Gelation as condensation frustrated by hydrodynamics and mechanical isostaticity
Hideyo Tsurusawa, Mathieu Leocmach, John Russo, Hajime Tanaka
GGelation as condensation frustrated by hydrodynamics and mechanical isostaticity
Hideyo Tsurusawa, ∗ Mathieu Leocmach, ∗ John Russo, and Hajime Tanaka † Department of Fundamental Engineering, Institute of Industrial Science,University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan Univ Lyon, Universit´e Claude Bernard Lyon 1, CNRS,Institut Lumi`ere Mati`ere, F-69622, Villeurbanne, France School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom Institute of Industrial Science, University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
Colloidal gels have unique mechanical and transport properties that stem from their bicontinousnature, in which a colloidal network is intertwined with a viscous solvent, and have found numer-ous applications in foods, cosmetics, construction materials, and for medical applications, such ascartilage replacements. So far, our understanding of the process of colloidal gelation is limited tolong-time dynamical effects, where gelation is viewed as a phase separation process interrupted bythe glass transition. However, this picture neglects two important effects: the influence of hydro-dynamic interactions, and the emergence of mechanical stability. With confocal microscopy exper-iments, here we successfully follow the entire process of gelation with a single-particle resolution,yielding time-resolved measures of internal stress and viscoelasticity from the very beginning of theaggregation process. First, we demonstrate that the incompressible nature of a liquid componentconstrains the initial stage of phase separation, assisting the formation of a percolated network.Then we show that this network is not mechanically stable and undergoes rearrangements driven byself-generated internal stress. Finally, we show that mechanical metastability is reached only afterpercolation of locally isostatic environments, proving isostaticity a necessary condition for the sta-bility and load bearing ability of gels rather than the glass transition. Our work reveals the crucialroles of momentum conservation in gelation in addition to the conventional purely out-of-equilibriumthermodynamic picture.
I. INTRODUCTION
A gel is a soft solid composed of two intertwinedphases: a solid network and a liquid solvent. They arean ubiquitous state of matter in every-day life, makingup most of the foods we eat, the cosmetics we use, con-cretes, and our own organs. In colloidal gels, the net-work is composed of colloidal particles bonded togetherby attractive forces. Such colloidal assemblies are outof equilibrium, as the thermodynamic ground state ofthe system involves the macroscopic separation betweena particle-rich and a particle-poor phase. Despite thethermodynamic driving force towards compactness, thegel persists due to the dynamical arrest of the network,often described as a glass transition [1–8]. This has ledto the popular physical picture that a gel is formed bydynamical arrest of bicontinuous spinodal decompositiondue to glass transition. The direct link between spon-taneous gelation and spinodal decomposition has beencarefully confirmed by combining experiments and the-ories [8]. This recently established scenario is certainlya large step towards a more complete understanding ofcolloidal gelation.However, this picture still leaves some fundamen-tal problems unanswered: (i) The knowledge of ordi-nary spinodal decomposition predicts that the minority ∗ These authors contributed equally to this work † [email protected] colloid-rich phase should form isolated clusters ratherthan the observed percolated network [9]. (ii) A col-loidal gel is sometimes formed by a network made ofthin arms, which are too thin to be regarded as glasses.This casts some doubt on the popular scenario of dy-namic arrest due to a glass transition. Indeed, the glasstransition is defined as a kinetic transition and has nodirect link to mechanical stability in a strict sense. Slowdynamics and mechanical stability are conceptually dif-ferent. In an extreme case, for example, a gel formedby bonds with a short lifetime can be ergodic and in anequilibrium state. (iii) A gel often displays superdiffusivebehavior, detected as the compressed exponential decayof a density correlation function, during ageing, as ob-served by time-resolved spectroscopy techniques [10–12]and recently simulations [13–15]. The origin of this phe-nomenon and its relation to problem (ii) are still elusive.Several mechanisms have been proposed to try to ra-tionalize some of these issues. Fluid momentum con-servation can play an important role in phase separa-tion, giving to hydrodynamics an active role in struc-tural evolution [4, 16]. There have been some numericalstudies on the role of hydrodynamics [17, 18] and me-chanics [10, 13, 15, 19–21] in colloidal gelation, however,there has so far been no experimental access to theseproblems due to the lack of a method to follow the wholekinetic processes with single-particle resolution in bothspace and time. More importantly, some questions onthe emergence of elasticity, which is the most fundamen-tal physical property of gels, have still remained unan-swered. The stability of gels is ascribed to the formation a r X i v : . [ c ond - m a t . s o f t ] A p r of locally favored structures, or local energy-miminumconfigurations [22], while the mechanics of the networkis being recognized to play a major role in the ageingbehavior of gels [13, 14, 23, 24]. Purely geometrical con-ditions for mechanical stability have also been proposed.Kohl et al. [25] have found that in dilute suspensions, afinal gel state could be obtained only for interaction po-tentials where directed percolation was observed. Hsiao et al. [26] have found that strain-induced yielding coin-cide with the loss of rigid clusters. Rigidity was definedusing Maxwell criterion for isostaticity, that is 6 neigh-bors per particle [27]. However, the relationship betweenlocal structures, dynamic arrest, and the emergence ofelasticity remains poorly understood even at a funda-mental level.In order to address these problems experimentally, inthis Article we study roles of hydrodynamics and me-chanics in colloidal gelation by dynamical confocal mi-croscopy observation of the entire gelation process, fromthe beginning to the end with particle-level resolution.To reveal the roles of momentum conservation in gela-tion, it is crucial to develop a special protocol to initi-ate phase separation without causing any harmful flow(see Methods). We stress that such flow not only makesthe initial state ill-defined, but also has permanent influ-ence on the kinetic pathway, as well as the final state.This protocol allows us to experimentally elucidate (a)the importance of hydrodynamic interactions in formingand stabilizing elongated clusters and their percolation,and (b) the crucial role of percolation of isostatic struc-tures in conferring mechanical stability to the network.These findings shed new light on the mechanisms of gelformation and coarsening, and also on the fate of gels. II. SINGLE-PARTICLE PHASE SEPARATIONA. System design
In our protocol, we first enclose a salt-free suspensionof sterically and charge-stabilized colloids and non ad-sorbing polymers in a thin microscopy cell sketched inFig. 1(a). The bottom wall of the cell is an osmotic mem-brane providing contact with a long channel full of thesame solvent mixture. Salt dissolution and subsequentmigration of the ions along the channel and through themembrane induce screening of the electrostatic repulsion,revealing the depletion potential well due to the poly-mers. The time needed for the ions to diffuse from themembrane across the cell thickness is of the same or-der of magnitude as the Brownian time of the particles τ B = 10 s. This relation between the two key timescalesenables us to switch instantaneously (physically) froma long-range repulsive to a short-range attractive sys-tem without any external solvent flow, which has neverbeen achieved experimentally before. This causes uni-form gelation starting from the homogeneous Wignercrystal state, allowing in situ confocal microscopy obser- ReservoirObjective lensObservation cellMembrane S a l t d i ff u s i o n a µ m b c FIG. 1. Reservoir cell. (a) Sketch of our experimental setup.The observation cell contains initially colloids, polymer andno salt. (b) Confocal slice of a gel formed in situ by ourmethod ( φ = 25 . c p = 1 . ex situ andimmediately pumped into a capillary. vation throughout the process from a well-defined initialtime.Figure 1(b) and (c) compare the final structures of twogels prepared at the same state point with the two differ-ent protocols: the first one by our salt-injection protocol,and the latter by the conventional approach, where a gelis formed in a capillary and then shear melted at thestart of the experiment. Already a visual inspection re-veals that the latter is coarser, highlighting that shakingor shear melting protocols [8, 28] are not equivalent toa quench. Our special quench protocol provides an idealexperimental platform to make a comparison with theoryand simulations. However, we note that Brownian Dy-namics simulations cannot reproduce our experimentalresults even with the same quench, because they neglectthe solvent mediated hydrodynamic interactions [17, 18].In Fig. 2, we show a computer reconstruction from ex-perimental coordinates of a typical gelation experimentat a relatively low volume fraction ( φ = 7 . t = 0), the suspensionis in a Wigner crystal state where the particles are farapart due to long range Coulomb repulsion. As soon asthe charges are screened, the particles begin to aggre-gate and form clusters that progressively connect to eachother while coarsening to finally percolate over the fieldof view around t = 35 min, see Methods. The absenceof macroscopic flow can be confirmed in SupplementaryMovie 1. B. Phase separation dynamics
In Fig. 3(a), we show the phase diagram, where wecan divide the state points into three regions based onthe final state obtained by our protocol: at low poly-mer concentration ( c p < . ReservoirObjective lensObservation cellFilterSalt diffusion a µ m b c FIG. 2. Snapshots from the entire gelation process reconstructed via particle-tracking of a typical sample close to the cluster-gelline ( φ = 7 . c p = 1 mg/g) using the salt-injection protocol. Particles are colored according to the size of the cluster theybelong to, going from blue for monomers to red for the percolated cluster. − − ( t − t ) /τ B h q i σ . . . . ¯ N C l m a x / L b c . . φ [%] c p [ m g / g ] spinodal gel fluid clusters a FIG. 3. Different regimes of gelation. (a) Phase diagram.Experimental points are categorised based on the final stateobtained in the reservoir cell. The spinodal line is obtainedfrom free volume theory in polymer dilute regime. The dashedline is the polymer overlap concentration in the free volume.State points analysed in b and c are circled. (b) Growthof the characteristic wave number. By increasing density: φ = 4 . , , ,
27 %, c p = 1 , . , . , (cid:7) , ◦ , (cid:4) and • respectively. The lines are possible scaling laws forthe intermediate coarsening regime. (c) Comparison of sys-tem evolution in terms of largest cluster extent and of meancoordination number for the same samples. ( φ < .
05) and high polymer concentration the parti-cles condense into long-lived well separated clusters asobserved in [29]; in the rest of the explored phase spacewe observe a long-lived space spanning network. In the phase diagram we also plot the spinodal (continuous) andthe polymer overlap concentration (dashed) lines as ob-tained from free-volume theory calculations [30]. Despitethe limitations of the theory, the agreement between thespinodal line and our experiments is rather satisfactory,with the only exception being the region of small col-loidal volume fractions and high polymer concentration(see, e.g., Ref. [8]).To confirm whether the different samples followspinodal-decomposition kinetics, we compute the timedependent static structure factor S ( q, t ). For all gel andcluster samples, we observe the appearance of a low q peak in S ( q ), see Supplementary Figure S1, which ischaracteristic of spinodal decomposition in a system witha conserved order parameter. Then, we compute thecharacteristic wave number (cid:104) q (cid:105) and show its temporalevolution in Fig. 3(b) for various colloidal volume frac-tions. The curves for all samples follow a master curvecoherent with spinodal decomposition kinetics: At shorttimes (cid:104) q (cid:105) ( t ) shows a plateau indicating that the low q peak builds up at a constant wave number correspond-ing to distances of about 2 σ . This plateau is character-istic of the early stage spinodal decomposition, which isdescribed by Cahn’s linear theory [9]. At intermediatetimes, on the other hand, we observe coarsening with (cid:104) q (cid:105) ∼ t − α , with an exponent α which is compatible withboth α = 1 /
3, typical of spinodal decomposition with-out dynamical asymmetry between the two phases, and α = 1 /
2, which is often observed in viscoelastic phaseseparation (see, e.g., Ref. [17]). Due to the narrow rangeof this power law regime and finite size effects, we can-not conclude definitely on the exponent value. Finally atlonger times each sample deviates from the master curveto form a plateau indicating arrest. The more dilutesamples arrest sooner, but reciprocal space informationdoes not allow to identify whether the origin of arrest isdifferent between clusters and percolated networks.To identify the origin of the dynamical arrest observedabove, we now take advantage of the single-particle levelresolution of our reconstructed trajectories. To char-acterise this path, we compute the instantaneous meannumber of neighbors ¯ N C , or coordination number, thatquantifies the compactness of the structure. We alsocompute the spatial extent of the largest cluster l max that we normalize by the size of the field of view L toobtain a measure of the distance to percolation of thesystem. Figure 3(c) shows a system trajectory in the( l max /L, ¯ N C ) plane for various colloidal volume fractions.All trajectories show a linear increase of both cluster size l max /L and number of neighbors ¯ N C at early times, cor-responding to the first plateau in Fig. 3(b). This is fol-lowed by the coarsening stage, which happens differentlydepending on the density. At high densities, coarsen-ing occurs after percolation, which happens within thefirst few τ B after charge screening by salt. At low den-sities, percolation never takes place and coarsening re-sults in the compaction of individual clusters, that keepstheir overall size l max /L , while increasing the number ofneighbors ¯ N C . We did not observe any Ostwald ripeningamong clusters, indicating that the diffusive evaporation-condensation coarsening mechanism is negligible com-pared to cluster collisions and coalescence, as expectedfor colloidal viscoelastic phase separation [16]. At in-termediate densities, we observe the process detailed inFig. 2: formation of low-compacity clusters that thenslowly connect together to build the percolating network.This process can take hundreds of τ B and is compet-ing with cluster compaction, as indicated by the obliquetrajectory (red open circles) in Fig. 3(c). Particle-levelquantities thus demonstrate that the path to gelation isnot universal and depends on the colloid volume fractioneven within the gelation region. By contrast, polymerconcentration, i.e. the depth of the attraction potential,has little effect on the path to gelation, see Supplemen-tary Figure S2.Our observations indicate that both the cluster andgel phases are due to viscoelastic spinodal decompo-sition [4, 16]: network-type spinodal for the gel, anddroplet-type spinodal for the clusters, where strong dy-namical asymmetry between colloids and the solventleads to unique roles of hydrodynamics and mechanicsin phase separation.In section III we will explore the role of hydrodynam-ics. To do so, we will restrict to dilute cases where per-colation occurs late or never, leaving enough time to ob-serve disconnected clusters. In section IV, we will ex-plore the precise mechanism of arrest and the emergenceof mechanical rigidity by studying the dynamics withinthe network of percolating samples. III. ROLE OF HYDRODYNAMICS
In Fig. 4(a), we follow the compaction of clusters madeof only three particles in a non-percolating sample. The . . . . ∆ t/τ B P e l expBD 10 . . . . λ /λ λ /λ t/τ B a s p e c t r a t i o gelcluster0 20 40 60 80 100 120 140 160 1800246 · − angle (degree) p r o b a b ili t y ( d e g r ee − ) existingfutureisolated a bc FIG. 4. Hydrodynamics. (a) Probability of staying elongatedfor a triplet in a non-percolating sample ( φ = 4 %, c p =1 mg/g, blue) and in corresponding BD simulations. Thecontinuous lines are the respective best exponential fits ofcharacteristic time 27 τ B and 5 τ B respectively. (b) Evolutionof the aspect ratios of clusters of 4 particles and more inthe same sample (dashed lines) and in a percolating sample( φ = 8 %, c p = 1 . time-averaged probability distribution of the radius ofgyration R g of these triplets shows two peaks on bothsides of R ∗ g = 0 . σ , see Supplementary Figure S3. For R g < R ∗ g the cluster is compact, with a structure close toan equilateral triangle. For R g > R ∗ g the three particlesare aligned and the cluster is elongated. We found thatjust after the quench triplets have a slightly higher prob-ability of being elongated. Afterwards, they either con-nect to other clusters or relax to the more stable compactstate. To follow this relaxation, we define the probabilityto stay elongated as P el (∆ t ) = (cid:104) P ( δ i ( t + ∆ t ) | δ i ( t )) (cid:105) t,i , (1)where δ i ( t ) is the probability for the triplet i to be elon-gated at time t . Figure 4(a) (blue line) shows that thedecay of P el (∆ t ) is exponential with a characteristic timeof 27 τ B . In the same figure we also plot (red symbols)the same quantity computed from Brownian Dynamicssimulations of short-range attractive colloids designedto match the experiments (see Methods), in which thetriplet compaction process is simulated in absence of hy-drodynamic interactions. For the simulations we observea considerably faster exponential decay compared to theexperiments, suggesting that the triplet compaction pro-cess is indeed slowed down significantly by hydrodynamicinteractions.The shape of clusters composed of more than 3 par-ticles cannot be followed in the same way. Instead, wecompute the principal moments of gyration of individualclusters λ j , ordered such that λ ≥ λ ≥ λ , with use theaspect ratios λ /λ and λ /λ to quantify the departurefrom sphericity. In Fig. 4(b), we show the evolution ofthe average value of these aspect ratios either for a non-percolating sample (dashed line), or before percolationfor a percolating one (continuous line). In both cases, weobserve that the clusters are originally not compact andbecome more isotropic over tens of τ B . As can be seen inFig. 2 and Supplementary Figure S5, structural isotropyis recovered only after the fusion of many anisotropicclusters into a branched structure that may or may notbe percolating.These observations can be understood as due to hy-drodynamic effects. Indeed in a solvent, particles can-not converge freely to form compact structures [4, 17].The compaction is delayed by the incompressibility ofthe solvent, which allows only divergence-free transverseflow fields. Furthermore, clusters influenced by hydrody-namic interactions tend to be more elongated, less com-pact. We can test this hypothesis by measuring at whichangle particles meet relative to existing neighbors. If in-fluenced by hydrodynamics, particles should avoid thedirection of existing neighbors and come from more openangles. In Fig. 4(c) we show the bond angle distributionfor three different sets of bonds: (i) existing bonds, (ii)bonds that will form within the next τ B ( future bonds ),(iii) future bonds where the newly attached particle is amonomer. As expected, existing bonds (i) are preferen-tially at a 60 ◦ angle, indicating stable packing, with sec-ondary peaks coherent with a mixture of tetrahedral andhexagonal packing. Future bonds (ii) have more acuteangles and almost never 180 ◦ , since they are mostly dueto particles attached to second neighbors, see sketch inFig. 4(c). Here hydrodynamics plays no role. By con-trast, future bonds (iii) involving isolated particles format more obtuse angles, with a clear peak around 180 ◦ .This confirms that hydrodynamics has a significant in-fluence on particle aggregation and explains why clustersare initially elongated.Consequently, long-lived elongated clusters have ahigher probability to meet via either rotational or trans-lational diffusion than compact spherical clusters. Hy-drodynamics explains why in a rather dilute regime wecan observe immediate formation of elongated clustersand then their slow, hydrodynamically-assisted aggrega-tion into a percolated structure. We stress that this isa direct consequence of large-size disparity between col-loidal particles and solvent molecules, which leads to thephysical situation where discrete solid objects are floating . . . . l m a x , l D [ L ] isotropicdirected N C ≥ N C ≥ t/τ B G , G , Σ [ m P a ] t/τ B Σ G G a bc d FIG. 5. Evolution of space-spanning microstructure and me-chanical response. (a) and (b): Percolation processes, for adilute ( φ = 8 % , c p = 1 . / g) and a dense ( φ = 27 % , c p =1 mg / g) sample. The processes of isotropic and directed per-colation of all particles ( N C ≥
1) are respectively plottedas thin orange curve and orange symbols. The processesof isotropic and directed percolation of isostatic particles( N C ≥
6) are respectively plotted as thick purple curve andpurple symbols. (c) and (d): Mechanical response for thesame samples. Elastic ( G (cid:48) ) and viscous ( G (cid:48)(cid:48) ) shear moduliat high frequency ( f = 0 . τ − ), obtained by two-particle mi-crorheology, are drawn respectively as filled and open circles.The thick grey curve is the internal stress Σ obtained fromthe measure of bond-breaking probability. The thin orange(left) and thick purple vertical lines show the isotropic per-colation times for all particles ( t perco ) and isostatic particles( t ) respectively. in a continuum liquid. IV. EMERGENCE OF MECHANICALSTABILITYA. Percolation
To better grasp the timing of space-spanning mi-crostructural changes in dilute and dense regime, weplot in Fig. 5(a) and (b) the time evolution of largestconnected cluster l max (orange curves), and define the isotropic percolation time ( t perco ) as the moment when l max > . L . In dense case, l max reaches the size L ofthe observation window in a few Brownian times, whereasit takes hundreds of τ B in the dilute case. We also dis-play the maximum spatial extent of directed paths l D (orange symbols in Fig. 5(a) and (b)). A directed pathis defined as a path with no loop or turning back, suchthat every step is in either the positive X, Y, or Z direc-tions. We thus define the directed percolation time ( t D )as the moment when l D > . L . Finally, in Fig. 5(a)and (b) we also consider isostatic clusters , defined as clus-ters which comprise only particles that have at least sixbonded neighbors. For isostatic clusters we then plotboth l max (purple curves) and l D (purple symbols) (seealso Supplementary Movie 2). We define t as the time ofpercolation of isostatic clusters. We observe that directedpercolation of all particles (at t D , see orange triangles)and isotropic percolation of isostatic particles (at t , seethick purple curve) occur simultaneously in the diluteregime. However the two time scales are well separatedin the dense regime. This separation of time scales offersthe opportunity to test the role of both type of spacespanning microstructures in the mechanical stability ofgels. B. Mechanical stability and percolations
The solid nature of a material is most often definedfrom linear mechanical response. However for colloidalgels, mechanical stability cannot be predicted withoutan understanding of internal stresses [14]. Here we areable to extract both information from our particle-levelexperiments. We use the particles themselves as passivemicrorheological probes to extract the elastic ( G (cid:48) ) andviscous ( G (cid:48)(cid:48) ) parts of the shear modulus, see Methods.We also extract the average value of the internal stress Σfrom the bond breaking rate, see Methods. Results areshown in Fig. 5(c) and (d) for direct comparisons withthe microstructure.The typical ranges of stresses and moduli we measureextend below 0 . G (cid:48)(cid:48) consistent withthe viscosity of a hard sphere suspension at their respec-tive volume fraction. Internal stresses are high at shorttime, reflecting the stretching due to hydrodynamic frus-tration. The emergence of mechanical stability is cap-tured simultaneously from both the linear viscoelasticitymeasurements, with the crossing between G (cid:48) and G (cid:48)(cid:48) , andthe internal stress, which exhibits a sharp drop in Σ. Thetiming of the emergence of elasticity is thus unambigu-ous and occurs well after isotropic percolation time t perco (see orange vertical lines in Fig. 5(c) and (d)). This gen-eralizes observations by Kohl et al. [25] on the final stateof dilute samples.In dilute samples, the elastic behavior occurs in thesame time scale as directed percolation of all particles.However, isotropic percolation of isostaticity also occurssimultaneously. Therefore we have to look at the denseregime to disentangle the two possible microstructuralcauses. Indeed in the dense regime the elastic behavioremerges well after directed percolation of all particles, inthe same time scale as isotropic percolation of isostaticity. φ [%] t / t D − . − . − . − . ( t − t perco ) /τ B h X i j ( t ) − X i j ( t p e r c o ) i / σ τ B τ B τ B Isostatic N C < a bcde FIG. 6. Directed percolation and isostaticity percolation. (a)Ratio of the time of isostaticity percolation by the time to di-rected percolation, function of colloid volume fraction. Hor-izontal dashed line shows when both times are equal. (b)Detail of a reconstruction from confocal coordinates at per-colation time in a dilute sample ( φ = 8 % , c p = 1 . / g).Isostatic particles are drawn to scale, non isostatic ones aredrawn smaller for clarity. The bond network is displayed inorange. (c) and (d) Same as (b) at later times. (e) Incrementof Euclidian distance between two isostatic clusters, averagedover all such pairs initially connected by a non-isostatic net-work strand. The reference time is the percolation time. This allows to lay the main result of this article: theemergence of rigidity is caused by isotropic percolationof isostatic clusters, able to bear stress across the sample.
C. Directed or isostaticity percolations
In Fig. 6(a), we compare across all our experimentsthe time to percolation of isostaticity t to the time todirected percolation t D . We confirm that at high volumefractions, typically φ >
14 %, the two phenomena aredecoupled, with 2 < t /t D <
20 depending on the statepoint. By contrast at lower φ , both percolations occursimultaneously, independently of the attraction strength.The reason for this coincidence can be understood bythe specific path to gelation in the dilute regime. Wehave seen in Fig. 3(c) that isotropic percolation occurredat a late stage, when the clusters had time to compact.Indeed, Fig. 5(a) shows that at percolation time, isostaticparticles already form clusters that reach up to a tenthof the observation window. Cluster-size distribution (seeSupplementary Figure S6) and three dimensional recon-struction in Fig. 6(b) show that these isostatic clustersare compact, typically 3 to 5 particles in diameter andlinked by non-isostatic bridges. The floppiness of thesebridges prevents directed paths to reach percolation.From this situation, percolation of isostaticity proceedsby the compaction of the floppy bridges. Importantly,this compaction takes place without adsorption of newparticles onto the bridge. Compaction is a local processthat involves no particle migration but only creation ofnew bonds, as shown in Fig. 6(b)-(d). Consistently, thiscompaction leads to a straightening and a shortening ofthe strands. We quantify this shortening by computingthe Euclidean distance X ij ( t ) between the centers of massof two isostatic clusters i and j . The increment of thisdistance with respect to percolation time, averaged overall cluster pairs connected by a floppy bridge is shown onFig. 6(e). The observed shortening is about 0 . σ or 25%of the initial length. Directed percolation becomes possi-ble when a percolating path has become straight enough,which implies isostaticity. That is why directed percola-tion and isostaticity percolation occur simultaneously inthe dilute regime. D. Stress-induced network breakup
In a dense system, after directional percolation of allparticles, the number of nearest neighbors monotonicallyincreases to minimize the energy of the structure, result-ing in the growth of isostatic configurations, as discussedabove. During this process the mechanical tension in-ternal to the network grows, driving it towards com-paction, which can lead to network coarsening accom-panying bond breakage (see Fig. 7(a) and (b)). Unlikein simulations [13], we cannot directly measure the lo-cal internal stress at this moment, but we can still seeits effects through the local stretching measured by thedegree of two-fold symmetry q (see the particle color inFig. 7(a)). From this, we may say that a bond break-age event is the consequence of stress concentration ona weak bond, leading to local stretching of the bond,and its eventual breakup. In other words, mechanicalstress acts against diffusive particle aggregation (or com-paction), which is the stress-diffusion coupling character-istic of phase separation in dynamically asymmetric mix-tures [4, 16]. This stress-driven aging is accompanied bymechanical fracture of the percolated network structureby the self-generated mechanical stress. The mechanicalstability can be attained only after the formation of apercolated isostatic structure, which is a necessary con-dition for a structure to be mechanically stable. Whenthe percolated isostatic structure can support the inter-nal stress everywhere, the system can attain mechanicalstability. V. DISCUSSION AND CONCLUSION
In summary, we have observed with particle-level res-olution the entire process of gelation at various statepoints for the first time. The early stages are charac-terized by the universal features of spinodal decompo-
16 min 30 . ab FIG. 7. Breakup of the network by internal stress. (a)Reconstruction from experimental coordinates ( φ = 29 %, c p = 0 . / g) of strand rupture event. Particles are drawnto scale and colored by a measure of two-fold symmetry q (seeSec. VI C 5 on its definition) from blue (low) to red (high). Wenote that q is a measure of the degree of local stretching. (b)Same event from a topological point of view. The red line in-dicates the shortest on-graph path between the two particlesof interest, whose drastic change clearly indicates the breakupevent. The meshed surface is a Gaussian coarse-graining ofthe network pattern. sition, with clusters emerging with a constant q vector.However, we have shown that hydrodynamic interactionshinder the formation of compact clusters, and give tothe coarsening process a non-universal behavior. At highvolume fractions, elongated structures immediately per-colate into a thin, mechanically unstable network thatundergoes stress-driven rearrangements enabling the for-mation of locally isostatic structures that finally perco-late. At low volume fractions, percolation is delayed, thusinitially elongated clusters have the time to compact be-fore eventually connecting into a percolating structure.Isostatic clusters thus already exist at percolation timebut are linked by floppy strands that have to compact toinduce isostaticity percolation. In both cases, microrhe-ology indicates that neither isotropic percolation nor di-rected percolation ensure mechanically solidity. Instead,we show that mechanical gelation is characterized by theisotropic percolation of isostatic structures.The picture of gelation that emerges from our ob-servations is far more rich than previously understood,and suggest that mechanical stability plays a more im-portant role than dynamical arrest. Firstly, our studyclearly shows that the spinodal decomposition in gela-tion is under a strong influence of mechanics. Dynamicasymmetry between big colloidal particles and small sol-vent molecules leads to hydrodynamically assisted per-colation and coarsening under self-generated mechanicalstress. Secondly, unlike the glass transition, which is ki-netically defined as the point above which the relaxationtime is slower than the observation time, the arrest isdue to percolation of isostaticity. Thus, we argue that akey feature of gelation is viscoelastic spinodal decompo-sition arrested by isostaticity percolation . Then, the me-chanical stability of a gel is determined by a competitionbetween the yield stress of the isostaticity network andthe internal stress towards network shrinking producedby the interface free-energy cost. Since a gel is not in anequilibrium state and the stress can be concentrated ina weak part of the network , perfect mechanical stabilitymay never be attained, resulting in slow ageing via eithersurface diffusion or bond breakage.In contrast to a purely out-of-equilibrium thermody-namic picture of gelation, an understanding based onthe mechanical equilibrium and isostaticity might pavethe way to a more operative description of colloidal gels,and allow complex issues to be addressed in terms ofmechanics and rheology. For example, stress-driven age-ing plays a fundamental role in the formation of porouscrystals [33]. The spontaneous delayed collapse of col-loidal gels [28, 34] could be viewed as the final overcomeof the mechanical frustration. Under small stresses alsoa delayed yielding is observed [35–37]. Despite a sus-tained attention, the yielding process of colloidal gels stilllacks a general consensus. For instance we do not knowwhy some colloidal gels display a yield stress fluid behav-ior, that is a reversible yielding and no fracture [35, 38],whereas others display a brittle solid behavior with theirreversible opening of fractures [37]. Understanding col-loidal gels as a both non-ergodic and mechanically sta-bilised state of matter may help solving these issues. VI. MATERIALS AND METHODSA. Experimental
1. Samples
We used colloidal particles made of pmma (poly(methyl methacrylate)) copolymerized withmethacryloxypropyl terminated pdms (poly(dimethylsiloxane)) for steric stabilisation [39], with 2 % ofmethacrylic acid to allow electrostatic repulsion, andwith (rhodamine isothiocyanate)-aminostyrene for fluo-rescent labelling [40]. Colloids are dispersed in a mixtureof cis-decalin (Tokyo Kasei) and bromocyclohexane(Sigma-Aldrich) that matches both optical index anddensity of the colloids.To induce short-ranged depletion attraction, we use8 . R g = 148 nm. In the absence of salt, the Debyelength is expected to reach several µ m and the (weakly)charged colloids experience a long range electrostatic re-pulsion [41]. We confirm that colloids never come closeenough to feel the short-ranged attraction and form aWigner crystal.
2. Special experimental protocol to initiate phase separationwithout harmful flow
Gelation of micron-size colloids suitable for quantita-tive confocal microscopy is usually induced by depletionattractions due to polymers in the solvent. The experi-mental protocols that have been used so far for studyingthe kinetics of phase separation and gelation are as fol-lows: (1) Colloidal suspensions and polymer solutionsare mixed just before an experiment, and after mixingtransferred to a capillary tube as quickly as possible.(2) A mixture, which is already in a final state point inthe phase diagram and intrinsically unstable, or phase-separated, is vigorously stirred just before an experimentto break pre-existing phase separated structures by shearmelting. However, these protocols have two common se-rious deficiencies. Firstly the initial state can never beenhomogeneous perfectly, and so there already exist parti-cle aggregates at t = 0. Secondly, the mixing inevitablyinvolves turbulent flow, which does not decay but remainswhen the observation is initiated. The gelation processobserved by these conventional protocols inevitably suf-fers from the influence of ill-defined initial static and dy-namic conditions, and it has been almost impossible toaccess the very initial stage of gelation without interfer-ence of pre-existing aggregates and/or turbulent flow.We overcome these limitations as follows. We use a col-loidal system that is charge stabilised at long range, hasa short range depletion attraction, and is also stericallystabilised causing nearly hard sphere repulsion at con-tact. We disperse colloidal particles and non-adsorbingpolymers in a mixture of organic solvents that matchesboth the refractive index and the density of the parti-cles. We realize a density matching of the order of 10 − between the density of the colloids and of the solvent,enough to observe the late stage of gelation with little in-fluence of gravity despite our large particle size (gravita-tional Peclet number Pe < − ). Because of the weaklypolar nature of the solvent mixture (its dielectric con-stant (cid:15) r = 5 ∼ κ − = 10 µ m, long enough for the large colloids (diame-ter σ = 2 . µ m) to form a homogeneous Wigner crystalin the mixture [42]. The short ranged ( ∼ σ/ × µ m) made of glass in contactwith an half-open glass channel approximately 400 timeslarger in volume, via a millipore filter with pore size of100 nm that allows the salt through but neither polymernor colloid (see Fig. 1(a)). The channel is filled with thesame solvents at density matching composition. At thebeginning of the experiment, solid tetrabutylammoniumbromide (Fluka) is introduced to the channel. Data ac-quisition starts within 30 s after salt introduction. Ourprocedure induces practically no solvent flow in the ob-servation cell. We confirmed the presence of undissolvedsalt several days after mixing, indicating that the obser-vation cell was brought to saturation concentration.Given the diffusion constants of Bromide and alkylcation (6 and 2 × − m s − [43]), we estimate thecharacteristic diffusion time of salt from top to bottom ofthe order of 10 s. Therefore, we reach uniform final saltconcentration into the observation cell within only a fewBrownian times of the colloids. Indeed we measured adelay of about 1 min between the aggregation at the bot-tom and at the top of the cell. We define the initiationtime of the aggregation process when the maximum ofthe g ( r ) jumps from the lattice constant of the Wignercrystal to the hard-core diameter σ .We collect the data on a Leica SP5 confocal micro-scope, using 532 nm laser excitation. The temperaturewas controlled on both stage and objective lens, allowinga more precise density matching. The scanned volumeis at least 82 × × µ m . The particle coordinatesare tracked in three dimensions (3D) with an accuracy ofaround 0 . σ [44]. B. Simulations
To simulate the process of triplet compaction in ab-sence of hydrodynamic interactions, we use Langevindynamic simulations, where the characteristic dampingtime of the velocities τ D is chosen to be equal to theBrownian time τ B , i.e. the time it takes a colloid todiffuse its diameter. We use a generalised LJ poten-tial (with exponent n = 100 and interaction strength (cid:15) = 8 K B T ) chosen to match the second virial coefficientof the Asakura-Osawa potential corresponding to exper-imental conditions (ratio of polymer to colloid diameter, q = 0 . (cid:15) = 8 K B T ). Following Ref. [8], theprocess of matching the second virial coefficient shouldensure equivalent dynamical behavior for all short-rangepotentials. The elongation probability Eq. (1) is com-puted by running two hundred independent simulationsand measuring the statistics of open and compact con-figurations of the triplets. C. Analysis
1. Characterisation of the system.
From direct confocal measurements [45, 46], we esti-mate the hard-core diameter of our colloids ( σ = 2 . µ m)and the range of the interaction potential (that confirmedour scaling of R g within 1%), leading to a polymer-colloidsize ratio q = 2 R g /σ = 0 .
2. Detection of bonds.
In principle the attraction well of the depletion ex-tends to σ + 2 R g , however, resolution-dependent track-ing imprecision and systematic errors do not give a pre-cise enough estimate of such short distance. Thereforewe consider two particles bonded when their distance isshorter the first minimum of g ( r ), i.e. 3 . µ m. Thisdefines the bond graph that we analyse using NetworkXlibrary [47]. We have checked that the precise choice ofthis distance does not affect significantly our results, inparticular percolation times. We consider that a bond iseffectively broken when it does not reform within 10 τ B .
3. Estimation of the internal stress.
To measure internal stresses, we model the bond break-ing rate using a Kramers approach [48]. In absence offorce F acting on a bond, the dissociation rate is k D ( F = 0) = ω exp (cid:18) − E A k B T (cid:19) , (2)where E A is the depth of the potential and ω an attemptfrequency that depends on the precise shape of the po-tential [49] and on the diffusion constant in the depletionshell [31]. For small forces, the rate becomes [32] k D ( F ) = k D ( F = 0) exp (cid:18) F δk B T (cid:19) (3)with δ the width of the potential, here 2 R g . If we canmeasure k D in absence of force, we can obtain the forceat all times: F ( t ) = k B Tδ log k D ( t ) k D ( F = 0) . (4)We convert the force into the internal stress Σ using thearea of contact between depletion shells Σ = 2 F / ( πσδ ).Here we measure k D ( F = 0) by supposing that at longtimes, once hydrodynamic stresses can be neglected, localrearrangements of the network are force-free, that is whenthe involved particles keep a common neighbor after bondbreaking, the long time limit of the blue curve in Fig. 8.
4. Microrheological measurements of viscoelasticity.
We perform two-particle microrheology followingRef. [50]. Briefly, we compute the two-point mean squaredisplacement (cid:104) ∆ r (cid:105) D ( t, ∆ t ), averaged over all couples ofparticles ( i, j ) so that σ < r ij < r max and particle i isfurther away than r max from any edge of the observationwindow. We chose r max as the fourth of the shortest di-mension of the observation window. Using a generalisedStokes-Einstein relation, we obtain the complex modulus G ( ω, t ).0 − − − t/τ B k D t/τ B totallocal a b FIG. 8. Bond breaking rates. for a dilute (a) ( φ = 8 % , c p =1 . / g) and a dense (b) ( φ = 27 % , c p = 1 mg / g) perco-lating sample. The thick grey curve shows the total breakingrate. The thin blue curve counts only breaking events afterwhich the two particles still have a common neighbor. In general the crossing of G (cid:48) and G (cid:48)(cid:48) depends on the fre-quency. That is why the gel point is defined as the timewhen G (cid:48) and G (cid:48)(cid:48) both scale as identical power laws offrequency which corresponds to a loss tangent G (cid:48)(cid:48) /G (cid:48) in-dependent of frequency [51]. In Supplementary Figure S4we shows the evolution of G (cid:48)(cid:48) /G (cid:48) in a percolating sam-ple, for three frequencies. Gelation point occurs at thevery end of the experimental time, slightly later than thecrossing of G (cid:48) and G (cid:48)(cid:48) at high frequency. We confirmedthis trend for all samples and we can safely conclude thatin all cases the gel point occurs soon after percolation ofisostaticity within the experimental window.
5. Characterisation of the degree of local stretching.
To detect local 2-fold symmetry and thus elonga-tion, we use Steinhardt bond orientational order param- eter [52, 53] for particle i , q ( i ) = (cid:118)(cid:117)(cid:117)(cid:116) π (cid:88) m = − | q ,m ( i ) | , (5) q ,m ( i ) = 1 N i N i (cid:88) j =1 Y ,m ( θ ( r ij ) , φ ( r ij )) , (6)where the Y (cid:96),m are spherical harmonics and r ij is one ofthe N i bonds involving particle i .
6. Fourier space analysis
Our experimental data do not have periodic boundaryconditions, so we must use a window function to ensurethe correct correlation, especially at small q . Here we usethe Hanning window, that significantly affects only thevalues of S ( q ) at the first lowest five q that we discard inthe rest of the analysis. We checked that our results arenot affected by other reasonable choices of the windowfunction.The main wavevector is defined as < q > = (cid:82) q min dq q S ( q ) (cid:82) q min dq S ( q ) , (7)where q min is fixed at all times at a value that correspondsto the minimum between the low-q peak and the hardsphere peak. ACKNOWLEDGMENTS
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Leocmach, doi:10.5281/zenodo.1066568 (2017),10.5281/zenodo.1066568. upplementary information forGelation as condensation frustrated by hydrodynamics and mechanical isostaticity Hideyo Tsurusawa, ∗ Mathieu Leocmach, ∗ John Russo, and Hajime Tanaka † Department of Fundamental Engineering, Institute of Industrial Science,University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan Univ Lyon, Universit´e Claude Bernard Lyon 1, CNRS,Institut Lumi`ere Mati`ere, F-69622, VILLEURBANNE, France School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom Institute of Industrial Science, University of Tokyo,4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
Supplementary Analysis of isostaticity percolation
We compute the cluster size distribution of all particlesat percolation time, and also the isostatic cluster sizedistribution at the percolation time of isostaticity. Wetake into account the finite size of the particles by addingone particle radius to the radius of gyration R g of eachcluster. The normalized cluster size is thus R g σ + 12 = 1 σ vuut s s X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~X i − s s X i =1 ~X i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 (1)where s is the cluster size. In that way a one parti-cle cluster has a size of 0 . σ . In that way the fractalrange is extended to small clusters. Figure S6(a) showthe resulting cluster size distributions for a dilute gel. Atusual percolation time, the central range of cluster sizesexhibit a fractal dimension compatible with diffusion-limited cluster aggregation ( D = 1 . D = 3) fractal dimension. Atsmall cluster sizes we also observe compactness, consis-tent with the scenario where compaction at small scalesproceeds before diffusion-limited percolation. This signa-ture of the original compaction remains when isostatic-ity percolates. However larger isostatic clusters exhibita fractal dimension more compatible with directed per-colation ( D = 2 . D = 2 .
53) at all cluster sizes. However whenisostaticity percolates the cluster size distribution of iso-static clusters displays a fractal dimension compatiblewith directed percolation ( D = 2 . ∗ These authors contributed equally to this work † Electronic address: [email protected]
To get more insight on the way directed percolationand isostaticity percolation are related in the dilute case,we take a look at what changes as isostaticity invadesthe network. We take as reference configuration the per-colation time t perco . For low volume fraction gels, localcompaction has already occurred at that time, and wedetect hundreds of small isostatic clusters that are em-bedded in a percolated but non isostatic network. Wedefine X i ( t ) the position of the center of mass at time t of the set particles that formed isostatic cluster i at t = t perco . We note X ij ( t ) = |X j ( t ) − X i ( t ) | the Euclid-ian distance between the centers of mass of i and j , and∆ X ij ( t ) = X ij ( t ) − X ij ( t perco ) its increment. When tak-ing an ensemble average over all pairs of clusters thatare not directly connected or far apart in the network h ∆ X ij ( t ) i is null. However when considering only pairs ofclusters that are less than 10 bonds away on the network,we can observe a shortening of the distance betweenthem, see Fig. 6(e). It means that, as loose strands areconverted to isostaticity, see Fig. 6(b)-(d), these strandsshorten and straighten. This change in morphology al-lows longer directed path and thus promotes directed per-colation. Supplementary Figures
Wigner Hard sphereGelation S ( q ) S ( q ) S ( q ) qσ S ( q ) abcd Fig. S1: Temporal change of the structure factor. Panels (a)-(d) are for the four samples shown on Fig. 3 of the main textby decreasing volume fraction. The thick black curve corre-sponds to the initial Wigner crystal before salt introduction(ill defined thus not shown in (d)). Thin curves from dark redto yellow are spaced by 150 s and display a peak correspond-ing to the hard sphere diameter as well as a growing peak atlow q indicating gelation. . . . . ¯ N C l m a x / L c p (mg/g)1.20.690.32 Fig. S2: Gelation path dependence on polymer concentration.Comparison of system evolution in terms of largest cluster ex-tent and of mean coordination number. By decreasing poly-mer concentration: φ = 16 , ,
14 % and c p = 1 . , . , . (cid:4) , (cid:3) and ◦ respectively. . .
55 0 . .
65 0 . .
75 0 . .
85 0 . .
95 1050100150200250 R g /σ τ B
10 to 20 τ B
20 to 30 τ B
350 to 500 τ B Fig. S3: Evolution of the population of triplets as a functionof their radius of gyration. The result is for a non percolatingsample ( φ = 4 %, c p = 1 mg / g). t/τ B G ( ω , t ) / G ( ω , t ) ( ωτ ) − Fig. S4: Loss tangent. Evolution of the ratio G ( ω, t ) /G ( ω, t )for three frequencies for the sample at φ = 27 %, c p = 1 mg/g.The dotted vertical lines show the percolation time of particleshaving at least 2, 4 and 6 neighbors respectively of particleshaving at least 2, 4 and 6 neighbors respectively, from left toright. Fig. S5: Cluster phase formation observed by our method. Experimental coordinates are reconstructed and colored by thenumber of particles in the cluster( φ = 4 . c p = 1 mg/g). s R g / σ + / N C ≥ N C ≥ s D = 3 D = 2 . D = 2 . D = 1 . a b Fig. S6: Cluster size distributions considering either all particles (orange) or isostatic particles (purple), at their respectiveisotropic percolation time. The later is shifted by a factor 2 for clarity. The thin black lines correspond to a fractal dimension of 3(compact), dark grey lines to a fractal dimension of 2.53 (random percolation), thick light grey lines to a fractal dimension of 2.27(directed percolation), and thick light blue lines correspond to a fractal dimension of 1.85 (DLCA). (a) φ = 8 % , c p = 1 . / g.(b) φ = 27 % , c p = 1 mg / g. Supplementary Movies
Supplementary Movie 1.
Reconstructions in athin slice from confocal coordinates of the whole processof gelation at φ = 7 . c p = 1 mg/g. The phase sep-aration is induced by salt injection. Particles are drawnto scale, and colored according to the radius of gyrationof the cluster they belong to. Time stamp is ( t − t ) /τ B .We can see that there is little macroscopic flow in thisprotocol. Supplementary Movie 2.