FIG. 15.

(A) Schematics of aggregates of particles interacting via short-range (i.e. contact) interactions and long-range repulsions, with a screening length κ⁻¹, which can be large or small compared to the aggregate size R. (B) Excess energy of spherical aggregates versus radius, showing a crossover from power-law growth for R ≪ κ⁻¹ to an asymptotically saturating bulk energy for R ≫ κ⁻¹. (C) Plot of the equilibrium finite radius R_* of the SALR model versus surface energy Σ, showing a continuous divergence to the bulk at a finite Σ = Σ_max. The second-order-like transition from finite-to-bulk states predicted by this model (solid curve) is consistent with the monotonically increasing form of the accumulant plotted in the inset, where this result assumes the bulk state has uniform density. Assuming instead, that the bulk state has a lower energy, non-uniform density (e.g. a periodic aggregate morphology (Zhang et al., 2019)), the transition will be first-order, as illustrated by the non-monotonic accumulant shown as a dashed line. (D) shows a confocal microscope image of clusters of charged colloidal particles (radius 660 nm) adapted from (Stradner et al., 2004). In these experiments short-range attractions are induced by depletion forces generated by inert polymers in suspension, while electrostatic repulsions are maintained at relative long range due to the low-concentration of mobile ions in host organic solvent (Van Schooneveld et al., 2009).