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Direct observation of a local structuralmechanism for dynamic arrest

C. PATRICK ROYALL1,2*, STEPHEN R. WILLIAMS3, TAKEHIRO OHTSUKA2 AND HAJIME TANAKA2*1School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK2Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan3Research School of Chemistry, The Australian National University, Canberra, ACT 0200, Australia*e-mail: paddy.royall@bristol.ac.uk; tanaka@iis.u-tokyo.ac.jp

Published online: 22 June 2008; doi:10.1038/nmat2219

The mechanism by which a liquid may become arrested, forming a glass or gel, is a long-standing problem of materials science. Inparticular, long-lived (energetically) locally favoured structures (LFSs), the geometry of which may prevent the system relaxing toits equilibrium state, have long been thought to play a key role in dynamical arrest. Here, we propose a definition of LFSs which weidentify with a novel topological method and directly measure with experiments on a colloidal liquid–gel transition. The populationof LFSs is a strong function of (effective) temperature in the ergodic liquid phase, rising sharply approaching dynamical arrest, andindeed forms a percolating network that becomes the ‘arms’ of the gel. Owing to the LFSs, the gel is unable to reach equilibrium,crystal–gas coexistence. Our results provide direct experimental observation of a link between local structure and dynamical arrest,and open a new perspective on a wide range of metastable materials.

When a liquid is cooled sufficiently, and is unable to accessthe thermodynamic equilibrium state, it may undergo dynamicalarrest, which broadly occurs in two forms, glasses and gels1,2.The former are associated with supercooling a liquid below itsfreezing temperature, whereas the latter are identified with arrestedphase separation3–5. Although possible dynamic mechanisms havereceived considerable attention, direct experimental evidence ofstructural mechanisms6,7 has proved elusive8–10. In the case of thecolloidal gels of interest here, it has been suggested that spinodal-like phase separation leads to the formation of a colloid-richpercolating network4. Furthermore, dynamic asymmetry betweenlarge colloidal particles and small solvent molecules is expected topromote the formation of a network structure of the slow (colloid-rich) component: viscoelastic phase separation11. We expect thatlocal arrest leads to rigidity in the ‘arms’ of the network and are thusmotivated to seek some local structural signature that underlies theoverall arrest. In the case of the colloidal gels of interest here, theunderlying thermodynamic ground state is crystal–gas coexistence.Thus, in addressing the question of why gelation occurs, that is,why phase separation is arrested and the locally dense ‘arms’ aregeometrically frustrated from crystallizing, some parallels may bedrawn with those glasses in which the thermodynamic ground stateis a crystal.

Colloidal gels are governed by their potential-energylandscape12, where the thermodynamic equilibrium state islocated in a deep valley, the global energy minimum (crystal–gascoexistence). However, many other local energy minima exist,corresponding to gels. In this picture, dynamical arrest occurs whenthe system is quenched sufficiently that it can only explore a limitednumber of (local) potential-energy minima on the experimentaltimescale. In other words, on cooling, the minima becomeseparated by energy barriers too high to surmount easily. Thesystem is then trapped in a local minimum in the potential-energy

landscape, unable to escape and reach equilibrium unless a veryrare event occurs, and is non-ergodic on any reasonable timescale13.This description is intuitively appealing and relevant for manyprocesses such as protein folding12, but direct interpretation ofthe potential-energy landscape in an experimental system remainsvery challenging.

The key lies in identifying these local minima in thepotential-energy landscape, and connecting them with dynamicalarrest. Such a mechanism was proposed by Frank6, who, byconsidering groups of 13 atoms of the Lennard-Jones model ofnoble gases, noted that five-fold symmetric icosahedral structuresare energetically favoured compared with face-centred-cubic(f.c.c.) structures and are therefore local potential-energy minima.Icosahedra are expected in the liquid phase, with the populationincreasing on cooling, and their five-fold symmetry may impedecrystallization6,7. Although isolated 13-atom structures wereconsidered, immersion in a bulk liquid has little effect on thepotential-energy considerations14.

Attempts to verify this link between (icosahedral) locallyfavoured structures (LFSs) and dynamical arrest took a significantstep forward using the bond-order parameter W6 (see theSupplementary Information), which is associated with five-foldsymmetry in computer simulations of the Lennard-Jones model15.Indeed, on approaching dynamical arrest, domains consistent withfive-fold symmetry have been found16. Some evidence of five-fold symmetry in liquids has also been found in experiments17,18,along with indirect measurements of five-fold symmetry on coolingbelow the glass-transition temperature19. However, unambiguous,direct, experimental evidence of an increase of five-fold symmetryon dynamical arrest seems to be lacking. In any case, bulk scatteringmeasurements that average over many particles seem unlikely toyield direct identification of LFSs. Colloidal suspensions, which,owing to their well-defined thermodynamic temperature20, may

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Figure 1 Structural and dynamic characterization of the colloidal liquid–gel transition. a–c, Confocal microscopy images of a hard-sphere-like colloidal suspensioncP = 0 (a), a colloid–polymer mixture cP = 7.0×10−5 (ergodic liquid) (b) and a dynamically arrested gel cP = 1.10×10−4 (c). Polymers are not shown in a–c. Scalebars= 10 µm. d, W6 bond-order parameter distribution. Colours correspond to polymer weight fractions ×10−4. Here, gelation occurred at cPG = 7.7±0.7×10−5. e, Phasediagram, in the φ− cP plane, where φ is the colloid volume fraction. Red data points denote ergodic liquids; blue data points denote colloidal gels. The solid line is taken fromfree-volume theory, which considers the equilibrium case where the gel undergoes phase separation to gas–crystal coexistence3. Dotted lines are tie lines of equal chemicalpotential; L and X are colloidal liquids and crystals, respectively. f, MSDs at various cP/cPG. The straight dashed line has a slope of 1 (ergodic liquid). Polymer concentrationsare quoted as cP/cPG. τB is the time to diffuse one radius at infinite dilution, 19 s. Here, we assume isotropic motion and track the colloids in two dimensions and time.

be treated as mesoscopic atoms, provide one promising meansto identify LFSs experimentally, as it is possible to determinethe coordinates of every particle with microscopy21. Despite someevidence for five-fold symmetry in dilute gels22 and hard-sphereglasses21, supercooled hard-sphere liquids show little change in W6

approaching dynamical arrest23.We propose that simply making a measurement such as

W6, which averages over many particles, is insufficient tounambiguously identify LFSs. Here, we argue that the icosahedronis just one example of an LFS, appropriate for Lennard-Jones-likematerials. We shall define LFSs as being the global potential-energyminimum of a group of m particles in isolation24, notingthat m is not fixed. Even for spherically symmetric potentialsat fixed m, various structures are found, depending on theinteraction range12,24. The global minimum structures we considerare shown in Supplementary Information, Fig. S1. Although otherenergy minima exist for these isolated clusters24, here we shallconsider only the global minimum for each m. Significantly,a number of these structures are not five-fold symmetric. Wedirectly identify these LFSs by the use of a novel algorithm, thetopological cluster classification (TCC). We consider a colloidalgel because it has a well-defined attractive interaction25,26 (seeSupplementary Information, Fig. S2), such that under ourdefinition LFSs are clearly and unambiguously defined, unlike,for example, athermal hard-sphere glasses. Our analysis revealsa massive rise in LFSs, forming a percolating network ondynamical arrest; in fact, the ‘arms’ of the gel predominantlycomprise LFSs.

Adding polymer to a colloidal suspension can induce effectiveattractions widely accepted as a larger scale analogue of simpleatomic systems. Colloid–polymer mixtures exhibit different phasesincluding colloidal gases, liquids and solids, along with dynamicallyarrested states such as glasses and gels3,27. The phase diagramof the system studied here is shown in Fig. 1e. The depletionattraction between the colloids, which drives the phase behaviour,results from the polymer entropy, because the polymer-free volumeis maximized when the colloids approach one another (seeSupplementary Information, Fig. S2)25. Controlling the state pointby adding polymer in this way allows the system to be quenchedfrom the high-temperature-like, hard-sphere limit, to dynamicalarrest, which at moderate colloid concentrations results in a gel.We observe the system with confocal microscopy, Fig. 1a–c, andfix the colloid volume fraction to φ = 0.35, comfortably belowφ = 0.494, the equilibrium freezing point for hard spheres (oursystem with no polymer). Similar results were obtained in a moredilute system (φ = 0.05). By adding salt26, we avoid long-rangedelectrostatic repulsions22. We determine the colloid coordinates21

and calculate the bond-orientational order parameter W6 (ref. 15),before analysing the LFSs. Our experiments are supplemented bybrownian dynamics computer simulations.

We now consider how to identify the LFSs. We seek structuresthat constitute the ground state for each m. We have carefullycharacterized the interaction potential in our model colloidalsystem26 and found it to be almost identical to the attractiveMorse potential (see Supplementary Information, Fig. S2 andequation (S2)). We therefore take those structures that constitute

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Figure 2 Coordinates identified as belonging to different LFSs. a, Liquid, cP/cPG = 0.73±0.04. b, (Ergodic) liquid close to gelation, cP/cPG = 0.92±0.04. c, Colloidalgel, cP/cPG = 1.08±0.04. d, Dilute gel, φ = 0.05, cP = 1.76×10−4, showing percolating LFSs. Particles are colour-coded as follows: grey, free (not in any cluster), shown0.4 actual size; white, m= 5, shown 0.6 actual size; yellow, crystalline, shown 0.8 actual size. Other particles are members of LFSs of size m given by the colours, shown0.8 actual size.

global potential-energy minima for clusters interacting via theMorse potential (with range parameter ρ0 = 30) to be LFSs24.Furthermore, we use the Morse potential in our browniandynamics simulations, and find a very similar behaviour (see theSupplementary Information). To identify these structures in thebulk system, we have developed a novel topological method, TCC,that identifies LFSs in terms of their bond network. We begin withthe bond network between all of the particles. The bond lengthis equated with the interaction range, that is, the polymer size,0.18σ, where σ = 2.4 µm is the colloid diameter25,26, leading topercolation in both the colloidal gel and the liquid for φ = 0.35.All of the shortest path three-, four- and five-membered rings inthe bond network are identified. These rings are then classified interms of those with an extra particle bonded to all of the particlesin the ring and those that have two or no such extra particles.We term these the basic LFSs, into which many of the larger LFSscan be decomposed. A given particle may be a member of morethan one basic LFS, that is, basic LFSs may overlap. We use thisstrategy to identify all of the LFSs with 13 or less particles. The LFSswe consider are shown in Supplementary Information, Fig. S1. Inaddition, we identify the f.c.c. and hexagonal close-packed (h.c.p.)13-particle structures in terms of a central particle and its 12 nearestneighbours. If a particle was found to be part of more than one LFSsize, it was labelled as part of the larger size, and the associationwith the smaller ignored. For more details, see ref. 28.

Having outlined our method, we proceed to the results.Real-space confocal microscope images are shown in Fig. 1a–c.At low polymer concentrations, a colloidal liquid is seen(Fig. 1a,b); higher concentrations lead to a dynamically arrestednetwork, or gel, (Fig. 1c), with large-scale structure consistentwith arrested spinodal decomposition4. The radial distributionfunction illustrates the change in structure resulting from theincreasing levels of attraction, with a rise in the first-, and higherorder maxima, accompanied by a shift to smaller separations,shown in Supplementary Information, Fig. S3. The mean squareddisplacement (MSD) is shown in Fig. 1f. This shows typicalcharacteristics of dynamical arrest, and leads us to a definitionof the polymer concentration cP required for gelation cPG. At lowpolymer concentration, cP/cPG < 1, we find a diffusive liquid;higher polymer concentration, cP/cPG > 1, leads to dynamicalarrest, where only very local displacements are observed.

Figure 1d shows the distribution of the bond-orientationalorder parameter W6. These exhibit little change, only a fractionalshift to negative values, consistent with a slight increase in five-foldsymmetry, for moderate polymer concentrations. Our data are inline with that of ref. 23 in that W6 may not change greatly on arrest.However, dynamical arrest certainly occurs (Fig. 1f).

We now consider direct measurement of LFSs using thetopological cluster classification. Figure 2a shows LFSs in a liquid,cP/cPG = 0.73 ± 0.04. The LFSs are readily identified, but, at

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this state point, with weaker interactions than those required forgelation (higher effective temperature), they are clearly isolated.Increasing the polymer concentration to cP/cPG = 0.92 ± 0.04(Fig. 2b), while still in the ergodic liquid phase, the population ofLFSs increases significantly, but they remain isolated. This is notthe case in the gel, where a percolating network comprising LFSsis seen, Fig. 2c, cP/cPG = 1.08±0.04. Thus, it seems that dynamicalarrest may be associated with the formation of a percolated networkof LFSs. In Fig. 2, LFSs with different sizes and different structuresare seen, underlining the importance of considering more thanone species.

The growth in LFS population is illustrated in Fig. 3. The moststriking feature is the strong increase approaching dynamical arrest,as suggested by Fig. 2. In general, smaller LFSs are more populous.We have shown that the LFSs become much more numerous ondynamical arrest. We recall that the equilibrium state is gas–f.c.c.crystal coexistence, and that the arms of the gel are mostly LFSsthat are geometrically incompatible with f.c.c. crystals (Fig. 2c,d,see Supplementary Information, Fig. S1), and therefore crystallizevery slowly. The high incidence of octahedral LFSs of size m = 6(W6 = −0.012, see Supplementary Information, Fig. S1) may helpto explain the lack of change in the W6 distribution on quenching.Although octahedra form part of the f.c.c. lattice, crystal growthwill be suppressed by the proximity of other popular clusters withmore negative W6 values such as m = 5 (triangular bipyramid)(W6 = −0.121), m = 9 (C2v) (W6 = −0.148) and m = 8 (Cs)(W6 = −0.144). Noting that, under our definition, icosahedraare not LFSs for this system, we stress that the nature of theLFSs, and hence the degree of five-fold symmetry, is a material-specific property.

Our results suggest that the LFSs form faster than large crystalscan nucleate and impede crystallization, see the inset of Fig. 3. Athigher polymer concentrations, this kinetic trapping is expectedto be more severe and indeed on deep quenching the smallpopulation of crystalline particles falls drastically (Fig. 3, inset),hints of which have been seen in computer simulation29. Thus,kinetic selection may win over global energetic considerations.

Note that the number of bonds per particle Nb/N rises muchmore slowly than does the LFS population, so the populationincrease is not merely due to the local densification associated withgelation. In the high-temperature-like ‘hard-sphere’ limit, wherethere are no attractions between the particles, we still find LFSs.This might seem surprising, because LFSs are energetically favouredstates. However, many LFSs result in efficient local packing30 andprovide an important avenue by which a dense system may lowerits free energy.

We now consider our earlier assertion that LFSs correspondto local energy minima, by comparing the coordination numberof the particles in clusters with the ‘free’ particles not identifiedwith any cluster. Figure 4a shows that particles in LFSs havesubstantially more neighbours than free particles. For this short-ranged attractive interaction, the number of neighbours providesa good measure of the potential energy of each particle. We thusconclude that the structures we define as LFSs represent low-energy states.

We have shown that dynamical arrest is correlated with thegrowth in the proportion of particles in LFSs, but a clear linkimplies a connected LFS network. We note that by consideringa range of LFSs, and allowing them to interpenetrate, our TCCmethod permits the formation of a percolating network, but, giventhe variety of structures considered, filling space with LFSs is likelyto be geometrically frustrated. Percolation is confirmed in Fig. 2d,where the structure of a dilute gel φ = 0.05, cP/cPG = 1.45±0.14, isshown. Here, the arms of the gel clearly comprise LFSs, forminga network. In the more dense system φ = 0.35, we test for LFSpercolation by measuring the dimension of the largest connectedregion of LFSs lc, and plotting this as a fraction of image size L inFig. 4b. Here, lc/L = 1 corresponds to percolation, and indeed wesee that the LFSs percolate around cPG.

We furthermore examine the dynamics of the LFSs. We expectthat, on gelation, as the particles become trapped in LFSs, thenthe LFS lifetime should increase strongly. We illustrate this withmovies, which show short-lived LFSs in the colloidal liquid (seeSupplementary Information, Movie S1, cP/cPG = 0.73 ± 0.04);approaching the gel state (see Supplementary Information,Movie S2, cP/cPG =0.86±0.04) the LFSs last longer, which stronglycontrasts with the gel state (see Supplementary Information,Movie S3, cP/cPG = 1.8±0.1), in which we find a largely arrestednetwork of LFSs, with some mobility (and subsequent LFS break-up) due to the more mobile surface particles. We note thatthis is consistent with computer simulations where dynamicalheterogeneity was connected to surface (‘fast’) particles and slowerparticles in the core of the ‘arms’31.

We now present evidence that the arrest is directly relatedto the LFSs. By tracking the colloidal particles in time, similarlyto ref. 32, we measure the effect of the clusters on the MSDfor an ergodic liquid just above gelation cP/cPG = 0.79 ± 0.06and 0.88 ± 0.07 (Fig. 4c). We see that the particles that remainin LFSs for a prolonged time diffuse considerably more slowlythan those that do not. As the effective temperature is loweredfurther, this effect becomes more pronounced. We also recall thatthe population of particles in LFSs increases markedly aroundthese state points. This shows how the formation of long-livedclusters, on gelation, is the mechanism responsible for the dynamicslowing. Whereas coordinate tracking errors complicate accuratemeasurements of displacement in the experimental data, browniandynamics simulation data in the gel state show a similar trend tothe ergodic liquid (Fig. 4d). The faster particles not in LFSs may beconnected to the fast surface population noted by Puertas et al.31.

The lifetime of the LFSs is expected to increase in the ergodicliquid, but to diverge on arrest. We measure LFS lifetime as thecorrelation of the residency of individual particles in an LFS,

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Figure 4 Static and dynamic characteristics of gelation. a, Coordination numbers NN/N as a function of polymer concentration. The average coordination number of allparticles is shown by the black line; the blue line denotes particles within LFSs and the red line denotes free particles that do not form part of an LFS. b, The length of thelargest connected low-energy region as a fraction of the image length lc/L as defined in the text. The low-potential-energy regions are defined as connected groups ofparticles identified as part of an LFS. The largest low-energy region is thus the largest group of connected particles in an LFS. Shaded areas are a guide to the eye. c, MSD asa function of time spent in an LFS for the ergodic fluid cP/cPG = 0.79 and 0.88. The measuring time here was τ = 0.37τB. d, MSD calculated from brownian dynamicssimulations as a function of time spent in an LFS for a gel, ε/εG = 1.43 (see the Supplementary Information), here τ = 1.2τB. e. LFS lifetime as characterized by〈cLFS (0)cLFS (t )〉 (see text for definition). Legend denotes cP/cPG.

〈cLFS(0)cLFS(t)〉, where cLFS(t) = 1 if a particle in an LFS at time0 is still in an LFS at a later time t , and 0 otherwise. Plotting〈cLFS(0)cLFS(t)〉 in Fig. 4d reveals a very considerable increase inthe LFS lifetime on gelation. However, at long times, coordinatetracking errors lead to LFSs being identified as breaking up. Note,however, that all state points for cP/cPG > 1 collapse onto the samecurve, indicating little change in the dynamics and that the decay isprobably due to the tracking errors. This is supported by browniandynamics simulations without such tracking errors, which showa clear divergence in the lifetime on gelation (see SupplementaryInformation, Fig. S6).

Combining our observations, we may present a mechanism forarrest. In the ergodic liquid, isolated LFSs have a short lifetime,but at some polymer concentration, percolation of LFSs occurs.Connecting LFSs results in a significant lowering in potentialenergy. The dilute gel in Fig. 2d, for example, in which the LFSs

are bonded to one another, should have a lower energy than anequivalent number of isolated LFSs. This formation of a percolatingnetwork of LFSs therefore, represents a deep local minimum inthe energy landscape, in which the system is kinetically trappedfor some time. We note that this mechanism is compatible withthe connection between arrest and phase separation4 rather thana ‘cluster glass transition’33.

In summary, we have presented clear experimental evidence ofa local structural mechanism for colloidal gelation. By carefullyidentifying relevant LFSs, we reveal that dynamical arrest isassociated with the formation of a percolating network of LFSs.In other words, dynamical arrest is driven by a network ofparticles, each in a local energy minimum, unable to reach theequilibrium state, gas–crystal coexistence. We have also shown thata high population of LFSs, expected in a dynamically arrestedsystem, is not necessarily correlated with a strongly negative W6.

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We have considered a colloidal gel, but because we focused onlocal structure, in particular the suppression of crystallization,our approach has very clear implications for a great manymetastable materials, from colloidal glasses, even to molecularand metallic glasses. We emphasize the importance of consideringmany different LFSs, and that in general these are material specific.Finally, we believe that our TCC method offers significant potentialto advance our understanding of the structure in amorphousmaterials, and may for example provide a useful means to studycrystallization pathways.

METHODS

EXPERIMENTAL DETAILSWe used polymethylmethacrylate colloids sterically stabilized with polyhydroxylsteric acid. The colloids were labelled with the fluorescent dye 4-chloro-7-nitrobenzo-2-oxa-1,3-diazol and had a diameter σ = 2.4 µm with 3%polydispersity. The polymer used was polystyrene, with a molecular weightof 3.1 × 107, here Mw/Mn = 1.3 (ref. 26). To closely match the colloiddensity and refractive index, we used a solvent mixture of cis-decalin andcyclohexyl bromide. Owing to the refractive index matching, the van derWaals interactions are reduced to a fraction of the thermal energy kBT andneglected. To screen any electrostatic interactions, we dissolved tetra-butylammonium bromide salt, to a concentration of 4 mM, so the Debye screeninglength is well below the characteristic depletion interaction range26. In detailedstudies of the same system, we found excellent agreement with the depletioninteraction25, assuming some polymer swelling, due to the ‘good’ solventused, resulting in a polymer–colloid size ratio of 0.18 (ref. 26). We conducteda number of experiments, in the case of φ = 0.35, in which the gel pointcPG = 1.0±0.2×10−2. We attribute this difference to weak charging effects, butstress that no further change in behaviour was detected.

Received 3 September 2007; accepted 19 May 2008; published 22 June 2008.

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Supplementary Information accompanies this paper on www.nature.com/naturematerials.

AcknowledgementsThe authors are grateful to A. van Blaaderen and D. Derks for particle synthesis help and gifts. Wewish to thank P. Bartlett, D. Derks, D. Head and R. Jack for critical reading of the manuscript, andT. Ichikawa for kind instrumentation support. This work was partially supported by a grant-in-aidfrom the Ministry of Education, Culture, Sports, Science and Technology, Japan. C.P.R. is grateful tothe Royal Society for financial support.

Author contributionsC.P.R., S.R.W. and H.T. conceived the project and wrote the manuscript, C.P.R. carried out theexperiments, simulation and analysis, S.R.W. wrote the TCC code and T.O. wrote the W6 analysis code.

Author informationReprints and permission information is available online at http://npg.nature.com/reprintsandpermissions.Correspondence and requests for materials should be addressed to C.P.R. or H.T.

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