Torsional Polymer Chains: Flexibility Models, Conformational Entropy, and Uniaxial–Biaxial Nematic Phase Transition

The ribbon-like shape of conjugated and ladder-like polymers makes their flexibility and orientational order more complex than what is captured by standard bending models, such as worm-like (persistent) or freely jointed chains. The key distinction is that, in addition to bending, backbone torsion (twisting) is allowed, enabling the formation of biaxial nematic phases. Herein, we examine the conformational statistics and orientational ordering of infinitely stiff, rodlike chains capable of torsional deformation. Two models are introduced: (i) the torsional persistent chain with continuous twistability and (ii) the torsional freely jointed chain with twistability localized at hinges connecting stiff board-like segments. (The first model corresponds to the recently proposed ribbon-like chain model in the limit of infinite bending rigidity). An analytical expression for the Lifshitz conformational entropy of the torsional persistent chain is derived. Assuming perfect uniaxial order, the uniaxial–biaxial nematic transition is described within the 2D Maier–Saupe theory. For both models of torsional flexibility, biaxial ordering occurs via a continuous second-order phase transition, in agreement with general symmetry considerations. The biaxial order parameter is found as a function of interaction strength, with analytical results in weak and strong ordering limits. Compared to freely jointed counterparts, torsional persistent chains require π2/3 times stronger interactions to undergo the transition; they also exhibit weaker biaxial ordering. Our findings offer fundamental insights into phase behavior and liquid crystallinity in conjugated polymers.