Abstract
Colloidal gels have unique mechanical and transport properties that stem from their bicontinuous nature, in which a colloidal network is intertwined with a viscous solvent, and have found numerous applications in foods, cosmetics, and construction materials and for medical applications, such as cartilage replacements. So far, our understanding of the process of colloidal gelation is limited to long-time dynamical effects, where gelation is viewed as a phase separation process interrupted by the glass transition. However, this purely out-of-equilibrium thermodynamic picture does not address the emergence of mechanical stability. With confocal microscopy experiments, we reveal that mechanical metastability is reached only after isotropic percolation of locally isostatic environments, establishing a direct link between the load-bearing ability of gels and the isostaticity condition. Our work suggests an operative description of gels based on mechanical equilibrium and isostaticity, providing the physical basis for the stability and rheology of these materials.
INTRODUCTION
Colloidal gels are soft solids composed of two intertwined phases: a solid network and a liquid solvent. They are a ubiquitous state of matter in everyday life, making up most of the foods we eat, the cosmetics we use, concretes, and our own organs. In colloidal gels, the network is composed of colloidal particles bonded together by attractive forces. These colloidal assemblies are out of equilibrium, as the thermodynamic ground state of the system involves the macroscopic separation between a particle-rich (liquid) and a particle-poor (gas) phase. Despite the thermodynamic driving force toward compactness, the gel persists because of the dynamical arrest of the network, often described as a glass transition (1–8). This has led to the popular physical picture that a gel is formed by dynamical arrest of bicontinuous spinodal decomposition due to glass transition. The direct link between spontaneous gelation and spinodal decomposition has been carefully confirmed by combining experiments and theories (8). This recently established scenario is certainly a large step toward a more complete understanding of colloidal gelation.
However, this picture still leaves some fundamental problems unanswered: (i) The knowledge of ordinary spinodal decomposition predicts that the minority colloid-rich phase should form isolated clusters rather than the observed percolated network (9). (ii) A colloidal gel is sometimes formed by a network made of thin arms, which are too thin to be regarded as glasses. This casts some doubt on the popular scenario of dynamic arrest due to a glass transition. The glass transition is defined as a kinetic transition and has no direct link to mechanical stability in a strict sense. Slow dynamics and mechanical stability are conceptually different. In an extreme case, for example, a gel formed by bonds with a short lifetime can be ergodic and in an equilibrium state. (iii) A gel often displays superdiffusive behavior, detected as the compressed exponential decay of a density correlation function, during aging, as observed by time-resolved spectroscopy techniques (10–12) and, recently, simulations (13–15). The origin of this phenomenon and its relation to problem (ii) are still elusive.
Several mechanisms have been proposed to try to rationalize some of these issues. Fluid momentum conservation can play an important role in phase separation of colloidal suspensions, giving to hydrodynamics an active role in network formation of the colloid-rich phase (4, 16). There have been some numerical studies on the role of hydrodynamics (17, 18) and mechanics (10, 13, 15, 19–21) in colloidal gelation. However, experimental investigations of these problems have been limited to observation of gels already formed (22, 23) because of the lack of a method to follow the whole kinetic processes with single-particle resolution in both space and time.
Moreover, some questions on the emergence of elasticity, which is the most fundamental physical property of gels, have still remained unanswered. It is well known that the isotropic percolation of a bond network is not sufficient to ensure mechanical stability (23–25). The stability of gels is ascribed to the formation of locally favored structures or local energy-minimum configurations (26), while the mechanics of the network is being recognized to play a major role in the aging behavior of gels (13, 14, 27, 28). Purely geometrical conditions for mechanical stability have also been proposed. Whether a network is rigid can be determined using a pebble game algorithm (29), but this method is limited to two-dimensional systems (30). A criterion on the average coordination number has been proposed (24) but has been recently invalidated (23). Kohl et al. (23) found that in dilute suspensions, a final gel state is obtained only after directed percolation was observed, which differs from isotropic percolation by taking into account the directionality of the clusters. Hsiao et al. (31) found that strain-induced yielding coincides with the loss of rigid clusters. Rigidity was defined using a local Maxwell criterion for isostaticity, that is, six neighbors per particle (32, 33). However, the relationship between local structures, dynamic arrest, and the emergence of elasticity remains poorly understood even at a fundamental level.
This issue is also related to the fundamental problem of distinguishing colloidal gels and glasses. In general, colloidal systems form two types of non-ergodic disordered states (34, 35), gels and glasses [hard sphere–like glasses (36), Wigner glasses (37, 38), and attractive glasses (39)], depending on the type of interactions and volume fractions. It is not straightforward to distinguish Wigner glasses and gels when long-range repulsion and short-range attraction are competing at low volume fractions (40, 41). A similar situation arises for gels and attractive glasses with short-range attractions at high volume fractions (42). The above difficulty in distinguishing gels and glasses conceptually originates from the picture that the non-ergodicity of gels emerges from dynamic arrest due to glass transition.
To address these problems, here, we study the role of mechanics in colloidal gelation by dynamical confocal microscopy experiments. Various experimental protocols have been proposed to obtain reproducible gel structures from the micrometer-sized particles necessary for particle-level resolution. Most of them involved shear (43–46), but, here, we use a protocol that does not involve external flow and that allows us to observe the entire gelation process, from the beginning to the end with particle-level resolution. Our results point to the crucial role of isostatic structures, which are clusters of particles that, according to the Maxwell criteria of stability, have a number of neighbors equal to the number of degrees of freedom. We observe that the emergence of solidity coincides with the appearance of a system-spanning cluster of isostatic particles, i.e., with the isotropic percolation of isostatic particles. Both the glass transition and directed percolation are necessary, but not sufficient, conditions for the emergence of mechanical metastability. In relation to this, we show that directed percolation and isotropic percolation of isostatic particles happen simultaneously only in dilute systems, while in concentrated systems, the two time scales decouple, allowing us to link rigidity with the appearance of the system-spanning cluster of isostatic particles. These findings shed new light on the mechanisms of gel formation and coarsening and also on the fate of gels.
RESULTS
Phase separation dynamics observed at a single-particle level
System design. In our protocol, we first enclose a salt-free suspension of sterically and charge-stabilized colloids and nonadsorbing polymers in a thin microscopy cell sketched in fig. S1. The bottom wall of the cell is an osmotic membrane providing contact with a long channel full of the same solvent mixture. Salt dissolution and subsequent migration of the ions along the channel and through the membrane induce screening of the electrostatic repulsion, revealing the depletion potential well due to the polymers. In contrast to similar designs used in our group and by others (47–49), here, the time needed for the ions to diffuse from the membrane across the cell thickness is of the same order of magnitude as the Brownian time of the particles τB = 10 s. This relation between the two key time scales enables us to switch instantaneously (physically) from a long-range repulsive to a short-range attractive system without any external solvent flow. This causes uniform gelation starting from the homogeneous state, allowing in situ confocal microscopy observation throughout the process from a well-defined initial time, as shown in Fig. 1 and movie S1.
Particles are colored according to the size of the cluster they belong to, going from blue for monomers to red for the percolated cluster. See also movie S1.
Here, we stress that after the salt concentration is homogenized, our system can be regarded as a standard model for sticky hard sphere systems. We confirmed this by comparing the experimental phase diagram with the one obtained with free-volume theory for sticky hard spheres (Fig. 2).
(A) Phase diagram with respect to colloid volume fraction ϕ and polymer concentration cp. Black symbols represent experimental points categorized from their final state obtained in the reservoir cell. The spinodal line (solid curve) is obtained from free-volume theory in polymer dilute regime, extended beyond the polymer overlap concentration as a guide for the eye (dotted curve). (B) Comparison of system evolution in terms of largest cluster extent (lmax/L) and of mean coordination number (
Phase separation dynamics. In Fig. 2A, we show the phase diagram, where we can divide the state points into three regions based on the final state obtained by our protocol: at low polymer concentration (cp < 0.2 mg/g), a sample fully relaxes to a fluid state; at very low colloid volume fraction (ϕ < 0.05) and high polymer concentration, the particles condense into long-lived well-separated clusters, as observed in (50); and in the rest of the explored phase space, we observe a long-lived space-spanning network. In the phase diagram, we also plot the spinodal line obtained from free-volume theory calculations (51) as the continuous and dotted curves for below and above the polymer overlap concentration, respectively. Despite the limitations of the theory, the agreement between the spinodal line and our experiments is rather satisfactory, with the only exception being the region of small colloidal volume fractions and high polymer concentration [see, e.g., (8)].
We confirm in the Supplementary Materials that the phase separation kinetics follows arrested spinodal behavior both in the cluster phase and in the percolating samples. We also confirm in the Supplementary Materials the crucial role of hydrodynamics in facilitating percolation.
To characterize the gelation path in real space, we compute the instantaneous mean number of neighbors
Emergence of mechanical stability
Percolation. As already noted by Kohl et al. (23), the average coordination number
Isotropic percolation is related to the appearance of a system-spanning network and can be determined by looking at the time evolution of the largest connected cluster lmax, as plotted in Fig. 3 (A and B) (orange curves). The isotropic percolation time (
(A and B) Percolation processes for a dilute (ϕ = 8%, cp = 1.5 mg/g) and a dense (ϕ = 27%, cp = 1 mg/g) sample. The processes of isotropic and directed percolation of all particles are plotted as thin orange curve and orange symbols, respectively. The processes of isotropic and directed percolation of isostatic particles are plotted as thick purple curve and purple symbols, respectively. (C and D) Mechanical response for the same samples. Elastic (G′) and viscous (G′′) shear moduli at the highest available frequency (
Directed percolation is related to the appearance of a directed path that spans the whole system. A directed path is defined as a path with no loop or turning back such that every step is in either the positive X, Y, or Z directions. The maximum spatial extent of directed paths lD is plotted in Fig. 3 (A and B) (orange symbols). We thus define the directed percolation time (
The concepts of isotropic and directed percolation can also be applied to a subset of particles. In particular, we focus here on isostatic particles, which are particles that have at least six bonded neighbors. For isostatic clusters, we plot both lmax (purple curves) and lD (purple symbols) in Fig. 3 (A and B) (see also movie S2). The isotropic percolation time of isostatic particles is
Figure 3 (A and B) shows the time evolution of the clusters in the dilute (ϕ = 8%) (Fig. 3A) and dense suspensions (ϕ = 21%) (Fig. 3B). We observe that directed percolation of all particles and isotropic percolation of isostatic particles occur simultaneously in the dilute regime,
Mechanical stability and percolations. The solid nature of a material is most often defined from linear mechanical response. For colloidal gels, however, mechanical stability cannot be predicted without an understanding of internal stresses (14). Here, we are able to extract both information from our particle-level experiments. We use the particles themselves as passive microrheological probes to extract the elastic (G′) and viscous (G′′) parts of the shear modulus (see the Supplementary Materials). Microrheology is a transformation (generalized Stokes-Einstein relation) applied on the (two-point) mean square displacement (52). It thus relates dynamical measurements with linear viscoelasticity. In particular, observing G′ > G′′ for a given frequency indicates the emergence of elasticity, or dynamical arrest, at the corresponding time scale. We denote this time as τgel, the gelling time (see the Supplementary Materials for a more accurate definition and a discussion on uncertainties). We also extract the average value of the internal stress Σ from the bond-breaking rate (see the Supplementary Materials). Results are shown in Fig. 3 (C and D) for direct comparisons with the microstructure.
The typical ranges of stresses and moduli that we measure extend below 0.1 mPa, well below sensitivity of conventional rheometers. That is why previous microscopic studies on the mechanics of colloidal gels have been restricted to the comparison of the structure before and after a large amplitude shear flow with no simultaneous measure of the stress response (31, 44, 53). From Fig. 3 (C and D), we see that, as expected, all samples are purely viscous at short times, with a value of G′′ consistent with the viscosity of a hard sphere suspension at their respective volume fractions. Internal stresses are high at short time, reflecting the stretching of the network, which is formed by hydrodynamic interactions in a mechanically frustrated state. The emergence of mechanical stability is captured simultaneously from both the linear viscoelasticity measurements, with the crossing between G′ and G′′, and the internal stress, which is accompanied by a sharp drop in Σ. The timing of the emergence of elasticity is thus unambiguous (vertical gray zone in Fig. 3, C and D) and occurs well after isotropic percolation time
In dilute samples, the elastic behavior occurs in the same time scale as directed percolation of all particles. However, isotropic percolation of isostaticity also occurs simultaneously,
Directed or isostaticity percolations. In Fig. 4A, we compare the time to percolation of isostaticity
(A) Ratio of the time of isostaticity percolation
The reason for this coincidence can be understood by the specific path to gelation in the dilute regime. We have seen in Fig. 2B that isotropic percolation occurs at a late stage, when the clusters have already become compact by squeezing out the solvent to form isostatic structures. Figure 3A shows that at the percolation time (
From this situation, percolation of isostaticity proceeds by the compaction of the floppy bridges. This compaction takes place without adsorption of new particles onto the bridge. Compaction is a local process that involves no particle migration but only creation of new bonds, as shown in Fig. 4 (B to D) and sketched in Fig. 5A. Consistently, this compaction leads to a straightening and a shortening of the strands. We quantify this shortening by computing the Euclidean distance 𝒳ij(t) between the centers of mass of two isostatic clusters i and j. The increment of this distance as a function of the time distance to the percolation, averaged over all cluster pairs connected by a floppy bridge, is shown in Fig. 4E. The observed shortening is about 0.7σ or 25% of the initial length. Directed percolation becomes possible only when a percolating path has become straight enough, which implies isostaticity. That is why directed percolation and isostaticity percolation occur simultaneously in the dilute regime. The simultaneity of directed percolation and emergence of rigidity in the dilute regime is thus a coincidence mediated by isotropic percolation of already isostatic clusters, in which the two different types of percolation can take place at the same time. That is, the emergence of rigidity in gels should not be linked to the universality class of directed percolation.
(A) Dilute path. (B) Dense path. Isostatic particles are shown in purple; non-isostatic particles are in gray.
Stress-induced network breakup. In a dense system, after directional percolation of all particles, the number of nearest neighbors monotonically increases to minimize the energy of the structure (mainly the interfacial energy cost), resulting in the growth of isostatic configurations (see Fig. 5B), as discussed above. During this process, the mechanical tension internal to the network grows, driving it toward compaction, which can lead to network coarsening accompanying bond breakage (see Fig. 6, A and B). Unlike in simulations (13), we cannot directly measure the local internal stress at this moment, but we can still see its effects through the local stretching measured by the degree of twofold symmetry q2 (see the particle color in Fig. 6A). From this, we may say that a bond breakage event is the consequence of stress concentration on a weak bond, leading to local stretching of the bond and its eventual breakup. That is, mechanical stress acts against diffusive particle aggregation (or compaction), which is the stress-diffusion coupling characteristic of phase separation in dynamically asymmetric mixtures (4, 16). This stress-driven aging is accompanied by mechanical fracture of the percolated network structure by the self-generated mechanical stress. The mechanical stability can be attained only after the formation of a percolated isostatic structure, which is a necessary and sufficient condition for a structure to be mechanically stable. When the percolated isostatic structure can support the internal stress everywhere, the system can attain mechanical stability.
(A) Reconstruction from experimental coordinates (ϕ = 29%, cp = 0.7 mg/g) of a strand rupture event. Particles are drawn to scale and colored by a measure of twofold symmetry q2 (74, 75) (see the Supplementary Materials for its definition) from blue (low) to red (high). We note that q2 is a measure of the degree of local stretching. (B) Same event from a topological point of view. The red line indicates the shortest on-graph path between the two particles of interest, whose drastic change indicates the breakup event. The meshed surface is a Gaussian coarse graining of the network pattern.
DISCUSSION
In summary, we have observed with particle-level resolution the entire process of gelation from the very beginning to the final arrested state at various state points. The early stages are characterized by the universal features of spinodal decomposition, with clusters emerging with a constant wave vector (in the Supplementary Materials, we also show that hydrodynamic interactions hinder the formation of isolated compact clusters, leading to the formation of a percolated network, and give a nonuniversal behavior to the coarsening process). At high volume fractions, elongated structures immediately percolate into a thin, mechanically unstable network that undergoes stress-driven rearrangements, enabling the formation of locally isostatic structures that finally percolate (see Fig. 5B). At low volume fractions, percolation is delayed; thus, initially elongated clusters have the time to compact before eventually connecting into a percolating structure (see Fig. 5A). Isostatic clusters thus already exist at the percolation time but are linked by floppy strands that have to compact to induce isostaticity percolation. Microrheological information reveals that the general mechanism responsible for mechanical solidity, which is signaled by the dominance of elastic over viscous modulus, is neither isotropic percolation nor directed percolation but instead the isotropic percolation of isostatic structures.
The picture of gelation that emerges from our observations is far more rich than previously understood and suggests that mechanical stability plays a fundamental role in addition to dynamical arrest. The glass transition is kinetically defined as the point above which the relaxation time is slower than the observation time, whereas mechanical stability is acquired with the percolation of isostaticity. Thus, we argue that a key feature of gelation is the arrest by isostaticity percolation of viscoelastic spinodal decomposition. Then, the mechanical stability of a gel is determined by a competition between the yield stress of the isostaticity network and the internal stress toward network shrinking produced by the interface free-energy cost. Because a gel is not in an equilibrium state and the stress can be concentrated in a weak part of the network, perfect mechanical stability may never be attained, resulting in slow aging via either surface diffusion or bond breakage.
An understanding based on the mechanical equilibrium and isostaticity might pave the way to a more operative description of colloidal gels and allow complex issues to be addressed in terms of mechanics and rheology. For example, stress-driven aging plays a fundamental role in the formation of porous crystals (49). Furthermore, an understanding of how hydrodynamic flows affect the microscopic structure of the gel and then how the isostatic network recovers could shed new light on thixotropy (i.e., a reversible decrease in viscosity) in colloidal gels (54). The spontaneous delayed collapse of colloidal gels (55, 56) could be viewed as the final step to overcoming the mechanical frustration. In addition, under small stresses, a delayed yielding is observed (57–59). Despite sustained attention, the yielding process of colloidal gels still lacks a general consensus. For instance, we do not know why some colloidal gels display a yield stress fluid behavior, that is, a reversible yielding and no fracture (57, 60), whereas others display a brittle solid behavior with the irreversible opening of fractures (59). Yielding, with the rapid increase in viscosity and the emergence of elastic behavior, can originate from multiple physical mechanisms. Together with the slowing down of the spontaneous relaxation of the system (as considered in the glassy paradigm of gelation), our work singles out the role of the formation of a network of isostatic particles, suggesting the possibility of a link with the jamming transition (5, 61).
MATERIALS AND METHODS
We used colloidal particles made of poly(methyl methacrylate) copolymerized with 25 kDa methacryloxypropyl-terminated poly(dimethylsiloxane) (Gelest) for steric stabilization (62), with 2% of methacrylic acid to allow electrostatic repulsion, and with (rhodamine isothiocyanate)-aminostyrene for fluorescent labeling (63). Particles [diameter σ = 2.75 μm estimated from direct confocal measurements (64, 65)] were dispersed in a mixture of cis-decalin (Tokyo Kasei) and bromocyclohexane (Sigma-Aldrich) that matches both the optical index and the density of the colloids (38). To induce short-ranged depletion attraction, we used 8.4 MDa polystyrene (Tosoh) as nonadsorbing polymer. We estimated the radius of gyration to Rg = 148 nm, leading to an attraction range of 1.10(6)σ. In the absence of salt, the Debye length reaches several micrometers and the (weakly) charged colloids experience a long-range electrostatic repulsion (66).
We enclosed this suspension in a thin (200 μm) microscopy cell sketched in fig. S1. The bottom wall of the cell is an osmotic membrane providing contact with a long channel full of the same solvent mixture. At the beginning of the experiment, we introduced solid tetrabutylammonium bromide (Fluka) into the channel and started data acquisition within 30 s. This salt injection screens the electrostatic interactions between colloids. Thus, after the salt injection, the colloid system can be regarded as sticky hard spheres (67).
We collected the data on a Leica SP5 confocal microscope, using 532-nm laser excitation. We controlled the temperature of both stage and objective lens, allowing a density matching of the order of 10−4 between particles and solvent, enough to observe the late stage of gelation with little influence of gravity despite our large particle size (gravitational Peclet number Pe < 10−6). We tracked the particle coordinates in three dimensions with an accuracy of around 0.03σ (68). We considered two particles bonded when their distance was shorter than the first minimum of g(r), i.e., 3.55 μm. This defines the bond graph that we analyzed using NetworkX library (69). The precise choice of this distance does not significantly affect our results, in particular percolation times (see fig. S4). A discussion of different criteria for defining a bond [e.g., (70)] is reported in the Supplementary Materials.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/5/eaav6090/DC1
Supplementary Text
Fig. S1. Reservoir cell.
Fig. S2. Temporal change of the structure factor.
Fig. S3. Temporal change in the characteristic wave number 〈q〉.
Fig. S4. Robustness of the bond network used in percolation analysis.
Fig. S5. Coordination number analysis for a dilute and a dense sample.
Fig. S6. Gelation path dependence on polymer concentration.
Fig. S7. Cluster phase formation observed by our method.
Fig. S8. Role of hydrodynamics on colloidal phase separation.
Fig. S9. Temporal change in loss angle.
Fig. S10. Cluster size distributions at respective percolation times.
Movie S1. Reconstructions from confocal coordinates of the whole process of gelation at ϕ = 7.7%, cp = 1 mg/g.
Movie S2. Reconstructions in a thin slice from confocal coordinates of the whole process of gelation at ϕ = 27%, cp = 1 mg/g.
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REFERENCES AND NOTES
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