Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction

The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with a master kinetic equation to show that cluster sizes and correlation lengths diverge with exponents ~1.6 and 0.8, respectively, consistent with percolation theory, while detailed balance in the particle attachment and detachment processes is broken. Cluster masses exhibit power-law distributions with exponents −3/2 and −5/2 before and after percolation, as predicted by solutions to the master kinetic equation. These results revealing a nonequilibrium continuous phase transition unify the structural arrest and yielding into related frameworks.

To put our system in the context of previously studied gelation, we compute the Baxter temperature τ as outlined in [1]. This allows us in particular to compare our state points with those of the adhesive hard sphere system, which should map onto our short-range attractive system following the Noro-Frenkel correspondence [2]. The reduced temperature τ is related to the range and depth of the attractive potential according to Eq. 1 of Ref [1].
Computing this parameter for our system, and plotting the mean coordination number as a function of τ , we find that the onset of gelation for the volume fraction φ = 0.12 presented in the manuscript occurs at τ ∼ 0.1, as shown in Supplementary Figure 1a dots. For φ = 0.12 investigated in the manuscript, the gelation transition is located slightly below the binodal, in good agreement with previous work on adhesive hard spheres [1].
Towards lower volume fractions, the fluid-gel boundary drops significantly, in qualitative agreement with [1], but the drop occurs more steeply. The difference may be partially explained by the different definition of the gelation onset, based on a constant autocorrelation function measured by dynamic light scattering in [1], and based on the percolation of the largest cluster in the present work. We also note that the indicated fluid-fluid binodal in Supplementary Figure 1 is known to become metastable with respect to the solid-liquid binodal (not shown) when the range of attraction is shorter than 0.14r. This is close to the investigated attraction range of the critical Casimir interaction studied; however, we do not observe any sign of solid-liquid transition in the range of attractions we investigated.
The relation to the fluid-fluid boundary is best shown in Supplementary Figure 2, where we indicate the onset of gelation in the phase diagram of simulated adhesive hard spheres [3].
The gelation onset is located below the phase boundary, and drops steeply towards lower volume fraction. As pointed out in [1], this is different from colloidal depletion systems, for which the gelation onset occurs at the fluid-fluid phase boundary [4]. The gelation of our system thus compares with that of adhesive hard spheres, and is also consistent with the gelation line in a recent paper on particles with short-range interacting random patches [5].

FRACTIONS
In the main manuscript, we have shown that gelation manifests as a nonequilibrium percolation transition. We observed that as a function of the mean coordination number, the largest cluster diverged. Here, we provide more evidence by exploring the percolation behavior in a wider range of volume fractions, and for a very different experimental system, namely protein microparticles, exhibiting short-range attractive interactions very different from those of the colloidal model system in the manuscript. Details of these protein microparticles, for which gelation was induced by slow acidificaton, are described below. For these different systems, the fraction f z of particles in the largest cluster as a function of average coordination number is shown in Supplementary Figure 3a. The data reveals divergence at a system and volume-fraction specific critical coordination number z c , in a way identical to the scaling reported in the manuscript. This remarkable collapse for the different systems and volume fractions suggests that there is a general mechanism underlying the gelation in all these systems.
We note that f z after gelation does not necessarily collapse: depending on the volume fraction (and attraction), the largest cluster may contain different fractions of particles. This is clearly shown in two reconstructions at different volume fractions of the simulated

SUPPLEMENTARY NOTE 3: PROTEIN MICROPARTICLES
We briefly outline some details of the protein microparticle system here. These particles with an average diameter of 1.9µm and a polydispersity of 15% are made by preclustering whey proteins using the double emulsification method as described in [6]. To achieve coldset gelation of the particles, the pre-clustering was accompanied by a mild heat treatment to ∼ 80 • C, close to the denaturation temperature of the proteins [7]. The particles were size-selected down to 15% polydispersity by repeated centrifugation. Sugar was added to the final aqueous solution to match the density and refractive index of the microparticles to that of the solvent. Subsequently, acid-induced gelation was achieved by adding 0.36% by weight of glucono δ-lactone (GDL), lowering the pH and adjusting it towards the isoelectric point of the proteins. This causes a reduction of the electrostatic repulsion of the protein particles, thereby rendering them unstable and inducing aggregation. We estimate the interaction potential in the framework of DLVO theory as a function of the zeta potential, taking into account the charge density of the proteins and the ion density in the solvent. Using these estimates, we find that close to gelation the interaction energy of protein microparticles is ∼ 30k B T , resulting in particle attachment with very low probability of detachment or rearrangement. The resulting approach to gelation is nevertheless very similar to both our critical Casimir colloidal model system and the simulations (both Mie and square-well potential), as shown in Supplementary Figures 3 and 4.