# 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.

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.

(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 $D_0 = k_bT /(6\pi\eta a)$.