Quantifying Chirality in Helical Polymers via a Geometric Extension of the Kremer-Grest Model

TL;DR

This work develops a transferable, geometry-driven framework to quantify chirality in helical polymers by extending the Kremer-Grest bead-spring model with explicit bond-angle and dihedral potentials that map onto intrinsic curvature $\kappa_0$ and torsion $\tau_0$. A normalized chirality characteristic $\chi$ is defined and related to the helix geometry via $h_0$ and $\rho_0$, bridging discrete KG helices with continuum HWLC descriptions and experimental measures. Molecular dynamics simulations reveal how excluded-volume interactions and thermal fluctuations modify $\kappa$, $\tau$, $h$, $\rho$, and $\chi$, clarifying conditions under which chirality is optimized. A coarse-graining procedure translates experimental helices into KG parameters, yielding good agreement across diverse polymers (e.g., PLLA, iPP, polyallenes, polyfurans, polypeptoids) and enabling predictive design of chiral materials and self-assembly pathways.

Abstract

Chirality in polymeric systems enables a wide range of emergent optical, mechanical, and transport phenomena, yet a unified framework that quantitatively connects molecular-scale geometry to chiral behavior remains lacking. Existing theoretical descriptions typically emphasize either continuum models, such as the helical wormlike chain (HWLC), which neglect intermolecular interactions, or mesophase-level theories, which obscure the role of molecular geometry. In this work, we introduce a comprehensive framework for quantifying chirality in helical polymers by extending the Kremer-Grest bead-spring model to explicitly map intrinsic curvature and torsion onto bond angle and dihedral potentials. We establish direct theoretical relationships between helical parameters such as pitch and radius, and connect them to a normalized, dimensionless chirality characteristic, $χ$ that captures local geometric correlations absent from conventional HWLC descriptions. Furthermore, using molecular dynamics simulations, we systematically quantify the influence of excluded volume interactions and thermal fluctuations on helical geometry and chirality, dispelling the common assumption that monotonic increases in chirality are associated only with decreasing pitch. Finally, we present a coarse-graining procedure that facilitates a direct comparison between experimental helical polymers and the Kremer-Grest helical chain, demonstrating quantitative agreement across a diverse set of polymer classes. This unified geometric and particle-based description provides a predictive roadmap for selecting and engineering chiral Kremer-Grest models and offers a general platform for designing polymeric materials with controlled and tunable chirality.

Figures (8)

• Figure 1: (A) The number of helical structures accessible is vast and can be parametrized by helical pitch $h$ and radius $\rho$. (B) This work establishes a framework that uses the tools of differential geometry ($\kappa_0$ and $\tau_0$) to develop a connection between the parametrization of an ideal helix and the Kremer-Grest helical chain, enabling a coarse-grained mapping between experimental polymers and bead-spring models.
• Figure 2: Theoretical values computed for (A) $\kappa_0$, (B) $\tau_0$, (C) $h_0$, (D) $2\rho_0$. Bond angles greater than $170^\circ$ result in undefined values of torsion due to high prevalence of collinearity.
• Figure 3: (A) Theoretical chirality characteristic ($\chi_0$) computed from Equation \ref{['eqn: chi tau kappa']}. (B) A representative helical residue similar to what would be extracted from molecular simulations. (C) The helical residue when projected along the $\mathbf{t}_{i-1}$ bond vector. (D) Perturbations to a parallelepiped volume when bonds in a helical residue are not mutually orthogonal.
• Figure 4: (A) Local curvature $\kappa$ derived from molecular dynamics simulations trajectories. (B) Deviation of local $\kappa$ (simulation) from theoretical $\kappa_0$. (C) Distance between 1-3 monomer pairs, with repulsive cutoff overlaid. (D) Local torsion $\tau$ derived from molecular dynamics simulation trajectories. (E) Deviation of local $\tau$ (simulation) from theoretical $\tau_0$. (F) Distance between 1-4 monomer pairs, with repulsive cutoff overlaid.
• Figure 5: (A) Bond angle ($\theta$) distributions and (B) Dihedral angle ($\phi$) distributions for a freely jointed chain, standard Kremer-Grest model, and select chiral models for $2K_\theta = K_\phi\equiv K=10k_BT$. (C) Impact of increasing $K/k_BT$ on $\kappa$. (D) Impact of increasing $K/k_BT$ on $\tau$.