Identifying structural flow defects in disordered solids using machine learning methods E. D. Cubuk,1, ∗ S. S. Schoenholz (Equal contribution),2, † J. M. Rieser,2 B. D. Malone,1 J. Rottler,3 D. J. Durian,2 E. Kaxiras,1 and A. J. Liu2 1
arXiv:1409.6820v1 [cond-mat.soft] 24 Sep 2014
Department of Physics and School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 2 Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 3 Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T1Z4, Canada We use machine learning methods on local structure to identify flow defects – or regions susceptible to rearrangement – in jammed and glassy systems. We apply this method successfully to two disparate systems: a two dimensional experimental realization of a granular pillar under compression, and a Lennard-Jones glass in both two and three dimensions above and below its glass transition temperature. We also identify characteristics of flow defects that differentiate them from the rest of the sample. Our results show it is possible to discern subtle structural features responsible for heterogeneous dynamics observed across a broad range of disordered materials. PACS numbers: 83.50.-v, 62.20.F-, 61.43.-j
All solids flow at high enough applied stress and melt at high enough temperature. Crystalline solids flow [1] and premelt [2] via localized particle rearrangements that occur preferentially at structural defects known as dislocations. The population of dislocations therefore controls both how crystalline solids flow and how they melt. In disordered solids, it has long been hypothesized that localized particle rearrangements [3] induced by stress or temperature also occur at localized flow defects [4–6]. Like dislocations in crystals [7], flow defects in disordered solids are particularly effective in scattering sound waves, so analyses of the low-frequency vibrational modes [8] have been used successfully to demonstrate the existence of localized flow defects [7, 9–17]. However, all attempts to identify flow defects [18, 19] directly from the structure, without using knowledge of the inter-particle interactions, have failed [18, 19]. Likewise, in supercooled liquids, purely structural measures correlate only weakly with kinetic heterogeneities [17], although correlations between structure and dynamics have been established indirectly [20–24]. Here we introduce a novel application of machine learning (ML) methods to identify “soft” particles that are susceptible to rearrangement or, equivalently, that belong to flow defects, from the local structural geometry alone. We apply the method to two very different systems–an experimental frictional granular packing under uniaxial compression and a model thermal Lennard-Jones glass in both two and three dimensions. The analysis of granular packing shows that our method succeeds even when previous methods based on vibrational modes [12] are inapplicable. The results for Lennard-Jones systems show that the correlation between structure and irreversible rearrangements does not degrade with increasing temperature, even above the dynamical glass transition, and is equally strong in two and three dimensions. Finally, we exploit the method to discover which structural properties distinguish soft particles from the rest of the sys-
tem, and to understand why previous attempts to identify them by structural analysis have failed. Physically-motivated quantities such as free volume or bond orientational order correlate with flow defects [12], but are insufficient to identify them a priori. We introduce a large set of quantities that are each less descriptive, but when used as a group provide a more complete and unbiased description of local structure. These quantities have been used to represent the potential energy landscape of complex materials from quantum mechanical calculations [25]. For a system composed of multiple species of particles, we define two families of structure functions for each particle i, GX Y (i; µ) =
X
2
e−(Rij −µ)
/L2
(1)
j
ΨX Y Z (i; ξ, λ, ζ) =
XX j
2
2
2
2
e−(Rij +Rik +Rjk )/ξ (1 + λ cos θijk )ζ
k
(2) where Rij is the distance between particles i and j; θijk is the angle between particles i, j and k; L, µ, ξ, λ, and ζ are constants; X,Y ,Z are labels that identify the different species of particles in the system, with the correspondence i ↔ X, j ↔ Y , k ↔ Z. By using many different values of the constants µ, ξ, λ, and ζ we generate many structure functions in each family that characterize different aspects of a particle’s local configuration; for a list, see supplementary information. The first family of structure functions G characterizes radial density properties of the neighborhood, while the second family, Ψ, characterizes bond orientation properties. The sums are taken over particle pairs whose distance is within a large cutoff RcS . Our results are qualitatively insensitive to the choice of RcS as long as it includes several neighbor shells. Having characterized local structure through GX Y (i; µ) and ΨX (i; ξ, λ, ζ), we introduce a method to infer from YZ this information the location of flow defects in disordered
2 solids. Generically, we begin with a set of N particles to be classified as “soft” or “hard.” Each particle is described by a set of M variables derived from the two X families of functions GX Y and ΨY Z by varying the constants µ, ξ, λ, and ζ (here M = 160); this is represented by the set of vectors {F1 , · · · , FN }, where Fi constitutes an embedding of the local environment of a particle i, constructed at a time ti , in RM . We select at random a subset of n of these particles (the “training set”) and categorize them a posteriori as being soft if they rearrange (the details of which will be discussed below) between time ti , when the structure is characterized, and time ti + ∆t. Otherwise the particles are labeled as hard. The next step is to use the particles in the training set, already classified as soft or hard, to construct a scheme to identify other particles as soft or hard. We use the support vector machine (SVM) method [26], which constructs a hyperplane in RM that best separates soft particles from hard ones. Once this hyperplane has been established for the training set, the rest of the particles (and any particles from similar systems) may be classified according to whether their local structures place them on soft or hard sides of the hyperplane. Generically, no exact separation is possible so the SVM method is adapted to penalize particles whose classification is incorrect; the degree of penalty is controlled by a parameter C where larger values of C allow for fewer incorrect classifications. We find that the quality of our classifications is insensitive to C for C > 0.1. The SVM algorithm was implemented using the LIBSVM package [27]. To identify rearrangements we calculate, for each particle i, the widely-used quantity [5] 1 X 2 (Rij (t + ∆t) − ΛRij (t))2 , (3) Dmin (i) ≡ min Λ z j which characterizes the magnitude of non-affine displacement during a time interval ∆t. Here the sum runs over neighbors j within a distance of RcD of particle i, Rij is the center-center displacement between particles i and j, z is the number of neighbors within RcD . The quantity is minimized over choices of the local strain tensor Λ. 2 We find Dmin is insensitive to the choice of RcD and ∆t D so long as Rc is large enough to capture the particles local neighborhood and ∆t is longer than the ballistic timescale. A particle is said to have undergone a rear2 2 2 . We choose Dmin,0 such that rangement if Dmin ≥ Dmin,0 approximately 0.15% of the particles from each species in each system has gone through a rearrangement although 2 the results depend little on the specific choice of Dmin,0 . We first test our approach on an experimental system of two-dimensional (d = 2) “pillars” of particles. A bidisperse pillar made up of grains (plastic cylinders resting upright on a horizontal substrate) is situated between two plates in a custom-built apparatus. The bottom
FIG. 1. Snapshot configurations of the two systems studied. 2 Particles are colored gray to red according to their Dmin value. Particles identified as soft by the SVM are outlined in black. (a) A snapshot of the pillar system. Compression occurs in the direction indicated. (b) A snapshot of the d = 2 sheared, thermal Lennard-Jones system.
plate is fixed and the top plate is driven into the pillar at a constant speed of v0 = 0.085 mm/s. The pillars are composed of a bidisperse mixture of approximately 1500 rigid grains with size ratio 3:4 and the large particles having a radius of dAA = 0.3175cm. These particles have elastic and frictional interactions with each other, as well as frictional interactions with the substrate, making the identification of flow defects using vibrational modes impossible. A camera is mounted above and captures images at 7 Hz throughout the compression. We construct our training set from compression experiments performed on ten different pillars. We select 6,000 particles at random from the entire duration of the experiments that undergo a rearrangement in the next 1.43 seconds and an equal number of particles that do 2 not. To identify rearrangements we calculate Dmin with 2 RcD = 1.5 · dAA and Dmin,0 = 0.25d2AA . Compression of the mechanical pillar from the top only affects particles above a certain “front” that starts at the top and advances towards the bottom with time. Our training set contains only particles within this activated front. Particles in a horizontal slice between y and y + δy are said to be within the activated front if the average speed of particles in the slice exceeds vthresh ∼ 0.04 mm/s. As a second test, we apply our approach to a model glass in both d = 2 and d = 3. We study a 65:35 binary Lennard-Jones (LJ) mixture with σAA = 1.0, σAB = 0.88, σBB = 0.8, AA = 1.0, AB = 1.5, and BB = 0.5 [28]. The LJ potential is cut off at 2.5σAA and smoothed so that both first and second derivatives go continuously to zero at the cutoff. The natural units for the simulation are σAA for distances, AA for energies, p 2 / and τ = mσAA AA for times. We perform molecular dynamics simulations using LAMMPS [29] with a time step of 5 × 10−3 τ at density ρ = 1.2, using a Nos´e-Hoover thermostat with a time constant of τ . Here, temperature is in units of AA /kB , where kB is the Boltzmann constant. In d = 2, we consider a system of 10,000 parti-
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2 FIG. 2. Probability that a particle of a given Dmin value
2 2 value FIG. is2.soft. Probability thatdashed a particle given Dmin The vertical lines of area corresponding Dmin,0 2 is soft. The(a) vertical dashed linespillar are system, corresponding Dmin,0 values. The result for the where dAA refers to (a) the The largeresult grain for diameter (since this iswhere a granular system values. the pillar system, dAA refers macroscopic grains, (since thermalthis fluctuations are negligible). to thewith large grain diameter is a granular system (b) The resultgrains, of using an SVM trained at are a temperature with macroscopic thermal fluctuations negligible).T (T = 0.1, 0.2, 0.3 andan 0.4SVM shown in di↵erent ) to classify (b) The result of using trained at a colors temperature T at the temperature for the d colors = 2 LJ glass. (c,d) (T = data 0.1, 0.2, 0.3 same and 0.4 shown in different ) to classify for species A and B, for respectively, theglass. d = 3 system data Results at the same temperature the d = for 2 LJ (c,d) at T 0.4, 0.5Aand Results for=species and0.6. B, respectively, for the d = 3 system at T = 0.4, 0.5 and 0.6.
cles at temperatures T = 0.1, 0.2, 0.3, and 0.4 at a single strain rate, ˙ = 10 4 /⌧ . In all cases data was collected cles at temperatures = 0.1,to 0.2, 0.3,steady and 0.4 at abysingle after allowing theTsystem reach state shear−4 strainingrate, γ ˙ = 10 /τ . In all cases data was collectedof up to 20% strain. In d = 3, we use a collection after 30,000 allowing the system to reach steady state particles at temperatures T = 0.4, 0.5, by 0.6 shearwith no ing up to 20% In dsystem = 3, has we use a collection strain. Thestrain. quiescent a glass transitionofat 30,000 = 0.4, with no TGparticles = 0.33 inatdtemperatures = 2 and TG =T0.58 in d0.5, = 30.6 [28]. Therefore,The in both dimensions we study system both above strain. quiescent system has athe glass transition at temperature. TG =and 0.33below in d its = 2glass andtransition TG = 0.58 in d = 3 [28]. Thereeach temperature construct trainingboth sets of 6,000 fore, inAt both dimensions wewe study the system above and 10,000 particles, in d temperature. = 2 and 3, respectively, seand below its glass transition at random from the entire training run that sets undergo a reAtlected each temperature we construct of 6,000 arrangement in the following t = 2⌧ units of time, and and 10,000 particles, in d = 2 and 3, respectively, sean equal number of particles that do not undergo a relected at random from the entire run that undergo a rearrangement. To identify rearrangements, we calculate arrangement in R the 2 D following ∆t = 2τ units of time, and Dmin with c = 2.5 AA to be the same as the range an equal of particles thatand do not of thenumber truncated LJ potential t =undergo 2⌧ . In ad re= 3 2 arrangement. To identify rearrangements, we the Dmin distributions of the species A andcalculate species B 2 Dmin with RcD = 2.5σ same as the range AA to be particles di↵er significantly andthe so are treated separately of thethroughout truncatedthe LJanalysis. potential and ∆t = 2τ . In d = 3 2 the Dmin of the species A and on species B We distributions now test our classifying hyperplane the three particles differ significantly and so are treated separately systems outlined above. For each system we construct a test set particles that were not used in training the throughout theofanalysis.
We now test our classifying hyperplane on the three systems outlined above. For each system we construct a test set of particles that were not used in training the
3
SVM; this test testset setconsists consistsofof100,000 100,000 partiSVM;for for the the pillar pillar this particle environments from ten additional compression expercle environments from ten additional compression exper7 7 iments. and dd==33LJ LJglasses glassesweweuse use 2×10 iments. For For the the dd = = 22 and 2⇥10 77 and particle environments environmentsrespectively. respectively. and75 75× ⇥10 10 unseen unseen particle 2 2 ), that a parFig. the probability, probability,PP(D (D Fig.22(a)-(d) (a)-(d) shows shows the ), that a parmin min 22 ticle with with an an observed observed value was identified as as ticle value ofof DDmin was identified min softby by our our classification classification aa priori 2 2 soft priorifor foreach eachsystem. system.FigFig (c)-(d)treat treat the the particles particles of BB sep(c)-(d) ofspecies speciesAAand andspecies species separately for for the the dd = = 33 LJ seesee that arately LJ glass. glass. InInall allcases caseswewe that 2 2 2min 2 P (D ) rises with increasing plastic activity, D . This min . This P (Dmin ) rises with increasing plastic activity, D min implies that that the the particles particles identified SVM implies identifiedasassoft softbybythe the SVM are more likely to be involved in plastic flow. For the are more likely to be involved in plastic flow. For the granular pillar, 21% of the particles are classified as soft, granular pillar, 21% of the particles are classified as soft, and these particles capture 80% of the rearrangements. and particles 80% of the rearrangements. For these the d = 2 and dcapture = 3 LJ systems, these numbers are For the d = 2 and d = 3 LJ systems, these numbers 26% and 73%, and 24% and 72%, respectively, at theare 26% and 73%, and 24% and 72%, respectively, highest temperatures studied. Thus, we consistently at findthe highest temperatures studied. weabout consistently find rearrangements to occur at soft Thus, particles 3-4 times rearrangements occur at soft soft particles particleswere about 3-4 times more frequently to than if the randomly 2 randomly more frequently the softshow particles were chosen. Finally,than Fig. if2(b,c,d) that P (Dmin ) col2 chosen. 2(b,c,d) that P (Dmin ) collapses forFinally, di↵erentFig. T both for 2Dshow and 3D systems, when 2 2 Dmin is by TT AA ; this arises for parlapses forscaled different both forscaling 2D and 3D since systems, when 2 2 not undergoing2 rearrangement, Dmin ⇠ hv 2 ifor ⇠ parT since Dticles min is scaled by T σAA ; this scaling arises 2 2 by the equipartition theorem [16]. ticles not undergoing rearrangement, Dmin ∼ hv i ∼ T Remarkably, our ability to identify by the equipartition theorem [16]. soft particles does not decrease with increasing temperature dimension. Remarkably, our ability to identify softorparticles does For the d = 2 LJ system over the same temperature not decrease with increasing temperature or dimension. range, the accuracy of the vibrational mode method deFor the d = 2 LJ system over the same temperature creases by over 50% [16]. The key di↵erence between range, the accuracy of the vibrational mode method dethe two methods is that we construct local environcreases by over 50% [16]. The key difference between ments from the actual particle positions in snapshots of the two methods is that we construct local environthe thermal system, while soft particles from vibrational ments the actual particle positions modesfrom are extracted from particle positionsininsnapshots the inher- of the system, while particles entthermal structures obtained by soft quenching to Tfrom = 0.vibrational modes extractedthat from particle in particles the inherTheare hyperplane divides softpositions from hard ent by the quenching to T = 0.environtellsstructures us which obtained features of local structural The are hyperplane divides soft from hard ment importantthat in distinguishing between theparticles hard and us softwhich particles. To illustrate this,structural we focus on the tells features of the local environsheared = 2 LJ glass T = 0.1. The first set ofthe strucment aredimportant inatdistinguishing between hard turesoft functions, GX µ) defined in Eq. (1),wehasfocus a familiar and particles. illustrate this, on the Y (i;To physicaldinterpretation of the radial sheared = 2 LJ glass in atterms T = 0.1. The firstdistribution set of strucX= limL!0 hGX function: g (r) (i; r)i/2⇡r. Thisa family XY Y in Eq. (1), has ture functions, GY (i; µ) defined familiar of structure functions isin essentially parameterization of physical interpretation terms of athe radial distribution a particle’s local contribution toXg(r). In Fig. 3 we show function: gXY (r) = limL→0 hGY (i; r)i/2πr. This family the approximations to various g(r) constructed in this of structure functions is essentially a parameterization of way, using L = 0.1 AA , for particles identified as hard a(black particle’s local contribution to g(r). In Fig. 3 we show lines) and soft (red lines). the Inapproximations various g(r) constructed in this all cases, we seetothat soft particles feature slightly way, using L = 0.1σ , for particles identified as hard AA lower peaks and higher troughs in g(r). To see whether (black lines) and soft (red lines). this di↵erence is sufficient to identify soft particles, we In all cases, we seebeyond that soft particles slightly expand our analysis average valuesfeature of GX Y (i; µ), lower and higher troughs in g(r). To see whether to thepeaks distributions of the di↵erent structure functions. this difference is sufficient identify softofparticles, In Fig. 4(a) we show thetodistribution values ofwe AB AB GB where rpeak is theaverage locationvalues of the of first expand our) analysis beyond GX (i; µ), A (i; rpeak Ypeak gBAdistributions (r), for soft of particles (red) and hard particles toof the the different structure functions. While the the distribution for soft In(blue/green). Fig. 4(a) we show distribution of particles values of B AB AB GA (i; rpeak ) where rpeak is the location of the first peak of gBA (r), for soft particles (red) and hard particles (blue/green). While the distribution for soft particles
4 4 1.0 1.0
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FIG.(black 3. Radial functions hard lines) ordistribution soft (red lines) particles.averaged gAB and gover BA of soft FIG. 3. or Radial (red distributionparticles. functionsgAB averaged over hard (black lines) andrefer gBA soft particles are soft not equal lines) to each other since they toofdi↵er(blackare lines) or soft (red lines) particles. gAB and gBA of soft particles not equal to each other since they refer to different kinds of regions: neighbors of soft particles from species particles are not equal to each of other since they refer tospecies di↵erent A kinds regions: soft and of neighbors of neighbors soft particles fromparticles species B,from respectively. ent kinds of regions: neighbors of soft particles from species A and neighbors of soft particles from species B, respectively. A and 1.0 neighbors of soft particles from species B, respectively. 1.0 0.75 P P
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8 12 16 0 0.2 0.4 0.6 0.8 1 8 12 16 0 B0.2(i;0.4 0.8 1 B GA (i; rpeak ) 2.070.6 AA , 1, 2) BA B B AB(i; 2.07 AA , 1, 2) GA (i; rpeak )of GA (i; rBA FIG. 4. (a) Distribution B peak ), proportional to the AB AB gaussian weighted densityof at r , for), soft (red) andtohard FIG. 4. (a) Distribution GA (i; r proportional the B peakpeak A ABAB AB (i; rpeak ), proportional tohard the FIG.(blue/green) 4. (a) Distribution of Gat particles. rpeak to the first peak of Bcorresponds gaussian weighted density rpeak , for soft (red) and AB B soft (red) and hard AB gaussian weighted at rcorresponds rpeak theAA first of g(blue/green) gBA . particles. (b)density Distribution of , for 2.07 , 1, peak 2), propeak AB or AB (i; to ABof neighbors portional to with2.07 small bond angles corresponds theAA first peak of (blue/green) particles. rpeak gAB or gBA .the (b)density Distribution of B , 1, 2), proAB (i;to B with near a particle i, for soft (red) and hard (blue/green) particles. portional to the density of neighbors small bond angles gAB or gBA . (b) Distribution of ΨAB (i; 2.07σAA , 1, 2), proThe shows examples of configurations withbond correspondnearinset a to particle i, for soft and hard (blue/green) particles. portional the density of(red) neighbors with small angles ing radial and bond orientation properties, where (light) The inset shows examples of configurations with correspondnear a particle i, for soft (red) and hard (blue/green)dark particles. grayradial neighbors are oforientation species A properties, (B). and bond dark (light) The ing inset shows examples of configurations where with correspondgray neighbors are of species A (B).
ing radial and bond orientation properties, where dark (light) gray neighbors are of species A (B). features a single peak, that for hard particles is bimodal. features a singlethe peak, that forof hard is bimodal. This indicates existence (atparticles least) two distinct This indicates the existence (at least) two into distinct populations of hard particles of which we divide two features a singleofpeak, that for hard particles is bimodal. AB populations particles divide into groups: one withhard GB (i; rpeak )/rwhich < 1/2we (blue) that we two will A B AB of (at least) two distinct Thisgroups: indicates the G existence one with rpeak )/r B < 1/2 AB (blue) that we will A (i;with call H0 -type, and one G )/r > 1/2 (green) A (i; rAB peak B populations of hard particles which we)/r divide two call H -type, and one with G (i; r > 1/2into (green) 0 will call H1 -type. Radial that we information therefore A peak B AB groups: one with G (i; r )/r < 1/2 (blue) that we will that we will call Radial information AH1 -type. peak particles distinguishes between soft and H1 hardtherefore particles B AB distinguishes between soft G particles and H1>hard callbut H0 -type, and one (i; r 1/2particles (green) not between softwith and H hard particles. A peak )/r 0 and Radial H0 hard particles. thatbut we willbetween call H1soft -type. therefore Wenot now consider theinformation distribution of BWe now consider theand distribution of0 distinguishes between soft particles H hard particles (i; 2.07 , 1, 2) for soft particles (red), H 1 AA AB B 2.07 AA , 1, 2) soft particles. particles (red), H0 AB (i; but not between soft and for H0 hard
We now consider the distribution ΨB (i; 2.07σ , 1, 2) for soft particles (red), AA AB
of H0
4 hard particles (blue), (blue), and andHH hard particles (green), hard particles 1 1hard particles (green), B BA BA shown in Fig. Fig. 3(c). 3(c). Physically, Physically, B Ψ (i;(i; 2.07r 1, 2) shown in 2.07r , 1, ,2) peak ABAB peak hard particles and Hmany particles (green), 1 hard is large large when(blue), there are aremany pairs of neighbors of when there pairs of neighbors of B BA shown in Fig.particle 3(c). that Physically, 2.07r , 1,with ABa(i;a peak⇠ the central central particle thatlieliewithin within distance ξ2) with distance is large when there are many pairs of neighbors of oneone is of species small angles angles between betweenthem, them,such suchthat that is of species the central particle that lie within a distance ⇠ with fallfall intointo A and and one one isis ofof species speciesB.B.The Thesoft softparticles particles small angles between them, such that one is of species a single category (one peak, red) while H0H -type and a single category (one peak, red) while -type 0 A and one is of species B. The soft particles fall into and H -type hard particles–defined from radial information 1 hard particles–defined information 1 -type category aHsingle (one peak, red)from whileradial H0 -type and above–have very di↵erent distributions (blue andand green above–have very different distributions (blue H1 -type hard particles–defined from radial informationgreen peaks). Unlike before, andand thethe peaks). Unlike before,here herethe thesoft softparticles particles above–have very di↵erent distributions (blue and green H hard particles have very di↵erent distributions while 0 H0 hardUnlike particles havehere verythe different distributions peaks). before, soft particles and thewhile soft particles and H1 hard particles have similar distriH particles distributions while soft particles andhave H1 very harddi↵erent particles have similar distri0 hard butions. Bond-angle information therefore distinguishes soft particles and H1 hard particles therefore have similar distributions. Bond-angle information distinguishes between Bond-angle soft particles and H0 -type harddistinguishes particles but butions. information therefore between softsoft particles andand H0H -type hard particles but not between particles -type hard particles. 1 between soft particles and H0and -typeH1hard particles but not between soft particles -type hard particles. To between fully distinguish between soft-type and hard hardparticles. particles, not soft particles and H 1 To fully distinguish between soft and hard particles, both radial and bond-angle information is needed. Soft To fully distinguish between information soft and hard both radial and bond-angle is particles, needed. Soft particles that–at aisminimum–have both radialhave and environments bond-angle information needed. Soft particles have inenvironments that–at shell a minimum–have fewer particles their nearest neighbor and larger particles have environments that–at a minimum–have fewer particles in their nearest neighbor shell and larger angles between adjacent particles. fewer particles in their nearest neighbor shell and larger angles between adjacent particles. In summary, we have presented a novel ML method for angles between adjacent particles. identifying flow defects in disordered solids. We note that In summary, we have presented a novel ML method In summary, we have presented a novel ML method for for we have focused on the short-time correlation of note strucidentifying flowdefects defects disordered solids. We identifying flow in in disordered solids. We note that that ture withfocused particleonrearrangements. However, our we have focused onthethe short-time correlation of strucwe have short-time correlation of method strucshould shed light rearrangements. on the connection between local ture with particle rearrangements. However, ourstrucmethod ture with particle However, our method tural evolution and the correlation of rearrangements in should thethe connection between locallocal strucshouldshed shedlight lightonon connection between structime and space [30] over longer time scales in glassformtural correlation of rearrangements in in turalevolution evolutionand andthethe correlation of rearrangements ing liquids. We also note that we cannot predict the time and [30] over longer time scales in glassformtime andspace space [30] over longer time scales in glassformspecific particles that note will participate in rearrangements ing liquids. We also that we cannot predict the ing liquids. We also note that we cannot predict the at a later time; that rather, we identify ainpopulation of parspecific will participate specificparticles particles that will participate rearrangements in rearrangements ticles that are likely to rearrange. The latter quantity at wewe identify a population of parat aalater latertime; time;rather, rather, identify a population of paris more useful in thermal and/or sheared systems, since ticles that are likely to rearrange. The latter quantity ticles that are likely to rearrange. The latter quantity to stochasticity in rearrangements. isfluctuations more usefullead in thermal and/or sheared systems, since is Our moremethod useful in thermal and/or sheared systems, since on local structure alone, and can fluctuations lead relies to stochasticity in rearrangements. fluctuations lead to stochasticity in rearrangements. beOur applied directly of experimental sysmethod relies to on snapshots local structure alone, and can Our relies on localmethods structure alone, tems, inmethod contrast toto previous [12]. Ourand ap-can be applied directly snapshots of experimental sysbe applied directly to snapshots of [12]. experimental proach scales linearly withmethods the number of Our particles, tems, inalso contrast to previous ap- systems, in vibrational contrast tomode previous [12]. N , while approaches scaleof asparticles, N 3 .Our Theapproach also scales linearly with themethods number proach scales linearly with the scale number efficient identification of flow defects is critical testN , whilealso vibrational mode approaches as Nof3to . particles, The 3 on efficient identification flow approaches defects critical to N testing, while phenomenological approaches to is plasticity based N vibrational ofmode scale as . The ing to plasticity based on testflowphenomenological defects [6, 31–33].approaches Previous applications of machine efficient identification of flow defects is critical to flow defects [6, 31–33]. Previous applications machine learning methods in physics have focused onofapproximaing phenomenological approaches to plasticity based on learning methods in physics have focused on approximation [25, 34, 35] or on optimization and design [36– flow defects [6, 31–33]. Previous applicationstools of machine tion [25, 34, 35] or on optimization and design tools [36– 39]. Our approach shows that such methods–designed for learning methods in physics have focused on approxima39]. Our approach shows that such also methods–designed forto detecting subtle correlations–can be used directly tion [25, 34, 35] or on optimization and design tools [36– detecting subtle correlations–can also be usedwith directly to gain Our conceptual understanding achieved conven39]. approach shows thatnot such methods–designed for gain conceptual understanding not achieved with conventional approaches. detecting subtle correlations–can also be used directly to tional approaches. E.D.C. and S.S.S. contributed equally to this work. We gain conceptual understanding not achieved with convenE.D.C. and Waterland, S.S.S. contributed equally toDaniel this work. We thank Amos Carl Goodrich, Sussman tionalAmos approaches. thank Waterland, Carl Goodrich, Daniel and Franz Spaepen for helpful discussions. ThisSussman work was E.D.C. and S.S.S. equally to this work. and Franz Spaepen for contributed helpful discussions. This work was We supported by the UPENN MRSEC, NSF-DMR-1120901 thank Amos Waterland, Carl Goodrich, Daniel Sussman supported by the UPENN MRSEC, NSF-DMR-1120901 (S.S.S., J.M.R.), NSF DMR-1305199 (D.J.D.), U.S. Deand Franz Spaepen helpful This work (S.S.S., J.M.R.), NSFfor DMR-1305199 (D.J.D.), U.S. De- was partment of Energy, Office of discussions. Basic Energy Sciences, partment ofby Energy, Office of Basic Energy Sciences, supported the UPENN MRSEC, NSF-DMR-1120901 Division of Materials Sciences and Engineering under Division of Materials and Engineering under (S.S.S., J.M.R.), NSFSciences DMR-1305199 (D.J.D.), U.S. Award DE-FG02-05ER46199 (A.J.L.), UBC Killam Fac-DeAward DE-FG02-05ER46199 (A.J.L.), UBC Killam Facpartment of Fellowship Energy, Office ofand Basic Energy ulty Research (J.R.), Harvard IACSSciences, Stuulty Research Fellowship (J.R.), andand Harvard IACS StuDivision of Materials Sciences Engineering under Award DE-FG02-05ER46199 (A.J.L.), UBC Killam Faculty Research Fellowship (J.R.), and Harvard IACS Stu-
5 dent Scholarship (E.D.C.).
∗ †
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
[14] [15]
[16] [17]
[email protected] [email protected] G. I. Taylor, Proc. R. Soc. Lond. A Math. Phys. Sci. 145, 362 (1934). A. M. Alsayed, M. F. Islam, J. Zhang, P. J. Collings, and A. G. Yodh, Science 309, 1207 (2005). A. S. Argon and H. Y. Kuo, Materials science and Engineering 39, 101 (1979). A. S. Argon, Acta metallurgica 27. M. L. Falk and J. S. Langer, Phys. Rev. E 57, 7192 (1998). L. Bocquet, A. Colin, and A. Ajdari, Phys. Rev. Lett. 103, 036001 (2009). J. Rottler, S. S. Schoenholz, and A. J. Liu, Phys. Rev. E 89, 042304 (2014). C. Brito and M. Wyart, J. Stat. Mech. , L08003 (2007). A. Widmer-Cooper, H. Perry, P. Harrowell, and D. R. Reichman, Nat. Phys. 4, 711 (2008). A. Widmer-Cooper, H. Perry, P. Harrowell, D. R. Reichman, et al., J. Chem. Phys. 131, 194508 (2009). R. Candelier, A. Widmer-Cooper, J. K. Kummerfeld, O. Dauchot, G. Biroli, P. Harrowell, and D. R. Reichman, Phys. Rev. Lett. 105, 135702 (2010). M. L. Manning and A. J. Liu, Phys. Rev. Lett. 107, 108302 (2011). K. Chen, M. L. Manning, P. J. Yunker, W. G. Ellenbroek, Z. Zhang, A. J. Liu, and A. G. Yodh, Phys. Rev. Lett. 107, 108301 (2011). G. M. Hocky and D. R. Reichman, J. Chem. Phys. 138, 12A537 (2013). K. Chen, T. Still, S. Schoenholz, K. B. Aptowicz, M. Schindler, A. C. Maggs, A. J. Liu, and A. G. Yodh, Phys. Rev. E. 88, 022315 (2013). S. S. Schoenholz, A. J. Liu, R. A. Riggleman, and J. Rottler, Phys. Rev. X 4, 031014 (2014). R. L. Jack, A. J. Dunleavy, and C. P. Royall, Phys. Rev. Lett. 113, 095703 (2014).
[18] J. Gilman, Phys. Today 28, 46 (1975). [19] P. Chaudhari, A. Levi, and P. Steinhardt, Phys. Rev. Lett. 43, 1517 (1979). [20] A. Widmer-Cooper, P. Harrowell, and H. Fynewever, Phys. Rev. Lett. 93, 135701 (2004). [21] A. Widmer-Cooper and P. Harrowell, J. Chem. Phys. 126, 154503 (2007). [22] D. Coslovich and G. Pastore, J. Chem. Phys. 75, 784 (2006). [23] D. Coslovich and G. Pastore, J. Chem. Phys. 127, 124504 (2007). [24] A. Malins, J. Eggers, H. Tanaka, and C. P. Royall, Faraday Discuss. 167, 405 (2013). [25] J. Behler and M. Parrinello, Phys. Rev. Lett. 98, 146401 (2007). [26] C. Cortes and V. Vapnik, Mach. Learn. 20, 273 (1995). [27] C.-C. Chang and C.-J. Lin, ACM Transactions on Intelligent Systems and Technology 2, 27:1 (2011). [28] R. Br¨ uning, D. A. St-Onge, S. Patterson, and W. Kob, J. Phys.: Condens. Matter 21, 035117 (2009). [29] S. Plimpton, J. Comp. Phys. 117, 1 (1995). [30] A. S. Keys, L. O. Hedges, J. P. Garrahan, S. C. Glotzer, and D. Chandler, Phys. Rev. X. 1, 021013 (2011). [31] M. L. Falk and J. S. Langer, Ann. Rev. of Cond. Matt. Phys. 2, 353 (2011). [32] V. Mansard, A. Colin, P. Chauduri, and L. Bocquet, Soft Matter 7 (2011). [33] J. Lin, A. Saade, E. Lerner, A. Rosso, and M. Wyart, Europhys. Lett. 105, 26003 (2014). [34] G. L. Hart, V. Blum, M. J. Walorski, and A. Zunger, Nat. Mater. 4, 391 (2005). [35] R. Z. Khaliullin, H. Eshet, T. D. K¨ uhne, J. Behler, and M. Parrinello, Nat. Mater. 10, 693 (2011). [36] C. C. Fischer, K. J. Tibbetts, D. Morgan, and G. Ceder, Nat. Mater. 5, 641 (2006). [37] C. J. Pickard and R. Needs, Nat. Mater. 9, 624 (2010). [38] B. Huskinson, M. P. Marshak, C. Suh, S. Er, M. R. Gerhardt, C. J. Galvin, X. Chen, A. Aspuru-Guzik, R. G. Gordon, and M. J. Aziz, Nature (London) 505, 195 (2014). [39] S. Curtarolo, G. L. Hart, M. B. Nardelli, N. Mingo, S. Sanvito, and O. Levy, Nat. Mater. 12, 191 (2013).
