Hunting the fractal vestiges of colloidal gelation

Published in Physics
Hunting the fractal vestiges of colloidal gelation
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(Left) Simulation snapshot of structural heterogeneity in a model particulate gel, providing
insight into its original rigid backbone (from our paper). The colors in the snapshot are based on the
local density of branching points. (Right) Snapshot of the cosmic wave background, providing insight
into the state of the early universe (Wikimedia Commons).

Particulate gels are composed of a small fraction of solid particles (colloids, proteins,...) embedded in a continuous liquid matrix. They are found in a wide range of products from food to cosmetics and even in the cementitious materials used extensively in the building industry.  From the application of shampoos and conditioners in the shower, to brushing our teeth with toothpaste, to eating cheese, yogurt, or jelly, we continuously manipulate these soft materials in our daily lives. When thinking of new smart inks for 3D printing or injectable hydrogels for delivering drugs and biologics it turns out that these gels are particularly interesting because they can be easily fluidized/resolidified.

As the microscopic-sized dispersed particles aggregate under the combined action of Brownian motion and weak attractive forces, they form clusters that eventually merge together to give rise to a percolated structure, which can trap and immobilize a large amount of fluid within. The presence of this sample-spanning network is essential, since it is responsible for the emergent solid-like behavior of the gel, and is a common feature of all particulate gels, suggesting that understanding the details of this scaffolding structure should contain the key to predicting the resulting gel mechanics. However, identifying the links between the gel microstructure and the macroscopic mechanical response has proven extremely difficult. These materials are extremely sensitive to variations in the processing conditions, the forces of attraction and the particle volume fraction: changing any of these may change the gel microstructure slightly but then produce profound differences in the gel's mechanical properties, which has so far frustrated attempts to develop a more systematic understanding of these materials.

Our collaboration started a few years ago and has been fuelled by the diversity and complementarity of perspectives and expertise in our group, combining statistical physics, microscopic models and numerical simulations with rheology, constitutive modeling and signal processing. A 2018 Workshop at KITP on the Physics of Dense Suspensions was a great opportunity to plant some of the initial seeds of this work, while the pandemic obviously confined our scientific discussions to Zoom for long periods. All along, however, we have spent hours and hours discussing, making hypothesis and breaking them down, building a common language, and eventually seeing an interesting new picture of the physics of particulate gels emerge from our rheological analysis and simulation data.

The breakthrough came from understanding the mechanical response of particulate gels to small deformations, i.e., the linear viscoelastic response, as obtained through a measurement protocol in which a gel is subjected to a gentle externally controlled oscillatory probe signal. Using this approach the viscoelastic response is encoded in a property known as the dynamic shear modulus G* of the gel comprising in-phase and out-of-phase components that capture both the elastic (storage) and viscous (lossy) characteristics of the material. Many soft particulate gels, especially when very soft, have a dynamic modulus exhibiting a power-law characteristic with changes in the imposed frequency of oscillation. This broad scale-free spectral response is indicative of the wide range of distinct relaxation processes that underlie the bulk material's response to deformation, and cannot be captured by simple constitutive models that combine a few representative mechanical elements and often work so well for other viscoelastic materials. We started by recognizing that the scale-free spectral response requires a scale-free, hierarchical organization of mechanical elements, and that there is a systematic way of shifting the mechanical spectra of a broad range of gels that are obtained in numerical simulations onto a unique rheological 'master curve'. This led us to hypothesize that during their formation, particulate gels develop a singular rigid structure that is fractal in nature, i.e. heiarchically organized, and which is progressively buried deeper into the microstructure by susequent aggregation of particles. The original delicate fractal nature of this rigid incipient network is therefore obscured by a denser and more complete structure, which is the one accessible to imaging or scattering. Through the numerical simulations, we have built evidence that these multiphase particulate mature gels retain a certain "memory" of their original fractal microstructure that characterizes the critical network when it first forms. This skeletal memory is encoded in the microstructure of the  mature gel in the form of interconnected regions (or "blobs") that control the macroscopic mechanics and are manifested precisely in the scale-free spectral response. The evidence built through the microscopic simulations, in turn, led us to construct an elegant ladder-like arrangement of viscoelastic elements, whose continuum limit takes the form of a compact fractional constitutive equation, highlighting the broad hierarchy of timescales that is directly connected to the vestigal memory of the self-similarity of the gel's rigid backbone when it was originally formed. The implications of the insight gained are multiple, and include a generalized scaling law that now allows us to predict how the gel mechanical response changes with changing the number density of particles over a broad range of conditions. As the fractal dimension of the self-similar skeleton is varied this general scaling law reduces to many of the diverse power-law results that have been reported in the literature for a wide range of different materials.

Within the framework we have built, the fractal signature at the point of gelation in the "critical gel" state is to rheology what the cosmic wave background (CMB) is to the big bang. In both cases, the resulting structures provide valuable insights into the history and properties of the underlying physical systems. Just as quantitative analysis of the CMB tells us about the state of the early universe and its properties, quantitative analysis of the rheological master curves obtained in the laboratory or in silico for a wide range of different particulate gels informs us about the stress-bearing microstructural skeleton that originally self-organized during the gelation process and its relationship to the mechanical properties of the fully-developed particle network that forms the skeletal backbone of the mature, solid-like, gelled materials we encounter every day in the kitchen, bathroom or construction site.

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