Brush-mediated angular constraints reshape structure, rigidity, and percolation in colloidal depletion gels

Abstract

Colloidal gels, like many other soft and disordered solids derive their mechanical properties not only from the strength of interparticle attraction, but also from the symmetry of the forces that constrain particle motion. While non-central interactions are known to profoundly alter rigidity and elasticity, they are typically introduced through particle anisotropy, surface roughness, or patchy interactions, obscuring their independent role. Here we demonstrate a minimal and geometry-preserving route to emergent non-central forces in colloidal gels by reducing the density of surface-grafted polymer brushes. At low brush density, partial brush interpenetration introduces an effective angular bending rigidity at particle contacts, despite fully isotropic particle geometry. This emergent constraint suppresses local densification, stabilizes low-coordination networks, and produces highly ramified gel structures with enhanced elasticity. Combining experiments, simulations, and mean-field theory, we show that these non-central constraints reorganize structure and mechanics across length scales, shifting gelation boundaries and increasing the elastic modulus by nearly a factor of three. Our results establish surface brush density as a generic control parameter for programming interaction symmetry in soft particulate matter, with implications for rigidity, percolation, and mechanical design in disordered systems.

Figures (18)

• Figure 1: Reducing brush-density introduces non-central constraints without changing particle geometry. (a) Brush density is reduced by decreasing the amount of 2-(2-bromoisobutyryloxy) ethyl acrylate inimer (black) and the total monomer feed for DMA-co-SPAm brush growth (color), while preserving particle size and morphology (SEM; scale bar: 1$\mu m$). (b) High brush density (top) favors compact, cluster-rich assembly, whereas lower brush density (bottom) introduces brush-mediated constraints that prevent compact clustering and yield thinner, more ramified gel strands. Confocal snapshots (right) show the corresponding microstructure under $c/c^* = 0.6$ and 10 mM NaCl and at the total volume fraction of $\phi=0.1$. (c) 3D network reconstructions from experimental z-stacks ($c/c^* = 0.6$, 10 mM NaCl). Lines indicate interparticle bonds and transparent spheres show particle positions.
• Figure 2: Brush-mediated interaction model coupling central force attraction and non-central, distance-activated bending rigidity. Schematic interaction potentials for high-brush (H, green) and low-brush (L, purple) colloids under the experimental condition of $c/c^* = 0.6$ and 10 mM NaCl. The brush layer (light gray) defines a finite overlap region where brush–brush interactions can activate non-central constraints. Dashed curves indicate the effect of changing electrostatic screening with salt concentration. In this model, low-brush particles carry less charge and exhibit stronger net attraction under identical conditions. In the overlap regime, a non-central angular bending rigidity (R) is applied by tracking three-body angles $\theta_0$ and penalizing deviations from this configuration with a bending stiffness $K$.
• Figure 3: Brush density reshapes gel morphology across depletant-salt phase space. Side-by-side comparison of simulated structures (left in each pair) and confocal images (right) for high-brush (green) and low-brush (purple) colloids. Rows correspond to increasing depletion strength via PAM concentration ($c/c^* = 0.2, 0.4, 0.8$), and columns correspond to increasing electrostatic screening via NaCl concentration (2.5, 5, 10 mM). Low brush density promotes earlier percolation and the formation of consistently more stringy, ramified networks. Under the same conditions, high-brush samples form dense, cluster-rich microstructures. Additional mapping is provided in Supplementary Fig.\ref{['fig:SI_total_mapping']}.
• Figure 4: Brush density controls local coordination and void size in arrested gels. Experimental (top, solid) and simulated (bottom, dashed) distributions comparing high-brush (green) and low-brush (purple) systems across PAM concentrations at 5 mM NaCl. (a) Contact number distributions shift to higher coordination with increasing attraction; high-brush gels (left) develop broader high-$Z$ tails, whereas low-brush gels (right) remain centered near $Z\approx3$. (b) Void diameter distributions broaden substantially for high-brush gels (left), indicating increased mesoscale heterogeneity, while low-brush gels (right) maintain smaller and more uniform void sizes. Data across additional salt concentrations are provided in Supplementary Figs.\ref{['fig:contact_all']},\ref{['fig:voidsize_exp']}.
• Figure 5: Multiscale structural metrics reveal earlier percolation and distinct arrested states at low brush density. Structural metrics for high-brush (green) and low-brush (purple) systems across depletant concentrations at 5 mM NaCl, showing experiment (filled symbol) and simulated (open symbol) data. (a) Average contact number $\langle Z \rangle$ rises sharply in the high-brush system but plateaus more gradually for the low-brush case. (b) Mean void diameter increases strongly in the high-brush system while remaining comparatively uniform at low brush density. (c) Fraction of particles in the largest connected component (LCC) and (d) percolation fraction, largest-cluster diameter normalized by the sample box diagonal, both rise sharply at lower depletant concentration in the low-brush system, consistent with the earlier formation of a system-spanning network. Dashed curves are provided as visual guides to trends in the experimental data. Additional salt concentrations are provided in Supplementary Fig\ref{['fig:full_data']}.