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MATER IALS SC I ENCE

1School of Chemical and Biomedical Engineering, Nanyang Technological Univer-sity, 62 Nanyang Drive, Singapore 637459, Singapore. 2Division of Physics andApplied Physics, School of Physical and Mathematical Sciences, Nanyang Techno-logical University, 21 Nanyang Link, Singapore 637371, Singapore. 3CNR-SPIN, Di-partimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy.*Corresponding author. Email: massimo@ntu.edu.sg (M.P.C.); r.ni@ntu.edu.sg (R.N.)

Lei et al., Sci. Adv. 2019;5 : eaau7423 25 January 2019

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Nonequilibrium strongly hyperuniform fluids of circleactive particles with large local density fluctuationsQun-Li Lei1, Massimo Pica Ciamarra2,3*, Ran Ni1*

Disordered hyperuniform structures are an exotic state of matter having vanishing long-wavelength density fluctua-tions similar to perfect crystals but without long-range order. Although its importance in materials science has beenbrought to the fore in past decades, the rational design of experimentally realizable disordered strongly hyperuniformmicrostructures remains challenging. Here we find a new type of nonequilibrium fluid with strong hyperuniformity intwo-dimensional systems of chiral active particles, where particles perform independent circularmotions of the radiusR with the same handedness. This new hyperuniform fluid features a special length scale, i.e., the diameter of thecircular trajectory of particles, below which large density fluctuations are observed. By developing a dynamicmean-field theory,we show that the large local density fluctuations canbe explained as amotility-inducedmicrophaseseparation, while the Fickian diffusion at large length scales and local center-of-mass-conserved noises are responsiblefor the global hyperuniformity.

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INTRODUCTIONPerfectly ordered structures, such as crystals or quasi-crystals at zerotemperature, are usually associated with some discrete symmetriesand exhibit long-range correlations, leading to the structure factorof the system S(q → 0) = 0 (1). Similarly, the local density variancein these structures ⟨dr2⟩ scales with the window size of observation Las ⟨dr2⟩ ~ L−l with l = d + 1, where d is the dimensionality of thesystem. In contrast, in normal disordered structures, e.g., conventionalgases, liquids and amorphous solids, and even thermalized crytals, thelong-wavelength density fluctuationmakes S(q→ 0) = const. > 0 andl = d. Recently, the concept of hyperuniformity was introduced tostudy the state of matter (2). A structure is hyperuniform when it hasvanishing long-wavelength density fluctuations, i.e., S(q→ 0) ~ qa→ 0witha > 0 and ⟨dr2⟩~L−lwith d< l≤ d+1 (2). It has been found in thepast two decades that, besides the ordered hyperuniform structures, i.e.,perfect crystals and quasi-crystals, a number of disordered structures arealso hyperuniform, including the maximally random jammed packing(3), avian photoreceptor patterns (4), and some nonequilibriumsystems (5–10).

Disordered hyperuniform structures have received an increasingamount of scientific attention, as some strongly hyperuniformdisorderedstructures with l = d + 1 exhibit similar properties as crystals with evenbetter performance, e.g., large isotropic photonic bandgaps insensitiveto defects (11) can be opened at low dielectric contrast (12). Althoughideal hyperuniform structures similar to perfect crystals are unavoidablyaffected by thermal excitation, it still shows promise in designing robustdisordered materials with novel functionalities (1, 13–16). By far, vari-ous protocols were developed to design particle interactions to formdisordered hyperuniform ground states in classical many-particlesystems. However, the resulting interactions normally have delicatelong-range or multibody terms (1), making their experimental realiza-tion highly challenging. An alternative approach is to use driven systems,e.g., self-organized colloidal suspensions under periodic shearing(17), to form nonequilibrium dynamic hyperuniform states, which can

effectively avoid the dynamic trapping, and, in principle, have a higherresistance to thermal perturbations (7). Now, experimentalists onlysucceeded in generating weakly hyperuniform structures with l ≃d + 0.5 (8) using this method. Nevertheless, this is certainly a direc-tion that deserves further investigation (10), as some self-drivensystems, or active matter systems, have produced a number of un-expected emergent phenomenanever found in correspondingequilibriumsystems (18–22). However, in conventional active matter systems, i.e.,active nematic systems and active Brownian particles systems, giantnumber fluctuations characterized by l < d (19) and motility-inducedphase separation (MIPS)with l≃ 0 (20–23) are usually observed. Theselarge density fluctuations seemingly prohibit the formation of hyperuni-form structures in active matter systems.

Recently, chiral active matter whose motion is chiral-symmetrybroken, e.g., active particles/swimmers with circular motion in two-dimensional (2D) (24–28) or active spinner fluids (29), has attractedconsiderable attention. Both experiments and simulations have shownmany interesting collective phenomena in these systems (25, 30–33). Inthis work, using computer simulations combinedwith analytic theories,we study a 2D system of circle active particles, which performindependent circular motion with the same handedness and randomcircling phases. We show that in the limit of strong driving or zerothermal noise, with increasing density of particles or radius of circularmotion R, the system undergoes an absorbing-active transition forminga nonequilibrium strongly hyperuniform fluid phase with density var-iance ⟨dr2⟩ ~ L−3 (L → ∞) the same as in perfect crystals. Furtherincreasing the density or R triggers the formation of dynamic clusters,which results in large local density fluctuations. These fluctuations are“confined” within the length scale of R, while the strong hyperunifor-mity persists at large length scales. This surprising coexistence of largelocal density fluctuations and the global hyperuniformity is explainedby dynamic mean-field theories at different length scales.

RESULTSModelAs illustrated in Fig. 1A, we consider a 2D suspension of N activecolloidal particles with diameter s. Each particle experiences an in-plane force Fp with random initial orientation and a constant torqueW perpendicular to the plane, which drive the particles to perform

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circular motion with the same handiness (30, 31). The dynamics ofparticle i at finite temperature T is governed by the over-dampedLangevin equations (26, 30)

r:iðtÞ ¼ g�1

t ½ � ∇iUðtÞ þ FpeiðtÞ� þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBT=gt

pxtiðtÞ ð1Þ

e:iðtÞ ¼ g�1

r Wþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kBT=gr

pxri ðtÞ

h i� eiðtÞ ð2Þ

where ri and ei are the position of particle i and its self-propulsionorientation, respectively, and kB is the Boltzmann constant. gt/r is thetranslational/rotational friction coefficient. For simplicity, we set gt =gr/s

2.xtiðtÞandxriðtÞ represent Gaussian noises with zeromean and unitvariance.We useWeeks-Chandler-Andersen (WCA) potential tomimicthe excluded volume interaction between colloidal particles U(t) (see

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Methods). The packing fraction of the system is defined as f =rs2p/4, where r is the particle density. The self-propulsion speed of par-ticle is v0 ¼ g�1

t Fp. The reduced noise strength in the system is definedas TR = kBT/(F

ps) that measures the strength of thermal noise com-pared with the self-propulsion. In the zero noise limit, i.e., TR = 0,isolated active particles perform circular motions with fixed radiusR = Fps2/W and period G = 2pgr/W. This athermal noise-free situationis the major focus of this work, and the effect of thermal noise isdiscussed later.

Absorbing-active transitionWe first simulate systems withN = 10,000 and TR = 0. Under low pack-ing fraction f and smallR condition, we find that the system falls into anabsorbing or arrested state, in which each particle performs anindependent circular motion without collisions and the mean squareddisplacement (MSD) ⟨Dr2⟩ of particles develops a plateau at long time

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Fig. 1. Absorbing-active transition. (A) Schematic of the 2D system of circle active particles. (B) Dynamic phase diagram in the representation of packing fraction f and circleradius R. (C andD) Typical snapshots of the active and adsorbing states near the critical point with f = 0.20with R=1.75s, where the color indicates the self-propulsionorientationof each particle. These two states are marked as magenta and cyan solid symbols, respectively, in (E). (E) MSD as functions of time for systemwith R = 1.75s started from randomconfigurations. Red line, active state (f = 0.22); blue line, absorbing state (f = 0.01); green line, system near the critical point (f = 0.20) in which the system ultimately falls into theabsorbing state after a long simulation time. (F andG) Diffusion constant as functions of f near the critical point fc for systems with different R. The dashed lines are the fitting ofpower law (f− fc)

b. (H and I) The structure factor S(q) and theorientation correlation functionC(r) of active (magenta) and absorbing (cyan) states asmarkedby solid symbols in (E).(J) The measured critical exponent b from (G) as a function of R. For all the calculations, N = 10,000 and TR =0.

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(blue line in Fig. 1E).With increasing f orR, the collisions between par-ticles become more frequent, making the system unable to find a non-interacting state. Thus, the system remains at an active diffusive statewith MSD ⟨Dr2⟩ ~ 4Dt (t → ∞) (red line in Fig. 1E). Here, D is thelong-time diffusion constant. The phase behaviors of the system aresummarized in the phase diagram Fig. 1B. In the following, we focuson the absorbing-active transition close to the boundary of two phases.

In Fig. 1F, we plotD as a function of f − fc for different R. Here, weobtain fc by fitting with the critical power law D ~ (f − fc)

b, whichdetermines the phase boundary in Fig. 1B. One can observe a sharptransition from the absorbing state [D(f) = 0] to the active state[D(f) > 0]when increasing f at a smallR=1.75s. The transition becomessmoother as R increases. In Fig. 1G, we show the log-log plot of D as afunction of f − fc. The obtained slope b is given in Fig. 1J.We find thatthe critical exponent b is about 0.15 for R = 1.75s, which is substan-tially smaller than the values in classical absorbing transitions, i.e., b =0.58 for directed percolation and b = 0.64 for conserved directed perco-lation (Manna type) (6–34). Such a small critical exponent isindependent of system size (see fig. S1). With increasing R, we find thatb increases to around 0.5 for R > 10s. A similar increase of the criticalexponent with increasing interaction range (in our case, R) has beenreported (35). In our system, fc would decrease to zero with increasingR. Hence, b at large R cannot be directly obtained in our system due tothe divergence of simulation time needed at the dilute limit.

To understand the physics behind the absorbing-active transition atsmall R, we choose a packing fraction f = 0.20 close to the critical point(fc = 0.195) for system with R = 1.75s. TheMSD for the system startedfrom random configuration is shown by the green line in Fig. 1E, inwhich one can see that the system ultimately falls into the absorbingstate after staying at the active state for a long time. In Fig. 1 (C andD), we show typical snapshots for the active state and absorbing statebefore and after the absorbing transition, as indicated by the magentaand cyan symbols in Fig. 1E.Movies for these two states can be found inmovies S1 and S2. One can notice a marked structural difference be-tween these two states. In the absorbing state, particles with similarorientation form finite synchronized clusters, while the active state ismore homogeneous without noticeable spatial heterogeneity. Thisstructural difference is also reflected in the structure factor S(q)and orientation correlation function C(r) = ⟨∑i≠j ei ⋅ ej d(rij − r)⟩/rN,as shown in Fig. 1 (H and I). Compared with the active state, S(q) forthe absorbing state develops a pronounced peak at qs ≃ 0.2, and thecorresponding C(r) also shows a stronger orientation correlation. Thesynchronized clusters formation in our circle active particle systemwithisotropic circling-phase distribution shares a similar mechanism withthe phase separation observed in an experimental bimodal phase-distributed system (see also fig. S4) (26). Both are a result of a crowding-induced attraction between particles with the same circling phase. Wefind that this distinct structural transformation during the absorbingtransition is absent in the conventional absorbing transition (5, 6, 17, 34),suggesting that the small critical exponentmeasured in our systemhas astructural origin. With increasing R, the structural difference betweentwo phases becomes weaker (see fig. S2), which occurs simultaneouslywith the increase of b. Further studies combining with finite-size analysisare necessary for determining whether the absorbing-active transition atsmall R is first order.

Hyperuniformity and large local density fluctuationsAs shown in Fig. 1H, for the active state near the critical point in thesystemwithR = 1.75s, the structure factor of the system exhibits hyper-

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uniform scaling S(q) ~ q(q → 0). In the random organization modelaiming to mimic the colloidal suspension under periodic shearing,a similar hyperuniform scaling was observed near the critical point(5, 6). However, for relative large R, the critical q scaling shifts to afaster q2 scaling (see fig. S2). To explore this further, we simulate asystem of N = 40,000 circle active particles. In Fig. 2A, we first plotthe MSD for active state systems with different R at f = 0.2. We findthat the diffusivity in the system rises slightly with increasing R. By fur-ther checking the scalings of the density variance ⟨dr2⟩ and S(q) for dif-ferent R (see Fig. 2, B and C), we observe a strong hyperuniformity inthe systems with R ≤ 25s, as indicated by the asymptotic behaviors:⟨dr2⟩ ~ L−3 (L→∞) and S(q) ~ q2(q→ 0). From ⟨dr2⟩, one can iden-tify an R-dependent length scale LHU, above which the systembecomes hyperuniform, while below which the system behaves likenormal fluids, i.e., ⟨dr2⟩ ~ L−2 (Fig. 2B). In S(q), a similar thresholdqHU can be found, which features the end of the hyperuniform scaling.With increasing R, LHU increases and qHU decreases until R ≥ 50s,where the hyperuniform scaling becomes less apparent due to the sys-tem finite-size effect as discussed below. This implies that R controlsthe length scales at which the system exhibits hyperuniformity.

Figure 2 (D to F) shows the result of an analogous investigation forhigher-density systems with f = 0.4. Compared with low-densitysystems, we observe pronounced enhancement of the diffusivity withincreasing R (Fig. 2D). The high-density systems also show clear hyper-uniform scaling at large length scales forR≤ 50s and the thresholdLHU(or qHU) increases (or decreases) with increasing R (Fig. 2, E and F).However, at f = 0.4, we find ⟨dr2⟩ ~ L−l with l < 2 when L ≪ LHUand l decreases with larger R. This implies that at length scales L ≪LHU, the system exhibits large density fluctuations, whose strengthand length scale are both controlled byR. This large fluctuation can alsobe identified by the scaling S(q) ~ q−2 for q > qHU as shown in Fig. 2F,which was reported as a signature of critical instability of active particlesystem undergoingMIPS (20). However, in most of our systems exceptR ≥ 100s, S(q) ~ q−2 does not diverge at qHU ≃ 0 as in MIPS (20) butstops at a finite qHU. This suggests that the large local density fluc-tuations observed in our system are due to clustering or microphaseformation. The crossover of two different scalings, i.e., large densityfluctuations and hyperuniformity, creates a peak in S(q) at qHU,whose height increases with larger R. Actually, for R ≥ 100s, wespeculate that in much larger systems, one can still observe the peakat finite qHU for f = 0.4 and the hyperuniform scaling for both f = 0.2and 0.4, as suggested by later theoretical analyses. In fig. S3, S(q) isshown for a larger system (N = 102,400) at f = 0.4. The result agreeswith our speculation. Typical snapshots of the system at f = 0.4 withvarious R are shown in Fig. 2G (also in movies S3 to S7 and fig. S3),and one can seemany finite-size clusters disappearing and reformingin the system. The average size of these dynamic clusters increaseswith increasing R, and at R = 1000s, because of the finite-size effect,the clusters percolate the simulation box. These findings are intriguing,as hyperuniformity and large density fluctuations induced by dynamiccluster formation are two seemingly opposite phenomena, which co-exist here in the same system at different length scales. In the following,we formulate dynamic mean-field theories to understand this new dy-namic hyperuniform fluid with large local density fluctuations.

Dynamic mean-field theoryStarting from the N-body Smoluchowski equation for active Brownianparticles (20, 21), one can prove (section S1) that in a homogeneouscircle active particles systemwith vanishing orientation order parameter

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Q ¼ ⟨eeT � 12 1⟩, the time-dependent local density field r(r, t) and the

local polarization field p(r, t) satisfy

∂tr ¼ �∇⋅½veðrÞp� De∇r� ð3Þ

∂tp ¼ � 12∇½veðrÞr� þ De∇2pþWr � p ð4Þ

where ve(r) = v0 + zr is a density-dependent effective velocity of par-ticles with a negative z reflecting the motility-induced “self-trapping”effect.De is the effective diffusion constant originated from the “evasive”motion of particles due to the collisions with neighboring particles(21). Wr ¼ g�1

r W is the reduced torque. The isotropic homogeneousstate, i.e., ½rðr; tÞ ¼ �r; pðr; tÞ ¼ 0�, is the solution to Eqs. 3 and 4. Bymaking a weak perturbation around this state, i.e., ½rðr; tÞ ¼ �rþdrðr; tÞ; pðr; tÞ ¼ dpðr; tÞ�, we obtain two linearized equations inthe Fourier space with the first-order approximation

iwþ Deq2

� �drq;w ¼ �iveq ⋅pq;w ð5Þ

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ðiwþ Deq2Þpq;w ¼ Wr � pq;w � iwqdrq;w ð6Þ

where [drq,w, pq,w] = ∫dr ⋅ e−iq⋅r∫dt e−iwt[dr, p], andw ¼ v0=2þ z�ris the parameter indicating the strength of self-trapping effect. By solvingEqs. 5 and 6, one obtains the dispersion relationship, which includes adiffusive mode w0 = iDeq

2 and two nondiffusive modes

w1;2 ¼ iDeq2±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivewq2 þW2

r

qð7Þ

The growth rate of the mode is k = Re(iw), whose sign determineswhether the perturbation dr ~ eiwt+iq⋅r grows or decays. One canprove that the mode 1 always decays, while the mode 2 may growfor w < 0 with

k2 ¼�Deq2 q <

v0ffiffiffiffiffiffiffiffiffiffiffi�vewp R�1

�Deq2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�vewq2 �W2

r

qq >

v0ffiffiffiffiffiffiffiffiffiffiffi�vewp R�1

8><>: ð8Þ

Fig. 2. Dynamic hyperuniform state. (A and D) MSD as functions of t for various R. (B and E) Density variances ⟨dr2⟩ as functions of window size L for various R. The L−3

asymptotic line indicates the hyperuniform scaling, which is the same as in perfect crystals. The L−2 scaling is for normal fluids, while L0 is for clustering- or phaseseparation–induced large density fluctuations. (C and F) Structure factor S(q) for various R. The q2 asymptotic line indicates the hyperuniform scaling, while the q−2 linerepresents clustering- or phase separation–induced large density fluctuations. (G) Typical snapshots for systems at f = 0.4 with various R. For all the calculations, N =40,000 and TR.

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In Fig. 3A, we plot three typical situations of k2 as functions of q athigh density (w < 0) by varying R (or Wr). One can see that for finiteR, k2 decreases from zero following the typical diffusive mode −Deq

2

from q = 0 to v0ffiffiffiffiffiffiffiffi�vewp R�1, above which k2 starts to increase and has the

chance to be positive at finite q > 0. This implies that the homogeneoussystem becomes unstable as a result of the growing fluctuation of finitewavelength. In the mean-field picture, this typically indicates that thesystem undergoes a microphase separation. This scenario is differentfrom the complete phase separation in which instability starts fromthe infinite wavelength at q = 0 (20, 21).

The instability point of the system is defined as kmax2 ðq�Þ ¼ 0 for

q* > 0, fromwhich we can obtain the relationship between the criticalpacking fraction f* and q* (see section S2)

f�

fc� 1

� �2� f�

fc

� �¼ 8De

v0Rð9Þ

q� ¼ffiffiffiffiffiffiffiffiv0RDe

rð10Þ

where fc ≃ 0.32 is the critical packing fraction for R = ∞ at which thesimulation shows complete MIPS (q* = 0). This suggests that MIPS canbe seen as a limit of the microphase separation in our system whenR→∞ at the mean-field level. For this special case, when f approachesfc, S(0) is supposed to jump from a finite value to∞, whichmarks the“spinodal” of MIPS. For systems with finite R, the first peak of S(q),which is located around q*, is also expected to jump up to a highervalue when the system crosses the “microphase separation point.” InFig. 3B, we plot the measured heights of the first peak in S(q) for var-ious combinations of R and f in computer simulations. We find thatwhen R > 50s, the height of the first peak in S(q) jumps from below 5to above 10 with increasing f over f*. The transition becomessmoother for systems of smaller R. Therefore, we choose S(q) = 7as the threshold to fit the phase boundary using Eq. 9 (the dotted linein Fig. 3B), which leads to De ≃ 0.62v0s. In the inset of Fig. 3B, weplot the location of q*measured in simulations and the prediction ofEq. 10 with the same De. The simulation results agree almost quan-titatively with the theoretical prediction. For small R cases (R < 50s),there is a noticeable inconsistency, suggesting the possible breakdownof

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the mean-field theory. In these cases, the size of clusters in the system iscomparable with the particle size. Therefore, microphase separation isno longer a proper concept to describe what happens in the system.Even for large R cases, which seem consistent with the mean-field mi-crophase separation scenario, it still remains open whether there existtrue sharp microphase separation points for finite R or if the sharpnessof the transition diverges at R → ∞.

To summarize, this theoretical analysis rationalizes the instabilityin the circle active particle system as a result of motility-induced self-trapping effect (negativew in Eqs. 7 and 8). The existence of nonzerotorque Wr restricts the growth of density fluctuations within a finitewavelength proportional to the circle radius R, which is the underlyingreason for the dynamic clusters formation and the resulting large localdensity fluctuations observed in our simulations.

Mechanism of global hyperuniformityTo understand the hyperuniformity at large length scales, we focus onthe spatial distribution of the instantaneous circling centers of activeparticles, which we treat as effective particles with radius R. For particlei at position ri, the circling center is at roi ¼ ri � s2ðFpi �WÞ=jWj2 ,which is a fixed position for an isolated circle active particle. Themotionof these effective particles is due to collisions with other particles. This issimilar to what happens in the random organization model (5, 6, 10),where only overlapped particles experience random kicks. As indicatedin the previous theoretical analysis, the self-trapping and the growingdensity fluctuations are confined within the length scale of R. At largerlength scales, fluctuations decay as the diffusive mode (Fig. 3A). Theseimply that the dynamic equation for the density field of these effectiveparticles ro(r, t) at large length scales can be described by the Fick’s lawof diffusion ∂tr = D∇2r in the q space

∂troq ¼ �Doeq

2roq þ xqðtÞ for q≪2pR

� �ð11Þ

with ½roq; xq� ¼ ∫dr⋅e�iq⋅r½ro; x�. Here,Doe is the diffusion coefficient

of effective particles and xq(t) is an additional noise term. Since we ne-glect the thermal noise, xq(t) comes from the chaotic multiparticle in-teraction (36), which obeys the center of mass conservation (CMC). Asproven in (10), the noise with additional CMC appears as a double-space derivative in the diffusion equation, i.e., xðtÞ ¼ ffiffiffi

�rp

∇2hðtÞ, with⟨h(r, t)h(r′, t′)⟩ = A2d(r − r′)d(t − t′) and A as the strength of the noise.This is different from the conventional single-space derivative noise thatarises solely from the particle number conservation (18, 37). Following(10), by adding such a double-space derivative noise term into Eq. 11 asthe perturbation, one obtains the dynamic equation for the density fluc-tuation dro in the Fourier space

iwdroq;w ¼ �Doeq

2droq;w � q2ffiffiffi�r

phq;w for q≪

2pR

� �ð12Þ

where hq,w is the Fourier transform of the noise h(r, t). From Eq. 12, weobtain the structure factor

SoðqÞ ¼ ∫∞

�∞1

2ptmaxN⟨droq;wdr

o�q;w⟩dw

¼ A2

2Deq2 for q≪

2pR

� � ð13Þ

Fig. 3. Dynamic microphase separation. (A) Growing rate k2 as functions of qfor systems at f = 0.35 with various R obtained from the dynamic mean-fieldtheory. (B) Measured heights of the first peak in S(q) for systems with differentcombinations of R and f in computer simulations indicated by the color of thesymbols. The dotted line is the fitting using Eq. 9 for the phase boundary. Insetshows the measured position of the first peak in S(q) in computer simulations(symbols) and the theoretical prediction (dotted line) based on the fitting in (B).

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where tmax is themaximum observation time (see section S3). Equation13 shows the same hyperuniform exponent in S(q) as observed in thesimulations and also shows the strongest hyperuniform scaling for den-sity variance ⟨dro2⟩ ~ L−3 at the length scales larger than R (2). In Fig.4A, we plot the scaled density variance ⟨dro2⟩R2 versus L/R forsystems with various R at f = 0.2, and one can see all points collapseinto a single curve. The curve consists of two distinct scalings with astronglyhyperuniformscalingL−3 atL>LHU≃2R andanormal fluid-likescaling L−2 at L < LHU. Similar collapse is shown in Fig. 4B for So(q)versus qR, where the hyperuniform scaling So(q) ~ q2 stops at qHU ≃2p/LHU. For high-density systems, i.e., f = 0.4, we obtain the samecrossover length scale LHU ≃ 2R as shown in Fig. 4 (C and D). In thiscase, the density variance and So(q) for differentR do not collapse into asingle curve because of cluster formation–induced large density fluctua-tions at L ≲ LHU. After obtaining the hyperuniform scaling for theseeffective particles, one can prove the existence of the same hyperuni-form scaling at similar length scales for the circle active particles (38).

From these analyses, one can see that, in systems of circle active par-ticles, there are two ingredients for the global hyperuniformity: (i) Thecircular trajectories of active particles localize their active motions,creating the Fickian diffusion condition for the active particles at large

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length scales; and (ii) the local density fluctuations originate from thechaotic multiparticle interactions, which obey the CMC. These argu-ments are based on (10), which produces the same hyperuniformscaling in S(q) using the random-organization model with CMC. Al-though the two systems share the similar mechanism of hyperunifor-mity at large length scales, there are also marked differences betweenthem. First, the driving force in our system is persistent, and the dynam-ics of the active particles is overdamped and deterministic. Second, oursystem has a characteristic length R, which separates two completelydifferent fluctuations. These cannot be described by the model in (10).

We notice that the two ingredients for the dynamic hyperuniformityabove seem to be satisfied in a recent experimental system, in whichspherical Janus colloids with additional soft repulsion perform almostperfect circling motion with the same chirality but two different phases,driven by external fields (26). The thermal noise was believed to be neg-ligible in this system, and the interactions between these nearly athermalactive particles produce an effective temperature, which controls bothkinetic and phase behavior of the system (26). To test whether the hyper-uniformity can exist in this circle active particle system with the bimodalcircling-phase distribution, we simulate a low density system (f = 0.2)with twodifferent circle radiiR=1s and3s using themodel (seeMethods),

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Fig. 4. Global hyperuniformity. (A and C) Scaled density variance ⟨dro2⟩R2 as functions of L/R and (B and D) So(q) as functions of qR, for various R at f = 0.2 (A and B)and 0.4 (C and D), respectively.

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which produced almost the same experimental result in (26). Our resultis shown in fig. S4, from which we find the bimodal-distributed systemfalls into a phase-separated absorbing state for R = 1s but stays in anactive mixing (lane) state for R = 3s, consistent with the previousfinding (26). In the active mixing state, we observe the same hyperuni-form scaling S(q → 0) ~ q2. This result demonstrates the robustnessof the hyperuniformity mechanism unveiled by this general model andsuggests a high possibility of realization in experiments. Moreover, italso suggests that other active systems satisfying the two ingredients,such as active spinner systems (29) or size/interaction oscillating particlesystems (39), may be used to produce the same hyperuniformity as well.

Effect of thermal noiseAccording to the definition, a perfectly hyperuniform state requires S(0)to exactly equal zero. However, in real experimental systems, thermalnoise is unavoidable. On the basis of the fluctuation-compressibilityrelationship

Sð0Þ ¼ kTrkBT ð14Þ

any thermal equilibrated systems with the positive isothermal compress-ibility kT at finite temperatures cannot be strictly hyperuniform due tothermal excitation (13). In crystals, thermal excitation appears as phononmodes, which cause background scattering or thermal diffuse scattering,whose effect can be measured by the Debye-Waller factor (13, 40).Nevertheless, the wide application of crystal materials indicates that ther-malization onlyweakens but does not destroymost physical properties ofthe ground-state crystal. Similarly, near-hyperuniformity (1, 13, 15) indisordered structures may also be enough to achieve some desiredfunctions, e.g., isotropic photonic/electronic bandgaps (14, 15). In Fig. 5,we analyze the influence of thermal excitation on the ground-state hy-peruniform system with f = 0.2 and R = 3s. We find that with a grad-ual increase of the reduced noise strength TR from zero, S(q → 0)begins to saturate at some nonzero value, which increases along withTR (Fig. 5A). In section S4, we introduce the thermal noise f = [ fx, fy]in Eq. 11 as

xðtÞ ¼ ffiffiffi�r

p∇⋅½∇hðtÞ þ fðtÞ� ð15Þ

Lei et al., Sci. Adv. 2019;5 : eaau7423 25 January 2019

where ⟨ fi(r, t)fj(r′, t′)⟩ = 2Dthermdijd(r − r′)d(t − t′) with Dtherm ¼kBTg�1

t ð1þ R2=s2Þ the self-diffusion constant of effective particlesdue to the thermal Brownian motion (25). Here, we assume that thethermal noise is a first-order weak perturbation on the chaotic noiseh(t), which leads to the decoupling of these two noise sources: ⟨ fi(r, t)h(r′, t′)⟩= 0. Then, we can estimate So(q) as a function of the reduced noisestrength TR for low-density systems as

SoðqÞ ¼ v0sDe

1þ R2

s2

� �TR þ A2

2Deq2 forðq < qHUÞ ð16Þ

In Fig. 5A, we plot the theoretical prediction of Eq. 16 as dashed linesfor differentTR values atR= 3s and f =0.2, by assuming them all acrossthe same So(qHU) point. In Fig. 5B, we compare the So(0) from the the-oretical predictionwith the So(0) obtained from the fitting of simulationresults (see Methods) for systems with R = 3s and 10s at f = 0.2. Onecan find quantitative agreements between simulation and theoreticalpredictions in Fig. 5, which suggest that the susceptibility of hyperuni-formity to the noise in our nonequilibrium system is similar to that inthermalized crystals, i.e., Eq. 14. However, we also emphasize thedifference: In our nonequilibrium system, the saturated value of So(0)is determined by the driving force as well. Therefore, in experiments, itis possible to observe a large range of hyperuniform scaling in S(q) ordensity variance as long as the driving force is much larger than thethermal noise, i.e., TR ≪ 1.

DISCUSSIONIn conclusion, by combining computer simulations with theoreticalanalyses, we investigate the dynamic phase behaviors in 2D systemsof circle active particles. In the zero-noise limit, we find that withincreasing the density of system or the radius of circular motion R,the system undergoes a transition from an absorbing state to an activefluid state, which is accompanied by a structural transformation forsmall R. In the active fluid state, we find a characteristic length scaleLHU. For L ≫ LHU, the system exhibits strong hyperuniformity withthe density variance scaling the same as in perfect crystals, while for L≲ LHU, we observe normal random fluctuations at low density and largedensity fluctuations (cluster formation) at high density. To understand themechanism of the phase behaviors of the system, we develop a dynamicmean-field theory. Linear stability analysis suggests that at the mean-fieldlevel, the large local density fluctuations at relative large R are a result ofmotility-induced microphase separation, which is confined within thelength scale of R. For the global hyperuniformity, we attribute it to theinterplay between the Fickian diffusion of active particles at large lengthscales and local particle collisions that conserve the center ofmass (10).Our work demonstrates that two extreme fluctuations, i.e., large densityfluctuations and hyperuniformity, can coexist in the same dynamic sys-tem at different length scales.We emphasize that this stable hierarchicalhyperuniform fluid is conceptually different from the disordered hyper-uniform solid or critical hyperuniform state. From a practical point ofview, our results suggest that even for exotic disordered hyperuniformstructures, there is still plenty of room at the “bottom,” i.e., one mayconstruct arbitrary local complex structures (ordered or disordered)with extra functionalities without harming the global hyperuniformity.This provides large freedom in designing hierarchical disordered hyper-uniform materials with unconventional properties.

Fig. 5. Effect of thermal noises on hyperuniformity. (A) Structure factor So(q)under different reduced noise strength TR for f = 0.2 and R = 3s. Open symbolsshow the simulation data, while the dashed lines are the theoretical predictions ofEq. 16. (B) Normalized So(0) as functions of TR from theoretical prediction (dashedline) and the fitting of simulation results (solid symbols) for systems with f = 0.2. Forall the calculations, n = 40,000.

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METHODSIn our simulations, we used a square simulation box with periodicboundary conditions in all directions, starting from initial configurationswith random particle positions and orientations tomake sure the initialstructure factor S(q) ~ 1. The time unit was chosen to be the time that aparticlemoves a distance of s in the dilute limit, i.e., t0 = s/v0. Tomimicthe excluded volume interaction between colloidal particles i and j, weused the WCA potential

UðrijÞ ¼ 4εσ

rij

� �12

−σ

rij

� �6

þ 14

" #ðrij < 21=6σÞ

0 ðrij > 21=6σÞ

8><>: ð17Þ

where rij is the center to center distance between particles i and j, with sas the diameter of particles. We chose e = Fps/24, which gives the typ-ical contact distance rc = s between particles based on force balanceFp ¼ ∂UðrijÞ

∂rijjrij¼rc. For systems at TR = 0, to exclude the noise generated

by discrete dynamic integrations, we used perfect convex polygons toapproximate the closed circle trajectories of active particles. This is rea-lized by finely tuning the integration step, which is around 10−3t0.

For systems with bimodal distributed circling-phase, following(26), we added an additional soft repulsion Us = A (r/s)−4 withcutoff distance rsc = 5s and A = 7.5e to model the isotropic dipoleinteraction between active particles. The self-propulsion force forthis system was reset to Fp = 27/6A/s to make the dipolar interac-tion balance the driving force at rc = 21/6s (26).

The density variance ⟨dr2⟩ of the system was calculated using aspherical window whose diameter is smaller than the half of simulationbox to avoid the finite-size effect. Under the periodic boundary condi-tion, the structure factor S(q) was calculated for some discrete q vectors,i.e., ½qx; qy� ¼ 2p

L0½i; j�ði; j ¼ 1; 2; 3…Þ , with L0 as the size of cubic

simulation box. The simulation time for the equilibrating and samplingprocesses depends on the system size and density. For hyperuniformsystem, the criterion for the system to reach equilibrium is whetherthe hyperuniform scaling of S(q) has fully extended to the smallestqmin = 2p/L0 without further change. The fitting function used in Fig.5B is Eq. 16, with an adjustable TR.

0

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/1/eaau7423/DC1Section S1. Derivation of the dynamic mean-field theory for 2D system of circle active particles.Section S2. Linear stability analysis.Section S3. Calculation of So(q) for system with CMC.Section S4. Effect of thermal noise on So(q).Fig. S1. Finite-size effect on the long-time diffusion coefficient D as a function of packingfraction f for systems with R = 1.75s.Fig. S2. Structural comparison of active and absorbing state at R = 10s near the criticalpoint fc = 0.0194.Fig. S3. Structure factor S(q) for large systems with different R at ϕ = 0.4 and TR = 0.Fig. S4. Hyperunifomity in an experimentally realizable system (26) with bimodal circling-phasedistribution.Movie S1. Active state in Fig. 1C.Movie S2. Absorbing state in Fig. 1D.Movie S3. Active state with R = 1000s in Fig. 2G.Movie S4. Active state with R = 100s in Fig. 2G.Movie S5. Active state with R = 50s in Fig. 2G.Movie S6. Active state with R = 25s in Fig. 2G.Movie S7. Active state with R = 23s in Fig. 2G.

Lei et al., Sci. Adv. 2019;5 : eaau7423 25 January 2019

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Acknowledgments: We thank H. Zhang in Shanghai Jiao Tong University for fruitfuldiscussions. We are grateful to the National Supercomputing Centre (NSCC) of Singapore

Lei et al., Sci. Adv. 2019;5 : eaau7423 25 January 2019

for supporting the numerical calculations. Funding: This work is supported by NanyangTechnological University Start-Up Grant (NTU-SUG; M4081781.120), the AcademicResearch Fund from Singapore Ministry of Education [M4011616.120, M4011873.120, andMOE2017-T2-1-066 (S)], and the Advanced Manufacturing and Engineering Young IndividualResearch Grant (A1784C0018) by the Science and Engineering Research Council of Agencyfor Science, Technology and Research Singapore. Author contributions: Q.-L.L. and R.N.conceived the research. Q.-L.L. performed the research. R.N. directed the research. Q.-L.L., M.P.C.,and R.N. analyzed the data and wrote the manuscript. Competing interests: The authorsdeclare that they have no competing interests. Data and materials availability: All dataneeded to evaluate the conclusions in the paper are presented in the paper and/or theSupplementary Materials. Additional data related to this paper may be requested from the authors.

Submitted 13 July 2018Accepted 12 December 2018Published 25 January 201910.1126/sciadv.aau7423

Citation: Q.-L. Lei, M. P. Ciamarra, R. Ni, Nonequilibrium strongly hyperuniform fluids of circleactive particles with large local density fluctuations. Sci. Adv. 5, eaau7423 (2019).

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fluctuationsNonequilibrium strongly hyperuniform fluids of circle active particles with large local density

Qun-Li Lei, Massimo Pica Ciamarra and Ran Ni

originally published online January 25, 2019DOI: 10.1126/sciadv.aau7423 (1), eaau7423.5Sci Adv 

ARTICLE TOOLS http://advances.sciencemag.org/content/5/1/eaau7423

MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2019/01/18/5.1.eaau7423.DC1

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