Fig. 4.

The internal cluster particle concentration. a The cluster densities ϕ_g (black) and volume fractions of clusters ϕ_c (red) determined from simulations and application of the graph theory. The lines are approximations of ϕ_g (black) and ϕ_c (red) found by substituting the hard sphere equation of state into the Cauchy-Born theory with 〈r²〉 = ξ²/15 and G′ given by the simulation results. Nearly identical values are found when using the experimentally determined modulus. b Initial volume fraction and reduced temperature of gel samples (open circles) and cluster densities ϕ_g determined in this work (closed circles, graph theory; circle-x, hard sphere equation of state). The square symbols reproduce the data of Ramakrishnan et al.¹⁰ with open squares the experimentally measured particle volume fraction and closed squares the cluster densities inferred from application of the Cauchy-Born theory with the hard-sphere equation of state. The attractive and repulsive glass lines with a bilinear shift from the mode coupling theory (MCT) are drawn in solid red^35,36 while an extrapolation through the experimental data is indicated with the dashed red line. The AHS phase boundaries with the fluid-solid coexistence lines for a/R_g = 10 ⁶ are given by short dashed black lines. Fluid-fluid coexistence is indicated by the solid black line and its extrapolation by the dashed black line. The local volume fraction of particle-rich regions (shown as open circle-plus) is compared with values from ref. ⁹: blue triangles are ϕ = 0.13 and δ = 0.059, green triangles are ϕ = 0.045 and δ = 0.018, and black triangles are ϕ = 0.045 and δ = 0.059, where δ is the range of the attraction relative to the particle radius