Turing patterns in Colloidal Systems with Competing Interactions in three dimensions

Self-organized periodic patterns - first described by Turing in reaction-diffusion systems - are a unifying theme across physics, chemistry, and biology. Yet, their three-dimensional manifestation in colloidal systems with competitive interactions remains under-explored. Here, we investigate three-dimensional Turing patterns in short-range attractive and long-range repulsive (SALR) colloidal systems using a nonlinear Density Functional Theory framework solved with pseudo-spectral methods. We demonstrate the emergence of diverse, interconnected structures, including cylindrical hexagonal, lamellar, and double gyroid phases, along with their inverted analogs. Notably, we also identify the formation of Frank-Kasper phases (σ and A15) and Laves phases (C14, C15), highlighting the system’s capacity to form complex polymorphic structures. Our phase diagram reveals a structural sequence σ→HEX→DG→L as the density increases, and dynamical simulations capture the evolution of these phases from a uniform initial state while capturing their free energy landscapes. Our results can improve our understanding of microphase separation in colloidal materials and establish new connections to self-assembly phenomena in micellar and polymeric systems, broadening the scope of Turing pattern formation in soft matter.