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تسجيل مستخدم جديدResponse to Comment on Repulsive contact interactions make jammed particulate systems inherently nonharmonic
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This is a response to the comment on our manuscript Repulsive contact interactions make jammed particulate systems inherently nonharmonic (Physical Review Letters 107 (2011) 078301) by C. P. Goodrich, A. J. Liu, and S. R. Nagel.
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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. Packings of disks with purely repu
lsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential 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 then studied deviations in the constant volume specific heat $C_V$ and deviations of the average disk positions $Delta r$ from their $T=0$ values in the temperature regime $T_{cb} < T < T_{r}$, 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 $Delta {overline C}_V^0/{overline C}_V^0$ relative to the zero temperature value ${overline C}_V^0$ can grow rapidly above $T_{cb}$, however, the deviation $Delta {overline C}_V^0/{overline C}_V^0$ 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 $Delta r^{ss}/Delta r^{ds}$ for single- and double-sided linear and nonlinear spring interactions. We find that $Delta r^{ss}/Delta r^{ds} > 100$ for linear spring interactions and is independent of system size.
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 $Delta phi=phi - phi_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Delta phi > Delta phi_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{rm iso}$) 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 $omega_k^d$ 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 $Delta phi > Delta phi^*_-(T_0)$, $Delta phi < Delta phi^*_+(T_0)$, and $Delta phi > Delta phi_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> approx z_{rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {it i.e} the density of vibrational modes $D(omega)$ is not a collection of sharp peaks at $omega_k^d$, and its precise form depends on the measurement method. For $Delta phi > Delta phi_{cb}(T_0)$ and $Delta phi < Delta phi^*_{-}(T_0)$, the system behaves as a hard-particle liquid.
There are deep, but hidden, geometric structures within jammed systems, associated with hidden symmetries. These can be revealed by repeated transformations under which these structures lead to fixed points. These geometric structures can be found in
the Voronoi tesselation of space defined by the packing. In this paper we examine two iterative processes: maximum inscribed sphere (MIS) inversion and a real-space coarsening scheme. Under repeated iterations of the MIS inversion process we find invariant systems in which every particle is equal to the maximum inscribed sphere within its Voronoi cell. Using a real-space coarsening scheme we reveal behavior in geometric order parameters which is length-scale invariant.
By calculating the linear response of packings of soft frictionless discs to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and non-affine deformations as a function of the distance to jam
ming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale $ell^*$ up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as $1/Delta z$, where $Delta z$ is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the non-affine displacements of the particles using the emph{displacement angle distribution}, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.
We compare the elastic response of spring networks whose contact geometry is derived from real packings of frictionless discs, to networks obtained by randomly cutting bonds in a highly connected network derived from a well-compressed packing. We fin
d that the shear response of packing-derived networks, and both the shear and compression response of randomly cut networks, are all similar: the elastic moduli vanish linearly near jamming, and distributions characterizing the local geometry of the response scale with distance to jamming. Compression of packing-derived networks is exceptional: the elastic modulus remains constant and the geometrical distributions do not exhibit simple scaling. We conclude that the compression response of jammed packings is anomalous, rather than the shear response.
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