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Two-dimensional polar liquid crystals have been discovered recently inmonolayers of anisotropic molecules. Here, we provide a systematic theoreticaldescription of liquid-crystalline phases for polar particles in two spatialdimensions. Starting from microscopic density functional theory, we derive aphase-field-crystal expression for the free-energy density which involves threelocal order-parameter fields, namely the translational density, thepolarization, and the nematic order parameter. Various coupling terms betweenthe order-parameter fields are obtained which are in line with macroscopicconsiderations. Since the coupling constants are brought into connection withthe molecular correlations, we establish a bridge from microscopic tomacroscopic modeling. Our theory provides a starting point for furthernumerical calculations of the stability of polar liquid-crystalline phases andis also relevant for modeling of microswimmers which are intrinsically polar.

Polar liquid crystals in two spatial dimensions: the bridge from microscopic to macroscopic modeling