The physics of polymer chains is often probed using molecular stretching experiments and various idealized single-chain models. The majority of these models consist of a discrete sequence of links, which may be treated as rigid or extensible. Although such models are well established and many specific extensible variants have been proposed, no generally applicable theory has been presented. Moreover, most existing treatments are heuristic rather than systematically and rigorously derived. This critical gap is closed here through the development of a generally applicable asymptotic theory for including link extensibility in a broad class of discrete models for single-chain thermodynamics. The theory is verified analytically using the freely jointed chain and validated numerically using the freely rotating chain. The resulting approximation is first-order accurate in inverse link stiffness, with quadratically decreasing error, and recovers extensible behavior across all link stiffnesses from a single rigid-link reference calculation.

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• Classical statistical mechanics
• Single polymer chains
• Approximation methods for many-body systems
• Freely-jointed chain
• Freely-rotating chain
• Numerical approximation & analysis