Downloaded by guest on October 2, 2021 mrhu solids amorphous local- law. soft scaling and non-Debye universal law another Debye follow the mix- that follow a modes that to ized modes converge pre- phonon modes law and vibrational of frequency, scaling the ture low limit, the at continuum violated that the is find in theory mean-field We the analysis solid. by mode amorphous dicted vibrational model large-scale Herein, a obey. extremely of vibrations an such that perform laws we the determine amorphous of and vibrations solids directly low-frequency the to of necessary nature absolutely the observe is devel- Considering it law. arguments, been scaling theoretical non-Debye recently these universal the have predict so to theories oped and mean-field vibrations, Microscopic localized on. soft phenomeno- which heterogeneities, of proposed, elastic been number are have assume A situation modes law. this vibrational for Debye explanations which logical the in elas- follow limit, homogeneous that continuum a phonons the as in behave body to all tic because expected counterintuitive are is materials situation solid thermal This those solids. from low-temperature 2017) crystalline different 1, markedly of June are and review solids for amorphous (received of vibrational 2017 properties 25, September low-frequency approved and PA, The Philadelphia, Pennsylvania, of University Liu, J. Andrea by Edited Japan 980-8577, Sendai a www.pnas.org/cgi/doi/10.1073/pnas.1709015114 2). follows (1, vibrational (vDOS) phonons the states that of of predict density terms theory follow phonon-gas in and solids explained theory Debye are Crystalline which physics. laws, matter universal condensed in mystery T law non-Debye Mizuno Hideyuki solids amorphous properties of vibrational the of limit Continuum esmdl(5.Ti hoyasmssf oaie vibrations localized soft assumes sys- theory two-level This tunneling famous (25). the model of tems extension an is which 24), and BP the predict 11). to solved (10, technique excitations is acoustic medium anomalous equation effective In the elasticity role. using central heterogeneous by the the plays approach, (16–18) nanoscale this the at mechan- response the in ical inhomogeneity One an differ. that assumes substantially (10–15) explanations approach these and proposed, been phonons to converge to expected are (6–9). modes limit, vibrational continuum its the to in and expected medium is elastic solid homogeneous amorphous, a any also as fre- behave but because crystalline low counterintuitive only very not highly material, at are even behaviors phonons These not are quency modes vibrational the At as (BP). peak increases boson excess the the as reflects around the directly than modes which larger (4), vibrational solids becomes crystalline solids At for amorphous (3). value of solids crystalline capacity of heat those the anoma- to are respect laws with these lous however, laws; universal by characterized (ω results experimental and with quency, agree indeed which systems, lows rdaeSho fAt n cecs h nvriyo oy,Tko1380,Jpn and Japan; 153-8902, Tokyo Tokyo, of University The Sciences, and Arts of School Graduate nte prahi h ocle otptnilmdl(19– model potential soft so-called the is approach Another have anomalies these for explanations theoretical of number A rpriso mrhu oishv enalong-standing a been have solids thermal amorphous low-temperature of and properties vibrational low-frequency he C ∝ ω T ∼ 3 0.1 n h hra odciiyfollows conductivity thermal the and , T κ ∝ | stmeaue.Smlry mrhu oisare solids amorphous Similarly, temperature). is H oeodro antd oe than lower magnitude of order (one THz otnu limit continuum T a,1 2 aaoShiba Hayato , ahrta as than rather ω BP T | . ∼ phonons g (ω 1 1 ,tetemlconductivity thermal the K, ) H 5,otnrfre to referred often (5), THz κ b ∝ n tuh Ikeda Atsushi and , ∝ ω | T 2 otlclzdmodes localized soft h etcpct fol- capacity heat the , 3 4,idctn that indicating (4), κ ∝ T T ∼ 3 sfre- is n3D in 10 ω a | BP K, ). 1073/pnas.1709015114/-/DCSupplemental lsls hi otcsa eopressure zero at contacts pressure their packing lose the cles As particles. the of 4.Ti rdcindfesfo h rdcino ey the- Debye of prediction the the predicts from instability) differs ory elastic prediction the This vDOS 44). to the (i.e., close of stability behavior is marginal characteristic system assuming the plateau, theory that this medium below effective located the of is anomalies low-frequency solids character- the amorphous a for region exhibits relevant model The clearly plateau. this have at of works plateau 50) vDOS istic (49, the numerical that to and established way 39) new Previous (38, (38–48). a theoretical solids amorphous providing of anomalies thereby the understand advancing, considerably now (37). transition (un)jamming where hsatcecnan uprigifrainoln at online information supporting contains article This 1 the under Published Submission. Direct PNAS a is article This interest. paper. of the conflict wrote no A.I. declare and authors H.M. The and data; analyzed A.I. and H.S., H.M., potential, pairwise tempera- zero the at through amorphous particles interacting jammed of ture randomly model is simplest which the (37), of solids studies on based is scenario (34–36). glasses of yielding supercooled of the dynamics and local- the (33) affect these liquids (26–31) to Interestingly, argued solids (32). also are glass amorphous modes spin ized model Heisenberg of the variety in numerically emer- and wide been the a have vibrations in and localized observed Soft conductivity BP. thermal the of anomalous gence the explain to uhrcnrbtos ..adAI eindrsac;HM n ..promdresearch; performed A.I. and H.M. research; designed A.I. and H.M. contributions: Author u-tokyo.ac.jp. owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To o xliigtetemlaoaiso mrhu solids. amorphous basis of theoretical anomalies thermal universal firm the a another explaining provide for follow findings law, Our modes Debye law. localized the non-Debye follow soft phonons the the whereas that local- Impor- discover soft solids. amorphous we with of tantly, coexist limit continuum phonons the that in modes reveal ized we limit. work, continuum this the in In even phonons, tem- from amor- low differ of solids properties very phous vibrational at the coun- emerge that suggesting crystalline to peratures, begin their anomalies to These respect terparts. are, that with laws anomalous universal by however, Amor- characterized phonons. also on are universal based solids theories follow phous by solids explained are crystalline that of laws properties thermal The Significance motnl,tema-edter nlsso hsmdlis model this of analysis theory mean-field the Importantly, eety ut ifrn cnrohsbe mrig This emerging. been has scenario different quite a Recently, g (ω H = ) (r b ) A nttt o aeil eerh oouUniversity, Tohoku Research, Materials for Institute steHaiiese ucinand function step Heaviside the is 0 ω NSlicense. PNAS 2 ω > ω φ(r eas h prefactor the because = ) ∗ where , 2 1 − . ω ∗ σ r steostfeunyo this of frequency onset the is 2 H g www.pnas.org/lookup/suppl/doi:10. (ω (σ p c = ) p 0 = ∝ NSEryEdition Early PNAS − slwrd h parti- the lowered, is ω r hc scle the called is which , c ∗ ), ω −2 σ 2 stediameter the is sconsiderably is at ω < ω ω ∗ ω First, . | ∗ f8 of 1 (41, [1]
APPLIED PHYSICAL PNAS PLUS SCIENCES −3/2 k k larger than the Debye level A0 ∝ ω∗ . Thus, this prediction O decreases in the other group. In the former group, O con- provides a new explanation of the BP in terms of marginal stabil- verges to almost 1 at ωex0 (we provide the precise definition of k ity (41, 44). Second, the model is analyzed by using replica the- ωex0 later); namely, these modes are phonons. (The O values of ory (40, 42, 45–48). This theory becomes exact in infinite dimen- these phonon modes are close to but not exactly 1, which indi- sions (45); thus, it can be a firm starting point for considering cates that they are very weakly perturbed. Exact O k = 1 may be the problem. Replica theory predicts that the transition from a realized only in the limit of ω → 0.) Second, the third image normal glass to a marginally stable glass occurs at a finite pres- from the top in Fig. 1A plots the participation ratio P k , which sure p = pG , which is called the Gardner transition (48). Near evaluates the extent of spatial localization of mode k (Eq. 9). the Gardner transition, replica theory predicts non-Debye scal- k 2 P takes values from 1 (extended over all particles equally) to ing of the vDOS g(ω) = cω at ω ω∗ (47), which perfectly 1/N 1 (localized in one particle) (26–29). As shown in this coincides with the prediction of the effective medium theory. image, P k also exhibits the division of modes into two groups: Remarkably, in the marginally stable glass phase, the region of k this scaling law extends down to ω → 0, which means no Debye One approaches P = O(1) with decreasing ω, and the other k regime even in the continuum limit (the limit of ω → 0) (47). approaches P = O(1/N ). The Inset shows that the nonphonon This non-Debye scaling law was recently shown to work at least modes (small O k ) are localized (small P k ), whereas the phonon k k near ω∗ (51). modes (large O ) are extended (large P ) at ω < ωex0. Third, Considering these different theoretical arguments, it is abso- the bottom image of Fig. 1A presents the normal and tangential lutely necessary to numerically observe the nature of the low- vibrational energies, δE kk, δE k⊥ (Eq. 10) (52). Again, the modes frequency vibrations of amorphous solids and determine the are split into two groups. The phonon modes follow the scaling laws that such vibrations obey. This task is not trivial because kk k⊥ 2 behavior δE ≈ δE ω as in the case of crystalline solids, the lower is the frequency that we require, the larger is the ∝ whereas the nonphonon localized modes are characterized by system that we need to simulate. Here, we perform a vibra- k⊥ tional mode analysis of the model amorphous solid defined by the ω-independent behavior of δE as in the case of the floppy Eq. 1, composed of up to millions of particles (N ∼ 106), which modes at ω > ω∗. These three coherent results demonstrate that enables us to access extremely low-frequency modes even far the phonon modes and the nonphonon localized modes coexist at ω < ωex0. below ωBP. Then, we investigate the nature of the modes in detail by calculating several different parameters. Notably, we We now perform more stringent tests of the nature of these g(ω) = cω2 two types of modes. In Fig. 2A, in the leftmost image, the phonon find that the non-Debye scaling is violated at low k k frequency, and in the continuum limit, the vibrational modes order parameter O is plotted against the frequency ω for each converge to a mixture of phonon modes that follow the Debye mode k in the system of N = 2,048,000. Because this system is law and soft localized modes that follow another universal in a finite box, phonons should have discrete energy levels. We non-Debye scaling law, the ω4 scaling law, which was recently calculate these energy levels from the macroscopic elastic mod- first observed in refs. 31 and 32 by suppressing the effects of uli and the linear dimension of the box L = 114 (Materials and phonons. Methods), and we display them as vertical lines in the figure. Indeed, the phonon modes (modes with large O k ) sit on these Results and Discussion levels. Conversely, the soft localized modes (modes with small We first study the vibrational modes of the 3D model system O k ) are located in the gaps between the different levels. Fur- at a pressure (density) above the unjamming transition (Mate- thermore, the two rightmost images in Fig. 2A show the eigen- rials and Methods). Fig. 1A (top image, red circles) presents vector field ek of the representative modes (highlighted as filled 2 −2 the reduced vDOS g(ω)/ω at p = 5 × 10 (packing fraction circles in the leftmost image), which demonstrates that the mode ϕ ≈ 0.73) together with the Debye level A0. The value of A0 on the level has a phonon structure, whereas the other mode is is independently determined from the macroscopic mechanical localized. These results unambiguously establish the distinction moduli (Eq. 5). The reduced vDOS clearly exhibits a maximum between phonon modes and soft localized modes. (i.e., the BP). Because our data cover a wide range of frequen- We note that the soft localized modes present an extended cies, we can precisely identify the position of the BP ωBP. In Fig. vibrational character. This character is best observed in the spa- 1, we place arrows that indicate ωBP and ω∗ (the onset frequency k tial correlation function Cˆq,σ(|qˆ · r|) (Eq. 8), as shown in second of the plateau). Please refer to Fig. S1 for the plateau and its image from the left of Fig. 2A. Here, we calculate the spatial cor- onset frequency ω∗ in g(ω). The value of ω is approximately BP relation of the eigen-vector field ek along the [100] transverse five times smaller than ω∗. As the frequency is further decreased 2 wave (Materials and Methods). This function exhibits a nice sinu- below ω , g(ω)/ω decreases toward but does not reach A0 BP soidal shape not only for the phonon mode, but also for the in the frequency region that we studied. We will carefully dis- cuss this result after characterizing the nature of the vibra- localized mode. This result indicates that the localized mode has tional modes. disordered vibrational motions in the localized region; however, To characterize the modes, we calculate three different param- these motions are accompanied by extended phonon vibrations eters (Materials and Methods). First, the second image from the in the background. This feature is very similar to the quasilocal- ized modes in defect crystals, which are produced by hybridiza- top in Fig. 1A presents the phonon order parameter O k , which is tion of the extended phonon and the localized defect modes defined as the projection onto phonons (Eqs. 6 and 7), for each k (2, 28). Consistent with this extended character, we observe mode k. O measures the extent to which the mode k is close to that the participation ratios of the localized modes are indepen- phonons, and it takes values from 1 (phonon) to 0 (nonphonon). dent of the system size N at a fixed ω (26, 28). However, as k At ω > ω∗, O is nearly zero, which confirms that these modes, the recent work (31) demonstrated, the localized modes lying called floppy modes (disordered extended modes) (38, 39), are below the lowest phonon mode exhibit N -dependent participa- largely different from phonons. As the frequency is decreased tion ratios. We speculate that these localized modes are affected k from ω∗ to ωBP, O smoothly increases to ≈ 0.3. This result indi- by phonon vibrations so weakly that only a tiny extended charac- cates that the modes around the BP have a hybrid character of ter is attached. Indeed, this is true for the localized modes (below phonons and floppy modes. Remarkably, as the frequency is fur- the lowest phonon) in defect crystals (28). ther decreased below ωBP, the modes are divided into two groups: A clear distinction between phonon modes and soft localized O k increases with decreasing frequency in one group, whereas modes enables us to separately consider the vDOSs of these
2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1709015114 Mizuno et al. Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 P define We with modes. of types two in presented also of plot the estv otecoc of S2 choice the to sensitive 1A (d system model 3D 1. Fig. egseatyt h ey behavior as Debye define we the to exactly verges Inset g iuoe al. et Mizuno vDOS full the that suggests then law result other the follow that law Debye the follow loc k .W plot We ). AB (ω < tetidiaefo h o) oee,terslsaenot are results the however, top); the from image third (the P ntescn mg rmtetp.A shown, As top). the from image second the in P ) k irtoa oe ntemdlaopossld lt of Plots solid. amorphous model the in modes Vibrational ∝ c > ee esttetrsodvleas value threshold the set we Here, . O ω P k 4 c vs. hs enwcnld httepoo oe that modes phonon the that conclude now we Thus, . ω n edefine we and , g ex0 A P ex k However, . (ω ntetpiaeand image top the in o h Dsse in system 3D the For . = h Dmdlsse (d system model 2D The (B) 3). ) and g ex (ω P g g = ) loc loc c g loc (ω for (ω g A loc (ω ) ) 0 (ω 5 ω nFg 1A Fig. in g ) ∝ × ex 2 olw ifrn cln law scaling different a follows ) Inset A, (ω n h otlclzdmodes localized soft the and 10 ω stevO fmdswith modes of vDOS the as 4 g A ) −3 ex ftescn mg rmtetop. the from image second the of oxs at coexist 0 stevO fmodes of vDOS the as (ω g ω < (ω ,tevO fetne oe (P modes extended of vDOS the ), 2 P tetpiaeand image top (the P tafinite a at = ) = c c < 10 = ) h akn rsueis pressure packing The 2). g 2 ex ω < ω × (ω −2 g 10 + ) ex si Fig. in as ω (ω ex0 −2 which , g(ω g ) This . loc ( con- )/ω Fig. (ω d ) −1 n fthe of and vnulycnegst h ey DSi h ii of limit the in vDOS Debye because the to converges eventually oe 1–4.Mroe,i a eetyrpre htthe that reported recently was it ω Moreover, (19–24). model the n bevdta hs otlclzdmdseatyfollow exactly modes localized soft these present spontaneously the that at modes the are observed localized modes In soft and and vibrational (53). modes the phonon mode into low-frequency that divided the phonon size found on we lowest system focusing work, the the or by below (31), tuning suppressed small regime (32), are sufficiently potential phonons be to of random effects a the introducing if glasses structural 4 p a a eosre nteHiebr pngasadin and glass spin Heisenberg the in observed be can law k ω ω = > 4 4 5 10 law. × cln stesm a rpsdi h soft-potential the in proposed law same the is scaling O g −2 10 loc k , and ) −2 (ω P k , . ) δ Inset E g ∝ k loc k and , (ω ω ntetidiaefo h o in top the from image third the in 4 ,wihi hto oaie oe (P modes localized of that is which ), easfse than faster decays δ E k ⊥ fec mode each of k NSEryEdition Early PNAS g sfntosof functions as ex (ω ) ∝ A ω and 2 k oethat Note . < B 10 ω ω < ω The (A) . presents −2 | ω f8 of 3 ,are ), → ex0 0
APPLIED PHYSICAL PNAS PLUS SCIENCES A
B
−2 Fig. 2. Vibrational modes in the low-ω regime. (A) ω < ωex0 of the 3D model system. (B) ω < ω0 of the 2D model system. p = 5 × 10 . The system sizes are N = 2,048,000, L = 114 (3D) and N = 128,000, L = 391 (2D). The leftmost image plots Ok vs. ωk for each mode k (symbols) and the energy levels of phonons (vertical lines). The remaining images provide more detailed information on the two modes highlighted by red and blue filled circles in the left image. The k k ˆ second image from the left presents the spatial correlation function of the eigen-vector field e , Cˆq,σ (q · r), along the [100] transverse wave. The third and fourth images from the left visualize the eigen-vector fields of these two modes. For the 3D system in A, the vector fields are plotted at a fixed plane of a thickness of particle diameter, and the localized region is emphasized by black arrows.
We repeat this analysis using different packing pressures (den- gloc(ω) against ωˆ in Fig. 4B. Remarkably, all of the data converge 4 sities). We observe that the basic features are unchanged (Fig. to another universal scaling law gloc(ω) = αlocωˆ with αloc = 58. S3A). Furthermore, we find that the vDOSs at different pressures This convergence occurs at ωˆ = 0.066 as in gex(ω). These two can be summarized by the scaling laws. To illustrate this result, results demonstrate that the full vDOS g(ω) = gex(ω) + gloc(ω) 2 4 we determine ω∗ for each pressure and introduce the scaled fre- can be expressed as g(ω) = A0ω + αlocωˆ at ω . ωex0. quency ωˆ = ω/ω∗. Here, we confirm the well-established prop- 1/2 erty of ω∗ ∝ p (38, 39, 49, 50). Then, we plot g(ω) and g(ω)/ωˆ2 against ωˆ at various pressures in Fig. 3. All of the data perfectly collapse around and above the BP. We emphasize that there are no adjustable parameters for this collapse. At ωˆ & 1, i.e., ω & ω∗, g(ω) exhibits a constant plateau g(ω) = α∗ = 0.37 (Fig. S1A). The BP is located at ωˆ = 0.21; thus, ωBP = 0.21ω∗ at all pressures. The vDOS around the BP can be fitted to the 2 non-Debye scaling αBPωˆ with αBP = 0.66, as predicted by the mean-field theory (44, 47) and observed in simulations (51). However, the data systematically deviate from this scaling law around ωˆ ≈ 0.1 at all pressures. This result can be contrasted with the results of replica theory, which predicts that the present system has a marginally stable glass phase in some finite region of the density above the unjamming point (48), and the region of 2 the non-Debye scaling law αBPωˆ extends down to ω → 0 in this phase (47). Our result is more consistent with the results in refs. 44 and 54. Rather, at lower frequency ω ωBP , we find that another scal- 2 ing law works. In Fig. 4A, we plot gex(ω)/A0ω∗ against ωˆ. We A ω A = α ω−3/2 observe that the Debye level 0 is related to ∗ as 0 0 ∗ Fig. 3. The vDOSs at different p in the 3D model system. g(ω) is plot- with α0 = 0.27. As ωˆ decreases, all of the data at different pres- 2 2 ted against the scaled frequency ωˆ = ω/ω∗ for several different p. Inset sures collapse and converge to gex(ω)/A0ω∗ =ω ˆ , which is exactly is the same as the main image, but for the reduced vDOSs g/ωˆ 2. Around 2 the Debye behavior gex(ω) = A0ω . This convergence occurs at ωBP = 0.21ω∗, the vDOSs are collapsed onto the non-Debye scaling law 2 ωˆ = 0.066; thus, ωex0 = 0.066ω∗ at all pressures. Next, we plot g(ω) = αBPωˆ with αBP = 0.66.
4 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1709015114 Mizuno et al. Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 n h irtoa ntblt 2–4,predicts (22–24), instability consider- model, vibrational potential soft the of ing picture the recent by the that controlled note We are modes, localized of physics soft the including tions, x ny nohrwrs h opoo otiuint h vDOS frequency the for reduced to as contribution expressed the nonphonon be the of can words, function other In universal only. a of form part phonon the and 58, with covers follows: that as vDOS limit the of continuum form the functional the write can we regions, (P modes localized the of modes vDOSs extended ω the the of of vDOSs Plot scaled (B) the of Plot (A) (P system. model 3D the 4. Fig. iuoe al. et Mizuno onto g ˆ ˆ ex . k B A . hrfr,b olcigterslsi l ftefrequency the of all in results the collecting by Therefore, / Insets > ω ˆ hsrsl mle htalo h opoo vibra- nonphonon the of all that implies result This 0.066 x g 10 ω 2 ex & BP in −2 (ω h DS fteetne oe n h otlclzdmdsin modes localized soft the and modes extended the of vDOSs The 1, r h aea h aniae,btfrterdcdvDOSs reduced the for but images, main the as same the are g 0 = A ) ), A = (ω ω 0 G g and ˆ ∗ A ex .21ω = = ) aey h ssaiiyadtemria stability. marginal the and isostaticity the namely, , (x 0 = ω α g = ) 2 g 0 loc and ∗ ex ω A , / (ω ∗ −3/2 α ω ω ˆ 0 A α α BP g 4 g )/A ω ex0 BP ∗ loc 0 nonphonon in 2 x ω 0 = (ω 0 = 2 2 ω Below B. with ) ω ∗ 2 + α ω for = gis h cldfrequency scaled the against , .066ω ∗ 0 α α ω (ω loc loc ∗ 1/2 α 2 x (ω/ω 0 = ) ∼ ∗ ω (ω/ω 0 = , ω ex0 0. ω α ∗ G ∗ and 21, ∗ tiigy xetfor except Strikingly, .27 = ) (ω/ω 4 0 = ∗ 4 0.066ω with ) 2 k h DStksthe takes vDOS the , . (ω (ω (ω 37, < ∗ α where ), 10 loc ∗ ∼ . & G α h DS collapse vDOSs the , −2 = g BP (x ω ω ω loc 58. ), ex0 ∗ BP = ) 0 = (ω ), g ), ), loc ) G α .66, (ω ∝ ω ˆ loc (x ,against ), ω = x = ) BP α −3 ω/ω ω/ω 4 loc ω [2] for α = 4 ∗ ∗ ∗ ; . ∝ pressure nFg 4A Fig. in Debye the by scaled col- amplitude The BP 56). the level, (6, that works indicates for also previous reported lapse in were vDOSs systems the Lennard–Jones of 2D collapses of that and phonons function to observing universal gence contribution by a nonphonon the rationalized becomes does also be it can is thus, vDOS result (phonon) Debye This univer- the a 5. as Fig. expressed spa- that are in of the is pressures have function various feature also at sal interesting they vDOSs Another full and phonons. the phonons, of of structures levels tial energy the on α At (1). approximation harmonic the within k predicted be can system. present r hnn.Ti euti oeeieti i.2B Fig. in evident more of is tem result This as phonons. by define are characterized we are that modes the frequency finite situation. a different at surprisingly a 1B encounter Fig. we where system, not framework. are mean-field they current although the stability, by (elas- marginal modes captured the localized transition soft from unjamming the originate that the also suggests when to the result modes This close that instability).] localized brought tic observe soft is [We exhibit system S3B). to the localized begins (Fig. soft system quelled the unstressed strongly Furthermore, are 47). (44, modes from stability originate modes marginal these the that confirms which S5), (Fig. system the dis- that for observe the mode region We whether stability. a scaling evaluate marginal whether can the from one observing originates system, by mode unstressed Thus, the 54). sta- in marginally 44, appears the from (39, model, far state be present the to ble the known which is In system in Methods). unstressed the and system (Materials the with springs replaced are as relaxed system original defined the of is contacts system. particle–particle system “unstressed” the unstressed of analysis The mode vibrational a perform provides system two-level 55). (25, the effects capacity; linear heat the con- the additional to provide tributions effects Anharmonic description. monic at C Finally, prediction. Debye of dependence pressure the predict also can ω theory This not. is of exponent the namdlaopossld I hswr,w aesuidan studied have modes we vibrational work, this the (In of solid. limit amorphous model continuum a the are in observe phonons to of tion effects the if the systems that amorphous suppressed. few 2D (31) so in work is even recent the The modes modes. that these phonon indicated below by of dominated even number are appear vDOSs the modes although localized soft 2D, some that note icse ntesf oeta oe 2) Around (20). model C potential soft the in discussed value h B BP BP (T (T sPac’ constant), Planck’s is ω ae ntevO nEq. in vDOS the on Based ial,w ou ntevbainlmdso h Dmodel 2D the of modes vibrational the on focus we Finally, we behaviors, nonphonon the of origin the discuss further To ncnlso,w aeue ag-cl ueia simula- numerical large-scale a used have we conclusion, In T (p ω ∗ −1/2 ) ) . ∗ −2 C ) ∝ ∝ g ~ω ex N (ω hw that shows hti,hwvr ifrn from different however, is, that α α slre than larger is (T p tteujmigtasto,a hw nEq. in shown as transition, unjamming the at ex0 BP BP ∗ 128, = T T n2.I otat n3,ti uniydvre as diverges quantity this 3D, in contrast, In 2D. in as see (also ) )/A ω dpnettr hog h unu anharmonic quantum the through term -dependent (k oeta hs rdcin r ihntehar- the within are predictions these that Note . ∗ ∝ −2 B ω 0 A ω/ω T l ftevbainlmdsat modes vibrational the of all 000; steBlzancntn,and constant, Boltzmann the is ω 0 scnitn ihorrsl,bttecoefficient the but result, our with consistent is g 3 BP g d T ω (ω −1 (ω ∗ 4 mre from emerges 3 Eq. = ) ) vrteentire the over a flclzdmdscnb observed be can modes localized of law and A 5) osntdpn ntepacking the on depend not does (56), C ovre mohyt h ey vDOS Debye the to smoothly converges 0 α (T , S3 C BP C loc ) ω ˆ (T and g ossso w em:teDebye the terms: two of consists (T 2 k (ω B O ssprse nteunstressed the in suppressed is ) ) T = ) h etcapacity heat the 2, g k ae xesvle vrthe over values excess takes is S7 Figs. ∝ nonphonon ∼ g ≈ α A (ω ~ω loc ω 0 aey hs modes these namely, 1; ω ω = ) ω ω NSEryEdition Early PNAS ∗ eie sillustrated as regime, 0 d BP , ∗ −4 Below . −1 (ω g and α ∝ T (ω BP iia conver- Similar ). with ω 5 ~ ω/ω = ) (ω/ω hspitwas point This . ∗ = S8 ∝ k h A α .However, ). ω ∗ B p o h sys- the for /2π ∗ 0 ∗ 0 T 1/2 ω < ω n2,as 2D, in ) otof most , ∝ 2 provides ∼ Since . where , ω | fthe of C 2 ~ω ω ∗ −d f8 of 5 0 0 (T and BP /2 sit in ) ; ,
APPLIED PHYSICAL PNAS PLUS SCIENCES denoted by σ). The particle mass is m. Length, mass, and time are measured 1/2 in units of σ, m, and τ = (mσ2/) , respectively. To access low-frequency vibrational modes, we consider several different system sizes N (number of particles), ranging from relatively small, N = 16,000, to extremely large, N = 2,048,000. We always remove the rattler particles that have less than d contacting particles. Mechanically stable amorphous packings are generated for a range of packing pressures from p ∼ 10−1 to 10−4. We first randomly place N par- ticles in a cubic (3D) or square (2D) box with periodic boundary conditions in all directions. The system is then quenched to a minimum energy state. Finally, the packing fraction ϕ is adjusted by compressive deformation (CO) until a target pressure p is reached. We also study the shear-stabilized (SS) system (58) by minimizing the energy with respect to the shear degrees of freedom. No differences between the CO and SS systems have been con- firmed for our systems of N ≥ 16,000. In this work, we present the results obtained from the CO system. In the packings obtained by using the above protocol, the interparticle forces are always positive −φ0(r) > 0. For this reason, we refer to this orig- inal state as the stressed system. In addition to the stressed system, we also study the unstressed system, where we retain the stiffness φ00(r) but drop Fig. 5. The vDOSs at different p in the 2D model system. g(ω) is plotted the force −φ0(r) = 0 in the analysis. Since the positive forces make the sys- against the scaled frequency ωˆ = ω/ω∗ for several different p. Inset is the tem mechanically unstable, dropping the forces makes the original stressed same as the main image, but for the reduced vDOSs g/ωˆ. The full vDOSs are system more stable (39, 44, 54). collapsed onto a universal function of the scaled frequency ωˆ over the entire ω regime. At ω0 = 0.066ω∗, g(ω) converges to the Debye vDOS g(ω) = A0ω. Vibrational Mode Analysis. We perform the standard vibrational mode anal- ysis (1, 2), where we solve the eigen-value problem of the dynamical matrix (dN × dN matrix) to obtain the eigen value λk and the eigen vector amorphous system of the harmonic potential defined by Eq. 1. h i ek = ek, ek, ..., ek for the modes k = 1, 2, ..., dN − d (d zero-ω, transla- However, we expect that our main results may persist in a vari- 1 2 N ety of amorphous systems, e.g., the Lennard–Jones glasses, which tional modes are removed). Here, the eigen vectors are orthonormalized as ek · el = PN ek · el = δ , where δ is the Kronecker delta function. i=1 i i k,l k,l √ requires further study.) In 3D, we have found that the vDOS k k 2 From the dataset of eigen frequencies, ω = λ (k = 1, 2, ..., dN − d), follows the non-Debye scaling g(ω) = αBP(ω/ω∗) only around we calculate the vDOS as and above the BP, and below the BP, the vibrational modes are dN−d divided into two groups: The modes in one group converge to 1 X 2 g(ω) = δ ω − ωk , [3] the phonon modes that follow the Debye law gex(ω) = A0ω , dN − d and the modes in the other group converge to the soft local- k=1 ized modes that follow another universal non-Debye scaling where δ(x) is the Dirac delta function. 4 gloc(ω) = αloc(ω/ω∗) . Strikingly, all of the nonphonon contribu- In the present work, we analyze several different system sizes, ranging tions to the vDOSs at different pressures can be expressed as a from N = 16,000 to N = 2,048,000. We first calculate all of the vibrational universal function of the reduced frequency ω/ω∗. In contrast, modes in the system of N = 16,000. We then calculate only the low-frequency modes in the larger systems of N > 16,000. Finally, the modes obtained completely different behaviors are observed in 2D: Vibrational k modes smoothly converge to phonons without the appearance of from different system sizes are combined as a function of the frequency ω . We find that the results from different system sizes smoothly connect with the group of soft localized modes. each other, which provides the mode information in the very low-frequency Our results, on the one hand, provide a direct verification regime (Fig. S9). The vDOS is calculated from these datasets, and the results of the basic assumption of the soft potential model (19–25). of these different system sizes are presented together in the figures. We showed the coexistence of phonon modes and soft local- The present system exhibits a characteristic plateau in g(ω) at ω > ω∗, ized modes, which is the central idea for explaining the low- where ω∗ is defined as the onset frequency of this plateau (Fig. S1) (49, 50). T anomalies of thermal conduction and the formation of BP Practically, we determine the value of ω∗ as the BP position of the unstressed in this phenomenological model. However, more quantitative system (39, 52). predictions of this theory, such as the pressure dependence of −3 4 Phonon and Debye vDOS. In an isotropic elastic medium, phonons are ωBP(p) and the coefficient of gloc(ω) ω ω , are inconsistent ∝ BP described as eq,σ = eq,σ , eq,σ , ..., eq,σ (1, 2), with our results. [The pressure dependence of ωBP(p) of the 1 2 N
Lennard–Jones glasses seems to be consistent with the predic- ˆq, σ q,σ P tion of soft potential model (57), which needs further confirma- ei = √ exp(iq · ri). [4] tion.] On the other hand, the violation of the non-Debye scaling N at the BP, the emergence of soft localized modes with another q is the wave vector, and qˆ = q/ |q|. Due to the periodic boundary con- non-Debye scaling, and the crucial difference between 2D and dition of the finite dimension L, q is discretized as q = (2π/L)(i, j, k) for 3D appear to be beyond the reach of the current mean-field 3D and as q = (2π/L)(i, j) for 2D (i, j, k = 0, 1, 2, 3, ... are integers). The theory. However, the fact that the nonphonon contributions to value of σ denotes one longitudinal (σ = 1) and two transverse (σ = 2, 3) the vDOS are expressed as a universal function of ω/ω∗ sug- modes for 3D, and it denotes one longitudinal (σ = 1) and one trans- ˆ gests that these features are linked to the isostaticity and the verse (σ = 2) modes for 2D. Pq, σ is a unit vector that represents the direc- ˆq,1 marginal stability, which are captured by the mean-field theory tion of polarization, and it is determined as P = qˆ (longitudinal) and Pˆq,2 · qˆ = Pˆq, 3 · qˆ = 0 (transverse). Note that the vectors eq,σ are orthonor- (38–48). 0 0 q,σ q0,σ0 PN q,σ q ,σ mal as e · e = i=1 ei · ei ≈ δq,q0 δσ,σ0 . Here, strictly speaking, q,σ q0,σ0 0 −1/2 0 Materials and Methods e · e = δσ,σ0 for q = q and = O(N ) for q 6= q , which becomes q,σ q0,σ0 System Description. We study 3D (d = 3) and 2D (d = 2) model amorphous exactly e · e = δq,q0 δσ,σ0 as N → ∞ (the thermodynamic limit). solids, which are composed of randomly jammed particles (37). Particles i, j In the low-ω limit, the continuum mechanics determine the dispersion q,σ σ σ 1 p interact via a finite-range, purely repulsive, harmonic potential φ(rij)[1], relation as ω = c |q|. c is the phonon speed; c = cL = (K + 4G/3)/ρ 2 3 p 1 p 2 where rij = |ri − rj| is the distance between the two particles. The 3D system and c = c = cT = G/ρ for 3D, and c = cL = (K + G)/ρ and c = p d is monodisperse with a diameter of σ, whereas the 2D system is a 50%-50% cT = G/ρ for 2D. Here, ρ = N/L is the mass density, and K and G are the binary mixture with a size ratio of 1.4 (the diameter of the smaller species is bulk and shear elastic moduli, respectively. In this study, we calculate K and
6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1709015114 Mizuno et al. Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 0 cimce ,Roc ,Soin 20)Aosi teuto ngassadits and glasses in attenuation Acoustic (2007) T Scopigno G, Ruocco W, Schirmacher 10. 1 arzoA cimce ,FaaociA ucoG(03 eeoeeu shear Heterogeneous (2013) G Ruocco A, Fratalocchi W, Schirmacher A, Marruzzo 11. 5 hn ,e l 21)Eprmna tde fvbainlmdsi two-dimensional a in modes vibrational of studies Experimental (2017) al. et corre- L, elastic Zhang and scattering 15. phonon Anomalous (2016) A Lemaitre heterogeneities H, elastic Tanaka and S, Gelin excitations Acoustic 14. (2014) JL Barrat S, Mossa H, ther- Mizuno and states, 13. vibrational heterogeneity, Elastic (2013) JL Barrat S, Mossa H, Mizuno 12. 7 smdsM agyA odnegC artJ 20)Lcleatct a n plas- and map elasticity Local (2009) JL Barrat C, Goldenberg A, Tanguy M, hetero- Tsamados Mechanical (2004) 17. JJ dePablo PF, Nealey K, VanWorkum TS, Jain K, Yoshimoto 16. C of choice the on depend not of do value conclusions the and results our that confirmed phonon particular one onto projection as expansion) series (Fourier vector eigen The follows. n rmpoos ntepeetstudy, present the In phonons. from ent O phonons; A overlapped large of number finite a phonons to close Parameter. Order Phonon where as vDOS the yield to phonons G iuoe al. et Mizuno parameter order phonon than the more define of we extent the by overlapped that note Here, h sition, as see (also where 2D, for .SiaH aaaY aaaiT i 21)Uviigdmninlt dependence dimensionality Unveiling (2016) K Kim T, Kawasaki Y, Yamada H, Shiba 9. .Mnc ,MsaS(09 nmlu rpriso h cutcecttosi glasses in excitations acoustic the of properties Anomalous (2009) S Mossa G, Monaco 8. Boissi F, amorphous Leonforte of limit 7. Continuum (2002) JL Barrat F, Leonforte JP, Wittmer A, Tanguy 6. noncrystalline of heat N specific U, and Buchenau conductivity Thermal 5. (1971) RO Pohl RC, Zeller 4. (1981) WA Phillips 3. (1978) N Breuer G, Leibfried 2. (1996) C Kittel 1. (18π ˆ q, k q,σ k k sn h amncfruain(2.Dbeter onstenme of number the counts theory Debye (52). formulation harmonic the using nadto,w aclt h nraie)sailcreainfunction correlation spatial (normalized) the calculate we addition, In ftemode the If eainwt h oo peak. boson the with relation relaxation. structural Lett inherent eclipses Rev fluctuation infinite 2D dynamics: glassy of lsiiyo lse:Teoii ftebsnpeak. boson the of origin The glasses: of elasticity scale. length mesoscopic the on systems. Three-dimensional iii. bodies. elastic amorphous vibrations. harmonic low-frequency of study 66:174205. finite-size A bodies: elastic silica. vitreous in vibrations frequency solids. Ed. 3rd 81. Vol Berlin), (Springer, Physics Modern in Tracts Springer Ed. 7th mrhu solid. amorphous solids. amorphous in lations solids. disordered in transition. amorphisation an across conductivity mal iiyi oe enr-oe glass. Lennard-Jones model a in ticity scales. length small at glasses polymer model in geneities σ = e (| o hnn whereas phonon, a for 1 q,σ 2 q ˆ ω ρ)/ ˆ A · D ee edfie“ag vrapd as overlapped” “large define we Here, . 0 hsRvB Rev Phys is S7 Figs. of r|) C = 117:245701. ∝ ˆ q, k c d N σ L −3 c /ω m T ρ ˆ e .
= k P nrdcint oi tt Physics State Solid to Introduction k + ∝ ce ,DaoxA 18)Nurnsatrn td ftelow- the of study scattering Neutron (1984) AJ Dianoux N, ucker ¨ D d q ˆ and rjce to projected a Commun Nat 4:2029–2041. N sapoo,then phonon, a is e steDbelvland level Debye the is · q,σ 2c r ,Tnu ,WtmrJ,Bra L(05 otnu ii of limit Continuum (2005) JL Barrat JP, Wittmer A, Tanguy R, ere ` mrhu ois o eprtr Properties Temperature Low Solids: Amorphous ω q,σ (r /L rcNt cdSiUSA Sci Acad Natl Proc ∗ 1/2 T −3 g i S8 O d O − D O yitouigtepoo re parameter order phonon the introducing by i q,σ k ). (ω stenme est.Coet h namn tran- unjamming the to Close density. number the is k q,σ k ∝ r = j eeaut h xett hc h mode the which to extent the evaluate We ) 1/3 ) e ≈
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O = 0 ; A ˆ q, ω k 3,4,5,5) and 52), 50, 49, (37, O D D q,σ k σ q,σ k = P d −1 hsRvE Rev Phys as hsRvLett Rev Phys P P o oeta scnieal differ- considerably is that mode a for 0
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q,σ k ω N k e m e k e ω ω i k D q,σ m i k as d = as e (r D oe.Cnieigteabove, the Considering modes. D (r −d d 80:026112. steDbefrequency; Debye the is = q,σ i 1. i ! c Rep Sci Jh ie n os e York), New Sons, and Wiley (John ) ) · = ) 53:2316–2319. 0 a sd oee,we however, used; was 100 e O e hn ecncluaethe calculate can we Then, . uohsLett Europhys ω k k O q,σ k h d
P 106:16907–16912. P = (8π −1 2 q,σ k ˆ q, ˆ q, . hsRvB Rev Phys hsRvLett Rev Phys . 3:1407. A σ P σ , ρ)/ ˆ 0 ≥ · · q,σ ∝ e e N i k j k (r ; (r m ω c O i L 104:56001. −2 /(dN j Srne,Berlin), (Springer, ∗ −d ) ) q,σ k E 72:224206. E /2 + 93:175501. ≥N , − hsRvB Rev Phys c ∝ m T −2 /(dN d p i ω O ,i.e., ), −d e D e k Phys −d k 1/2 q,σ [8] [5] [7] [6] /4 as = is ) , 8 coe R ucoG(04 ieefcsadqaioaie vibrations. quasilocalized and effects Size (2004) G in Ruocco vibrations HR, atomic Schober of localization 28. and Anharmonicity (1999) SR Elliott model SN, a Taraskin in motion 27. atomic Low-frequency (1996) M Sampoli thermal G, low-temperature Ruocco V, Anomalous Mazzacurati (1972) 26. CM Varma BI, Halperin PW, Anderson systems, two-level instability, 25. Vibrational (2007) VL Gurevich HR, Schober DA, Parshin 24. glasses. in waves sound and modes soft of Interaction potentials (1992) Anharmonic al. et (1991) U, HR Buchenau Schober 21. VL, Gurevich YM, Galperin anomalies U, low-temperature Buchenau the of 20. Theory (1983) FN Ignat’ev MI, elastic Klinger local VG, the Karpov of distribution spatial 19. Measuring (2013) JL Barrat S, Mossa H, Mizuno 18. 4 aoe E Lema reorga- CE, Irreversible Maloney (2008) 34. DR Reichman P, Harrowell H, Perry A, Widmer-Cooper 33. Mart M, Baity-Jesi 32. D E, Lerner 31. crite- Ioffe-Regel and peak Boson (2016) A Tanguy in DA, vibrations Parshin C, quasi-localized Fusco and YM, Anharmonic Beltukov (2010) SR 30. Nagel AJ, Liu V, Vitelli N, Xu 29. peak boson the of dependence Pressure (2005) HR Schober instability, DA, vibrational Parshin Anharmonicity, VL, Gurevich (2003) HR 23. Schober DA, Parshin VL, Gurevich 22. nSIAtxIEX tteIsiuefrSldSaePyis h University The Physics, State Solid for Institute the at XA Tokyo. ICE of Scien- Altix performed for partly SGI were Grant-in-Aid calculations on numerical and The 16H04034. 17H04853, 17K14369, B B A Research Society tific Scientists Scientists Young Japan Young for by use- for Grant-in-Aid supported Science for Grant-in-Aid was of Miyazaki work Promotion K. the This and for suggestions. Bouchbinder, and E. Lerner, discussions E. ful Berthier, L. Zamponi, F. ACKNOWLEDGMENTS. h op oe h tangential the δ mode, floppy the of choice the results to the ( However, sensitive modes. localized not the to are assigned are particles total the with modes study, present and equally, cases, extreme As ratio participation the Ratio. Participation mode hnn then phonon, a where o h rsn eusv system, repulsive present the For energy tional e tial particles vector The Energy. Vibrational i.S2). Fig. E ij k k k ouu nglasses. in modulus irossilica. vitreous glass. glasses. spin and glasses of properties glasses. in peak boson the and glasses. in localization vibrational and structures. amorphous of properties thermal the in hsRvE Rev Phys modes. soft localized from originates liquid 711–715. supercooled a in nization glasses. spin in systems two-level and glasses. structural in modes vibrational directionality. bond of effect The 93:023006. materials: siliconlike amorphous in rion failure. mechanical for solids-modes jammed 84:1361–1372. glasses. in glasses. in peak boson the and B Rev ⊥ e = ij k ⊥ ∝ e hi k e irtoswt epc otebn vector bond the to respect with vibrations uohsLett Europhys 46:2798–2808. ω i sapoo,then phonon, a is ij k eoe h vrg vralpiso particles of pairs all over average the denotes , 0 j · e hc a edcmoe notenormal the into decomposed be can which , (52). hsRvB Rev Phys n ij k 74:016118. rn ,BuhidrE(06 ttsisadpoete flow-frequency of properties and Statistics (2016) E Bouchbinder G, uring ¨ δ ij P δ = E k E hsRvB Rev Phys k = n k e = = δ ij i k P = nMyrV aiiG ee-aioS(05 otmds localization, modes, Soft (2015) S Perez-Gaviro G, Parisi V, ´ ın-Mayor 1/N and , E k teA(06 mrhu ytm nahra,qaittcshear. quasistatic athermal, in systems Amorphous (2006) A ˆ ıtre − δ hsRvE Rev Phys emaueteetn fvbainllclzto using localization vibrational of extent the measure We ial,w aclt h irtoa energy vibrational the calculate we Finally, X k (i 34:681–686. = ω E k ,j 71:014209. e k k ) P k o nielmd hr l fteprilsvibrate particles the of all where mode ideal an for 1 j k etakH kd,Y i,L .Slet .Charbonneau, P. Silbert, E. L. Jin, Y. Ikeda, H. thank We and 2 o oeta novsol n atce nthe In particle. one only involves that mode a for P " k a edcmoe as decomposed be can /2 ersnstevbainlmto ttecnatof contact the at motion vibrational the represents − e 59:8572–8585. k (26–29), φ ij k = ⊥ δ 00 C P δ 2 E ˆ q, (r k N 87:042306. 1 E k hsRvB Rev Phys hsRvB Rev Phys k = ij σ ⊥ k " ) ⊥ (| < . X i e q ˆ =1 N e ij k r ohpootoa to proportional both are ij k δ hsRvLett Rev Phys · P hlspia Mag Philosophical k hsRvB Rev Phys δ − E P c xiisasnsia curve. sinusoidal a exhibits r|) hsRvLett Rev Phys E k c = e ⊥ k 2 76:064206. 67:094203. i k ⊥ o 5 for 10 + e · exhibits ij k uohsLett Europhys e sawy oiie fmode If positive. always is −2 φ · i k 2r n 0 (r 43:5039–5045. × ij 2 htivlels hn1% than less involve that ij # ij 115:267205. o hsJETP Phys Sov ) −1 10 117:035501. n NSEryEdition Early PNAS n ij ω e −3 . codnl,tevibra- the Accordingly, . ij idpnetbehavior, -independent ij k 25(1):1–9. ⊥ = 90:56001. 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