Cluster-resolved associating-fluid theory for patch-geometry-dependent phase behavior in patchy particle mixtures

The spatial arrangement of binding patches is a key design variable in programmable self-assembly because it determines which local motifs, network connectivities, and ordered states are accessible at equilibrium. Yet this geometric information is often lost in analytical theories of associating fluids, which typically distinguish particles by valence and interaction strength but not by local patch arrangement. This limitation is especially pronounced in quasi-two-dimensional and interfacial assemblies, where reduced coordination enhances the thermodynamic impact of steric and bonding constraints. Here, we present a cluster-resolved statistical associating fluid theory (SAFT-P) that promotes a chosen local cluster (superparticle) to an explicit thermodynamic degree of freedom. This approach preserves an analytical free-energy description while retaining local motif information. Using binary and ternary patchy mixtures on two-dimensional square lattices with identical valence but different patch arrangements, we show that local geometry can shift phase boundaries and bias demixing in ways that conventional monomer-level SAFT does not capture. In particular, SAFT-P predicts an excess-composition coexistence curve for a mixture of four-patch geometric isomers that conventional monomer-level SAFT cannot distinguish at matched valence and interaction strengths. We also test the theory on two-dimensional triangular and three-dimensional cubic lattices, which are included primarily to demonstrate the extensibility of the framework. SAFT-P therefore provides a practical route to connect nanoscale patch placement with mesoscopic assembly and macroscopic thermodynamics in patchy colloids and related interfacial assemblies.