Local structure of percolating gels at very low volume fractions

Response to "Comment on 'Communication: Is directed percolation in colloid-polymer mixtures linked to dynamic arrest?' " [J. Chem. Phys. 148, 241101 (2018)]

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On the dynamically arrested states of equilibrium and non-equilibrium gels: a comprehensive Brownian dynamics study

In this work a systematic study over a wide number of final thermodynamic states for two gel-forming liquids was performed. Such two kind of gel formers are distinguished by their specific interparticle interaction potential. We explored several thermodynamic states determining the thermodynamic, structural and dynamic properties of both liquids after a sudden temperature change. The thermodynamic analysis allows to identify that the liquid with short range attraction and long range repulsion lacks of a stable gas-liquid phase separation liquid, in contrast with the liquid with short range attractions. Thus, although for some thermodynamic states the structural behavior, measured by the static structure factor, is similar to and characteristic of the gel phase, for the short range attractive fluid the gel phase is a consequence of a spinodal decomposition process. In contrast, gelation in the short range attraction and long range repulsion liquid is not due to a phase separation. We also analyze the similarities and differences of the dynamic behavior of both systems through the analysis of the mean square displacement, the self part of the intermediate scattering function, the diffusion coefficient and theαrelaxation time. Finally, using one of the main results of the non-equilibrium self-consistent generalized Langevin equation theory (NE-SCGLE), we determine the dynamic arrest phase diagram in the volume fraction and temperature (φvsT) plane.

Microscopic structural origin behind slowing down of colloidal phase separation approaching gelation

The gelation of colloidal particles interacting through a short-range attraction is widely recognized as a consequence of the dynamic arrest of phase separation into colloid-rich and solvent-rich phases. However, the microscopic origin behind the slowing down and dynamic arrest of phase separation remains elusive. In order to access microscopic structural changes through the entire process of gelation in a continuous fashion, we used core–shell fluorescent colloidal particles, laser scanning confocal microscopy, and a unique experimental protocol that allows us to initiate phase separation instantaneously and gently. Combining these enables us to track the trajectories of individual particles seamlessly during the whole phase-separation process from the early stage to the late arresting stage. We reveal that the enhancement of local packing and the resulting formation of locally stable rigid structures slow down the phase-separation process and arrest it to form a gel with an average coordination number of z = 6–7. This result supports a mechanical perspective on the dynamic arrest of sticky-sphere systems based on the microstructure, replacing conventional explanations based on the macroscopic vitrification of the colloid-rich phase. Our findings illuminate the microscopic mechanisms behind the dynamic arrest of colloidal phase separation, the emergence of mechanical rigidity, and the stability of colloidal gels.

Location of the gel-like boundary in patchy colloidal dispersions: Rigidity percolation, structure, and particle dynamics

During the past decade, there has been a hot debate about the physical mechanisms that determine when a colloidal dispersion approaches the gel transition. However, there is still no consensus on a possible unique route that leads to the conditions for the formation of a gel-like state. Based on gel states identified in experiments, Valadez-Pérez et al. [Phys. Rev. E 88, 060302(R) (2013)PLEEE81539-375510.1103/PhysRevE.88.060302] proposed rigidity percolation as the precursor of colloidal gelation in adhesive hard-sphere dispersions with coordination number 〈n_{b}〉 equal to 2.4. Although this criterion was originally established to describe mechanical transitions in network-forming molecular materials with highly directional interactions, it worked well to explain gel formation in colloidal suspensions with isotropic short-range attractive forces. Recently, this idea has also been used to account for the dynamical arrest experimentally observed in attractive spherocylinders. Then, by assuming that rigidity percolation also drives gelation in spherical colloids interacting with short-ranged and highly directional potentials, we locate the thermodynamic states where gelation seems to occur in dispersions made up of patchy colloids. To check whether the criterion 〈n_{b}〉=2.4 also holds in patchy colloidal systems, we apply the so-called bond-bending analysis to determine the fraction of floppy modes at some percolating clusters. This analysis confirms that the condition 〈n_{b}〉=2.4 is a good approximation to determine those percolating clusters that are either mechanically stable or rigid. Furthermore, our results point out that not all combinations of patches and coverages lead to a gel-like state. Additionally, we systematically study the structure and the cluster size distribution along those thermodynamic states identified as gels. We show that for high coverage values, the structure is very similar for systems that have the same coverage regardless the number or the position of the patches on the particle surface. Finally, by using dynamic Monte Carlo computer simulations, we calculate both the mean-square displacement and the intermediate scattering function at and in the neighborhood of the gel-like states.

Real space analysis of colloidal gels: triumphs, challenges and future directions

Colloidal gels constitute an important class of materials found in many contexts and with a wide range of applications. Yet as matter far from equilibrium, gels exhibit a variety of time-dependent behaviours, which can be perplexing, such as an increase in strength prior to catastrophic failure. Remarkably, such complex phenomena are faithfully captured by an extremely simple model-'sticky spheres'. Here we review progress in our understanding of colloidal gels made through the use of real space analysis and particle resolved studies. We consider the challenges of obtaining a suitable experimental system where the refractive index and density of the colloidal particles is matched to that of the solvent. We review work to obtain a particle-level mechanism for rigidity in gels and the evolution of our understanding of time-dependent behaviour, from early-time aggregation to ageing, before considering the response of colloidal gels to deformation and then move on to more complex systems of anisotropic particles and mixtures. Finally we note some more exotic materials with similar properties.

Diffusion, phase behavior, and gelation in a two-dimensional layer of colloids in osmotic equilibrium with a polymer reservoir

The addition of enough non-adsorbing polymers to an otherwise stable colloidal suspension gives rise to a variety of phase behaviors and kinetic arrest due to the depletion attraction induced between the colloids by the polymers. We report a study of these phenomena in a two-dimensional layer of colloids. The three-dimensional phenomenology of crystal-fluid coexistence is reproduced, but gelation takes a novel form, in which the strands in the gel structure are locally crystalline. We compare our findings with a previous simulation and theory and find substantial agreement.

The influence of surface roughness on the adhesive interactions and phase behavior of suspensions of calcite nanoparticles

We investigate the impact of nanoparticle roughness on the phase behaviour of suspensions in models of calcium carbonate nanoparticles. We use a Derjaguin approach that incorporates roughness effects and interactions between the nanoparticles modelled with a combination of DLVO forces and hydration forces, derived using experimental data and atomistic molecular dynamics simulations, respectively. Roughness effects, such as atomic steps or terraces appearing in mineral surfaces result in very different effective inter-nanoparticle potentials. Using stochastic Langevin Dynamics computer simulations and the effective interparticle interactions we demonstrate that relatively small changes in the roughness of the particles modify significantly the stability of the suspensions. We propose that the sensitivity of the phase behavior to the roughness is connected to the short length scale of the adhesive attraction arising from the ordering of water layers confined between calcite surfaces. Particles with smooth surfaces feature strong adhesive forces, and form gel fractal structures, while small surface roughness, of the order of atomic steps in mineral faces, stabilize the suspension. We believe that our work helps to rationalize the contrasting experimental results that have been obtained recently using nanoparticles or extended surfaces, which provide support for the existence of adhesive or repulsive interactions, respectively. We further use our model to analyze the synergistic effects of roughness, pH and ion concentration on the phase behavior of suspensions, connecting with recent experiments using calcium carbonate nanoparticles.

Universal Gelation of Metal Oxide Nanocrystals via Depletion Attractions

Nanocrystal gelation provides a powerful framework to translate nanoscale properties into bulk materials and to engineer emergent properties through the assembled microstructure. However, many established gelation strategies rely on chemical reactions and specific interactions, e.g., stabilizing ligands or ions on the nanocrystals' surfaces, and are therefore not easily transferable. Here, we report a general gelation strategy via nonspecific and purely entropic depletion attractions applied to three types of metal oxide nanocrystals. The gelation thresholds of two compositionally distinct spherical nanocrystals agree quantitatively, demonstrating the adaptability of the approach for different chemistries. Consistent with theoretical phase behavior predictions, nanocrystal cubes form gels at a lower polymer concentration than nanocrystal spheres, allowing shape to serve as a handle to control gelation. These results suggest that the fundamental underpinnings of depletion-driven assembly, traditionally associated with larger colloidal particles, are also applicable at the nanoscale.

Computer simulations of colloidal gels: how hindered particle rotation affects structure and rheology

The effects of particle roughness and short-ranged non-central forces on colloidal gels are studied using computer simulations in which particles experience a sinusoidal variation in energy as they rotate. The number of minima n and energy scale K are the key parameters; for large K and n, particle rotation is strongly hindered, but for small K and n particle rotation is nearly free. A series of systems are simulated and characterized using fractal dimensions, structure factors, coordination number distributions, bond-angle distributions and linear rheology. When particles rotate easily, clusters restructure to favor dense packings. This leads to longer gelation times and gels with strand-like morphology. The elastic moduli of such gels scale as G'∝ω^0.5 at high shear frequencies ω. In contrast, hindered particle rotation inhibits restructuring and leads to rapid gelation and diffuse morphology. Such gels are stiffer, with G'∝ω^0.35. The viscous moduli G'' in the low-barrier and high-barrier regimes scale according to exponents 0.53 and 0.5, respectively. The crossover frequency between elastic and viscous behaviors generally increases with the barrier to rotation. These findings agree qualitatively with some recent experiments on heterogeneously-surface particles and with studies of DLCA-type gels and gels of smooth spheres.

Plasticity in colloidal gel strands

Colloidal gels are space-spanning networks of aggregated particles. The mechanical response of colloidal gels is governed, to a large extent, by the properties of the individual gel strands. To study how colloidal gels respond to repeated deformations, we perform Brownian dynamics simulations on single strands of aggregated colloidal particles. While current models assume that gel failure is due to the brittle rupture of gel strands, our simulations show that gel strands undergo large plastic deformations prior to breaking. Rearrangement of particles within the strands leads to plastic lengthening and softening of the strands, which may ultimately lead to strand necking and ductile failure. This failure mechanism occurs irrespective of the thickness and length of the strands and the range and strength of the interaction potential. Rupture of gel strands is more likely for long and thin strands and for a long-ranged interaction potential.

Opposed flow focusing: evidence of a second order jetting transition

We propose a novel microfluidic "opposed-flow" geometry in which the continuous fluid phase is fed into a junction in a direction opposite to the dispersed phase. This pulls out the dispersed phase into a micron-sized jet, which decays into micron-sized droplets. As the driving pressure is tuned to a critical value, the jet radius vanishes as a power law down to sizes below 1 μm. By contrast, the conventional "coflowing" junction leads to a first order jetting transition, in which the jet disappears at a finite radius of several μm, to give way to a "dripping" state, resulting in much larger droplets. We demonstrate the effectiveness of our method by producing the first microfluidic silicone oil emulsions with a sub micron particle radius, and utilize these droplets to produce colloidal clusters.

Coupling between criticality and gelation in "sticky" spheres: a structural analysis

We combine experiments and simulations to study the link between criticality and gelation in sticky spheres. We employ confocal microscopy to image colloid-polymer mixtures and Monte Carlo simulations of the square-well (SW) potential as a reference model. To this end, we map our experimental samples onto the SW model. We find an excellent structural agreement between experiments and simulations, both for locally favored structures at the single particle level and large-scale fluctuations at criticality. We follow in detail the rapid structural change in the critical fluid when approaching the gas-liquid binodal and highlight the role of critical density fluctuations for this structural crossover. Our results link the arrested spinodal decomposition to long-lived energetically favored structures, which grow even away from the binodal due to the critical scaling of the bulk correlation length and static susceptibility.

Vitrification and gelation in sticky spheres

Glasses and gels are the two dynamically arrested, disordered states of matter. Despite their importance, their similarities and differences remain elusive, especially at high density, where until now it has been impossible to distinguish them. We identify dynamical and structural signatures which distinguish the gel and glass transitions in a colloidal model system of hard and "sticky" spheres. It has been suggested that "spinodal" gelation is initiated by gas-liquid viscoelastic phase separation to a bicontinuous network and the resulting densification leads to vitrification of the colloid-rich phase, but whether this phase has sufficient density for arrest is unclear [M. A. Miller and D. Frenkel, Phys. Rev. Lett. 90, 135702 (2003) and P. J. Lu et al., Nature 435, 499-504 (2008)]. Moreover alternative mechanisms for arrest involving percolation have been proposed [A. P. R. Eberle et al., Phys. Rev. Lett. 106, 105704 (2011)]. Here we resolve these outstanding questions, beginning by determining the phase diagram. This, along with demonstrating that percolation plays no role in controlling the dynamics of our system, enables us to confirm spinodal decomposition as the mechanism for gelation. We are then able to show that gels can be formed even at much higher densities than previously supposed, at least to a volume fraction of ϕ = 0.59. Far from being networks, these gels apparently resemble glasses but are still clearly distinguished by the "discontinuous" nature of the transition and the resulting rapid solidification, which leads to the formation of inhomogeneous (with small voids) and far-from-equilibrium local structures. This is markedly different from the glass transition, whose continuous nature leads to the formation of homogeneous and locally equilibrated structures. We further reveal that the onset of the attractive glass transition in the form of a supercooled liquid is in fact interrupted by gelation. Our findings provide a general thermodynamic, dynamic, and structural basis upon which we can distinguish gelation from vitrification.

Structural covariance in the hard sphere fluid

We study the joint variability of structural information in a hard sphere fluid biased to avoid crystallisation and form five-fold symmetric geometric motifs. We show that the structural covariance matrix approach, originally proposed for on-lattice liquids [P. Ronceray and P. Harrowell, J. Stat. Mech.: Theory Exp. 2016(8), 084002], can be meaningfully employed to understand structural relationships between different motifs and can predict, within the linear-response regime, structural changes related to motifs distinct from that used to bias the system.