A dual 4-chain model of heterogeneous polymer networks for exploring entanglement and chirality

This study develops a dual 4-chain model to investigate structural heterogeneity, entanglement topology, and chirality in polymer networks. The framework extends classical p-chain models by constructing a periodic dual-unit architecture through lattice reconfiguration of tetrahedral 4-chain cells, thereby enabling explicit analysis of moment-induced distortions arising from mechanically disparate chains at equivalent cross-linking points. The model distinguishes between homochiral configurations that retain uncompensated moments and heterochiral arrangements that achieve energetic equilibrium through dipole cancellation. By integrating Langevin chain statistics with moment equilibrium analysis, the strain energy density is decomposed into contributions from affine cross-linked network deformation and non-affine heterogeneity-induced rotation. The heterogeneity modulus emerges as a key parameter quantifying energy accumulation from localized moment effects, which proves essential for capturing entanglement dynamics and chiral interactions. Theoretical predictions demonstrate that the model degenerates to classical Langevin behavior for sufficiently long chains while predicting pronounced stress enhancement and premature failure for short-chain systems with significant heterogeneity. Experimental validation encompasses multiaxial mechanical responses of vulcanized natural rubber, stereocomplex-induced property modulation in polylactic acid systems with varying enantiomeric ratios, and uniaxial tensile behavior of highly entangled elastomers and double-network hydrogels. The framework successfully reconciles molecular-scale chirality with macroscopic mechanical performance, establishing a predictive platform for heterogeneous deformation mechanisms in stereoregular polymer architectures.