Bananas for Tonks' gas: structural transitions in bent-core molecule bi-layers

We study the role of particle geometry in the ordering of bilayers of bent-core particles using a simplified two-dimensional model. Particles are confined to two parallel one-dimensional layers and can adopt only two discrete orientations of their bent cores. By mapping the system exactly onto a pair of coupled Ising chains in the thermodynamic limit, we may obtain closed-form expressions for all correlation functions. The coupling between orientational and layer degrees of freedom produces remarkably rich behaviour, including crossovers between ferromagnetic and antiferromagnetic orientational order, and states with polarised layers. Despite its simplicity, this exactly soluble model reproduces qualitative features of bent-core smectic liquid crystals and confined colloids, providing insight into the subtle role of particle geometry in driving structural correlations.