Assessing the role of static length scales behind glassydynamics in polydisperse hard disksJohn Russo1 and Hajime Tanaka1

Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

Edited by James S. Langer, University of California, Santa Barbara, CA, and approved April 24, 2015 (received for review January 29, 2015)

The possible role of growing static order in the dynamical slowingdown toward the glass transition has recently attracted consider-able attention. On the basis of random first-order transitiontheory, a new method to measure the static correlation length ofamorphous order, called “point-to-set” (PTS) length, has been pro-posed and used to show that the dynamic length grows muchfaster than the static length. Here, we study the nature of thePTS length, using a polydisperse hard-disk system, which is amodel that is known to exhibit a growing hexatic order upondensification. We show that the PTS correlation length is decoupledfrom the steeper increase of the correlation length of hexatic orderand dynamic heterogeneity, while closely mirroring the decaylength of two-body density correlations. Our results thus provide aclear example that other forms of order can play an important rolein the slowing down of the dynamics, casting a serious doubt on theorder-agnostic nature of the PTS length and its relevance to slowdynamics, provided that a polydisperse hard-disk system is a typicalglass former.

glass transition | structural length scales | pinning | hexatic order |slow dynamics

When we supercool a liquid while avoiding crystallization,dynamics becomes heterogeneous (1, 2) and slows down

significantly toward the glass transition, below which a systembecomes a nonergodic state. Now there is a consensus that thisslowing down accompanies the growth of dynamical correlationlength (3). Several different physical scenarios have been pro-posed, yet the origin is still a matter of serious debate: althoughsome scenarios describe the glass transition as a purely kineticphenomenon (4), others posit a growing static order (5) or a lossof configurational entropy (6) behind dynamical slowing down.Among this last category, we will focus here on two distinct ap-proaches. The first one is random first-order transition (RFOT)theory (7–9), which is based on a finite dimensional extension ofmean-field models with an exponentially large number of meta-stable states. The second approach, recently proposed by some of us(10, 11), ascribes the growth of the dynamical correlation lengthwith the corresponding growth of the static correlation length. Here,we focus on these two scenarios based on static order and considerwhich is more relevant to the origin of glassy slow dynamics, using asimple model glass former, 2D polydisperse hard disks (12, 13).In RFOT, metastable states are thought to have amorphous

order, whose correlation length diverges toward the ideal glasstransition point. It was recently suggested that the so-calledpoint-to-set (PTS) length, which is the correlation length ofamorphous order, can be extracted by pinning a finite fraction ofparticles and studying the dependence of the overlap function onthe pinning particle concentration. According to the RFOT theory,amorphous order develops in any glass-forming liquids and thismethod is thought to be able to pick up the static correlation lengthwhatever the order is, i.e., the method is claimed to be order ag-nostic (14). Thus, the use of pinning fields has been considered tobe a promising new direction in the study of the glass transition.Within the RFOT theory, it was shown that freezing the positions ofa finite concentration of particles shifts the ideal glass transition tohigher temperatures, potentially granting access to the glass state in

equilibrium (15–22). Moreover, the average distance betweenpinned particles at the liquid-to-glass transition represents a di-rect measure of the PTS correlation length. PTS correlationlengths aim at measuring hidden static length scales by looking atthe extent of the perturbation induced by frozen particles on therest of the liquid. It is intuitively defined as the average distancebetween pinned particles that forces the system to stay in anamorphous configuration with a vanishing configurational en-tropy. The reasons behind the popularity of PTS correlationlengths in the study of the glass transition are at least twofold:(i) they are expected to provide an “order-agnostic” method tomeasure static correlations (23, 24); (ii) it is theoretically establishedthat no divergence of the relaxation time of a glass at finite tem-perature can occur without the concomitant divergence of the staticcorrelation length (25).On the other hand, it was recently noted (10, 11, 13) that, for

moderately polydisperse hard disks, an increase in the areafraction of particles ϕ, hexatic (or sixfold bond orientational)order grows and its correlation length ξ6 is supposed to diverge,obeying the Ising-like power law, toward the ideal glass transitionpoint ϕ0, where the structural relaxation time τα diverges fol-lowing the Vogel–Fulcher–Tammann law. We have also con-firmed that the dynamical correlation length ξ4 is proportional tothe hexatic correlation length ξ6, and furthermore there is almosta one-to-one correspondence between the degree of hexatic or-der and the slowness of dynamics. These results suggest an in-timate link between static order and dynamics: the dynamicalslowing down is accompanied by an increase in both size andlifetime of hexatic ordered regions. We also found that 3D poly-disperse hard and Lennard–Jones spheres exhibit essentially thesame behavior (11). These results suggested that the dynamicalslowing down is a consequence of the growing activation energyassociated with the Ising-type power-law growth of the correlation

Significance

The origin of dynamical slowing down toward glass transition is afundamental unsolved problem in condensed matter physics. Acrucial question is whether this slowing down has a structuralorigin. Recently, a method to detect hidden order within the fluidwas proposed, based on the idea that freezing a fraction of theparticles in a system causes a transition akin to glass transition.Here, we show that a glass former, polydisperse hard disks, has astrong increase of structural order, well correlated with slow dy-namics, which goes undetected by the pinning method. This castsdoubt on the order-agnostic qualities of the pinning length scaleand keeps static length scales in the race for plausible explana-tions of the glass transition problem.

Author contributions: H.T. designed research; J.R. performed research; J.R. analyzed data;and J.R. and H.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. Email: tanaka@iis.u-tokyo.ac.jp or russoj@iis.u-tokyo.ac.jp.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1501911112/-/DCSupplemental.

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length of critical-like fluctuations of static order toward theideal glass transition point (10, 11). Recently, a theory for theoccurrence of such criticality in disordered systems with topo-logically ordered cluster of particles was also proposed byLanger (26).The role of local order on the dynamics of glassy systems re-

mains controversial at least for two reasons. The first problem isthat the local order that one needs to measure is system de-pendent, and up to now the relevance of bond-orientationalorder was demonstrated only for polydisperse particle systemsand a spin liquid, and not for bidisperse systems (10, 11). Thesecond problem is conceptual: are static correlations reallyresponsible for the dynamical slowing down? The PTS cor-relation length is often described as a remedy to both prob-lems, because it should be able to detect static correlationswithout a detailed knowledge of the local order involved in thesecorrelations. In studies of binary mixtures of hard spheres, the PTScorrelation length was shown to grow only modestly in the regimeaccessible to computer simulation (14, 23), differently from thedynamical correlation length, which grows much more rapidly.These results suggested that no link exists between a singlestatic length scale and the dynamical slowing down (the onlyexception would be close to a possible ideal glass transitiontemperature) (5, 14). On the other hand, measures of the PTScorrelation length (27) have shown that it correlates well with theaverage dynamics of the system, and with dynamic heterogeneities

(28). The PTS correlation length is in agreement with measures ofthe density of plastic modes (29), providing support for the ideaof a fundamental length scale controlling the dynamics of thesupercooled liquids.Unlike previous studies, in this work we measure the PTS

correlation length in a system for which a growing local orderwas previously found, i.e., polydisperse hard disks. This will allowus to compare the growth of PTS correlation lengths with that ofbond-orientational lengths. We will consider several pinningstrategies (random pinning, uniform pinning, and cavity pinning)(Fig. 1 A–C), and then look for the underlying structural featuresthat are captured by the PTS correlation length. In principle,each different pinning geometry probes a different length scale(30). The PTS static length scale ξPTS was first introduced in thespherical cavity geometry (31). Both random and uniform pin-ning are expected to express the same static length scale, herecalled ξK, and the RFOT theory predicts a temperature scalingrelation between ξPTS and ξK, which in its simplest form is writtenas ξK ðTÞ∼ ξPTSðTÞ1=2 (17). If the PTS correlation method is in-deed order agnostic, it should be able to pick up the correlationlength of hexatic order in a polydisperse hard-disk system, pro-vided that it is a typical glass former. Thus, it is a main interest ofthis article to reveal whether the PTS length is the same as thehexatic correlation length. This question is of crucial importanceto reveal the origin of slow glassy dynamics.

A

B

C

D

E F

Fig. 1. Growth of static and dynamic correlation lengths. Different pinning strategies, with pinned particles colored in black: (A) random pinning,(B) uniform pinning, and (C) cavity pinning. (D) Correlation lengths as a function of density ρ: red squares for bond-orientational order ξ6, blue triangles for thetwo-body correlations ξ2, black circles for ξK for random pinning, green diamonds for ξK for uniform pinning, black-filled circles for ξPTS, and pink triangles forthe dynamic correlation length ξ4. The dynamical correlation length is scaled with ξ04 ∼ 1.7 to ease the visual comparison with the static length scales. The Insetshows the scaling of the structural relaxation time τα with the hexatic correlation length ξ6 (points), and the fit with the relation τα = τ0 expðDξ=ξ0Þ (dashedline). (E) Snapshot of a configuration at ρ= 0.92 in which the disks are colored according to the following criteria: white, low mobility and high order; black,high mobility and low order; cyan, low mobility and low order; and magenta, high mobility and high order. (F) Same as in the previous panel, but for ρ= 0.97.

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Results: Unpinned CaseThe system studied is composed of N = 10,000 polydisperse harddisks with disk-size polydispersity Δ= 11% (Materials and Meth-ods). We start by considering the case without an external pin-ning field, c= 0. We focus on the following number densitiesfrom ρ= 0.92 to ρ= 0.97, which correspond to area fractionsranging from ϕ= 0.73 to ϕ= 0.77. As described in SI Appendix,we extract the correlation length for bond-orientationally or-dered regions (SI Appendix, Fig. S1) by fitting the exponentialdecay of the peaks of the correlation function g6ðrÞ=gðrÞ (SIAppendix, Fig. S2). We use an exponential function instead of a2D Ornstein–Zernike function to avoid a priori assumptions onthe origin of the growth of the correlation length. The correla-tion length ξ6 is plotted in Fig. 1D, together with other lengthscales that we will derive later. The two-body correlation func-tion ξ2 is obtained by fitting with an exponential law the decay ofgðrÞ− 1 (SI Appendix, Fig. S3). The results of this fit are alsosummarized in Fig. 1D. These results confirm that, for poly-disperse glass-forming systems, the growth of many-body corre-lations associated with bond-orientational order is much fasterthan the growth of two-body correlations (11, 13, 32).As was shown in refs. 11 and 13, there is a link between dy-

namic heterogeneities and regions of high hexatic order in thefluid. We provide here further support to this scenario. Wemeasure dynamical length scales by performing event-drivenmolecular dynamics simulations and extracting the four-pointdensity correlator, with the procedure described in SI Appendix,Figs. S4 and S5. In Fig. 1D, we plot the dynamical correlationlength ξ4, showing that indeed ξ4 scales with density like ξ6. Toshow that there is a causal relation between slow dynamical re-gions and regions of high hexatic order, in Fig. 1 E and F, we plotsnapshots of configurations prepared at ρ= 0.92 and ρ= 0.97,respectively. For each snapshot we run event-driven moleculardynamics simulations in the isoconfigurational ensemble (Mate-rials and Methods), where 200 trajectories are started from thesame initial configuration but with different initial velocities. Thedegree of structural order is investigated by taking the averagehexatic field over these N = 200 trajectories (also called iso-configurational average) at a time t= τα=10, where τα is thestructural relaxation time measured through the intermediatescattering function (SI Appendix, Fig. S6). The dynamics is in-stead investigated through the isoconfigurational average of thedisplacement jΔrij between t= 0 and t= τα, which is also ap-proximately the time at which the heterogeneities are maximum(as measured by the four-point susceptibility shown in SI Ap-pendix, Fig. S4). All disks in the snapshots of Fig. 1 E and F arethen grouped into sets of high and low mobility/order, dependingon whether their mobility/order is higher or lower than the 50thpercentile. We can then identify four different sets of particles:low mobility and high order (white); high mobility and low order(black); low mobility and low order (orange); and high mobilityand high order (magenta). Our results show that, for all densitiesconsidered in this work, 66% of particles are in the first two sets(33% in each), demonstrating a high degree of correlation be-tween structural ordered regions and immobile regions (or, viceversa, between disordered regions and mobile regions) even at aparticle level. Moreover, the remaining two sets (each account-ing for the 17% of particles) are located at the interface betweenmobile and immobile extended regions. Magenta disks are lo-cated on the surface and in between black clusters, whereas or-ange disks are located on the surface and in between whiteclusters. In other words, disks that are next to a low/high mobilityregion will also have low/high mobility. Here, we note that theembedded fractal nature of order parameter fluctuations ischaracteristic of critical fluctuations. In SI Appendix, we also notethat a higher degree of correlation can be obtained if the relativedisplacement jΔRij is used instead of the absolute displacement

jΔrij. The relative displacement jΔRij is defined as the dis-placement between time t= 0 and t= τα, of particle i with respectto itsM neighbors, Ri = ri − ð1=MÞPM

j rj. The coherent motion of aparticle with its neighbors does not contribute to the stress re-laxation as no bonds are broken in this process. In this case, thefirst two sets account for 76% of the particles (SI Appendix,Fig. S7).

Results: PinningHaving characterized the static and dynamic properties of theunperturbed system, we now introduce the pinning field. First,simulations are fully equilibrated with cluster-moves algorithms(Materials and Methods). A representation of the pinning fieldsis given in Fig. 1 A–C. In the “random-pinning” geometry,Np particles are chosen randomly and pinned, i.e., their posi-tion is kept fixed during the course of the simulations. Foreach density, we introduce pinning fields with concentrationsc= 0.01,  0.06,  0.10,  0.15,  and  0.20, and for each concentrationwe average over nine different realizations of the fields. In thisscheme, the distance between pinned particles is defined only asan average over a broad distribution, as both clusters of pinnedparticles and extended regions without pinned particles are likelyproduced. For this reason, random pinning is expected to bemore sensitive to finite size effects, as was observed in ref. 19,where it is noted that random pinning can smear out the Kauz-mann transition in very small systems. The pinning geometry canalso have strong effects on the dynamics (21, 30, 33, 34). To limitthe fluctuations in the distance between pinned particles, we alsoadopt a “uniform-pinning” geometry, where a simple cubic lat-tice is overlaid to the equilibrated configuration, and the closestparticle to each lattice point is pinned. Particles are pinned atthe following average distances: a= 2.5,  2.75,  3,  4.25,  6,  8,  and  10,and each distance is averaged over seven realizations of the field.Finally, in the “cavity-pinning” geometry, all particles outside acavity of radius R are pinned. Because the static length scalescurrently accessible to simulations are expected to be small, wellwithin 10σ, simulations with cavity pinning involve a small numberof particles, thus requiring extensive average over differentrealizations of the field (here 100 simulations for each cavitydiameter).Because the pinning field is applied to equilibrium configu-

rations, the static properties should be unchanged with respect tothe c= 0 case. We check this by computing both positionaland hexatic order for different concentrations c. All results areconsistent with the c= 0 case and the SD between simulations atdifferent c is represented with the error bars for ξ6 in Fig. 1D.Correlation lengths are extracted from all pinning geometries,following the procedure outlined in SI Appendix, Figs. S8–S12. Inall cases, the physical idea is to detect the characteristic length(average distance between pinned particles in the random- anduniform-pinning geometries, or the size of the cavity in the cavitygeometry), which produces a high localization of the mobileparticles, as measured by overlap functions. We plot the random-pinning correlation length ξK,random, the uniform-pinning corre-lation length ξK ,uniform, and the PTS length scale from cavitypinning ξPTS in Fig. 1D (see SI Appendix, Figs. S8–S12, on thedetails of its estimation). We see that the growth of the PTSlength scale, irrespective of the pinning strategy, is significantlyslower than the growth of bond-orientational order ξ6, whilebeing comparable to the growth of pair correlations ξ2 in thesystem. We also confirm that estimating the PTS length fromuniform and random pinning through the relation ξPTSðTÞ∼ ξ2KðTÞstill produces a much weaker growth than that of ξ6. The Inset ofFig. 1D shows the scaling between the ξ6 length scale and the re-laxation time τα, τα = τ0 expðDξ=ξ0Þ, where D is a measure of thefragility of the system (10). This scaling also supports a directconnection between the growth of structural correlation and slowdynamics, but its origin is still unclear. A justification of this relation

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in the context of a theory of topological ordered clusters has beengiven in ref. 26.All static length scales can also be obtained by considering

coarse-grained variables, dividing the simulation box into smallerboxes of side length l= 0.3σ, ensuring that each box can be oc-cupied at most by one disk at any time during the simulation. Theproblem is then mapped on a set of discrete variables, defined asnαi = 1 when the center of a disk is in box i in configuration α, andnαi = 0 otherwise. The average occupancy for a particular re-alization of the pinning field is defined as ni, where the overlinedenotes an average over thermal fluctuations for a fixed re-alization of the pinning field. Fig. 2 shows the field ni for threedifferent concentrations of the pinned particles for the case ofuniform pinning at ρ= 0.92 (top row) and ρ= 0.97 (bottom row).Going from the low to high concentration (Left to Right in thefigure), the amount of localization progressively increases. Lo-calization occurs when the occupancy field becomes stronglypeaked in discrete locations in space, surrounded by regions ofnegligible probability occupancy. For ρ= 0.92 (top row in Fig. 2),the displayed pinning distances a are always bigger than thepinning correlation length (Fig. 1D), and the occupancy field ni ishomogeneous. Instead, for ρ= 0.97 (bottom row in Fig. 2), re-gions of high localization progressively appear and form a con-nected network close to a= 3, which is the pinning correlationlength at this density (ξK,uniform ∼ a∼ 3). So the pinning correla-tion length ξK is the length scale at which localized regions areclose to percolation. In SI Appendix, Fig. S13, we show that, atthe level of undercooling that we can reach, the transition from ahomogenous state to a localized state is continuous.A visual inspection of Fig. 2 (bottom row) shows that localized

particles have a high degree of hexatic order (SI Appendix, Fig.S13), already suggesting that ξK is smaller than the hexatic cor-relation length ξ6. From Fig. 1D, we know that the pinning lengthscale ξK is close to the pair correlation length ξ2. Each pinnedparticle generates an oscillatory perturbation of the ni field,which originates from the two-body static correlations betweenthe pinned particle and the mobile particles in the liquid, andwhose extent is thus given by ξ2. The length scales measured inFig. 1D thus indicate that the localization occurs when the av-erage distance between pinned particles is comparable to therange of two-body correlations, ξ2, below which the number ofparticle arrangements drastically decreases, and the configurational

entropy vanishes. The extent of the regions with high localizationof particles is exactly what is being measured by the PTS correla-tion function.The results thus show that the growth of the pinning correla-

tion length is similar to the growth of two-body correlations. Onthe other hand, the growth of bond-orientational order is muchfaster and clearly decoupled from the pinning correlation length.We also checked that the same is true for coarse-grainedquantities. This strongly indicates that the PTS correlation lengthis not order agnostic but targets the growth of a particular orderin the system, that is, the size of the regions where particles arelocalized due to the pinning field. The growth of these regionsfollows the growth of two-body correlations: particles are local-ized due to the perturbation that pinned particles introduce tothe ni field, and the length scale of this perturbation is given bytwo-body correlations. In other words, at least in the densityrange considered here, the localization transition due to pointpinning requires that the average distance between pinned par-ticles is smaller than the two-body correlation length.

Discussion and ConclusionsIn this article, we have extracted several static length scales fromsystems of polydisperse hard disks with polydispersity Δ= 0.11, inthe range ρ∈ ½0.92; 0.97�. The results confirmed that the lengthscale associated with bond-orientational order grows more rap-idly than the length of pair correlations (11, 13, 32). The use ofpinning fields enabled the calculation of the PTS correlationlength, showing that it grows only moderately with increasingsupercooling, a result that is in agreement with measures of thePTS length in binary mixtures (23). For polydisperse systems, thePTS correlation length is not coupled to that of bond-orienta-tional order, which is directly linked to the dynamical correlationlength (10, 11): the growth of the former is considerably slowerthan the latter. For different glass-forming systems, this suggeststhat also other forms of order originating from many-body in-teractions could go undetected by PTS measures.The PTS length captures a localization transition that occurs

in presence of pinned particles. This localization transitionoriginates when the occupancy field, n, has extended regions ofhigh probability due to neighboring pinned particles. The per-turbation that a single pinned particle produces in the n field isdue to “pair correlations.” In absence of strong nonlinear effects,

Fig. 2. Development of ni field with a decrease in a for uniform pinning. The particle density is ρ= 0.92 (top row) and ρ= 0.97 (bottom row), and the averagedistance between particles a= 10 (Left), a= 4.25 (Middle), and a=3 (Right). The color code is mapped to the occupancy probability according to the color bar.The Inset inside some of the panels shows the magnification of the Bottom Left corner of the image of an area of one-eighth of the total image. Distances arein units of the average diameter hσi.

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the length scale of the localized regions extends no further thantwo-body correlations. This is the case in the density intervalaccessible to our simulations, where pinned particles need to beplaced closer than the pair correlation length to produce local-ized regions in the fluid. A second requirement that our resultssuggest is that pinned particles should be in positions compatiblewith high local hexatic order.The localization transition that occurs with increasing con-

centration of pinned particles happens continuously in the den-sity range we could access in equilibrium. The results do not ruleout the possibility that strong nonlinearities will produce a lo-calization transition that extends beyond pair correlations forhigher (but yet unreachable) densities.To summarize, the PTS length measured by particle pinning

simply reflects pair correlation and fails in detecting the corre-lation of bond-orientational order (more precisely, hexatic or-der), which intrinsically originates from many-body interactions.Although the PTS length is decoupled from the dynamical cor-relation length, the hexatic order correlation is strongly coupledto it. This implies that slow dynamics in our system is controlledby the development of hexatic ordering, and not by translationalorder detected by the PTS correlation. Although the generalityof this conclusion needs to be checked carefully, our study sug-gests that the PTS length is not order agnostic and the growth ofthe PTS length is not responsible for glassy slow dynamics, atleast for our system.Differently from the fluid-hexatic transition, the results in refs.

11 and 13 have provided evidence that the glass transition inpolydisperse hard disks involves a power-law growth of thehexatic correlations that follows the Ising universality class, but,differently from ordinary critical phenomena, is accompanied by

a strong divergence [logðταÞ∼ ξd=2] of the relaxation time. Thisbehavior could originate from frustration and disorder effects,which cause the energy barriers for viscous flow to increasewith decreasing temperature (10). Understanding the relationbetween power-law growth of the ξ6 and the divergence ofdynamics represents the main challenge to be addressed infuture work.

Materials and MethodsWe study 2D polydisperse hard disks with Monte Carlo simulations. Thediameter σ of the disks is extracted from a Gaussian distribution, and the

polydispersity is defined as the SD of the distribution, Δ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÆσ2æ− Æσæ2

q=Æσæ.

In the present work, we fix Δ= 11% for which no transition to a hexatic

phase is observed. The unit of length is set by the average disk diameter Æσæ.All simulations are run at fixed densities ρ=N=V, with N= 10,000, with the

event-chain algorithm (35, 36), which allows for a fast equilibration even atvery high densities. After the equilibration run, we activate the pinning fieldand switch to Metropolis dynamics with swap moves between randomlyselected pairs of nonpinned particles.

The connection between static and dynamic length scale shown inFig. 1D was obtained with event-driven simulations (37) in the isoconfigura-tional ensemble (38), where 200 trajectories are started from an equili-brated configuration at ρ= 0.97 but with a different assignment of initialvelocities.

Details are given in the SI Appendix.

ACKNOWLEDGMENTS. We are grateful to the following individuals forvaluable comments and constructive criticisms: Ludvic Berthier, Gulio Biroli,Patrick Charbonneau, Walter Kob, Jim Langer, David Reichman, Gilles Tarjus,and Sho Yaida. This study was partly supported by Grants-in-Aid forScientific Research (S) and Specially Promoted Research from the JapanSociety for the Promotion of Science.

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6924 | www.pnas.org/cgi/doi/10.1073/pnas.1501911112 Russo and Tanaka

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