Torsional Polymer Chains: Flexibility Mechanisms, Lifshitz Entropy, and Biaxial Orientational Order

Conjugated and ladder-like polymers exhibit complex orientational behavior due to their ribbon-like geometry, in which torsional deformations play a central role. We examine rodlike chains that are rigid against bending but allow twisting, introducing two different models. The torsional persistent chain treats the backbone as an inextensible rod that is rigid to bending but continuously twistable; its orientation evolves as a diffusion in angle (a first-order Markov process), and we derive an analytical Lifshitz conformational entropy functional. In contrast, the torsional freely jointed chain discretizes the backbone into rigid, board-like torsional Kuhn segments connected by ideal torsional hinges with independent, uniformly distributed angles; twist is localized as jumps at the hinges (a zeroth-order Markov process). Assuming perfect uniaxial alignment and using a 2D Maier–Saupe framework, both models exhibit a continuous (second-order) uniaxial-biaxial transition. These findings provide fundamental theoretical insights into biaxial nematic phases and liquid-crystalline behavior in ribbon-like polymers.