic vibrational spectra in systems with fluctuating contact networks?
• RQ3 Why do different measurement methods yield different $D(\omega)$ in the hypo-coordinated regime?
• RQ4 How do thermal fluctuations affect the jamming transition and the stability of the contact network?
• RQ5 Can the nonharmonic vibrational response in HCS be collapsed into a universal scaling form?

Key findings

• Iso-coordinated solids (ICS) exist for $\Delta\phi > \Delta\phi_c(T_0) \sim N^{0.85} \sqrt{T_0}/A$, where $\langle z\rangle = z_{\rm iso}$ and the vibrational response is harmonic with sharp peaks at $\omega_k^d$.
• Hypo-coordinated solids (HCS) occur in the region $\Delta\phi > \Delta\phi_{-}^*(T_0)$, $\Delta\phi < \Delta\phi_{+}^*(T_0)$, and $\Delta\phi > \Delta\phi_{cb}(T_0)$, with $\langle z\rangle/z_{\rm iso} \approx 0.5$, and exhibit strongly nonharmonic vibrational response.
• In HCS, $D(\omega^v)$ and $D(\omega^s)$ differ significantly: $D(\omega^v)$ has a stronger low-frequency peak and opposite curvature at high frequencies compared to $D(\omega^s)$.
• For $\Delta\phi > \Delta\phi_{cb}(T_0)$ and $\Delta\phi < \Delta\phi_{-}^*(T_0)$, the system behaves as a hard-particle liquid (HPL), with $\langle z\rangle/z_{\rm iso} \sim 0$ and $D(\omega)$ resembling that of a liquid.
• The generalized cage size $l_c$ scales as $\Delta\phi$ below jamming and $\sim (\Delta\phi)^\lambda$ with $\lambda \gtrsim 2$ above jamming.
• The mode softening frequency $\omega_m^s$ collapses under scaling $\omega_m^s \sim \alpha \sqrt{T_0} (\Delta\phi)^\nu$, with $\nu \approx 0.25$ above jamming and $\nu = 1$ below, indicating distinct scaling regimes.