Observation of liquid glass in suspensions of ellipsoidal colloids
Jörg Roller, Aleena Laganapan, Janne-Mieke Meijer, Matthias Fuchs, Andreas Zumbusch
OObservation of liquid glass in suspensions ofellipsoidal colloids
J ¨org Roller, Aleena Laganapan, Janne-Mieke Meijer, , Matthias Fuchs, † Andreas Zumbusch ∗ University of Konstanz, Dep. of Chemistry, Germany University of Konstanz, Dep. of Physics, Germany University of Amsterdam, Institute of Physics, The NetherlandsTo whom correspondence should be addressed;E-mail: † [email protected], ∗ [email protected] Despite the omnipresence of colloidal suspensions, little is known about the in-fluence of shape on phase transformations, especially in nonequilibrium. Todate, real-space imaging results are limited to systems composed of sphericalcolloids. In most natural and technical systems, however, particles are non-spherical and their structural dynamics are determined by translational androtational degrees of freedom. Using confocal microscopy, we reveal that sus-pensions of ellipsoidal colloids form an unexpected state of matter, a liquidglass in which rotations are frozen while translations remain fluid. Image anal-ysis unveils hitherto unknown nematic precursors as characteristic structuralelements of this state. The mutual obstruction of these ramified clusters pre-vents liquid crystalline order. Our results give unique insight into the interplaybetween local structures and phase transformations. This helps to guide ap-plications such as self-assembly of colloidal superstructures and also gives firstevidence of the importance of shape on the glass transition in general.
Introduction
Suspensions of colloidal particles are widely spread in nature and technology and have beenstudied intensely over more than a century. When the density of such suspensions is increasedto high volume fractions, often their structural dynamics are arrested in a disordered, glassy1 a r X i v : . [ c ond - m a t . s o f t ] M a y tate before they can form an ordered structure. This puts limits on technical applications suchas the formation of superstructures from colloidal particles by self-assembly processes. Yet,it also means that knowledge gained from investigations of phase transformations of colloidalsuspensions to ordered structures and glasses provides insight into similar phenomena in a broadrange of complex glass forming materials, ranging from metals to biological cells ( ).The colloidal glass transition has therefore been considered a model that features many ofthe glass transition phenomena found in atomic and molecular systems ( ). Being big enoughto allow their real-space observation using optical microscopy, but small enough to remain sus-pended over extended periods, colloids in such model systems are employed as big atoms ( )which can be investigated on an individual particle basis (
6, 7 ). By analyzing the trajectories ofthousands of particles, detailed insights into glass phenomena, such as dynamic heterogeneitiesof collectively rearranging structures have been obtained (
8, 9 ). Apart from their use as modelsystems, however, synthetic colloids have increasingly also been perceived as interesting mate-rial building blocks in their own right (
10, 11 ). The recent growth of this field of research hasbeen supported by a multitude of novel techniques for the synthesis of colloidal particles withspecific geometries and interactions ( ).The availability of shape-anisotropic particles allows investigations of their phase transfor-mations in dense suspensions which promises to give unique insight into structural dynamics ofcomplex systems. This is especially important for the investigation of steric effects, which needto be controlled in the self-assembly of colloidal building blocks into materials with specificcollective properties. The simplest deviation from spherical symmetry is uniaxial stretching toprolate ellipsoids ( ). Already for these simple particles, theory and simulations predict a richphase diagram ( ) and complex glass formation ( ) due to the presence of translationaland rotational degrees of freedom. However, the few investigations of suspensions of ellipsoidshave all been focused on static structures obtained by driving particles in external gravitationalor electric fields ( ). The only investigations on steric effects of ellipsoid shape have beenperformed in 2D films (
25, 26 ). To date, hardly any experimental data on the influence of stericfactors on the phase transformation in 3D ellipsoidal suspensions exist.Here, we present the first particle-resolved studies of the structural dynamics of ellipsoidalcolloid suspensions. The experiments were performed on a large range of different volume frac-tions. Mode coupling theory (MCT) predicts that in systems of this type, a liquid glass shouldexist in which particle rotation is frozen whereas their translation is still liquid ( ). We usedquantitative optical microscopy in combination with a novel type of core-shell particles ( )to simultaneously track translational and rotational particle motion with high precision ( ).Upon increasing the volume fraction φ , we find that rotational degrees of freedom undergo aglass transition before translational dynamics are arrested such that a liquid glass is formed. Tounderstand the nature of the observed rotational and translational glass transitions, we furthercorroborate our results with an MCT analysis and Brownian dynamics simulations and find thatlong-range correlations are the cause for the emergence of this unique state. Detailed imageanalysis reveals that hitherto unknown nematic precursors exist as the characteristic structuralelements of the liquid glass. They consist of ramified aligned regions intersected by differently2rdered or disordered regions and appear to mediate the long-range correlations present in theliquid glass state of particle suspensions. Our experiments give a first impression of the com-plex behavior of colloidal suspensions arising from the introduction of the simplest geometricaldistortion of the particles’ shape from sphericity. They show how sterical factors lead to theemergence of peculiar local structures mediating long range spatial correlations which result inthe formation of amorphous states preempting the globally ordered state. Results and Discussion
In our experiments, we used ellipsoidal polymethylmethacrylate (PMMA) colloids with a longsemi-axis of a = 4 . µ m and a short semi-axis of b = 1 . µ m, i.e. an aspect ratio of a/b = 3 . (Fig. 1A). The particles were sterically stabilized and suspended in a density- and refractive-index-matched solvent mixture ( ). Tracking of particle positions and orientations was facil-itated by using particles with core-shell geometry where the spherical core and the ellipsoidalshell were labeled with different fluorphores ( ). Using confocal laser scanning microscopy,we typically recorded the temporal development of the 3D particle positions and orientationsfor more than 6000 particles with accuracies of nm and ◦ , respectively ( ) (Fig. 1B).Since volume fractions φ are the pivotal variable in all the experiments, we took utmost carein their determination (see Methods for details). In order to validate the experimental results,we performed event-driven Brownian dynamics (ED-BD) simulations that model overdampeddynamics of hard ellipsoids ( ). The system consisted of N non-overlapping ellipsoids in acubic simulation box of length L that was varied depending on the desired φ . Results fromexperiments and computer simulations were analyzed using MCT. Structural correlations and lack of order
To gain insight into the systems’ behavior, we first extracted the pair correlation functions g ( r ) and equilibrium structure factors S ( q ) from the 3D particle positions at different φ (Fig. 1C,D).With increasing φ , we observed the emergence of next-neighbor and second next-neighborpeaks in g ( r ) while long range correlations were absent. At the same time in S ( q ) , the in-verse peak position π/q max , indicating the average particle separation, became smaller withincreasing density. All observations are typical for liquid-like structural correlations and showthat the systems remained in a disordered state without translational order for all investigated φ . Fig. 1 also reveals the excellent coincidence between the measured g ( r ) and S ( q ) withthose obtained by the ED-BD simulations. This confirms that in the suspensions, our ellipticalcolloids interact like hard particles.The equilibrium phase diagram of ellipsoids predicts the transition to a nematic state withincreasing φ ( ), thus we determined the orientational order of the whole sample. It wasprobed by calculating the nematic order parameter S ( ), which is the eigenvalue possessing3 AC D zy x
Fig. 1. Characterization of ellipsoidal colloids and their packings:
Scanning electronmicroscope image ( A ) of the ellipsoidal colloids with aspect ratio a/b = 3 . . The inset showsa confocal microscope image, highlighting the core-shell structure. Scale bar is µ m. ( B )Computer rendered 3D reconstruction of a subset of a sample volume at φ = 0 . with the RGBvalue of the color indicating the particle orientations. Scale bar is µ m. ( C ) Pair correlationsfunction g ( r ) and ( D ) structure factor S ( q ) for φ as labeled from experiment (green points) andsimulation data (black lines). Distances r are rescaled by the ellipsoid width b .4 B Fig. 2. Nematic order parameter S obtained from the nematic tensor Q ij . Panel ( A ) shows S for different φ , indicating the absence of a global particle alignment in the system. ( B ) Timeevolution of the nematic order parameter for different φ , again showing no signs of an evolvingdirectional order.the largest absolute value of the nematic order tensor Q ij . The nematic order tensor is given by Q ij = (cid:28) u i u j − δ ij (cid:29) , (1)where (cid:126)u is an eigenvector representing the orientation of the ellipsoid. The order parameter S for different measured φ shows no indications for nematic order in the probed systems (Fig. 2A).For all φ , it remains below a value of S < . which is the commonly accepted criterion for theisotropic-nematic transition in simulations ( ). The absence of nematic order is confirmed byplotting the changes of the order parameter S with time (Fig. 2B) for different φ . It shows notemporal evolution for all φ , indicating the stability of the observed isotropic states. Dynamical signatures of glass formation
The absence of structural order even in very dense suspensions hints at the possible formationof a glass. To verify this, it is necessary to investigate the structural relaxation dynamics ofthe suspensions. The translational and rotational dynamics of the system are contained in thetemporal evolution of the self part of the density correlation function F s ( q, t ) = 1 N N (cid:88) i (cid:104) exp [ i(cid:126)q · ( (cid:126)r i ( t (cid:48) + t ) − (cid:126)r i ( t (cid:48) ))] (cid:105) (2)and of the orientational correlation functions L n ( t ) = 1 N N (cid:88) i (cid:104) P n (cos( θ i ( t (cid:48) + t ) − θ i ( t (cid:48) )) (cid:105) , (3)respectively. Here, (cid:126)r i is the position of the center of particle i , θ i its orientation relative to a fixedlaboratory direction, and N the total particle number. The P n are the Legendre polynomials of5rder n , and (cid:104)(cid:105) denotes averaging over t (cid:48) . For the calculation of F s ( q, t ) and L n ( t ) , the wavevector q was set to different values . < q · b < . and n = 2 , was chosen, respectively.The left panels of Fig. 3 describe the translation behavior and depict the experimentallydetermined F s ( q, t ) values for each φ . In panel A, the wavevector is set to q = 2 . µ m − (viz. bq = 6 . ) which corresponds to the maximum in the static structure factor and thus tothe average particle distance. In panel C, a larger wavevector, q = 3 . µ m − ( bq = 7 . ) isselected, where the relaxation is faster and the final decay can be resolved well in fluid states.Correlation functions obtained for different values of q can be superimposed by scaling with log( tq ) for low φ as is expected for the dynamics of isolated ellipsoids (see Supplementarytext). For a liquid state, one expects the translation correlators F s ( q, t ) to decay completely overtime. During the measurement time, this was observed for volume fractions up to φ = 0 . .The curves obtained for φ = 0 . are already stretched and thus show a slowing down of thedecay dynamics, but one can still assume their complete decay albeit at times longer than thoseprobed. States at φ ≤ . thus are also fluid-like. By contrast, at the highest volume fraction φ = 0 . the translational correlator does not decay anymore which is see by the clear plateaufor translational motion typical for a glass state appears. From these observations we concludethat for translational dynamics, a glass transition occurred between φ = 0 . and φ = 0 . .Fig. 3 B and D depict the temporal relaxation behavior of the rotation correlators L n ( t ) for n = 2 and n = 4 , respectively. A comparison with their translational counterparts illustratedin Fig. 3 A and C, clearly shows the differences in the translational and rotational relaxationof the suspensions. For volume fractions φ ≤ . , a complete relaxation of orientationalcorrelation is observed even if the curves for φ = 0 . are already stretched significantly.However, for the two highest φ a clear plateau indicative of incomplete relaxation appears inboth orientational correlation functions. These frozen orientational correlations are clear signsof glass-like behavior. Plateaus in the orientational correlations functions could also arise fromnematic ordering. The observed values for the plateau heights, however, are too large to becompatible with the negligible nematic order of our samples. In nematic states, the relation L ( t → ∞ ) = S holds ( ), and the small values of the order parameter S from Fig. (2) areincompatible with the high amplitudes of frozen-in orientational correlations. We thus concludethat the data show a glass transition in the rotational dynamics occurring between φ = 0 . and φ = 0 . .The inspection of Fig. 3 leads to the conclusion that two different glass transitions, one inthe orientational and a second one in the translational motion, exist. To gain more insight intothis phenomenon, we turn to a quantitative analysis. In fully relaxing systems, the long-timedecay, often termed α -relaxation, can be fitted with a Kohlrausch function ( ): Φ( t ) = f Φ exp( − ( t/τ ) β ) , (4)where f Φ is an amplitude which was set to f Φ = 1 , τ is the relaxation time, and β is the so-called stretching exponent. Using this function we obtain fits that agree very well with our datafor curves showing a clear decay within the experimental time window (Fig. 3 C and D; fitparameters are collected in the Supplementary text). Results for the translational and rotational6 t [s] F s ( q ,t ) q = 2.6 µm -1 t [s] L ( t ) t [s] F s ( q ,t ) q = 3.2 µm -1 t [s] L ( t ) A BC DE F
Fig. 3. Temporal correlation functions capturing translational and rotational dynamicsfor different φ . ( A ) and ( C ) depict the self part of the density correlation F s ( q, t ) at wave-lengths comparable to the average particle separation, viz. bq = 6 . (A) and bq = 7 . (B), respectively. ( B ) and ( D ) show the orientational correlation function L n ( t ) for n = 2 and n = 4 ; the legend in panel B gives the packing fractions for panels A-D. Close to the glasstransition, the decay of the curves is too slow to be captured within the measurement times. ( C )and ( D ) include fits to the correlators in fluid states using the Kohlrausch function Eq. (4). ( E )and ( F ) show the relaxation times obtained from the fits. The colored regions mark the glasstransitions φ tc and φ rc which were obtained by the MCT glass stability analysis (see Materialsand Methods). 7 -relaxation times τ t and τ r are depicted in Fig. 3E,F. The rise of the α relaxation time τ r for rotations sets in at lower φ than the rise in the corresponding translational τ t , and τ r alsoexceeds the observation time at a lower φ than τ t . This reaffirms the presence of two differentglass transitions. We also note that the correlators for translational relaxation at φ = 0 . andfor rotational relaxation at φ = 0 . are well fitted, which confirms the fluid-like behavior atthese φ , which are the packing fractions of the fluid states closest to the transitions.A quantitative determination of the φ at which the glass transition for translation φ tc androtation φ rc occur is obtained by the glass stability analysis of MCT ( ) (cf. Materials andMethods). Performing such an analysis, we find that satisfactory fits to the data (Fig. 3E andF) can only be obtained assuming two different glass transitions φ tc = 0 . and φ rc = 0 . .Therefore, also this analysis shows the existence of two separate glass transitions for rotationand translation in the experiment.In summary, the analysis of the systems’ dynamics reveals that we observed a density regionwhere orientational motion in the sample was frozen while translational motion persisted. Asthis state lacks global nematic order, it is properly described as a liquid glass ( ). Event driven Brownian dynamic simulations pointing to nematic correla-tions -1 t [s] F s ( q , t ) -1 t [s] L ( t ) A B
Fig. 4. Glass transition analysis of simulation data. ( A ) Translational F s ( q, t ) and ( B ) orien-tational L ( t ) correlation functions from ED-BD simulations, where long-range nematic fluc-tuations are suppressed by rough walls. A MCT glass stability analysis (black lines for the twohighest densities, cf. Materials and Methods) finds a single glass transition density in this case.The value φ c ≈ . is also supported by a power law fit (with γ = 2 . ) to the Kohlrauschrelaxation times for both translation and rotation.The interpretation of experimental results obtained from colloidal suspensions often is com-plicated by the fact that the particles possess a certain polydispersity and might carry residualcharges. Since these problems are non-existent in simulations, the latter are an important tooleven if the number of particles which can be studied in this manner is typically rather small. Tounderstand the origin of the formation of liquid glass in our system, we performed additional8D-BD simulations. We found that nematic order sets in quickly using monodisperse ellip-soids and periodic boundary conditions in accordance to previous simulation studies (
18, 35 ).Therefore, we tested different scenarios that could explain the formation of the experimentallyobserved glass states. First, since the experimental system is inherently polydisperse, we in-cluded polydispersity in our simulations. However, we found that a polydispersity of − ,well above that of the experimental particle system used, did not affect the structure and dynam-ics of the system. Second, since Letz et al. ( ) predicted that the formation of a liquid glass iscaused by long-wavelength fluctuations which may not be properly modeled in simulations dueto finite size effects, we introduced rough walls in our simulations. The approach of confiningglass-forming liquids has been used experimentally ( ) and in simulations ( ) for studies ofpolydisperse spheres. Indeed, we found that nematic order is suppressed in systems with roughwalls. Results for the dynamics of this system are shown in Fig. 4 for F s ( q, t ) and L ( t ) . Here, L is shown as it exhibits glassy dynamics more clearly than L . The lines represent the MCTfitting curves obtained from Eqs. 6 and 7. Up to φ = 0 . both correlation functions decayedto fluid states, while at φ = 0 . and φ = 0 . the system was already very close to the glassphase. However, in contrast to the experiments, the simulations still result in a single value forthe glass transition of φ tc = φ rc ≈ . .While the discrepancy between the experimental system and simulations appears dissatis-fying, it allows us to formulate a working hypothesis on the prerequisites for the formation ofa glass and a liquid glass, respectively: It is known from experiments on suspensions of spher-ical colloids that the glass transition at the higher φ tc results from isotropic caging ( ). Thisis accessible by simulation if nematic order is suppressed by the introduction of rough walls.Under this condition, the build-up of local neighbor shells leading to caging is possible alreadyin small systems ( ). The growing fluid structure is reflected in the evolution of the structurefactor S ( q ) shown in Fig. 1D. By contrast, the formation of a liquid glass above φ rc and below φ tc requires long range correlations, which cannot be captured in the simulation box sizes cur-rently accessible. This is especially true for shape anisotropic particles for which it has beenshown that simulations are susceptible to finite size effects ( ). Solvent mediated interactions,which were neglected in the simulations, could also play a role, but have been shown to be ofminor importance in the structural dynamics of spherical colloids ( ). We therefore conjecturethat the discrepancy between experiment and simulations reveals that the nature of the observedliquid glass state is relying on long-range correlations. Nematic precursor analysis
To identify these long-range correlations and to test our hypothesis, we analyzed the spatialcorrelations in the samples. A close inspection of the imaging data revealed clusters of similarlyoriented particles in the system. Two particles were assigned to the same precursor cluster, ifthey were next neighbors sharing a face of their individual Voronoi-cells and had an angulardifference in orientation which was less than ∆ α ≤ ◦ . The choice of ∆ α was based on thehalf-width of the peak in the probability distribution P (∆ θ ) of the orientation of all ellipsoids in9
200 400 600 N c P ( N c ) N c P ( N c ) A B C ϕ =0.46 ϕ =0.55 ϕ =0.57 D E FG H I ϕ =0.55 ϕ =0 57 Fig. 5. Analysis of nematic precursors. ( A-C ) 3D representation of the 20 largest clusterswith similar orientation, defined as described in the text, for different φ showing the existenceof nematic precursors which are intercepting each other. Boxsize is × × µ m each. Allparticles are scaled by a factor of . for better visibility.( D ) Probability distribution P ( θ ) of theorientation of all ellipsoids in the system for different φ . For increasing φ , a favored orientationis emerging, indicating nematic precursors but no apparent nematic order. ( E ) The green parti-cles belong to one cluster with similar orientation, while the surrounding blue particles are in anamorphous arrangement or belong to clusters of different orientations. The hindrance betweenthe different precursors prevents global nematic order. Boxsize is × × µ m. ( F ) Orienta-tional structure factor S ( q ) obtained from spatially correlating the orientations of all particles.Its dominating small- q peak records the growth of the nematic precursors. ( G )Normalized clus-ter size N c /N tot for high φ , where N c is the number of particles within a cluster and N tot thetotal number of particles within the image volume. The cluster size is increasing for higher φ .( H,I ) The distributions of cluster sizes for the two highest φ .10ystems at high φ (Fig. 5D). Clusters were defined as nematic precursors, if they contained morethan 30 particles. The choice of 30 particles was based on the largest cluster sizes found for φ =0 . , which corresponds to an isotropic sample. Since the clusters do not form nematic regions,we term them nematic precursors. Fig. 5A-C depicts the 20 largest nematic precursors for thethree highest φ studied. In general, we found that the number of particles in nematic precursorsincreased with growing φ , as shown in Fig. 5G-I. Also the cluster size distributions becamebroader with increasing packing fraction. Our findings show that with φ increasing to densitieswhere we find glassy states, more and more particles are found in local structures with a peculiarorder. In these, no nematic order is detected even on small length scales, because other particlesand clusters intersected the nematic precursors. This is exemplified in Fig. 5E, where the greenparticles belong to a nematic precursor whereas the blue particles are disordered surroundingparticles intersecting the cluster. In the vicinity of the glass transition, particles tended to possesssimilar orientations (see Fig. 5D), but the nematic order parameter remained negligible (seeFig. 2). Nematic order formation is found only for particles with larger aspect ratios ( ). Theexistence of the nematic precursors is also visible in the orientational structure factor, S ( q ) ,which reflects the density fluctuations of quadrupolar symmetry, (cid:37) l =2 ,m =0 ( q ) proportional tothe spherical harmonic for l = 2 and m = 0 ( ) (see Supplementary Materials). It is sensitiveto the growth of nematic structures, which shows up as a peak for small wavevectors, and hasbeen identified as correlation driving the glass transition at φ rc in MCT ( ). Fig. 5F (notethe log-linear plotting) shows that the large- q peak in S ( q ) , which records the orientationalalignment of neighboring particles, remains rather unchanged while, upon increasing φ , thenematic precursors cause a growing peak at small wavevectors. Its width measures the averagesize of the clusters, which exceeds the average particle separation by roughly a decade.Our finding of a liquid glass state and of nematic precursors in a system of colloidal ellip-soids in 3D is in stark contrast to experimental and theoretical results for colloidal ellipsoidsconfined in 2D (
25, 41 ). While for an aspect ratio of a/b = 3 . the 2D studies also show theappearance of a state in which rotational dynamics is frozen and translational mobility persists,the structural order of this state is very different from the 3D case, as in 2D domains with highnematic order are found. It has been suggested that the driving force for these nematic domainsin the 2D system could be long-wavelength fluctuations, as first described by Mermin and Wag-ner ( ). However, as Mermin-Wagner fluctuations are absent in 3D, we conclude from ourexperiments and simulations that in the 3D case long-wavelength fluctuations are supported bylong-range nematic correlations, induced by nematic precursors that are intersected by differ-ently oriented particles. Thus, nematic precursors appear to be prerequisite for the formation ofbulk liquid glass states. Conclusions
Hard ellipsoidal particles are a relatively simple system posing an entropic packing problemonly slightly more intricate than the one of spheres. Considering this, the richness of the states11iagram of equilibrium and kinetically arrested states is striking. As is well known since On-sager’s work on thin needles, above a specific density hard elongated particles align locally forentropic reasons. Unexpectedly, for an intermediate aspect ratio of the ellipsoidal colloids, localalignment gets frustrated on intermediate length scales and ramified, differently oriented regionsresult. Our simulations show that polydispersity plays no major role and we thus expect the for-mation of glass driven by differently aligned cooperative regions to exist in other glass-formingsystems as well. This effect thus needs to be considered in the active concerted efforts to formstructured materials from colloidal constituents. Our findings suggest that obtaining high orderwill require strong enthalpic contributions. The emerging nematic precursors contain hundredsof particles and it is the mutual obstruction of these cooperative regions that leads to the forma-tion of liquid glass. We showed that the critical long-range fluctuations connected with nematicordering, which is a weakly first order phase transition, are involved in the formation of thisnovel glassy state. This should be contrasted with the familiar glass transition which tracks thecrystallization line, where a strongly first order phase transition takes place. Thus, the liquidglass state may give the long sought paradigm where the interplay between equilibrium criticalcorrelations and critical slowing down versus glass-formation can be studied microscopically.This promises to shed light on the origins of dynamic heterogeneity in molecular systems. Ad-ditionally, it could also result in an intriguing venue for future studies concerning the formationand evolution of topological defects. While on a molecular scale, these dominate the phase-ordering dynamics in liquid crystals ( ), they also serve as a model system for the evolution ofmatter in the universe according to the Kibble-Zurek mechanism ( ). In both cases, the glassyarrest observed in our results will have an important influence on the resulting structures. Materials and Methods
Experimental methods
In a first step, spherical PMMA/PMMA core-shell particles dyed with different fluorophoresin the cores and the shells were synthesized according to a route described in (
28, 45 ). Asfluorophores for the cores we used the Bodipy dye ((4,4-difluoro-8-(4-methacrylatephenyl)-3,5-bis-(4-methoxyphenyl)-4-bora-3a,4a-diaza-s-indacene), which was synthesized accordingto ( ). The dye Quasar 670 (LGC Biosearch Technologies) served as a marker for the shellsand was used in form of a 4-vinylbenzylester. For core particle synthesis, the dye was linkedto methacrylic acid (Fluka) and copolymerized. The cores were cross-linked using ethyleneglycol dimethylacrylate. The resulting particle cores had a diameter of 1 µ m. Several seededdispersion polymerization steps were then used to grow a PMMA shell of the desired thicknessonto these cores. The PMMA in the shell material was not cross-linked.In a second step, the particles were thermo-mechanically stretched as reported by Kevilleet al. ( ) with modifications for the PMMA/PMMA core-shell particles as described in ( ). After the stretching, the ellipsoidal particles were restabilized with polyhydroxystearic12cid (PHSA) and dispersed in a mixture of 85:15 (w/w) cyclohexylbromide and cis-decalin tomatch both density and refractive index of the PMMA colloids ( ). Since PMMA particlesare known to charge in organic solvents, tetrabutylamoniumbromide was added to screen thecharges,The resulting suspensions were filled into a home-built sample chamber. Samples werestudied using a confocal laser scanning microscope (TCSP5, Leica Microsystems) equippedwith a resonant scanner (8 kHz, bidirectional mode) and a glycerol objective (63x, NA 1.30).An argon laser (wavelength λ = 514 . nm) and a helium neon laser ( λ = 632 , nm) were usedfor excitation of the dyes. The temperature of the whole microscope system was stabilized at . ± . ◦ C (Ludin Cube 2, Life Imaging Services), since at this temperature the density matchbetween particles and solvent was optimal. Volume fractions up to φ = 0 . were prepared bycentrifugation at higher temperature and removing solvent. The highest φ were obtained byslightly raising the temperature in a dense sample directly in the measurement chamber for afew days to achieve density mismatch. This led to the slow enrichment of particles at the bottomof the sample.The temperature was then changed back to ◦ C. After placing a sample onto the micro-scope it was allowed to equilibrate for h to minimize drift effects. Remaining drift, whichin all cases was small compared to the particle movement, was corrected with a routine ap-plied after detection and tracking. To minimize wall effects, all measurements were performedin a depth of − µ m into the sample. Positions and orientations of all particles in oneimage volume (144 µ m × µ m × µ m for structural analysis and 144 µ m × µ m × µ mfor measurements on the dynamics) containing more than 6000 particles were identified andlinked to trajectories using self-written and tested Matlab algorithms. The core-shell structurefacilitates tracking of the ellipsoid position and orientation with accuracies of nm and ◦ ,respectively ( ) (Fig. 1B). Experimental time-scales were limited by the bleaching of thefluorophores after long laser exposure.Volume fractions are the pivotal variable influencing the structural dynamics of the colloidalsuspensions. For their determination, particles were detected and counted in each measuredsample and the related Voronoi-volume was calculated to obtain the exact number fraction ρ .Multiplying this with the volume of a particle gives φ : φ = 43 π (cid:32) d (cid:33) ρ. (5)The determination of particle volumes, however, is intricate since PMMA particles are known toswell in CHB. Therefore, we determined the particle diameter by calculating the pair correlationfunction for a dense sample of spherical particles before stretching. The volume of a particleis conserved upon stretching, which was proven by crosschecking the obtained value with thewidth of an ellipsoid acquired by the pair correlation of the elliptical particles.13 lass stability analysis The values for the glass transitions for translation φ tc and rotation φ rc are found by the glassstability analysis of MCT in which the collection of all measured correlation functions is used.Close to the glass transition φ c , a critical decay onto the plateau occurs in a correlation function Φ j ( t ) (e.g. the translation or the rotation correlator functions F s ( q, t ) and L n ( t ) , respectively).Using first order correction terms ( ), the decay is given by Φ cj ( t ) = f cj + h j (cid:32)(cid:18) tt (cid:19) − a + ( k j + κ ( a )) (cid:18) tt (cid:19) − a (cid:33) (6)where t and κ ( a ) are system dependent constants and h j , k j and f cj are parameters, whichdepend on the correlator j . t [s] L ( t ) t [s] L ( t ) t [s] F s ( q , t ) A BC
Fig. 6. Glass stability analysis of translational (A) and (B) rotational relaxations:
Datafrom experiment (colored lines) and fits (solid lines) according to equations 6 and 7. Twodifferent glass transitions for translation φ tc = 0 . and rotation φ rc = 0 . were necessary to fitthe data. In panel ( C ), the grey dash-dotted lines indicate the result, when an identical transitiondensity was assumed for translation and rotation. For this condition, poor agreement with thedata at φ = 0 . was observed.Below φ c , the late plateau and the decay from the plateau is given by the von Schweidlerlaw: Φ j ( t ) = f cj − h j (cid:32) tτ φ (cid:33) b + k j (cid:32) tτ φ (cid:33) b (7)14here τ φ is a time scaling depending sensitively on φτ φ = t B /b (cid:32) C ( φ c − φ ) φ c (cid:33) − γ (8)and determines the final relaxation time τ close to and below φ c . Importantly, the exponents a , b and γ are related and determine all universal aspects close to the glass transition. Here, weemploy the exponents a = 0 . , b = 0 . (therefore γ = 2 . ) and B = 0 . and C = 1 . torepresent a repulsive glass-forming system ( ). For the glass stability analysis, both translationand rotation correlator functions were fitted simultaneously using Eqs. (6) and (7) for the threehighest volume fractions φ = 0 . , φ = 0 . , and φ = 0 . considering all measured F s ( q, t ) in the range . ≤ q · b ≤ . and L n for n = 2 , . Results are shown in Fig. 6A,B (again L is shown because the glassy two-step relaxation is more visible than in L ). The best fittingresults were obtained by keeping κ ( a ) = 1 and t = 0 . fixed and having f j , h j , k j , and τ φ asfree fitting parameters. Satisfactory fits to the data were only obtained assuming two differentglass transitions φ tc = 0 . and φ rc = 0 . . The grey dotted lines in Fig. 6C indicate the resultwhen only one glass transition density was assumed for translation and rotation. Additionalbackground on the MCT states diagram and the stability analysis is given in the Supplementarytext. Simulation details
We performed event-driven Brownian dynamics (ED-BD) simulations ( ) that model over-damped Brownian dynamics of hard ellipsoids. The system consisted of N non-overlappingellipsoids in a simulation box of length L and volume V = L that was varied depending on thedesired φ . Similar to De Michele et al. ( ), we set the moment of inertia to I = 2 mb / . Forvolume fractions of φ = 0 . , . , . , ellipsoids were placed randomly in the simulation box,while for volume fractions between φ = 0 . and φ = 0 . , the Lubachevsky-Stillinger ( )technique was used. All systems were equilibrated by Newtonian dynamics allowing transla-tional and rotational correlation functions to decay to zero. Two types of boundary conditionswere modeled. The first case corresponds to systems with periodic boundary conditions (PBC)with N = 504 . The second case corresponds to systems confined within rough walls. For theconfined systems, we started from a random configuration of ellipsoids and pinned theoutermost particles in the range < d < ∆ and L − ∆ < d < L to serve as the wall parti-cles. The remaining particles were free to move. The ED-BD model was first benchmarkedfor a dilute system of φ = 0 . , where we reproduced the ratio of translational and rotationaldiffusion coefficients found in the experiments: D rot b /D trans = 1 . setting the ratio of Brow-nian time steps ∆ t rot / ∆ t trans = 0 . (with ∆ t rot = 2 I/k B T D rot = 0 . short enough to giveangular diffusion). This ratio was then applied to all φ . At all densities, the time scale whenmatching to the experiments is taken from D trans ( φ ) . All presented results were averaged from20 initial configurations of the ellipsoids. 15 eferences
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We gratefully acknowledge the financial support by Deutsche Forschungsgemeinschaft, SFB1214,TP B5. We thank S. Schtter and F. Rabold for particle synthesis, Dr. P. Pfleiderer for his contri-bution on particle detection, Dr. M. Gruber for calculations on the schematic MCT model andProfs. Dr. H. Lekkerkerker and Dr. G. Maret for helpful discussions. J.-M. M. acknowledgesfunding provided by the Alexander von Humboldt Foundation.19 upplementary materials
Supplementary text to the following figures:Fig. S1: Superposition of correlation functions F s ( q, t ) and L n ( t ) from experimentFig. S2: Mean squared displacement, mean squared angular displacement and diffusion coeffi-cients from experiment for different φ Fig. S3: Correlation functions F s ( q, t ) and L n ( t ) and nematic order parameter Q of the Brown-ian simulated system with periodic boundary conditionsFig. S4: Correlation functions F s ( q, t ) and L n ( t ) of the Molecular dynamics simulated systemwith wallsFig. S5: S ( q ) of the Brownian simulated systemFig. S6: State diagram of a schematic MCT model Control experiments and fit parameters:
As expected for dilute systems, the curves for all determined correlation functions can be su-perimposed for volume fractions φ below the glass transition. This requires that the densitycorrelators F s ( q, t ) for several values of q are plotted versus log(∆ tq ) and the orientationalcorrelators L n ( t ) are plotted against log(∆ tn ) ) (Fig. S1). log( tn ) L n log( tq ) F s ( q ,t ) A B
Fig. S1. Superposition of the correlation functions.
For packing densities below the glasstransition, the values obtained for the density correlator F s ( q, t ) at different values of q can besuperimposed if they are plotted versus log(∆ tq ) ( A ). The four depicted wavevectors lie in therange . < q · b < . . Superposition is also possible when the orientational correlators L n ( t ) are plotted against log(∆ tn ) ( B ) for φ < φ c . Here, correlators for n = 2 and areshown.The data on the systems’s dynamics shown in the main text were measured several times foreach volume fraction. A combination of all the measured data is shown in Fig. S2A,B, wherewe plotted the mean squared displacement (MSD) (cid:104) ∆ r (cid:105) and the mean squared angular dis-placement (MSAD) (cid:104) ∆ θ (cid:105) for all measurements. Since the data nicely collapse onto one curvefor each φ , for clarity plots containing just single curves are shown in the main text. To evaluate20 q = 3 . µm − ) n = 4 ) 0.77396 0.74768 0.53068 0.33530 Table 1:
Fit parameters for the fits shown in Fig. 3 B and D.the homogeneity of the particle movements along the axes parallel (cid:104) ∆ r (cid:105) || and perpendicular (cid:104) ∆ r (cid:105) ⊥ to the orientation axis of an ellipsoid, we separately analyzed the respective MSDs(Fig. S2C,D). From fits to the data, we obtained short time diffusion coefficients D || and D ⊥ .As expected, particles tend to move faster along the orientation axis than perpendicular to it asis shown in the plot D || /D ⊥ . While this ratio slightly increases with φ and have a maximumin the liquid glass state at φ = 0 . , they hardly deviate from the values expected for a freelydiffusing ellipsoidal particle indicated by the dashed line in Fig. S2E (39) .As described in the main text, we fitted the obtained correlation functions to the Kohlrausch-function (Eq. (4). Only curves which showed a clear decay within the measured time windowwere fitted; representative examples are shown in Fig. 3C,D. Fit parameters are collected inTable 1. Details on the simulation results:
Fig. S3 shows the correlation functions, F s ( q, t ) , L ( t ) , L ( t ) and the order parameter S forthe systems with periodic boundary conditions. Similar to De Michele et al. ( ), the isotropic-nematic threshold was set to S = 0 . , hence the nematic transition is at φ ≈ . . For nematicsystems, F s ( q, t ) monotonically decreases to 0 without significant stretching, while L ( t ) tendsto form a plateau that corresponds to the orientational ordering. We verified the formation of anematic state in these systems by calculating the plateau height of L ( t ) and the final value of S ( ), lim t →∞ L ( t ) = S (S1)For the nematic systems obtained in the simulations, the final values of L ( t ) are close tothe S values despite the finite time and finite size of the box.Fig. S4 shows F s ( q, t ) , L ( t ) and L ( t ) for the systems with rough walls. In this figure, thecharacteristic plateau formation for glasses can be observed. Unlike the systems with periodicboundary conditions (Fig. S3), the corresponding S values for the systems with rough wallsare all below 0.3. Plateaus are clearly observed starting at φ = 0 . . The L ( t ) plateau valueis ≈ . and Eq. (S1) is no longer obeyed. Instead, the plateau values are used for the vonSchweidler fits for glassy systems as discussed in the main text. S lm ( q ) is an orientation-dependent structure factor that can be used to analyze the correlationof the orientations of the ellipsoids. It is defined as S lm ( q ) = 1 N (cid:104) ρ ∗ lm ( q ) ρ lm ( q ) (cid:105) , (S2)21 D / D t [s] -3 -2 -1 r [ µ m ] t [s] -2 -1 r [ µ m ] C DE t [s] -3 -2 -1 [ r ad ] t [s] -2 -1 r [ µ m ] A B
Fig. S2. Experimental data showing the mean squared displacement (MSD) (cid:104) ∆r (cid:105) (A)and mean squared angular displacement (MSAD) (cid:104) ∆ θ (cid:105) (B) for all measured samples(grey lines). Very good coincidence is found for subsequent measurements at the same φ .Therefore only one measurement for each φ is shown in the main text which appear to be goodrepresentatives (colored lines). The dotted line indicates the slope m = 1 which shows the freediffusive behaviour. Additional calculation of the mean squared displacement perpendicular (cid:104) ∆ r (cid:105) ⊥ ( C ) and parallel (cid:104) ∆ r (cid:105) || ( D ) to the orientation axis of an ellipsoid from the experiments.( E ) The ratio of the obtained diffusion coefficients, the vertical dashed line illustrates the valuefor a free particle derived in Ref. (39) .with the microscopic density defined as ρ lm ( q ) = √ πi l N (cid:88) j =1 e i q · r j Y lm ( Ω j ) , (S3)where Y lm ( Ω ) is the spherical harmonic function for Euler angles Ω ( θ, φ ) . Note that we22 B -1 F s ( q = . ,t ) t (sec).55.54.53.52.51.50.49.48.47.46.40.20.05 0.00.20.40.60.81.010 -1 L ( t ) t (sec) C -1 L ( t ) t (sec) 0.00.20.40.60.81.010 -1 final L valuesL → → → → → → → S t (sec)0.00.20.40.60.81.010 -1 F s ( q = . ,t ) t (sec).55.54.53.52.51.50.49.48 D E
Fig. S3. Data from BD simulations with periodic boundary conditions for F s ( q , t ) (A), L ( t ) (B) and S (C) where the plateau formation is attributed to nematic ordering startingat φ = . . Figure legend of part A also applies to B,C . Additional data for F s ( q, t ) ( D ) and L ( t ) ( E ) A B -2 -1 L ( t ) t (mb /k B T)0.00.20.40.60.81.010 -2 -1 F s ( q = . ,t ) t (mb /k B T)0.570.560.550.540.530.520.510.500.490.480.470.460.45 -2 -1 L ( t ) t (mb /k B T)0.570.560.550.540.530.520.510.500.490.480.470.460.45 C Fig. S4. Data from MD simulations with rough walls for F s ( q , t ) (A), L ( t ) (B) and L ( t ) (C) where the plateau formation for high packing fractions is attributed to glass formationat φ ≈ . . Figure legend of part B also applies to C .23nly consider the diagonal terms of the orientation-dependent structure factor. The results for l = 2 , m = 0 in the simulations (using periodic boundary conditions) are shown in Fig. S5.The transition to a nematic phase in the simulations is reflected in S in two ways. First, forsmall q -vectors, S increases as φ increases, signaling the formation of long-range order thatis limited to the size of the simulation box. Second, the neighbor peak of S starts to becomemore visible and shifts to the right as φ increases. That is, the favored alignment of neighboringellipsoids is such that their axes of symmetry ( a ) are parallel to each other. q 2b S ( q ) Fig. S5. Simulation counterpart of Fig. 5F of the main text.
Calculated S ( q ) shows thenematic transition for φ > . , indicated by the right shift of the first peak. Mode coupling theory
We argue that the region of liquid glass in phase space is bounded by two glass transitions atfixed aspect ratio, providing the basis for our MCT glass stability analysis in the main text andextending the previous MCT calculations ( ); there the possibility of a glass-to-glass transitionwas not explored.The transition of a fluid of hard ellipsoids to a liquid glass was found by MCT in a fully mi-croscopic calculation. It was shown to be driven by long-range nematic correlations which ariseclose to the equilibrium isotropic-nematic transition (see Fig. 5F recording these correlations inthe samples). The transition line meets another line of fluid to glass transitions, which extendsto the MCT hard sphere transition for aspect ratio approaching unity. It is driven by an increaseof liquid short range structure seen in S ( q ) (’cage effect’; see Fig. 1D). As MCT glass tran-sitions are fold bifurcations in a nonlinear algebraic system, transition lines do not stop whenthey meet but rather intersect generically. A schematic model shows that the latter transitionline extends into the glassy region above the transition to the liquid glass. There it gives a lineof liquid glass to glass transitions which is in agreement with the experimental observations.Starting point is the model by Bosse and Krieger (40) (BK). It describes the generic couplingof two degrees of freedom in the case of a single (discontinuous or generic (41) ) glass transition.Their correlators Φ ( t ) and Φ ( t ) shall correspond to L n ( t ) and F s ( q, t ) , respectively. The Φ i ( t ) ig. S6 Schematic MCT model capturing the generic glass transition scenario of twomodes: The left panel shows the states diagram at fixed v = 0 . and w = 3 , where twodiscontinuous bifurcations exist. They separate fluid ( f = 0 , f = 0 , blue) from liquid glass( f > , f = 0 , orange) and glass ( f > , f > , green) states. Parameters v and v mim-ick aspect ratio and density. The other panels show the correlators: The middle panels shows Φ ( t ) modeling orientational motion (viz. L n ( t ) ), and the right panel Φ ( t ) modeling transla-tional motion (viz. F s ( q, t ) ); the overdamped MCT equations of motion are solved for the pointsmarked by stars in the states diagram (precise values are v = 3 . , 4.1, 4.5 at v = 0 . ).obey Zwanzig-Mori equations with memory kernels m i ( t ) (for i = 1 , ) given as a generalquadratic form. The slowing-down of the correlators’ relaxation arises from the feedback in theretarded friction kernels modeled by m ( t ) = v Φ ( t ) + v Φ ( t ) and m BK2 ( t ) = v Φ ( t )Φ ( t ) .We generalize the model by including a parameter w modeling the coupling of the seconddynamical mode to itself; viz. m ( t ) = v Φ ( t )Φ ( t ) + w Φ ( t ) . This allows for a secondgeneric glass transition. The two glass transitions of the model will correspond to the B and B (cid:48) transitions introduced by Letz et al. ( ). The parameters v and v encode the increasingorientational friction arising due to slow orientational and translational motion, respectively.Thus v should correlate with the aspect ratio and v with the density. The cross-coupling term v parametrizes the translational friction arising from rotation-translation coupling, while w captures the feedback within the translational motion only. The glass parameters f i = Φ i ( t →∞ ) obey the equations f i / (1 − f i ) = m i ( t → ∞ ) , where glass transitions appear as bifurcations.Since the model lacks a quadratic coupling of the first mode into the second kernel, it containsa type B (cid:48) transition at v = 4 and v small enough, where f jumps from zero to a finite value,while Φ ( t ) remains fluid like. For parameter sets with a second transition from fluid to glass,which is continuous in the BK-model ( w = 0 ) and discontinuous for w > , the schematicmodel shows that this B line cannot terminate at the intersection with the B (cid:48) -line. Rather, itcontinues into the glass region, so that there exist two different glass states separated by a lineof glass-to-glass transitions. Fig. S6 gives the pertinent states diagram of the model. Choosingan overdamped dynamics with initial time-scale τ in both correlators (see Eq. (4.34), p. 20325f Ref. (41) ), typical correlation functions for the fluid (blue), liquid glass (orange), and glass(green) state are depicted in Fig. S6 as well. In liquid glass states, Φ ( t ) arrests at a finiteplateau while Φ ( t ) decays to zero. References for Supplementary material
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