Order-disorder transition in soft and deformable particle assembly with dynamic size-dispersity in two dimensions

Soft and deformable objects are widespread in natural and synthetic systems, including micellar domains, microgel particles, foams, and biological cells. Understanding their phase behavior at high concentrations is crucial for controlling long-range order. Here, we employ a Voronoi-based model to study the packing of deformable particles in two dimensions under thermal fluctuations. Particles are represented as interconnected polygons, with the system energy comprising penalties for deviations in area and perimeter from preferred values. The strengths of these penalties capture two key features of packing: dynamic size dispersity, mimicking chain exchange in block copolymer micelles or solvent exchange in microgels, and particle line tension, reflecting the energy cost of shape changes. The model exhibits an order-disorder transition (ODT): low perimeter penalties yield disordered states, while higher penalties produce a hexagonal crystal lattice. Large dynamic size dispersity shifts the ODT to higher perimeter penalties. We explain this by analyzing particle sizes, defect formation barriers, and Voronoi entropy, which show that defect formation is easier when the area penalty term is smaller, providing a mechanistic basis for the ODT trends. In regimes far from ODT, deviations from the hexagonal lattice are accurately described by normal mode displacement fields, confirming that thermal fluctuations rather than defects govern the structure.