The macroscopic properties of gels arise from their slow dynamics and load-bearing network structure, which are exploited by nature and in numerous industrial products. However, a link between these structural and dynamical properties has remained elusive. Here we present confocal microscopy experiments and simulations of gel-forming colloid–polymer mixtures. They reveal that gel formation is preceded by continuous and directed percolation. Both transitions lead to system-spanning networks, but only directed percolation results in extremely slow dynamics, ageing and a shrinking of the gel that resembles synaeresis. Therefore, dynamical arrest in gels is found to be linked to a structural transition, namely directed percolation, which is quantitatively associated with the mean number of bonded neighbours. Directed percolation denotes a universality class of transitions. Our study hence connects gel formation to a well-developed theoretical framework, which now can be exploited to achieve a detailed understanding of arrested gels.
Amorphous solids are ubiquitous in natural and engineered materials: sand piles, window glass, ceramics, plastics, nanocomposites, gelatin and living cells are just a few examples of solid materials with an amorphous structure. Glassy systems are amorphous solids in which the solid-like mechanical properties are linked to a dramatic slowdown of the microscopic dynamics when decreasing the temperature or increasing the particle density1,2,3,4,5,6,7. Gels also show arrested dynamics8,9,10,11,12,13,14,15,16,17, but their structure is different. They exhibit a heterogeneous load-bearing network structure that is formed by cross-linking (chemical gels) or attraction-induced aggregation (physical gels)18,19,20,21,22,23,24,25,26,27,28,29,30,31. Dynamical arrest was associated with different mechanisms, for example, arrested phase separation21, cluster aggregation21, the occurrence and spatial organization of locally favoured structures18 and the onset of rigidity percolation32. Furthermore, a progressive slowdown of the dynamics after sample preparation, that is, ageing, is observed. However, the relation between the slowdown and arrest of the dynamics, and the structural properties of these systems are still under debate8,9.
Model systems of physical gels have been established, for example, dispersions of spherical colloidal particles, in which attractive forces are induced by a temperature variation or depletion effects due to the addition of linear non-adsorbing polymer chains30,31. Nevertheless, there are only few studies on the relation between the slowdown of the dynamics and the microscopic structure.
Here we show that the slowdown of the particle dynamics and the onset of ageing can be related to the microscopic structure of colloidal gels, namely a directed percolation (DP) transition. This is demonstrated in experiments on charged colloid–polymer mixtures30,33 and in Brownian dynamics simulations of particles interacting through a combination of coulombic repulsion and depletion-induced attraction modelled by the Asakura–Oosawa potential34. The onset of DP is investigated along different paths in which either the attractive or the repulsive parts of the interactions are varied. We find that in any case continuous percolation precedes DP and that both can be associated with characteristic values of the mean number of bonds per particle. Furthermore, DP is found to be linked to the onset of ageing effects.
States of the system
In charged colloid–polymer mixtures, two particle–particle interactions compete on different length scales: First, screened electrostatic interactions, which are long ranged for low salt concentrations and become shorter ranged on addition of salt; second, depletion interactions, which are short ranged, typically at most one tenth of the colloidal particle diameter, and controlled through the radius of gyration of the polymers. We experimentally investigated charged colloidal polymethylmethacrylate (PMMA) spheres with volume fraction Φ≈0.2 and diameter σ=1.72 μm in the presence of non-adsorbing linear polystyrene with radius of gyration rg=65 nm, where the effective polymer-colloid size ratio ξeff≈0.03 (Supplementary Table 1).
In experiments and simulations, we consistently observe four states that differ in the structural arrangements of the colloidal particles (Fig. 1, left). Which state occurs depends on the salt concentration csalt, and therefore the screening length κ−1 characterizing the range of the repulsion and the polymer concentration cp, that is, attraction strength. We consider fixed cp and vary csalt, which will be referred to as ‘path B’. In the absence of salt, mainly individual particles but also very few clusters are distributed homogeneously throughout the sample (sample C1; note that cp of this sample is higher than the one of the other samples, Supplementary Table 1). Adding salt, particles aggregate into isolated small clusters (sample B1). At large salt concentrations, larger clusters are observed which form a network in three dimensions (sample B2). Furthermore, a heterogeneous network structure with thick strands consisting of even larger, dense clusters are formed at the largest salt concentration studied (sample B3).
These states show different pair correlation functions g(r) (Fig. 1, right). Without salt (C1), g(r) is dominated by a peak located at the mean particle distance ρ−1/3. Furthermore, a small peak at particle contact reflects the presence of a small number of aggregated particles. For small amounts of added salt (B1), the increase in the contact peak and a corresponding decrease of the second peak of g(r) is consistent with the presence of doublets, triplets or small string-like clusters (visible in Fig. 1a). The size of the clusters is limited by the repulsive contribution to the potential22,35,36. For sample B2, the large peak at contact reflects the large fraction of particles forming clusters. Its g(r) is reminiscent of a percolated gel-like network structure32,37,38. Finally, the radial distribution function of sample B3 with its very high first peak at contact reflects the presence of a large number of bonded particles. Furthermore, its deep first minimum and pronounced split second peak can arise from triplet structures and local close packed arrangements. These features of the g(r) are observed in the experiments as well as in the simulations. While there is qualitative agreement between experimental and simulation results, small quantitative differences indicate that the potential and its parameter values used in the simulations are not perfectly describing the interactions present in the samples. (The mapping of the simulation and experimental parameters is described in Supplementary Note 1.) Nevertheless, both, experiments and simulations, reveal different states on the path towards a gel; homogeneously distributed individual particles, small clusters, networks with thin and thicker strands consisting of larger, more compact and denser clusters, respectively.
Characterization of the different states by number of bonds
The different structures are distinguished by the degree of particle aggregation. This can be linked to the mean number of bonds per particle, 〈N〉. Individual particle–particle bonds cannot be determined unambiguously because the first peak in g(r) is broadened due to polydispersity (about 7%). Hence, two particles are defined to be bonded if they are closer than the position of the first minimum in g(r), which is located between 1.10 and 1.22σ (Fig. 1, right).
The mean number of bonds per particle, 〈N〉, is determined as a function of the interactions, which depend on the salt concentration csalt, determining the screening length κ−1, and the polymer concentration cp, controlling the depletion part of the potential VD,min=VD(σeff) evaluated at the overall potential minimum Vmin=V(σeff) (Fig. 2). At low polymer, cp, and/or salt, csalt, concentrations, only very few bonds are found (white to purple), whereas high cp and/or csalt result in a bonded state (red). In contrast to previous results23, 〈N〉 monotonically grows without a re-entrant transition. Even for large screening and strong attraction, 〈N〉 hardly exceeds seven and hence the highly bonded state appears not to be in equilibrium. For a purely repulsive interaction39,40 or in spring lattice models41 such a change is expected once the system becomes isostatic, that is, globally stable with six bonds per particle.
The structures along two paths are analysed in more detail (Fig. 3). Along path A, the salt concentration csalt and hence the screening length κ−1 is kept constant while the polymer concentration cp is increased and hence the value VD,min of the potential at its minimum is decreased, that is, the attraction strength is increased. (Note that path A implies a high salt concentration, which cannot be achieved in the organic solvent mixture used in the experiments.) Instead, along path B, the screening length κ−1 is reduced at constant attraction strength. On increasing attraction on path A, 〈N〉 increases and the distribution of the number of bonds per particle, p(N), first broadens, indicating a more heterogeneous structure, but then narrows again indicating that the probability to find monomers is reduced to zero, consistent with the observed network structure (Figs 1a,3a). Along path B, that is, increasing csalt (inset), the distributions also shift to larger N and broaden, again indicating the coarsening of the structure. Compared with path A, the p(N) are sharper, indicating that monomers are suppressed at low salt concentrations. Under these conditions, repulsions are long ranged and still substantial at the mean monomer separation. The system avoids this unfavourable situation by forming clusters. Although the repulsion between the clustered monomers is even larger, cluster formation is favoured by the short-ranged attractions and the larger mean cluster separation that reduces cluster–cluster repulsion.
We characterize p(N) by its mean 〈N〉, variance 〈ΔN2〉 and maximum Nmax. With increasing attraction, that is, |VD,min|, they all sharply increase at Vmin≈−3.0 kBT, where kBT is the thermal energy (Fig. 3b). Furthermore, we consider the distribution of angles between two successive bonds, . The distribution is normalized by the solid angle covered by the angle , that is, such that . It reveals a qualitative change that also occurs at Vmin≈−3.0 kBT (Fig. 3c). While the peak at , which is related to locally dense tetrahedral packings, grows monotonically with |Vmin|, a second peak at develops and becomes pronounced for . Furthermore, for these large potential depths, a peak at emerges, which indicates the formation of straight strings of connected particles. Therefore, based on the mean number of bonds per particle 〈N〉, the different structures on the path towards a gel can be distinguished and quantitatively characterized.
Continuous and DP transitions
We will now show that the particle networks undergo two percolation transitions, a continuous percolation transition and a DP transition. In both cases, percolated networks span the whole system. In continuous percolation, a path along a percolating cluster may contain steps in all directions, including backward steps (Fig. 4a, P). Instead, in the case of DP, only paths along an arbitrarily chosen direction are considered whereas possible backward steps in this direction are not considered (Fig. 4a, DP). In the limit of an infinite system size, percolating clusters occur in all directions in the sample. Therefore, on a macroscopic scale no preferred direction exists in a directed percolated network.
The (cumulative) probabilities p(Δx) that a particle participates in paths with a projected length of at least Δx and in directed paths with a length of at least Δx along any arbitrarily chosen direction are shown in the inset and main part of Fig. 4b, respectively, where the solid lines represent simulation data along path A and symbols indicate results from experiments along path B. For both percolation conditions, the probabilities decay to zero in the case of weak attractions (small |Vmin|). In contrast, for strong attractions plateaus develop, indicating large clusters that span the whole system. The height of the plateaus correspond to the fractions p(lbox) of particles that are part of clusters with the maximum possible length, that is, the box size lbox (Fig. 4c). The system size determines lbox and also affects the simulations. Nevertheless, within the investigated system sizes only a weak dependence on system size, quantified by the total number of particles M, is observed. In particular, the occurrence of the sharp increases of p(lbox) and their positions are not significantly affected. These sharp increases indicate the continuous and DP transitions.
We determine the continuous and DP transitions in simulations for the complete parameter space of Fig. 2. The locations are chosen to coincide with the percolation probability p(lbox)=0.2 for systems with 9,856 particles (Fig. 4c). Using this condition, along path A, the continuous percolation transition occurs at VP=−2.3 kBT and the DP transition is found at VDP=−3.0 kBT. The latter coincides with the significant changes observed in other parameters; 〈N〉 (Fig. 2, dark solid lines) as well as p(N), 〈ΔN2〉, Nmax and (Fig. 3). Furthermore, this indicates that the continuous percolation transition occurs at 〈N〉≈2 and the DP transition at 〈N〉≈3 (Fig. 3b). This suggests a connection between the percolation transitions and 〈N〉. Previous work32,37 on particles with attractive square-well interactions also observed continuous percolation for 〈N〉≈2. This work moreover suggested that rigidity percolation is associated with gelation, which was found to occur at 〈N〉≈2.4, that is, just in between our estimates for the onsets of continuous and DP. The possible link between rigidity and DP deserves further investigation.
Furthermore, close to the transitions the values of p(lbox) obey critical power law scaling with VP−Vmin and VDP−Vmin, respectively (Fig. 4c,d). Note that up to leading order the probability for a bond between neighbours is proportional to Vmin and therefore scaling laws for p(lbox) as a function of this probability also hold for p(lbox) as a function of Vmin. Our data are consistent with theoretical predictions for the critical exponents of the power law scalings, that is, βP=0.42 for continuous percolation42,43 and βDP=0.58 for directed percolation44 (black lines in Fig. 4c,d; only the intercepts are fitted and the data with almost constant p(lbox) at large VP−Vmin and VDP−Vmin, respectively, are not included in the fits). These findings strongly suggest that, on the path towards a gel, continuous and DP networks occur with well-defined transitions to other states as well as between them. Furthermore, this links the formation of gels to a universality class of transitions, namely DP, and hence a well-developed theoretical framework.
Onset of slowdown and ageing in directed percolated systems
The dynamics of the different structures are quantified by the self-intermediate scattering function F and mean squared displacement 〈Δr2〉 (Fig. 5). Diffusive dynamics is observed for individual particles and small clusters, characteristic for fluids. In continuous percolated samples, open network structures are formed (Fig. 1, B2) that are rearranged by local motions requiring only slightly more time of the order of the Brownian time (Fig. 5). In contrast, a marked slowdown of the dynamics is observed beyond the directed percolation (DP) transition. DP is characterized by dense clusters with thick strands (Fig. 1, B3) and thus rearrangements are expected to involve long-range, global motions and the breaking of several bonds. Correspondingly, they require long times. The dynamics of the different states hence are consistent with our structural observations. They suggest that the slowdown associated with gelation sets in at the DP transition and the dynamics eventually arrests deeper in the gel region.
The simulation data presented so far have been determined for a fixed waiting time of 300 after the initial quench. The effect of the waiting time is illustrated for the dependence of the mean number of bonds per particle, 〈N〉, on the potential depth Vmin (Fig. 6), which has been linked to the continuous and DP transitions (Fig. 3b). For small attraction strengths, , 〈N〉 is independent of the waiting time within the examined time range. However, for and thus for DP, 〈N〉 is found to increase with waiting time. This indicates that ageing effects are important in directed percolated systems and equilibrium is reached only very slowly. In continuous percolation, in contrast, equilibrium is attained quickly. These findings also indicate the importance of the above-mentioned local and global rearrangement processes in continuous and directed percolated systems, respectively. In addition, it also supports the above conclusion that the DP transition is an equilibrium transition just before the onset of dynamical arrest.
Synaeresis in directed percolated systems
We also investigate the vicinity of walls (Fig. 7). Initially, the particles fill the whole volume, also close to the wall. In fluids and continuously percolated samples this remains like this at least for several weeks (about 106 ). In contrast, in directed percolated systems, the vicinity of the walls becomes depleted of particles in less than a day (about 3 × 104 ), except for very few clusters which remain attached to the cover slip in the experiments, likely due to depletion and van der Waals attractions45. The depletion close to the walls indicates that directed connections between particles tend to compact the network. This resembles synaeresis, that is, the macroscopic expulsion of fluid from a gel due to the shrinking of the network. This has been observed in a variety of materials, like gelatin46, polysaccharide gels47,48, protein gels49, organogels50, microgels51 and also weakly attractive colloidal gels52. The link between network shrinkage and DP might provide new aspects for the understanding of synaeresis.
We investigate gel formation in a system with competing attractive and repulsive interactions. Confocal microscopy experiments on charged colloid–polymer mixtures are combined with Brownian dynamics simulations of particles interacting via the Asakura–Oosawa and Coulomb potentials. Depending on the overall potential minimum Vmin, which can be varied through the attractive and/or repulsive component of the interactions, different states are identified; fluids of individual particles or clusters as well as continuous and directed percolated networks. The transitions between these states are associated with changes in structural and dynamical parameters, in particular, the number of bonds per particle with significant increases in the mean, variance and most probable value of the distribution of the number of bonds.
The effect of continuous percolation on the dynamics is small. However, DP leads to a significant slowdown of the dynamics. This is attributed to the large number of bonds and concomitant strong confinement of the particles in the attractive potentials of their neighbours, which is also reflected in significant ageing observed in directed percolated systems. It suggests that equilibration is very slow in directed percolated systems, but occurs quickly in continuous percolated systems. Hence, our findings suggest that the slowdown associated with gelation already sets in at the DP transition. Deeper in the gel region, the dynamics then becomes arrested without any additional structural transition occurring.
Our finding that DP precedes the formation of arrested gels, establishes a relation between structural and dynamic features of gels. While previous studies linked gelation in adhesive hard spheres to continuous percolation28,29,31, we argue that in colloid–polymer mixtures with competing attractive and repulsive interactions directed rather than continuous percolation indicates the transition to gel states. Moreover, DP is a universality class of transitions. Thus gel formation can be linked to critical behaviour and the corresponding theoretical formalism53, which can be exploited to obtain a deeper understanding of gel formation. Moreover, maybe gel collapse under gravity30 is also initiated by percolation transitions. Our results hence contribute to an improved understanding of the relation between structural and dynamic features of gel-forming systems and hint at their importance for gel collapse and rheological properties.
Furthermore, in directed percolated systems the dense and directed networks appear to lead to a contraction. This results in a detachment from the walls and hence a layer that is depleted of particles appears. This phenomenon resembles synaeresis and might have important implications for applications and industrial products but also, for example, gel collapse and rheological measurements, in particular concerning wall slip45,54,55,56.
The samples consist of spherical PMMA particles, sterically stabilized with poly(12-hydroxystearic acid) polymers and fluorescently labelled with 7-nitrobenzo-2-oxa-1, 3-diazole-methyl-methacrylate, as well as linear polystyrene. The particles have a diameter σ=1.72 μm and a polydispersity of 7%, which are determined by static and dynamic light scattering of a very dilute suspension (volume fraction Φ=0.005). They are dispersed in a mixture of cis-decalin and cycloheptylbromide, which matches the density and almost matches the refractive index of the particles. We do not observe any indication of anisotropy, sedimentation or creaming of the samples during the observation time (days), indicating a high degree of density matching between solvent and particles. In some samples, creaming is observed after several weeks. In this solvent mixture, PMMA particles acquire a charge33,57.
To prepare the colloid stock solution, the particles are sedimented using a centrifuge. The sediment is assumed to be random close packed with a volume fraction ΦRCP=0.64 and used to prepare a colloid stock solution with Φ≈0.40. The polystyrene has a molar mass Mw=3 × 106 kg mol−1 resulting in a radius of gyration rg=65 nm (ref. 58) and polydispersity characterized by Mw/Mn=1.17. For a dilute mixture, the polymer-colloid size ration 2rg/σ=0.076. Since rg depends on concentration, an effective polymer-colloid size ratio ξeff≈0.03 is estimated based on the Generalized Free Volume Theory59,60,61 (Supplementary Table 1). Different polymer stock solutions are prepared by dispersing dry polymer in the solvent mixture to yield concentrations 4.3 mg ml−1≤cp=mpρs/ms≤18.9 mg ml−1 with mp the mass of the dry polymer, ρs the density and ms the mass of the solvent mixture, respectively. Furthermore, the overlap concentration is calculated using with NA Avogadro’s constant. To obtain colloid–polymer mixtures with Φ≈0.2, colloid and polymer stock solutions are mixed with a volume ratio of 50:50 for samples B1–B3, C1, C3 and D1, while dry polymer is added to the diluted colloid stock solution in the case of samples C2, C4 and C5. (The exact composition of each sample is reported in Supplementary Table 1.) To homogenize the samples, the mixtures are left on a vortex mixer for some minutes and then on a flask shaker for at least one day. To vary the screening length κ−1, different amounts of salt (tetrabutylammoniumchloride, Supplementary Table 1)33 are added and the mixture gently moved on the flask shaker for three days to help the dissolution of the salt in the organic solvent mixture, which is slow and only very limited. Therefore, unfortunately, the exact amount of dissolved salt is unknown. For a quantitative comparison with the simulations, hence a fitting procedure is applied (Supplementary Note 1).
Within 3 h after mixing, the samples are transferred to a home-built sample cell62 and imaged with a confocal unit (Visitech VT-Eye and Nikon A1R-MP for the structural and dynamic measurements, respectively) that is mounted on an inverted microscope (Nikon Ti-U and Nikon Ti-E, respectively) equipped with an oil immersion objective (Nikon Plan Apo VC × 100). In the structural measurements, for each sample, 25–32 image stacks are recorded, each consisting of 151 frames (512 × 512 pixels) with z steps of 200 nm, corresponding to a volume of 54 × 54 × 30 μm3, which contains about 7,000 particles. With a rate of 30 slices per second and averaging of 3 slices to obtain 1 frame, a sampling time smaller than 20 s per stack is achieved. For the determination of the dynamics, two-dimensional slices (512 × 512 pixels) corresponding to an area 63.5 × 63.5 μm2 are recorded. For each sample, at least two series with 10,000 slices at a rate of 15 slices per second and 4,000 slices at 1 slice per second are taken.
The particle coordinates are extracted using standard methods63 and, where necessary, are refined62. The mean squared displacement is calculated from the particle coordinates according to where the average is taken over particles i and starting times t0. For the intermediate scattering function the magnitude of the scattering vector is chosen as k=2πρ1/3≈2.37 μm−1. The delay time is scaled with the Brownian time for our system with the short-time self-diffusion coefficient D0=kBT/γ and the friction coefficient of a particle, γ. Note that we correct for possible drift in the set-up by subtracting the centre-of-mass motion of all particles in the observation volume.
We simulate a system consisting of M particles with mean diameter σ0 and a polydispersity of about 7%. The polydispersity is realized by shifting the interaction potentials multiple times corresponding to 17 different particle diameters σi, whose numbers Mi are normally distributed with a standard deviation of the diameter of 0.07σ and . If not stated otherwise, M=9,856.
Particles interact with a potential V(rij)=VC(rij)+VD(rij) that is composed of a repulsive screened Coulomb part64
Motivated by the experiments, ξeff=0.03 is fixed although in our model this does not necessarily imply that the position of the potential minimum is fixed. The hard interaction of particles at contact is approximated by (ref. 66). The prefactor W0 defines the energy scale of the attractive part and the prefactor H0 the energy scale of the hard-sphere-like potential and is kept constant at H0=0.25 kBT. We cut the potential and the corresponding force at 4σ and shift them to zero at the cut-off distance. In our model, the range of the attractive part in the potential may vary while other parameters are changed. The interaction potential exhibits a minimum with the effective diameter slightly smaller than σij. The effective packing fraction , with V the volume of the system, is kept constant. For bulk simulations we employ periodic boundary conditions to the cubic simulation boxes. For simulations with parallel flat walls, particles are exposed to an external force, which is defined for the bottom wall as
where we fix Fwallσ=104kBT. An analogous force is introduced for the top wall. These forces imply interactions close to hard wall interactions but with a small soft contribution to be suitable for simulations. In our Brownian dynamics simulations particle trajectories are governed by their individual Langevin equations67
with the friction coefficient γi and , where is the transpose of fj, I3 the three-dimensional unity matrix, γi the friction coefficient of a particle with diameter σi, and δ(t−t′) and δij the Dirac delta and the Kronecker delta, respectively.
As in the experiments, the Brownian time =σ2/(3πD0), with the short-time self-diffusion coefficient D0=kBT/γ and the friction coefficient of a particle with average diameter, γ, is used as time scale. The time steps are 10−4 or less. Starting from an initially randomly distributed ensemble, the systems are equilibrated for 300 before statistics are gathered, unless stated otherwise.
The data from experiments as well as the data of the simulations are available from the corresponding author.
How to cite this article: Kohl, M. et al. Directed percolation identified as equilibrium pre-transition towards non-equilibrium arrested gel states. Nat. Commun. 7:11817 doi: 10.1038/ncomms11817 (2016).
We thank Andrew Schofield (University of Edinburgh) for providing the PMMA particles. This work was supported by the International Helmholtz Research School of Biophysics and Soft Matter (IHRS BioSoft) and by the DFG within the Emmy-Noether programme (Schm 2657/2) and grant INST 208/617-1 FUGG funding the confocal microscope.
Supplementary Figure 1, Supplementary Table 1, Supplementary Notes 1-2 and Supplementary References.
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