= Solution
Write the particle <volume fraction> as $\eta=(4\pi a^3/3)n_p$. In a stable, repulsion-dominated <colloid>, low $\eta$ gives a disordered fluid. Increasing $\eta$ makes the interaction between neighboring particles important and can produce an ordered <crystal>, separated from the fluid by a <phase coexistence curve>. Increasing the <Debye–Hückel screening length> at fixed charge strengthens the repulsion at a given separation and allows freezing at smaller geometrical <volume fraction>.
The geometric intuition is that particles behave as though they occupy more than their bare diameter: roughly, an effective exclusion diameter $d_{\mathrm{eff}}$ extends out to the separation where $U_{\mathrm{el}}$ is of order $k_BT$. A <hard sphere> comparison then uses $\eta_{\mathrm{eff}}\sim\eta[d_{\mathrm{eff}}/(2a)]^3$. This explains the trend; it is not an exact freezing criterion.
At short <Debye–Hückel screening length> and negligible attraction, the diagram approaches <hard sphere> fluid–crystal coexistence. For longer-ranged soft repulsion, <body-centered cubic lattice> and <face-centered cubic lattice> crystals can both be stable, with the former often appearing towards the softer, lower-density freezing boundary. Screened-Coulomb simulations exhibit both these crystal structures.https://doi.org/10.1103/PhysRevLett.57.2694[Kremer, Robbins and Grest, Phase Diagram of Yukawa Systems: Model for Charge-Stabilized Colloids]{ref}
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2022-q6-phase-diagram.png]
{title=Schematic <phase diagram> for a stable repulsive <colloid>}
{description=Original qualitative fluid–crystal diagram at fixed radius, <temperature> and charge. The two curves bound a fluid–crystal coexistence band. Increased screening length allows freezing at smaller bare particle volume fraction. Crystal-structure boundaries and numerical coexistence positions are deliberately not assigned. The dashed arrow shows adding salt at fixed volume fraction; the second panel shows the direction of the boundary shift when surface charge increases.}
No universal quantitative diagram follows from $(\eta,\lambda_D)$ alone: the charge, <Hamaker constant> and surface conditions must also be fixed. If salt suppresses the stabilizing barrier enough for <Van der Waals force> attraction to dominate, a colloid-poor/colloid-rich separation or aggregation can replace part of this repulsive reference diagram. Dense <colloidal glasses> or <colloidal gels> may also be observed after kinetic arrest; they should not be mistaken for additional <thermodynamic equilibrium> <thermodynamic phases>.
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