We investigate two-dimensional colloidal assemblies governed by a modified SALR (“mermaid”) potential under logarithmic confinement. The modified interaction introduces a short-range null-force region in addition to the intermediate attraction and long-range repulsion, enabling structural symmetries that are inaccessible to the traditional mermaid model. Using overdamped Langevin dynamics with simulated annealing, we map ground states across particle number \(\varvec{N}\) and confinement steepness \(\beta _c\) . Local and global order are quantified through a geometric classification algorithm, correlation functions \(\varvec{S}(\textbf{k})\) and \(g_2(r)\) , and complementary shape and connectivity measures based on the moment of inertia tensor and the statistical counts of clusters, stripes, and voids. We find that logarithmic confinement combined with the null-force core suppresses radial symmetry and stabilizes heterogeneous configurations—square, rhomboidal, and mixed motifs—over broad \((\varvec{N},\beta _c)\) ranges. With increasing \(\beta _c\) , clusters merge and the system crystallizes into compact structures; at large \(\varvec{N}\) or under strong confinement, voids are suppressed and a single ordered aggregate forms. The resulting phase diagram organizes cluster, stripe, void, gear-like, and crystalline regimes as a function of density and confinement, highlighting qualitative differences from the traditional mermaid case. We discuss experimental routes to emulate the three-scale interaction and realize these patterns using steric, depletion, and electrostatic contributions combined with soft, finite logarithmic traps.
