Modelling the force chains in emulsion gels

Force chains are networks of stresses that propagate in complex, distinctive patterns across disordered media. They have been successfully quantified in granular materials and have been a useful concept to rationalise the flow behaviour of such systems. They can be visualised in granular experiments for example with the help of photo-elastic polymers.

Granular and colloidal materials mainly differ due to the scale at play: in granular systems, the constituents (particles) are too large for thermal fluctuations to play a role, while in soft materials, it is precisely those fluctuations that govern the interesting spontaneous processes, such as self-assembly. This distinction can be quantified by adimensional numbers, for example the Péclet number for sedimentation is

Pe = \frac{\mathrm{convection}}{\mathrm{diffusion}}=\frac{aU_0}{D_0}=\frac{4\pi\Delta \rho g a^4}{k_BT} (1)

where a is the particle radius. This means that typically particles well above the micron size immersed in a fluid at ambient temperature will lead to a granular response.

How do force chains change when we move from the granular to the colloidal scale? I was involved in devising a means to characterise and model an interesting experiment performed by J Dong (supervisor CP Royall) in collaboration with M Faers from Bayer and RL Jack from Cambridge. Jun imaged the contact regions between the droplets of an emulsion gel, during its formation, with droplet sizes of about 3 microns in diameter that are well inside the colloidal regime.

I constructed a model using an effective interaction potential, calibrated on the pair-wise static correlations, and then studied the evolution of the gel network as it forms. The experiments are able to identify short chains of a few particles where the compressive stresses are concentrated. These appear to start the formation of a gel backbone, but the experiments are limited in the accessible time window.

In the simulations, I can track the process and its evolution and compare the statistics of the forces and the length of the chains with the experiments. One can even access very late times and find that the compressive chains are well inside the backbone of the arms of the formating gel, as illustrated below.

Force chains (red strings) inside the arms of a model emulsion gel (transparent particles are the rest of the gel).

Several questions remain open: the effective interaction potential appears to point to the existence of very attractive forces at very short range, which are very difficult to measure directly; the arrested structure of the gel can be affected by kinetic pathways modified by — for example — hydrodynamic interactions; the repulsive forces of the emulsion droplet are large and hard to calibrate precisely. Yet we have demonstrated that an effective, particle-based model can capture the essence of the force distributions in gels, including the shape of the force-chain length distributions.

More details can be found in the full article :

Jun Dong, Francesco Turci,, Robert L. Jack, Malcolm A. Faers, and C. Patrick Royall
Direct imaging of contacts and forces in colloidal gels, J. Chem. Phys. 156, 214907 (2022), doi: 10.1063/5.0089276


(1) The Stokes terminal velocity of a sphere of radius a in a fluid with viscosity \eta is U_0 = 2\Delta\rho g a^2/(9\eta) and its Stokes-Einstein diffusion constant is D_0  = k_bT /(6\pi\eta a).

Morphology analysis of bone malformation

Soft matter is a broad field, ranging from colloidal particles to micelles, from proteins to cellular tissues. As I mentioned in a previous post, a physics-based approach to characterising the morphology of soft biological matter can be very insightful. It provides simple, geometrical and structural metrics to identify variation in tissues. We recently demonstrated this approach in a rich article in Nature’s Bone Research, a journal dedicated to the quantitative study of bone properties in different species.

A few years ago, Erika Kague, from Physiology at the University of Bristol, asked me to think about her problem. By inducing genetic mutations, she can generate individual zebrafish (Danio rerio) – a common model organism in biology – with various malformations. These she can investigate to understand the effect of bone mineral density on the insurgence of osteoporosis. The issue is that one may want to rapidly, systematically and quantify the amount of malformation.

To do that, I worked with talented PhD student Yushi Yang to automatise a computer vision workflow of segmentation of three-dimensional images, detection of relevant anchor points and determination of several geometrical and structural metrics. Some of these quantities have been inspired by thermal soft matter analogues, such as the porosity of gels; others rely on graph theory metrics. The entire manually tagged database can be augmented using deep learning U-nets that Stephen Cross (Wolfson Bioimaging Facility, Bristol) optimised for our use case.

The result is a detailed characterisation of which genes promote certain kinds of changes in the bone structure. The data reveal a delicate balance between too little and too much bone mineral density to minimise the chances of osteoporosis.

Bone structure in wild type (wt) and mutant (sp7) zebrafish. The mutant clearly displays multiple signs of malformations.

The full article is here

Erika Kague, Francesco Turci, et al., 3D assessment of intervertebral disc degeneration in zebrafish identifies high and low bone density linked to disc disease, Bone Research 9, 39 (2021), doi: 10.1038/s41413-021-00156-y

Dynamical solid–liquid transition through oscillatory shear

During the summer of a few years ago Eric Brillaux, a student of the Ecole Normale Superieure de Lyon, visited Bristol for a summer project. Thinking of something moderately ambitious that could (in theory) be achieved in a few months, we started to explore how simple crystals respond to oscillatory shear.

The original motivation of the work was rather speculative: I was wondering if it could be possible to transform mechanically an ordered system (a crystal) into an amorphous system (a glass) whilst preserving some degree of local order. To this purpose, the ideal model to consider was an atomistic binary mixture whose supercooled liquid state presents local structural motifs that are identical to the repeated units of its crystalline state (as we have previously shown, see here) .

In particular, we considered an oscillatory shear protocol. This is interesting not only because it mirrors more closely actual experimental methods, but also because it allows the system to behave either reversibly or irreversibly: if the oscillations are small, the crystal survives; if the deformations are large, amorphisation takes place.

In our article just out on Soft Matter, we discuss how this dynamical transition between the reversible and irreversible regime takes place in actual three dimensional crystals and how it depends on the crystal composition. For example, single-component crystals can transform structurally, from face-centred cubic structures to more body-centred cubic structures before becoming disordered. Instead, crystals of two components either transform reversibly or become amorphous at a critical oscillation amplitude.

The dynamics is also rather interesting: for instance, the growth of amorphous regions follows a coarsening pattern that is reminiscent of spinodal decomposition in non-driven systems.

In the end, the amorphous states that we obtain are not as rich in local structural motifs as I hoped, but the dynamical transition in itself has appeared to be very intriguing!

More details in the original article

Eric Brillaux, Francesco Turci, Soft Matter, 2019, doi 10.1039/C8SM01950A

Progress of the amorphisation of a binary crystal upon several oscillations of period τ0.

Coupling of sedimentation and liquid structure: Influence on hard sphere nucleation

Colloidal hard spheres at high volume fractions (beyond ~0.49) can crystallise: up to to 0.54, they coexist with a fluid phase, and at even higher densities they completely crystallise. The way the hard sphere fluid transforms into the solid is called crystal nucleation: due to thermal fluctuations, every now and then locally denser and more ordered regions appear and disappear; occasionally, these are large enough to further grow irreversibly, and form a crystalline region.

Nucleation is a generic process: in hard spheres, it should present its simplest traits, as it can be driven only by entropic forces. Yet, nucleation rates in colloidal hard spheres are rather different from what can be predicted from theory and numerical simulations. In particular, the discrepancy between the two increases of many orders of magnitude with decreasing the volume fraction from, for example, 0.55 to 0.53. Simulations normally consider idealised hard spheres in an ideal solvent. What could possibly go wrong?

Nick Wood, a talented PhD student here in Bristol, has taken care of some potential origins of the discrepancy analysing the effect of sedimentation on local order. Together with John Russo and Paddy Royall, we have considered in our recent paper how the flow induced by sedimentation may, via hydrodynamics interactions, transform the structure of the liquid, compared to the case in absence of sedimentation, and how such changes would impact on the nucleation barriers. The result is that some structural signatures clearly vary as a function of the density mismatch between colloids and solvent and that this leads to an estimated correction of the rates in the right direction, but by an amount that is not sufficent to address the entire discrepancy.

The complete article can be found here:
Nicholas Wood, John Russo, Francesco Turci and C. Patrick Royall J. Chem. Phys. 149, 204506 (2018);

Structure of the colloidal hard-sphere fluid with (left) and without (right) sedimentation. In green are highlighted motifs that hinder the crystal formation and anti-correlate with it, as demonstrated here.

Long-lived non-equilibrium interstitial solid solutions in binary mixtures

We just published on the Journal of Chemical Physics the experimental and computer simulation work of a Ioatzin Rios de Anda (PhD student in the Royall group) on kinetically arrested crystalline phases in colloidal binary mixtures.

Despite slightly different conditions (presence/absence of confinement or polydispersity in the particle sizes) the experiments and simulations match in the fundamental message of the work: when we have two species rather different in sizes, crystallisation of the bigger species can occur before the reordering of the smaller species, forming long-lived kinetically arrested structures (interstitial solid solutions), characterised by a high density of imperfections and vacancies. This shows how challenging the formation of a well ordered binary crystal is and, on the other hand, how they can potentially be considered for the realisation of partially ordered porous matrices.

Full reference:
Rios de Anda, I., Turci, F., Sear, R., & Royall, P.  Journal of Chemical Physics, 147, 124504 (2017).

Effects of vertical confinement on gelation and sedimentation of colloids

Disordered systems under confinement may show very specific properties, such as enhanced density fluctuations or flow instabilities.

Azaima Razali (Bristol) and Christopher Fullerton (Bath, now in Montpellier) have performed experiments and simulations on the effect of extreme confinement in colloidal gels and their work (to which I have the pleasure to add my contribution) has just been published in Soft Matter.

The notable result is that while gelation is often employed in bulk systems in order to slow down sedimentation, in strongly confined systems the opposite appears to be true, with sedimentation facilitated by the formation of a percolating network.

The full article can be found here:

A. Razali, C. J. Fullerton, F. Turci,  J. E. Hallett, R. L Jack and C. P. Royall, Effects of vertical confinement on gelation and sedimentation of colloids, Soft Matter, (2017), doi:10.1039/C6SM02221A

Local structure of percolating gels at very low volume fractions

We have recently published on the Journal of Chemical Physics the study resulting from the work of a Master Student in Bristol Chemistry: via numerical simulations, we explore the very low volume fraction regime of a colloidal gel and find striking structural signatures related to the compactness of the gel arms. Moreover, we find that the only limit for gel formation truly is the accessible observation time.

Full reference: S. Griffiths, F. Turci and C. P. Royall,   The Journal of Chemical Physics 146, 014905 (2017); doi: