Contact Forces in Microgel Suspensions

TL;DR

This work develops a liquid-drop model for microgels to study contact forces between soft micron-scale particles, combining a compressible bulk energy with surface-tension contributions. Using numerical minimization and two simple variational shapes (truncated superballs for attractive contacts and spheropolyhedra for repulsive ones), it shows that the deformation energy scales as a power law with indentation, $\Delta F \propto u^\alpha$ with $\alpha\approx2$, and that interactions become effectively pairwise additive in the large-tension microgel regime. The variational shapes accurately reproduce the exact energies across regimes and provide a computationally efficient framework for simulating large suspensions. These results justify Hertz-like and pairwise-additive descriptions in microgel contexts and offer practical shapes for large-scale simulations and theoretical investigations of soft-particle assemblies.

Abstract

Within a model where micrometer-size soft colloidal particles are viewed as liquid drops, we theoretically study the contact interaction between them. We compute the exact deformation energy across a broad range of indentations and for various model parameters, and we show that it can be reproduced using truncated superball and spheropolyhedral variational shapes in the attractive and the repulsive regime, respectively. At large surface tensions representative of microgels, this energy is pairwise additive well beyond small indentations and can be approximated by a power-law dependence on indentation with an exponent around 2.

Figures (4)

• Figure 1: Snapshot of two interacting microgels obtained using monomer-resolved numerical simulations (a; image courtesy of L. Rovigatti and E. Zaccarelli). Panel b shows the relationship between the reference and the resting drop, and panel c emphasizes the difference of the contact and non-contact tension.
• Figure 2: Deformation free energy per contact $\Delta F/z\gamma_FR_0^2$ in $\Psi=10^{-3}$ and $\Psi=1$ drops in the SC lattice vs. dimensionless engineering indentation $u$ (a), comparing exact (symbols) and variational results (solid lines) at $\omega=0.8,1,$ and 1.2. Insets show a few representative shapes at $\Psi=10^{-3}$ and $\omega=0.8$ and 1.2. In panel b, we plot $\Delta F/z\gamma_FR_0^2$ vs. $u$ for $\Psi=10^{-3}, 10^{-2},\ldots, 10^2$ at $\omega=1$, again comparing exact (symbols) and variational results (solid lines). Also included are isolines showing the ratio of contact and total area $A_C/A=0.3, 0.6,$ and $0.9$ (light gray lines) and the Morse--Witten result (red line).
• Figure 3: Relative difference of the truncated-sphero-polyhedron variational and exact deformation free ener-gies for $\Psi=10^{-3},10^{-2},\ldots 10^2$ (top to bottom in the left half of the diagram) and $\omega=1$; points are connected for clarity. The arrow shows the onset of complete faceting for $\Psi=10^{-3}$ and $10^{-2}.$
• Figure 4: Relative difference of the deformation free ener-gies in the DC and SC lattice $\Xi$ at $\omega=1$ for $\Psi=10^{-3},10^{-2},\ldots10^2$; symbols (left panel) represent exact results whereas curves (right panel) are computed using truncated spheropolyhedra. The red line is the prediction of the Morse--Witten theory.