The melting transitions of a colloidal lattice confined to a two-dimensional $\left(2\text{D}\right)$ periodic substrate of square symmetry are studied using Monte Carlo simulations for the case when the number of colloidal particles per unit cell of the substrate is one. When the strengths of interparticle and particle-substrate interactions are comparable, the incommensurate nature of square and triangular ordering leads to the formation of a partially pinned solid with only one of the smallest G vectors of the substrate present. This low-temperature phase has true long-range order. By varying the lattice parameter of the substrate while keeping the number of colloidal particles per unit cell of the substrate constant, it is seen that the transition from this low-temperature solid to a high-temperature modulated liquid phase can happen via either a single crossover transition or by a two-stage melting process. Single crossover is observed when the substrate-particle interaction becomes dominant over the particle-particle interaction. This can happen either via an explicit increase in substrate strength or as the substrate lattice parameter increases. For the two-stage melting scenario, the intermediate phase is found to be hexatic. The transitions observed in this work differ in certain aspects from the predictions of the KTHNY theory. The study reveals how constraints from substrate periodicity can fundamentally alter melting dynamics, offering insights into the design of tunable colloidal systems and advancing the understanding of phase transitions in two-dimensional particle systems.

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• Classical statistical mechanics
• Phase transitions
• Colloidal crystal
• Monte Carlo methods