June 29th (Fri.), 2018 Yukawa Institute for Theoretical Physics, Kyoto University, Japan Rheology of disordered particles — suspensions, glassy and granular materials 10:15-11:05, 40mins. talk and 10mins. discussion
Vibrational properties and phonon transport of amorphous solids
Hideyuki Mizuno, Hayato Shiba, Atsushi Ikeda
The University of Tokyo, Tohoku University Background 2/27 Vibrational properties of amorphous solids
Crystals (lattice structure) Amorphous solids (amorphous structure)
Molecules vibrate around lattice structure Molecules vibrate around amorphous structure
Tanguy et al., EPL 2010 Vibrational modes are phonons Some modes are localized -> These modes are non-phonons Background 3/27 Vibrational properties of amorphous solids
Crystals (lattice structure) Amorphous solids (amorphous structure)
Vibrational density of states (one component Lennard-Jones system)
Transverse Longitudinal Background 4/27 Excess low-ω vibrational modes in amorphous solids Excess over Debye-theory prediction (Boson peak) is observed in many glasses (amorphous solids) DOS Crystal
Crystal
Yamamuro et al., JCP 1996
Buchenau et al., PRL 1984 Background 5/27 Low-T thermal properties of amorphous solids Specific heat Thermal conductivity
SiO2 Crystal
SiO2 Glass
SiO2 Crystal SiO2 Glass
Zeller and Pohl, PRB, 1971 Background 6/27 Extension of Debye theory: Elastic heterogeneities
Crystals Debye theory Amorphous solids: Extended Debye theory
Ø Homogeneous elastic media Ø Heterogeneous elastic media Ø Elastic mechanics predict phonons or Ø Vibrational modes are deformed to non- acoustic waves as vibrational modes phonon modes by elastic heterogeneities Ø This explains heat capacity of crystals Ø This predicts heat capacity larger than T cubed behavior Debye prediction ⇒ Thermal properties of amorphous solids (Boson peak)
Heterogeneous modulus
Schirmacher et al., EPL 2006, PRL 2007 Background 7/27 Multi-scale structure of amorphous solids
Molecules Local elastic modulus Homogeneous media?
Micro-scale Meso-scale Macro-scale At meso-scale, local elastic modulus fluctuates in the space. -> Elastic heterogeneities control thermal properties of amorphous solids Mizuno, Mossa, Barrat, EPL 2013, PNAS 2014, PRB 2016
At macro-scale (continuum limit), amorphous solids behave as homogeneous elastic media? Mizuno, Shiba, Ikeda, PNAS 2017 Background 8/27 Low-T thermal properties of amorphous solids Specific heat Thermal conductivity
SiO2 Crystal
SiO2 Glass
SiO2 Crystal SiO2 Glass
Zeller and Pohl, PRB, 1971 9/27 Contents
Questions l What is nature of low-frequency vibrations of amorphous solids? l What laws do they obey?
We perform molecular-dynamics simulations on a simple model amorphous solid ü Molecules interact through harmonic pair potential ü System size : up to 4,096,000 ü Periodic boundary condition in all the directions Background 10/27 Low-T specific heat of amorphous solids Specific heat Within the harmonic approximation
vDOS
Low-temperature thermal properties are controlled by low-frequency vibrations:
SiO2 Crystal SiO2 Glass
Zeller and Pohl, PRB, 1971 Background 11/27 Two level system / Soft potential model Two level system
Debye theory -> Phonons
Additional, non-phonon modes exist in amorphous solids -> Two level system, presumably considered as localized motions of particles -> C T is explained Anderson, et al., Phil. Mag., 1972
Density of two level system with very low energy barriers 1
1 κ = c(ω)v(ω)ℓ(ω)g(ω)dω 1 ! = v L c(ω)g(ω)dω = v LC T 3 0 0 ∝ ! κ = c(ω)v(ω)ℓ(ω)g(ω)dω = v c(ω)ℓ(ω)g(ω)dω κ = c(ω)v(ω)ℓ(ω)g(ω)dω ! ! ! 3 = v0L c2(ω)g(ω)dω = v0LC =Tv L c(ω)g(ω)dω = v LC T 3 !ω !ω ∝ 0 0 ∝ c(ω) 3kBρ ! exp , ! ≃ k T −k T " B # = v" c(Bω)ℓ(#ω)g(ω)dω = v c(ω)ℓ(ω)g(ω)dω 1 !ℓ2(ω)=L ! !ω !ω 2 c(ω) 3k 1ρ 1 1exp , !ω !ω ℓ = ℓ−B + ℓ− − ℓ c(ω) 3kBρ exp , ≃ anh kBdisT dis−kBT ≃ k T −k T " # ≃" # " B # " B # $v = ω/q % Γ = v/ℓℓ(ω)=L κ = c(ω)vℓ(ω)=)ℓ(ωL)g(ω)dω 1 1 1 − 1 ! 1 1 v(ωℓ)== vℓ0anh− =+ ℓdis−M/ρ ℓdis ℓ = ℓ− + ℓ− − ℓ ≃ anh dis ≃ dis 3 v = ω/q 4Γ = v/ℓ = v0L c(ω)g(ω)dω = v0LC T 12/27 ℓ(ω)=$ ℓR& ω%− Vibrational modes$v = ω/q analysis% Γ = v/ℓ ∝ ∝ ! v(ω)=v0 =1 M/ρ ℓt ω− v(ω)== vv0 =c(ω)M/ℓ(ωρ)g(ω)dω Spring∝-mass model 4 4 ℓ(ω)=ℓR& ω− ℓ(ω)=! ℓR& ω− k =1, 2,...,3N ∝ 2 ∝ 1 !ω 1 !ω ℓt ω−c(ω) 3k ρ ℓexpt ω− , u = r r0 ∝ B ∝ − ≃ kBT −kBT 2 k =1, 2,...,3N " k =1# , 2,...,"3N # ∂ Φ ℓ(ω)=L u¨ = ü Linearizedu =uequationr r0 of motion u = r r0 − − 1 − ∂r∂r 0 2 1 2 1 " # ∂ Φ ℓ = ℓ− ∂+Φℓ− − ℓdis u¨ = u u¨ = anh dis u ≃ − ∂r∂r − ∂r∂r " #0 $v "= ω/q #%0 Γ = v/ℓ 2 ∂ Φ 2 ü Diagonalize Hessian matrix : k v(ω)=k v0 =k M/ρ e e = ω (3N 3N matrix) ∂r∂r 0 4 " # ℓ(ω)=ℓR& ω− ∝ ü Vibrational modes (normal modes, eigen modes) 1 ℓ ω− Eigen frequencies: t ∝ k =1, 2,...,3N Eigen vectors: 13/27 Vibrational modes in Lennard-Jones crystals
Crystals (lattice structure) Eigen vectors: Transverse phonon
Vibrational density of states
Transverse Longitudinal Longitudinal phonon 14/27 Vibrational modes in simple amorphous solid
Silbert, Liu, Nagel, PRE 2009 Ø A, Debye-like regime : Elastic-wave-like modes Ø B, Plateau regime : Disordered extended modes Ø C, High frequency regime : Highly localized modes
Elastic-wave-like mode Disordered extended mode Highly localized mode 15/27 Vibrational density of states
Debye level
ü We observe the boson peak around ω = 0.1. ü Reduced vDOS goes towards the Debye level, but does not converge in our frequency regime. Characterize each vibrational mode: 16/27 Phonon order parameter Phonons in isotropic medium under periodic boundary:
: wave vector
Expansion by phonon modes (Fourier expansion) of vibrational mode k:
Phonon order parameter: : Phonon
: Non-phonon Characterize each vibrational mode: 17/27 Participation ratio
Participation ratio of vibrational mode k: Mazzacurati, Ruocco, Sampoli, EPL 1996
1. Only one particle vibrates:
2. All the particles vibrate equivalently:
Fraction Pk of total particles participate in the vibrational mode k 18/27 Vibrational modes
At ω > ω* : Non-phonon, extended vibrational nature -> Disordered extended mode
At ω ωBP < ω* : Hybrid character of phonon and disordered mode
At ω < ωBP : Mixture of phonon mode and soft localized mode -> Consistent with the idea of two-level system 19/27 Close up to the low-frequency regime
N = 2,048,000, L = 114 Phonon mode
Localized mode
ü Non-phonon, localized modes sit between different phonon energy levels 20/27 Vibrational density of states 21/27 Vibrational density of states
Debye law Non-Debye law of localized modes 2
These results are clearly different from those of the har- B. Simulations monic potential [16]. In contrast, a recent study using the Kob-Andersen binary LJ system exhibited the soft We performed molecular dynamics simulations in d = localized modes following the universal non-Debye scal- 4 3dimensionalcubicboxesunderperiodicboundarycon- ing law g(ω) ω [18]. These modes are distinct from ditions. Particles are monodisperse and interact via the phonons and the∝ results of the harmonic potential agree LJ potential with a mass m and a truncation length rc. with this results [16]. Hence, no one has given the con- We used the two types of the potentials, LJ0 and LJ1 as clusive results about the continuum limit of LJ glasses. mentioned above. In this paper, we will measure length, To resolve this problem, we performed exhaustive anal- energy, and mass scale in units of σ, ϵ,andm.The ysis of low-frequency vibrational modes of LJ glasses. density was fixed at ρ =0.997 for any cases and the sys- According to our results, the vibrational modes of LJ tem sizes are from N =8000toN =256000forLJ0 glasses are similar to those of the harmonic potential, and from N =8000toN =1024000forLJ1.Inor- the coexistence of extended phonon and localized non- der to get amorphous configurations, we used LAMMPS, phonon modes [16]. We also discovered the reason of the the package for molecular dynamics, and performed the seemingly contradicting results reported thus far. That simulations as follows. Firstly, we prepared configura- is truncation of the potential during calculation and we tions of the equilibrium normal liquid at T =2bymeans evaluate this effect by simple linear stability analysis. of the microcanonical ensemble. Then, we carried out the steepest descent to get stable glassy configurations (inherent structure). This protocol equals instantaneous quench from T =2andwechoseitasthesimpleststart- II. METHODS ing point to explore the very low frequency region.
A. LJ potential C. Analysis 22/27 4 We explored theVibrational low-frequency modes vibrational in Lennard properties-Jones glass of glasses interacting via the LJ potential, We calculated dynamical100 matrices of amorphous con-
0.003 −1 All modes figurations by the harmonic10 approximation and diago- 12 6 k −2 σ σ 0.0025 P > 10 P k < 10−2 nalize them to get their eigenvalues λk and eigenvectors φ(r)=4ϵ , 2 0.002 (1) −2 ω k k k k 10 / )
r − r 0.0015 e =(e1 , e2 , , e )[2].Here,k k =1, 2, ,dN and
ω N P ! $ (
g 0.001 ··· −3 ··· " # " # they areA0 sorted in ascending10 order of eigenvalues. Eigen- 0.0005 k k k k ω0 ωBP by molecular dynamics simulations. When we use short-0 vectors are normalized as−4 e e = e e 0=1. 10 i i ω ) ∝ 0.8 ω 10 ω & cate the potential at a fixed value r [19]. Namely, we ( ω0 c 0.6 g 00.20.40.60.81 k k ignore interactions with particles at larger distancesO 0.4 than 0.30.4 ω 1. DOS O FIG. 2. The plot of the phonon order parameter versus the rc [19]. However, when the potential is truncated,0. it2 has participation ratio for the systems of LJ1. We divide all 0 modes into three groups based on their frequency: ω < ω0, adiscontinuityatr = rc and this discontinuity may0 lead ω0 < ω < ωBP,andωBP < ω. 10 Given eigenfrequencies of dynamical matrices ωk = to errors [19]. Some methods are used to avoid such−1 er- 10 √λk,thevibrationalDOSisdefinedas − rors [19]. Firstly, we can truncate and shift the potential,10 2 k P −3 such that it vanishes at r = rc [19]. We call this10 type of dN−d − the LJ potentialMixture LJ0, of phonons and soft localized 10 4 1 − g(ω)= δ(ω ωk). (4) 10 5 0.002 modes is observed in LJ glass − dN d − 10 1 100 101 k=1 0−.0015 2 ' Shimada, Mizuno, Ikeda, PRE 2018 ω ω φ(r) φ(rc)(r − ( A0 We also use the reducedg DOS (rDOS) g(ω)/ω to see %0(r>rc) FIG. 1. The vibrational modes of amorphous solids interact- 0.0005 2 ing via LJ1. The panels aredi rDOS,fference the phonon from order the parame- Debye law g(ω)=A0ω for crystalline ter, and the participation ratio from above. From the phonon 0 ω0 ωBP order parameter and thesolids, participation where ratio, weA can0 seeis the called the Debye level and is written bifurcation of modes into extended phonon and localized non- 1 Another method is to shift a force, first derivative of the k phonon modes. The rDOSas of extended modes (P > 0.01) 0.8 seems to converge to the Debye level A0. The DOS of local- potential, as well as the potential itself [19]. we call itk 0.6 ized modes (P < 0.01) (inset to the second panel) is fitted k 4 O 1 3 LJ1, by ω ,asreportedinref.[16,18]. d 0.4 18πρ / 0 0.2 A = d , ωD = −3 −3 , (5) ωD 0 c +2c ′ 0 ( L T ) φ(r) φ(rc) (r rc)φ (rc)(r Questions l What is nature of low-frequency vibrations of amorphous solids? l What laws do they obey? ü Mixture of phonons and soft localized modes ü Phonons follow Debye law, while localized modes follow non-Debye scaling law (ω4 law) ü This seems consistent with TLS/SPM picture ü In the continuum limit, glasses do NOT behave as homogeneous elastic media, but rather they behave as elastic media with ``defects” Mizuno, Shiba, Ikeda, PNAS (2017) 24/27 Conclusion Molecules Local elastic modulus Homogeneous media with defects Micro-scale Meso-scale Macro-scale At macro-scale (continuum limit), amorphous solids are essentially different from normal solids (elastic media). ü プレスリリース (東大・東北大)、ガラスと通常の固体の本質的な違いを発見 Low-T specific heat 25/27 Can we explain linear-T term by localized modes? Specific heat Continuum limit Phonon modes Localized modes + SiO2 Crystal SiO2 Glass Low-T specific heat 26/27 Can we explain linear-T term by localized modes? Localized modes Two level system l Do two-level tunneling transitions happen in the localized modes? l What is role of non-Debye law? Anderson, et al., Phil. Mag., 1972 4 4 FIG. 3. Comparison of the vDOS between numerical value and prediction from phonon transport properties. Plot of the d 1 reduced vDOS g(ω)/ω − as a function of ω. a,3Dmodelsystem(d =3). b,2Dmodelsystem(27/27 d = 2). The symbols present the numerical value of vDOSLocalized provided mode in Ref.is [50]. key to The understand line shows amorphous the prediction solids calculated from cα(ω)andΓα(ω)(α = T,L) by Eq.FIG. (2). 3. The Comparison prediction of of theEq. (2) vDOS is composed between of numerical two terms, value one andrelated prediction to dispersion from curve phonon (term1, transportf1[cα(ω properties.)]) and the Plot of the d 1 k 2 otherreduced to damping vDOS rateg(ω (term2,)/ω − f2as[cα a(Localω function), elasticΓα( modulusω)]). of Forω. a the,3Dmodelsystem( 3DHomogeneous system media in witha, defectsgexd(ω=3).), theb vDOS,2Dmodelsystem( of extended modesd = (P 2).> The10− symbols), present is alsothe plotted numerical for comparison. value of vDOS The horizontal provided in line Ref. indicates [50]. The the Debye line shows value theA0 calculated prediction from calculated the macroscopic from cα(ω moduli.)andΓα(ω)(α = T,L) by Eq. (2). The prediction of Eq. (2) is composed of two terms, one related to dispersion curve (term1, f1[cα(ω)]) and the k 2 other to damping rate (term2, f2[cα(ω), Γα(ω)]). For the 3D system in a, gex(ω), the vDOS of extended modes (P > 10− ), work,is also for the plotted first for time comparison. to our knowledge, The horizontal offers line clear indicatesRef. the [13] Debye demonstrated value A0 thatcalculated the bulk from modulus the macroscopicK(ω) can moduli. understanding of these two phonon transportsMeso-scale based on Macronot-scale be approximated by frequency-independent value. the vibrational properties. We therefore can consis- We therefore argue that the similarity between trans- tently identify the crossoverDisordered frequencyextended betweenmodes these twoPhonon modesverseLocalized andmodes longitudinal transport properties depends on work, for the first time to our knowledge, offers clear Ref. [13] demonstrated that the bulk modulus K(ω) can regimes as the continuum limit, ω = ωex0, that has amorphous systems and preparation procedures. beenunderstanding never achieved of before. these two We phonon remark that transports previous based onIn+ ordernot to be explore approximated role of the soft by localized frequency-independent modes, we value. worksthe [9, vibrational 10, 13] studied properties. the phonon We transport therefore at finite can consis-next calculateWe therefore the vDOS argue within that the the phononic similarity approxi- between trans- temperatures,tently identify where the the crossover crossover frequency frequency between is located thesemation two (generalizedverse and longitudinal Debye model) transport [7, 8, 10, properties 11, 41, 43]. depends on aroundregimes the boson as the peak continuum regime, ω limit,ωBPω.Thisresult= ωex0, thatContinuumAs has suggested limitamorphous in Ref. systems [11], we can and formulate preparation the procedures. reduced ∼ d 1 suggestsbeen that never the achieved temperature before. can shift We the remark crossover that fre- previousvDOS g(ω)In/ω order− as: to explore role of the soft localized modes, we quency to the higher ω side. In addition, the crossover works [9, 10, 13] studied the phonon transport at finite nextg(ω) calculate the vDOS within the phononic approxi- behavior is consistent with the prediction of the mean- temperatures, where the crossover frequency is located mationd 1 = (generalizedf1[cα(ω)] + f Debye2[cα(ω), model)Γα(ω)], [7, 8, 10,(2) 11, 41, 43]. field theories [7, 8, 10, 11, 41, 43]. However, as we will ω − around the boson peak regime, ω ωBP.Thisresult As suggested in Ref. [11], we can formulate the reduced demonstrate below, the soft localized modes∼ play an im- which is composed of twod 1 terms: f1[cα(ω)] is related to suggests that the temperature can shift the crossover fre- vDOS g(ω)/ω − as: portant role in the phonon transport, that is beyond cur- the dispersion curve, and f2[cα(ω), Γα(ω)] is to the sound quency to the higher ω side. In addition, the crossover 2 rent mean-field theories. broadening. In thisg(ω formulation,) g(ω)/ω is predicted Itbehavior is worth to is mention consistent that with both the the transverse prediction (α of= theT ) mean-from the data of c (ω) and= fΓ[c(ω()ω provided)] + f [c in(ω Fig.), Γ 1a.(ω)], (2) αωd 1 1α α 2 α α andfield longitudinal theories (α [7,= 8,L) 10, waves 11, show 41, 43]. the similarHowever, trans- as weDetailed will formulations− of f1[cα(ω)] and f2[cα(ω), Γα(ω)] portdemonstrate properties with below, the the identical soft localized crossover modes frequency play anare im- givenwhich in the is Methods composed section of two (see terms: Eq. (8)f for1[c 3Dα(ω and)] is related to ωex0portant, that has role been in the also phonon observed transport, in Ref. [9, that 11]. is beyond The cur-Eq. (9) forthe 2D). dispersion curve, and f [c (ω), Γ (ω)] is to the sound 2 α α 2 mean-fieldrent mean-field theory [7, theories. 8, 10, 11] assumes that the shear Figurebroadening. 3a compares theIn this reduced formulation, vDOS g(ωg)/(ωω)/pre-ω2 is predicted modulus heterogeneity is crucial compared to the bulk dicted by Eq. (2) and the numerical value presented in It is worth to mention that both the transverse (α = T ) from the data of c (ω) and Γ (ω) provided in Fig. 1a. modulus heterogeneity, that is true for the present amor- Ref. [50]. Although the phononicα approximationα is con- phousand system longitudinal [66]. In (α this= framework,L) waves show the shear the similar mod- trans-sidered toDetailed be valid formulations upto the Ioffe-Regel of f1[c limit,α(ω)]ω andωf2[cα(ω), Γα(ω)] ! IR ≈ ulusport heterogeneity properties induces with the the anomalous identical crossoverbehaviors for frequencyωBP, it seemsare given to produce in the Methods the numerical section value (see upto Eq. the (8) for 3D and bothωex0 the, transverse that has been and longitudinal also observed waves. in Ref. To [9, check 11]. Thehigher frequencyEq. (9) for regime, 2D). ω ! ω . In particular, it well thismean-field validity, we theory calculate [7, the 8, 10, longitudinal 11] assumes sound that speed the shearcaptures theFigure boson 3a peak compares (excess∗ vibrational the reduced modes vDOS overg(ω)/ω2 pre- 2 as modulusK/ρ +4c heterogeneityT (ω) /3 that excludes is crucial effects compared of the bulkto the bulkthe Debyedicted prediction) by Eq. around (2) andωBP. the Previous numerical works value [9, 63] presented in modulus heterogeneity. The inset to the top of Fig. 1a explain that the boson peak originates from the deforma- !modulus heterogeneity, that is true for the present amor- Ref. [50]. Although the phononic approximation is con- demonstrates that this value is close to the true longitu- tion of dispersion curve, i.e., from the first term f1[cα(ω)] phous system [66]. In this framework, the shear mod- sidered to be valid upto the Ioffe-Regel limit, ω ! ωIR dinal speed cL(ω), that supports this scenario. However, in Eq. (2). However, our result shows that f1[cα(ω)] ex- ≈ ulus heterogeneity induces the anomalous behaviors for ωBP, it seems to produce the numerical value upto the we see that although the difference is small, the minimum hibits only a tiny peak around ωBP and can not explain both the transverse and longitudinal waves.2 To check higher frequency regime, ω ! ω . In particular, it well dip of cL(ω) is larger than that of K/ρ +4cT (ω) /3. the boson peak. We therefore need to consider∗ the sec- Thisthis suggests validity, that we the calculate bulk modulus the longitudinal heterogeneity sound also speedond termcapturesf2[cα(ω), theΓα(ω boson)] coming peak from (excess the sound vibrational broad- modes over 2 ! makesas someK/ρ eff+4ectscT on(ω) the/3 longitudinal that excludes waves. effects Indeed, of the bulkening, asthe has Debye been pointed prediction) out in around Ref. [11].ωBP. Previous works [9, 63] modulus heterogeneity. The inset to the top of Fig. 1a explain that the boson peak originates from the deforma- ! demonstrates that this value is close to the true longitu- tion of dispersion curve, i.e., from the first term f1[cα(ω)] dinal speed cL(ω), that supports this scenario. However, in Eq. (2). However, our result shows that f1[cα(ω)] ex- we see that although the difference is small, the minimum hibits only a tiny peak around ωBP and can not explain 2 dip of cL(ω) is larger than that of K/ρ +4cT (ω) /3. the boson peak. We therefore need to consider the sec- This suggests that the bulk modulus heterogeneity also ond term f2[cα(ω), Γα(ω)] coming from the sound broad- makes some effects on the longitudinal! waves. Indeed, ening, as has been pointed out in Ref. [11].