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for P. At this QCP, b(T)/T, C(T)/T,andG(T) diverge with unusual power-law exponents. Stoichiometric FM metals with a Curie tem- T Ferromagnetic Quantum Critical perature C below 1 K are very rare. The Kondo- lattice system YbNi4P2 is one, with a remarkably low TC = 0.17 K (21). This is mainly a result of Point in the Heavy-Fermion substantial Kondo screening (with a Kondo tem- 3+ perature TK ≈ 8 K) of the magnetic Yb ions Metal YbNi4(P1−xAsx)2 [Ni is not magnetic (22)], leaving a FM ground state with a small ordered moment of ~0.05 Bohr 1 1 1 1 1 Alexander Steppke, * Robert Küchler, Stefan Lausberg, Edit Lengyel, Lucia Steinke, magneton (mB)(23) and strongly enhanced electron- 1 1 1,2 1 Robert Borth, Thomas Lühmann, Cornelius Krellner, Michael Nicklas, electron correlations (21). It is, therefore, a partic- 1 1 1 Christoph Geibel, Frank Steglich, Manuel Brando * ularly suitable system to search for and explore FM quantum criticality. Unconventional and other previously unknown phases of matter exist in the A preliminary study on YbNi4P2 polycrys- vicinity of a quantum critical point (QCP): a continuous phase change of matter at absolute zero. tals, in which the low-temperature transition at Intensive theoretical and experimental investigations on itinerant systems have shown that metallic TC was discovered (21), indicated peculiar non- ferromagnets tend to develop via either a first-order or through the formation Fermi liquid (NFL) properties in transport and of intermediate superconducting or inhomogeneous magnetic phases. Here, through precision thermodynamic quantities above TC: in partic- low-temperature measurements, we show that the Grüneisen ratio of the heavy fermion metallic ular, a stronger-than-logarithmically diverging ferromagnet YbNi4(P0.92As0.08)2 diverges upon cooling to T = 0, indicating a ferromagnetic QCP. Sommerfeld coefficient C(T )/T and a linear Our observation that this kind of instability, which is forbidden in d-electron metals, occurs in a heavy T-dependence of the electrical resistivity r(T). fermion system will have a large impact on the studies of quantum critical materials. These results suggest the proximity of a FM QCP; TC could be tuned smoothly to zero through the luctuations of quantum fields persist to ab- diverge upon approaching the QCP, but the di- isoelectronic substitution of larger As for P, ex- solute zero. In matter, when such quantum vergence of b(T)/T is stronger, leading to a di- panding the crystal lattice and causing, in Yb sys- G T b T C T T T on July 10, 2013 Ffluctuations are tuned by a nonthermal con- vergence of the Grüneisen ratio ( )= ( )/ ( ). tems, an increase of K and a reduction of C trol parameter such as pressure, they may become If the divergence of the Grüneisen ratio follows (24). We report thermodynamic and transport sufficiently large to cause a quantum phase tran- a power law, G(T) º T −l, the critical exponent measurements on single crystals of stoichiometric sition into a new ground state. If the transition is yields insight into the nature of the QCP (19). YbNi4P2 (23)aswellasYbNi4(P1−xAsx)2, with x continuous, it gives rise to a quantum critical point For instance, in CeNi2Ge2 l = 1 is observed, in- between 0.04 and 0.13. YbNi4P2 has a crystalline (QCP) (1–3). QCPs have been observed in a num- dicating a three-dimensional (3D) spin-density- structure of the tetragonal ZrFe4Si2 structure type 3+ ber of antiferromagnetic (AFM) d-andf-electron wave QCP (20). For the unconventional quantum (P42/mnm), with the Yb ions located between metallic systems (4, 5) but are thought not to exist critical material YbRh2Si2, a fractional exponent, chains of edge-connected Ni tetrahedra. The Yb in ferromagnetic (FM) metals for several reasons. l ≈ 0.7, was found (20). ions form chains along the crystallographic c www.sciencemag.org Theoretical considerations for itinerant (such In this report, we show that in single crys- direction, with the lattice parameter a = 7.0565 Å as d-electron) systems exclude a FM QCP at field tals of the weak 4f-derived metallic ferromagnet being almost twice as large as c = 3.5877 Å H = 0 in clean materials and rather point to a FM YbNi4P2, a FM QCP can be induced by applying (Fig. 1A, inset) (22). Band-structure calculations of first order (6–9). Ex- negative chemical pressure—substituting 10% As show that three sheets of the Fermi surface have periments indicate that FM QCPs are preempted by the development of superconducting (10, 11), modulated (12) or spin-glass-like (13) phases, or 14 a first-order FM quantum phase transition ( ). Downloaded from In f-electron systems, Kondo screening competes with magnetism and permits QCPs, at which the Kondo effect may even be destroyed, as shown in the AFM heavy-fermion metal YbRh2Si2 (15–17). For a FM f-derived Kondo-lattice system, it has been shown theoretically that the Kondo screen- ing may completely disappear deep inside the ferromagnetically ordered phase (18). It remains to be shown, however, whether the FM QCP con- curs also with a Kondo-destroying one. Anidealmethodtoprobetheexistenceof a heavy fermion QCP (which is generically sensi- tive to pressure) is to perform combined measure- ments of the volume thermal expansion, b(T), and specific heat, C(T). Both b(T)/T and C(T)/T

Fig. 1. Temperature dependence of the magnetic and electrical properties of YbNi4P2.(A)Realpartof 1 the magnetic susceptibility c′ plotted as a function of temperature, measured perpendicular (c′⊥) and Max Planck Institute for Chemical Physics of , Nöthnitzer c′ c T −0.66 c′ T Strasse 40, 01187 Dresden, Germany. 2Institute of Physics, parallel ( ‖)tothe axis. The orange dashed line emphasizes the behavior of ‖ versus Goethe University Frankfurt, Max-von-Laue-Strasse 1, 60438 below 10 K. (Inset) Unit cell of YbNi4P2.(B and C) Electrical resistivity r plotted as a function of Frankfurt am Main, Germany. temperature and measured with current j (B) perpendicular and (C) parallel to the crystallographic *To whom correspondence should be addressed. E-mail: c axis. The kinks at TC disappear at fields m0H ≥ 0.05 T in both field directions. At a field of 0.1 T, 2 [email protected] (A.S.); [email protected] (M.B.) both resistivities follow a temperature dependence close to T below TFL (arrows).

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Fig. 2. Magnetic phase diagrams of YbNi4P2.(A) C/T versus T of YbNi4P2 measured at magnetic fields H ‖ c. The dashed orange line emphasizes the T−0.43 power- law behavior observed above 0.2 K up to about 6 K. The dotted black line is the specific-heat contribu- tion calculated within the single- ion S =1/2Kondomodelof(25), with TK =8KandC/T scaled by a factor of 0.5 for better visibility. (B) C/T versus T of YbNi4P2 measured at magnetic fields H ⊥ c.In(A)and (B), the nuclear Schottky contribu- tion has been subtracted from the raw specific-heat data (32). (C)Mag- netic phase diagram for H ‖ c,de- rived from equal-entropy construction for the phase-transition anomaly in the specific-heat coefficient [C(T)/T], kink in the linear magnetostriction ∆L(T)/L (fig. S10), inflection point of the ac-susceptibility, and first derivative of the resistivity [r(T)]. The paramagnetic (PM) region is separated from the ferromagnetic (FM) region, where the moments on July 10, 2013 are aligned within the (a,b) plane, by a phase transition of the second ic phase diagram for H ⊥ c derived from C(T), r(T), and c′(T). The vertical order ( black line). The dashed lines in (C) and (D) indicate the cross- green line at H = 0 indicates that the FM order exists at H = 0 only. The over between the PM and the ferromagnetically polarized (FP) regions, orange circles mark the increase of c′(T) below the crossover temperature determined by the maxima in the ac-susceptibility (fig. S8). (D)Magnet- (fig. S8).

a pronounced 1D character (21), which may be relevant to the understanding of the unusual NFL www.sciencemag.org behavior of YbNi4P2. Because adjacent Yb chains are shifted by c/2, Yb3+ ions experience a crys- talline electrical field (CEF) that is rotated by 90° from chain to chain, although the symmetry on each Yb site is the same. This causes frustration between neighboring chains, which together with strong electron-electron correlations may enhance

the strength of quantum fluctuations. Downloaded from We first focus on stoichiometric YbNi4P2.The strongest evidence of the FM order is provided by the real-part of the ac-susceptibility, c′(T), mea- suredbothwithfieldH ‖ c (c′‖)andH ⊥ c (c′⊥) (Fig. 1A). Although the CEF anisotropy results Fig. 3. Tuning TC to zero by As substitution. (A)SpecificheatC(T)dividedbyT of YbNi4(P1−xAsx)2 for in a much larger ground-state moment along the c indicated values of x at H =0.Withincreasingx, the FM transition temperature is tuned to zero. For x = −0.43 −0.36 directionthaninthe(a,b) plane, the FM ordering 0.08, C/T follows a T power law with a deviation below 0.25 K to T (orange dashed lines). (B) a T T x is perpendicular to c.AsT is lowered toward Linear thermal-expansion coefficient ( ) divided by of YbNi4(P1−xAsx)2 with = 0, 0.08, and 0.13 measured in zero field along (a‖c) and perpendicular (a⊥c)tothec axis. Here, the orange dashed TC = (0.150 T 0.005) K, c′⊥ increases sharply − T −0.65 −T−0.66 a T T c up to a value of ~200 × 10 6 m3/mol (≈4inSIunits). lines emphasize the and power-law behavior of / versus along the crystallographic a a T T Below T , a pronounced increase of the imag- and directions, respectively. For YbNi4(P0.87As0.13)2, the coefficient ( )/ tends to saturate below 1 K, C indicating noncritical behavior. inary part c″⊥ of cac(T)(H⊥c) is observed, repre- sentative of energy dissipation (fig. S7). In the same T region, c′⊥ is only weakly T-dependent, possibly because the coercive field of the mag- seen at TC in the resistivity curves r⊥ (T)and sufficiently extended temperature range, is com- netic hysteresis is very small (21). c′|| increases r‖ (T), with current j perpendicular (Fig. 1B) monly considered a hallmark of quantum criti- −0.66 with decreasing T, following a T power law or parallel (Fig. 1C) to the c axis. Below TC,the cality (1, 2, 15, 16). Because of the relatively low n n below 10 K down to TC, where it saturates at a resistivity follows a T power law, with n⊥ =2.9 Kondo temperature, TK ≈ 8 K, a proper r ∼ T −6 3 value of ~17 × 10 m /mol. This anisotropy of and n‖ = 2.5, very likely because of charge-carrier behavior is observed only within the rather nar- the susceptibility suggests that the spins align scattering from FM . As in the poly- row temperature window 0.15 K ≤ T ≤ 1K.In n perpendicularly to the c direction, and c′‖ mea- crystals, the resistivity is quasi-T-linear just above this range, r = r0 + AT ,withn⊥ =1andn‖ =1.2, sures the transversal fluctuations. Clear kinks are TC. Such a NFL property, when observed over a and the coefficient A⊥ = 21.6 microhm·cm/K is

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Fig. 4. Quantum critical behav- a‖ and a⊥ being the linear expansion coefficients ior in YbNi4(P1−xAsx)2.(A)Low- along c and a, respectively) and the specific heat temperature ac-susceptibility ofthesamecrystal,withx as close as possible to x H for = 0.08, measured at =0, xc (19, 20). The Grüneisen ratio G(T)atxc should where a rounded maximum is diverge as T −1/nz with parameters n, the spatial T c′ T visible at C =0.028Kin ⊥( ). correlation-length exponent, and z, the dynamical T − x (B) phase diagram: The FM exponent (19). x ≈ QCP is located at c 0.1. (C) Whereas for YbNi P , a change of sign in C T 4 2 Temperature dependence of / a(T)/T is clearly visible at T (Fig. 3B), as ex- and b/T for the YbNi (P As ) C 4 0.92 0.08 2 pected at a second-order phase transition (30), sample, which is the sample for YbNi (P As ) ,nosignchangecan closest to the FM QCP. Both quan- 4 0.92 0.08 2 tities diverge with decreasing be seen down to 0.06 K, in accordance with − − T a T T as C(T)/T º T 0.43 (º T 0.36 be- C = 0.028 K. Upon cooling, the ratios ||/ − a T low 0.25 K) and b(T)/T º T 0.64. and ⊥/ diverge with similar exponents, yielding b T T º T −0.64T0.02 (Inset) Temperature dependence ( )/ . Although this is consistent of the absolute value of the di- with 3D FM spin fluctuations (19), the power- mensionless Grüneisen ratio G = law divergence of the Sommerfeld coefficient C T T Vmol/kT ·b/C for x =0.08[using ( )/ is at odds with the expected logarithmic the isothermal compressibility behavior, C(T)/T º log(1/T), for an itinerant −12 −1 19 x a T T kT =5.3×10 Pa of YbRh2Si2 3D FM QCP ( ). For =0.13, ( )/ saturates (33)] which reveals the presence below 1 K, indicating the crossover to a heavy − of quantum critical fluctuations at the FM QCP. The dashed red line emphasizes the T 0.22 divergence of G(T) Fermi liquid phase (31). in zero field. The key evidence for the FM QCP in YbNi4(P1−xAsx)2 is summarized in Fig. 4C. At x =0.08—very close to xc—not only do both 1.2 much larger than A‖ = 3.8 microhm·cm/K .This To approach the QCP, we used arsenic C(T)/T and b(T)/T diverge with power-law ex- on July 10, 2013 indicates a Fermi velocity that is much higher substitutions of x = 0.04, 0.08, and 0.13 in ponents T −0.43 (T −0.36 below 0.25 K) and T −0.64, along the Yb chains than within the (a,b)plane, YbNi4(P1−xAsx)2. The QCP is achieved at xc ≈ respectively, but the dimensionless Grüneisen which is in accordance with the anisotropic Fermi 0.1, as demonstrated by calorimetric and dila- ratio diverges as G(T) º T−0.22T0.04 (Fig. 4C, surface (21). tometric results in Fig. 3, A and B, and in the inset). This strongly indicates the existence of a Additional thermodynamic evidence of the phase diagram of Fig. 4B. For x = 0.04, the QCP at xc ≈ 0.1. The critical exponent l = −0.22 phase transition at TC and of the in-plane FM phase transition remains FM and second order rules out an itinerant QCP scenario because it order is presented in Fig. 2, A and B, in which the (Fig. 3A, green diamonds); in fact, C(T)/T of leads to nz ≈ 5, which is much larger than for T and H dependences of the specific heat are YbNi4(P0.96As0.04)2 has been measured at H ⊥ c, itinerant FM systems (n =1/2,z =3). shown together with the result of the single-ion and the transition was found to be shifted to The unexpected power-law exponents in all www.sciencemag.org Kondo model (25). At H =0,asharpl-type phase higher T with increasing H (fig. S6). For x = 0.08, thermodynamic quantities, indicating the pres- transition in C/T versus T is observed, associated C(T)/T increases continuously down to 0.06 K, ence of strong FM quantum critical fluctuations, with a small change of the molar 4f-derived en- and no phase transition is detected. C(T)/T can require new theoretical concepts beyond the itin- tropy (1% of Rln2, with R = 8.31 J/molK). A mag- be described by the same T-dependence as for erant spin-fluctuation theory (2). These concepts netic field applied parallel to the c axis shifts YbNi4P2 down to 0.25 K, where a slight devi- must take into account the localized nature of the transition temperature to lower T (Fig. 2A), ation from this power law (to T −0.36) is visible the Yb-4f states in the presence of very strong whereas for H ⊥ c, the crossover to the polarized (Fig. 3A, orange dashed lines). Pronounced spin-orbit coupling when compared with the CEF. T C T T f state is shifted to higher temperatures (Fig. 2B). changes of the -dependence of ( )/ and Further, the Kondo coupling of the 4 states to the Downloaded from This is corroborated by the H dependence of of the Grüneisen ratio G(T) were observed in conduction electrons must be considered. Iso- c′(T) (fig. S8, A and B): For H ⊥ c, a sharp peak YbRh2Si2 below 0.3 K (5, 15, 20); in our case, thermal measurements of the Hall coefficient on appears already at very low fields, where the sys- this deviation results in only a minor change of YbNi4(P1−xAsx)2 with x > ~0.1 under hydrostatic tem directly enters the polarized state, and then G(T) (figs. S12 and S13). Measurements of c′(T) pressure would greatly help to investigate the broadens at higher fields. The resulting magnetic down to 0.02 K picked up a transition at 0.028 K, exciting possibility that the FM QCP concurs phase diagrams are displayed in Fig. 2, C and D. which is similar to the behavior observed in with a localization-delocalization transition of These observations point to in-plane FM order in YbNi4P2 (Fig. 4A and fig. S9). This implies that, the Yb-4f states, similar to what was reported YbNi4P2, although neutron-diffraction experiments also for x = 0.08, the transition is FM and sec- for YbRh2Si2 at its AFM QCP (5, 16, 17). The are required to determine the exact magnetic struc- ond order. For x = 0.13, no phase transition is extent to which the FM instability in As-doped ture, which very likely is not collinear. observed, and C(T)/T versus T seems to follow YbNi4P2 is affected by the 1D character of the At H =0,C(T)/T increases below 6 K, fol- the same T dependence down to 0.1 K, but with electronic structure must also be clarified. T −0.43T0.02 lowing a power law over a notably wide smaller absolute values. This indicates that the References and Notes range down to 0.2 K, where classical thermal same power-law divergence observed in C(T)/T 1. P. Coleman, A. J. Schofield, Nature 433, 226 (2005). fluctuations associated with the phase transi- for the pure system is retained across xc. 2. H. Löhneysen, A. Rosch, M. Vojta, P. Wölfle, Rev. Mod. tion set in. A similar power law [C(T)/T º T −0.4] The observed NFL features suggest the pres- Phys. 79, 1015 (2007). 12 13 26–29 3. G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001). has been observed at the field-induced QCP of ence of strong quantum fluctuations ( , , ) 4. A. Yeh et al., Nature 419, 459 (2002). YbRh2Si2, an AFM system with strong FM fluc- but do not directly imply the existence of a FM 5. J. Custers et al., Nature 424, 524 (2003). Phys. Rev. Lett. tuations (15). For YbNi4P2, a pronounced l-peak QCP, that these fluctuations become critical at 6. D. Belitz, T. R. Kirkpatrick, T. Vojta, 82, at T indicates a second-order phase transition. xc ≈ 0.1 (13, 29). One key experiment to provide 4707 (1999). C Phys. Rev. Lett. This, and the distinct NFL properties above T , direct evidence of quantum critical fluctuations 7. A. V. Chubukov, C. Pépin, J. Rech, 92, C 147003 (2004). suggest that this compound is located in the close is the combined measurement of the volume- 8. G. J. Conduit, A. G. Green, B. D. Simons, Phys. Rev. Lett. vicinity of a QCP, where TC → 0 smoothly. thermal-expansion coefficient b = a‖ +2a⊥ (with 103, 207201 (2009).

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9. D. Belitz, T. Kirkpatrick, T. Vojta, Rev. Mod. Phys. 77, 22. C. Krellner, C. Geibel, J. Phys. Conf. Ser. 391, 012032 (2012). Acknowledgments: WeareindebtedtoD.Belitz, 579 (2005). 23. J. Spehling et al., Phys. Rev. B 85, 140406 (2012). A. V. Chubukov, P. Coleman, R. Daou, S. Friedemann, 10. E. Hassinger, D. Aoki, G. Knebel, J. Flouquet, J. Phys. 24. A. L. Cornelius, J. S. Schilling, D. Mandrus, M. Garst, P. Gegenwart, F. M. Grosche, S. Kirchner, Soc. Jpn. 77, 073703 (2008). J. D. Thompson, Phys. Rev. B 52, R15699 (1995). G.G.Lonzarich,B.Schmidt,Q.Si,P.Thalmeier,M.Vojta, 11. S. S. Saxena et al., Nature 406, 587 (2000). 25. K. D. Schotte, U. Schotte, Phys. Lett. A 55, 38 (1975). T. Vojta and S. Wirth for useful discussions and 12. M. Brando et al., Phys. Rev. Lett. 101, 026401 (2008). 26. N. T. Huy et al., Phys. Rev. B 75, 212405 (2007). M. C. Schuman for language editing. Part of the work 13. T. Westerkamp et al., Phys. Rev. Lett. 102, 206404 27. R. P. Smith et al., Nature 455, 1220 (2008). was supported by the DFG Research Unit 960 “Quantum (2009). 28. S. Jia et al., Nat. Phys. 7, 207 (2011). Phase Transitions.” 14. D. Belitz, T. R. Kirkpatrick, http://arxiv.org/abs/1204.0873. 29. C. Pfleiderer, P. Böni, T. Keller, U. K. Rössler, A. Rosch, 15. P. Gegenwart, Q. Si, F. Steglich, Nat. Phys. 4, 186 (2008). Science 316, 1871 (2007). Supplementary Materials 16. Q. Si, F. Steglich, Science 329, 1161 (2010). 30. M. Garst, A. Rosch, Phys. Rev. B 72, 205129 (2005). www.sciencemag.org/cgi/content/full/339/6122/933/DC1 17. H. Pfau et al., Nature 484, 493 (2012). 31. P. Thalmeier, B. Lüthi, in Handbook on the Physics and Materials and Methods 18. S. J. Yamamoto, Q. Si, Proc. Natl. Acad. Sci. U.S.A. 107, Chemistry of Rare Earths,K.A.GschneidnerJr., Supplementary Text 15704 (2010). L. Eyring, Eds. (North-Holland, New York, 1991), vol. 14, Figs. S1 to S13 19. L. Zhu, M. Garst, A. Rosch, Q. Si, Phys. Rev. Lett. 91, chap. 96, pp. 225–341. References (34–50) 066404 (2003). 32. Materials and methods are available as supplementary 20. R. Küchler et al., Phys. Rev. Lett. 91, 066405 (2003). materials on Science Online. 21 September 2012; accepted 17 December 2012 21. C. Krellner et al., New J. Phys. 13, 103014 (2011). 33. J. Plessel et al., Phys. Rev. B 67, 180403 (2003). 10.1126/science.1230583

tuations in vibrated rods (3), and phase separation Living Crystals of Light-Activated from self-induced diffusion gradients (4). Bio- logical (5–7) and artificial active particles (8–11) Colloidal Surfers also exhibit swarm patterns that result from their interactions (12–15). Jeremie Palacci,1* Stefano Sacanna,1 Asher Preska Steinberg,2 David J. Pine,1 Paul M. Chaikin1 In order to study active, driven, collective phenomena, we created a system of self-propelled Spontaneous formation of colonies of bacteria or flocks of birds are examples of self-organization particles where the propulsion can be turned on in active living matter. Here, we demonstrate a form of self-organization from nonequilibrium and off with a blue light. This switch provides on July 10, 2013 driving forces in a suspension of synthetic photoactivated colloidal particles. They lead to rapid control of particle propulsion and a con- two-dimensional “living crystals,” which form, break, explode, and re-form elsewhere. The dynamic venient means to distinguish nonequilibrium ac- assembly results from a competition between self-propulsion of particles and an attractive tivity from thermal Brownian motion. Further, interaction induced respectively by osmotic and phoretic effects and activated by light. We the particles are slightly magnetic and can be measured a transition from normal to giant-number fluctuations. Our experiments are quantitatively stabilized and steered by application of a mod- described by simple numerical simulations. We show that the existence of the living crystals is est magnetic field. Our system consists of an intrinsically related to the out-of-equilibrium collisions of the self-propelled particles.

1Department of Physics, New York University, 4 Washington elf-organization often develops in thermal arises in driven, dissipative systems, far from 2

Place, New York, NY 10003, USA. Department of Physics and www.sciencemag.org equilibrium as a consequence of entropy equilibrium. Examples include “random organi- Chemistry, Brandeis University, Waltham, MA 02453, USA. ” 1 Sand potential interactions. However, there zation of sheared colloidal suspensions ( )and *To whom correspondence should be addressed. E-mail: are a growing number of phenomena where order rods (2), nematic order from giant-number fluc- [email protected]

Fig. 1. (A) Scanning electron microscopy (SEM) of the bimaterial : a TPM colloidal sphere BC with protruding hematite cube (dark). (B) Living crys- tals assembled from a homogeneous distribution (inset) Downloaded from under illumination by blue light. (C) Living crystals melt A by thermal diffusion when light is extinguished: Image shows system 10 s after blue light is turned off (inset, after 100 s). (D to G) The false colors show the time evolution of particles belonging to different clusters. The clusters are not static but rearrange, exchange particles, merge (D→F), break apart (E→F), or become unstable and explode (blue cluster, F→G). For (B) to (G), the scale bars indicate 10 mm. The solid area fraction is Fs ≈ 0.14.

DEFG213s 241s 304s 338s

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