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Schirmacher W, Paoluzzi M, Mocanu FC, Khomenko D, Szamel G, Zamponi F, Ruocco G. The nature of non-phononic excitations in disordered systems. Nat Commun 2024; 15:3107. [PMID: 38600083 PMCID: PMC11258284 DOI: 10.1038/s41467-024-46981-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2023] [Accepted: 03/18/2024] [Indexed: 04/12/2024] Open
Abstract
The frequency scaling exponent of low-frequency excitations in microscopically small glasses, which do not allow for the existence of waves (phonons), has been in the focus of the recent literature. The density of states g(ω) of these modes obeys an ωs scaling, where the exponent s, ranging between 2 and 5, depends on the quenching protocol. The orgin of these findings remains controversal. Here we show, using heterogeneous-elasticity theory, that in a marginally-stable glass sample g(ω) follows a Debye-like scaling (s = 2), and the associated excitations (type-I) are of random-matrix type. Further, using a generalisation of the theory, we demonstrate that in more stable samples, other, (type-II) excitations prevail, which are non-irrotational oscillations, associated with local frozen-in stresses. The corresponding frequency scaling exponent s is governed by the statistics of small values of the stresses and, therefore, depends on the details of the interaction potential.
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Affiliation(s)
- Walter Schirmacher
- Institut für Physik, Staudinger Weg 7, Universität Mainz, D-55099, Mainz, Germany.
- Center for Life Nano Science @Sapienza, Istituto Italiano di Tecnologia, 291 Viale Regina Elena, I-00161, Roma, Italy.
| | - Matteo Paoluzzi
- Istituto per le Applicazioni del Calcolo del Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131, Napoli, NA, Italy
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Carrer de Martí i Franquès 1, 08028, Barcelona, Spain
- Dipartimento di Fisica, Universita' di Roma "La Sapienza", P'le Aldo Moro 5, I-00185, Roma, Italy
| | - Felix Cosmin Mocanu
- Dept. of Materials, Univ. of Oxford, Parks Road, Oxford, OX13PH, UK
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France
| | - Dmytro Khomenko
- Dipartimento di Fisica, Universita' di Roma "La Sapienza", P'le Aldo Moro 5, I-00185, Roma, Italy
| | - Grzegorz Szamel
- Dept. of Chemistry, Colorado State University, Fort Collins, CO, 80523, USA
| | - Francesco Zamponi
- Dipartimento di Fisica, Universita' di Roma "La Sapienza", P'le Aldo Moro 5, I-00185, Roma, Italy
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France
| | - Giancarlo Ruocco
- Center for Life Nano Science @Sapienza, Istituto Italiano di Tecnologia, 291 Viale Regina Elena, I-00161, Roma, Italy.
- Dipartimento di Fisica, Universita' di Roma "La Sapienza", P'le Aldo Moro 5, I-00185, Roma, Italy.
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Xu D, Zhang S, Tong H, Wang L, Xu N. Low-frequency vibrational density of states of ordinary and ultra-stable glasses. Nat Commun 2024; 15:1424. [PMID: 38365816 PMCID: PMC11258317 DOI: 10.1038/s41467-024-45671-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2023] [Accepted: 02/01/2024] [Indexed: 02/18/2024] Open
Abstract
A remarkable feature of disordered solids distinct from crystals is the violation of the Debye scaling law of the low-frequency vibrational density of states. Because the low-frequency vibration is responsible for many properties of solids, it is crucial to elucidate it for disordered solids. Numerous recent studies have suggested power-law scalings of the low-frequency vibrational density of states, but the scaling exponent is currently under intensive debate. Here, by classifying disordered solids into stable and unstable ones, we find two distinct and robust scaling exponents for non-phononic modes at low frequencies. Using the competition of these two scalings, we clarify the variation of the scaling exponent and hence reconcile the debate. Via the study of both ordinary and ultra-stable glasses, our work reveals a comprehensive picture of the low-frequency vibration of disordered solids and sheds light on the low-frequency vibrational features of ultra-stable glasses on approaching the ideal glass.
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Affiliation(s)
- Ding Xu
- Hefei National Research Center for Physical Sciences at the Microscale and CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei, 230026, P. R. China
- Department of Physics, University of Science and Technology of China, Hefei, 230026, P. R. China
| | - Shiyun Zhang
- Hefei National Research Center for Physical Sciences at the Microscale and CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei, 230026, P. R. China
- Department of Physics, University of Science and Technology of China, Hefei, 230026, P. R. China
| | - Hua Tong
- Department of Physics, University of Science and Technology of China, Hefei, 230026, P. R. China
| | - Lijin Wang
- School of Physics and Optoelectronic Engineering, Information Materials and Intelligent Sensing Laboratory of Anhui Province, Anhui University, Hefei, 230601, P. R. China.
| | - Ning Xu
- Hefei National Research Center for Physical Sciences at the Microscale and CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei, 230026, P. R. China.
- Department of Physics, University of Science and Technology of China, Hefei, 230026, P. R. China.
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Wang L, Fu L, Nie Y. Density of states below the first sound mode in 3D glasses. J Chem Phys 2022; 157:074502. [DOI: 10.1063/5.0102081] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
Glasses feature universally low-frequency excess vibrational modes beyond Debye prediction, which could help rationalize, e.g., the glasses’ unusual temperature dependence of thermal properties compared to crystalline solids. The way the density of states of these low-frequency excess modes D( ω) depends on the frequency ω has been debated for decades. Recent simulation studies of 3D glasses suggest that D( ω) scales universally with ω4 in a low-frequency regime below the first sound mode. However, no simulation study has ever probed as low frequencies as possible to test directly whether this quartic law could work all the way to extremely low frequencies. Here, we calculated D( ω) below the first sound mode in 3D glasses over a wide range of frequencies. We find D( ω) scales with ω β with β < 4 at very low frequencies examined, while the ω4 law works only in a limited intermediate-frequency regime in some glasses. Moreover, our further analysis suggests our observation does not depend on glass models or glass stabilities examined. The ω4 law of D( ω) below the first sound mode is dominant in current simulation studies of 3D glasses, and our direct observation of the breakdown of the quartic law at very low frequencies thus leaves an open but important question that may attract more future numerical and theoretical studies.
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Affiliation(s)
- Lijin Wang
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Licun Fu
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Yunhuan Nie
- Beijing Computational Science Research Center, Beijing 100193, China
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V Krishnan V, Ramola K, Karmakar S. Universal non-Debye low-frequency vibrations in sheared amorphous solids. SOFT MATTER 2022; 18:3395-3402. [PMID: 35416828 DOI: 10.1039/d2sm00218c] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We study energy minimised configurations of amorphous solids with a simple shear degree of freedom. We show that the low-frequency regime of the vibrational density of states of structural glass formers is crucially sensitive to the macroscopic stress of the sampled configurations. In both two and three dimensions, shear-stabilised configurations display a D(ωmin) ∼ ω5min regime, as opposed to the ω4min regime observed under unstrained conditions. In order to isolate the source of these deviations from crystalline behaviour, we also study configurations of two dimensional, strained amorphous solids close to a plastic event. We show that the minimum eigenvalue distribution at a strain 'γ' near the plastic event occurring at 'γP' assumes a universal form that displays a collapse when scaled by , and with the number of particles as N-0.22. Notably, at low frequencies, this scaled distribution displays a robust D(ωmin) ∼ ω6min power-law regime, which survives in the large N limit. Finally, we probe the properties of these configurations through a characterisation of the second and third eigenvalues of the Hessian matrix near a plastic event.
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Affiliation(s)
- Vishnu V Krishnan
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500046, India.
| | - Kabir Ramola
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500046, India.
| | - Smarajit Karmakar
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500046, India.
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Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media. Proc Natl Acad Sci U S A 2022; 119:2119288119. [PMID: 35169078 PMCID: PMC8872781 DOI: 10.1073/pnas.2119288119] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 01/18/2022] [Indexed: 11/18/2022] Open
Abstract
We present a theory, based on an unstable elasticity model with thermally fluctuating elastic constants, which explains the salient features of the instantaneous normal mode (INM) spectrum of a numerically simulated liquid over a large range of temperatures. The INM spectrum of a liquid is obtained in a molecular dynamics (MD) simulation by considering the harmonic eigenvalue spectrum of the potential energy, taken at a given instant (snapshot). Because the INM spectrum records the curvatures of the fluctuating potential landscape of the liquid, an understanding of this spectrum may pave the way toward understanding the liquid to glass transformation process. We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.
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Lerner E, Bouchbinder E. Low-energy quasilocalized excitations in structural glasses. J Chem Phys 2021; 155:200901. [PMID: 34852497 DOI: 10.1063/5.0069477] [Citation(s) in RCA: 23] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
Abstract
Glassy solids exhibit a wide variety of generic thermomechanical properties, ranging from universal anomalous specific heat at cryogenic temperatures to nonlinear plastic yielding and failure under external driving forces, which qualitatively differ from their crystalline counterparts. For a long time, it has been believed that many of these properties are intimately related to nonphononic, low-energy quasilocalized excitations (QLEs) in glasses. Indeed, recent computer simulations have conclusively revealed that the self-organization of glasses during vitrification upon cooling from a melt leads to the emergence of such QLEs. In this Perspective, we review developments over the past three decades toward understanding the emergence of QLEs in structural glasses and the degree of universality in their statistical and structural properties. We discuss the challenges and difficulties that hindered progress in achieving these goals and review the frameworks put forward to overcome them. We conclude with an outlook on future research directions and open questions.
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Affiliation(s)
- Edan Lerner
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - Eran Bouchbinder
- Chemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel
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Qiu Q, Bao JD. Debye Brownian oscillator and Debye-type noise: A series solution versus Monte Carlo simulation. Phys Rev E 2021; 104:014114. [PMID: 34412352 DOI: 10.1103/physreve.104.014114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2021] [Accepted: 06/21/2021] [Indexed: 11/07/2022]
Abstract
For the Debye Brownian oscillator, we present a series solution to the generalized Langevin equation describing the motion of a particle. The external potential is considered to be a harmonic potential and the spectral density of driven noise is a hard cutoff at high finite frequencies. The results are in agreement with both numerical calculations and Monte Carlo simulations. We demonstrate abnormal weak ergodic breaking; specifically, the long-time average of the observable vanishes but the corresponding ensemble average continues to oscillate with time. This Debye Brownian oscillator does not arrive at an equilibrium state and undergoes underdamped-like motion for any model parameter. Nevertheless, ergodic behavior and equilibrium can be recovered concurrently using a strong bound potential. We give an understanding of the behavior as being the consequence of discrete breather modes in the lattices similar to the formation of an additional periodic signal. Furthermore, we compare the results calculated by cutting off separately the spectral density and the correlation function of colored noise.
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Affiliation(s)
- Qian Qiu
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
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Lerner E. Finite-size effects in the nonphononic density of states in computer glasses. Phys Rev E 2020; 101:032120. [PMID: 32289945 DOI: 10.1103/physreve.101.032120] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 02/27/2020] [Indexed: 06/11/2023]
Abstract
The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency ω in structural glasses, D(ω), was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, D(ω) increases from zero frequency as ω^{4}, independent of spatial dimension and of microscopic details. However, it has been shown [Lerner and Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)2470-004510.1103/PhysRevE.96.020104] that the preparation protocol employed to create glassy samples may affect the form of their resulting D(ω): glassy samples rapidly quenched from high-temperature liquid states were shown to feature D(ω)∼ω^{β} with β<4, presumably limiting the degree of universality of the ω^{4} law. Here we show that exponents β<4 are seen only in small glassy samples quenched from high-temperature liquid states-whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations-while larger glassy samples made with the same protocol feature the universal ω^{4} law. Our results demonstrate that observations of β<4 in the nonphononic density of states stem from finite-size effects, and we thus conclude that the ω^{4} law should be featured by any sufficiently large glass quenched from a melt.
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Affiliation(s)
- Edan Lerner
- Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam, the Netherlands
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