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Register a new userRigidity-based approach to the boson peak in amorphous solids: from sphere packing to amorphous silica
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Matthieu Wyart
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Glasses have a large excess of low-frequency vibrational modes in comparison with continuous elastic body, the so-called Boson Peak, which appears to correlate with several crucial properties of glasses, such as transport or fragility. I review recent results showing that the Boson Peak is a necessary consequence of the weak connectivity of the solid. I explain why in assemblies repulsive spheres the boson peak shifts up to zero frequency as the pressure is lowered toward the jamming threshold, and derive the corresponding exponent. I show how these ideas capture the main low-frequency features of the vibrational spectrum of amorphous silica. These results extend arguments of Phillips on the presence of floppy modes in under-constrained covalent networks to glasses where the covalent network is rigid, or when interactions are purely radial.
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We present a random matrix approach to study general vibrational properties of stable amorphous solids with translational invariance using the correlated Wishart ensemble. Within this approach, both analytical and numerical methods can be applied. Using the random matrix theory, we found the analytical form of the vibrational density of states and the dynamical structure factor. We demonstrate the presence of the Ioffe-Regel crossover between low-frequency propagating phonons and diffusons at higher frequencies. The reduced vibrational density of states shows the boson peak, which frequency is close to the Ioffe-Regel crossover. We also present a simple numerical random matrix model with finite interaction radius, which properties rapidly converges to the analytical results with increasing the interaction radius. For fine interaction radius, the numerical model demonstrates the presence of the quasilocalized vibrations with a power-law low-frequency density of states.
The origin of boson peak -- an excess of density of states over Debyes model in glassy solids -- is still under intense debate, among which some theories and experiments suggest that boson peak is related to van-Hove singularity. Here we show that boson peak and van-Hove singularity are well separated identities, by measuring the vibrational density of states of a two-dimensional granular system, where packings are tuned gradually from a crystalline, to polycrystals, and to an amorphous material. We observe a coexistence of well separated boson peak and van-Hove singularities in polycrystals, in which the van-Hove singularities gradually shift to higher frequency values while broadening their shapes and eventually disappear completely when the structural disorder $eta$ becomes sufficiently high. By analyzing firstly the strongly disordered system ($eta=1$) and the disordered granular crystals ($eta=0$), and then systems of intermediate disorder with $eta$ in between, we find that boson peak is associated with spatially uncorrelated random flucutations of shear modulus $delta G/langle G rangle$ whereas the smearing of van-Hove singularities is associated with spatially correlated fluctuations of shear modulus $delta G/langle G rangle$.
We study the structural origin of the Bauschinger effect by accessing numerically the local plastic thresholds in the steady state flow of a two-dimensional model glass under athermal quasistatic deformation. More specifically, we compute the local residual strength, $Deltatau^{c}$, for arbitrary loading orientations and find that plastic deformation generically induces material polarization, i.e., a forward-backward asymmetry in the $Deltatau^{c}$ distribution. In steady plastic flow, local packings are on average closer to forward (rather than backward) instabilities, due to the stress-induced bias of barriers. However, presumably due to mechanical noise, a significant fraction of zones lie close to reverse (backward) yielding, as the distribution of $Deltatau^{c}$ for reverse shearing extends quasilinearly down to zero local residual strength. By constructing an elementary model of the early plastic response, we then show that unloading causes reverse plasticity of a growing amplitude, i.e., reverse softening, while it shifts away forward-yielding barriers. This result in an inversion of polarization in the low-$Deltatau^{c}$ region and, consequently, in the Bauschinger effect. This scenario is quite generic, which explains the pervasiveness of the effect.
The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of elasticity from local mechanical constraints offers a new paradigm for systems with no broken symmetry, analogous to emergent gauge theories of quantum spin liquids. Specifically, our $U(1)$ rank-2 symmetric tensor gauge theory of elasticity translates to the electromagnetism of fractonic phases of matter with the stress mapped to electric displacement and forces to vector charges. We corroborate our theoretical results with numerical simulations of soft frictionless disks in both two and three dimensions, and experiments on frictional disks in two dimensions. We also present experimental evidence indicating that force chains in granular media are sub-dimensional excitations of amorphous elasticity similar to fractons.
While crystalline two-dimensional materials have become an experimental reality during the past few years, an amorphous 2-D material has not been reported before. Here, using electron irradiation we create an sp2-hybridized one-atom-thick flat carbon membrane with a random arrangement of polygons, including four-membered carbon rings. We show how the transformation occurs step-by-step by nucleation and growth of low-energy multi-vacancy structures constructed of rotated hexagons and other polygons. Our observations, along with first-principles calculations, provide new insights to the bonding behavior of carbon and dynamics of defects in graphene. The created domains possess a band gap, which may open new possibilities for engineering graphene-based electronic devices.
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