Self-consistent chain-length scaling of curvature

Derive the first-principles chain-length dependence of the effective per-bond angular increment and self-consistent bond angle for monopolar autophoretic colloidal chains, thereby characterizing the asymptotic scaling of their polar order and curvature as the chain length increases.

Background

The paper observes that the steady-state polar order grows sub-linearly with chain length while the chain curvature decreases. Under an externally fixed bond-angle reduction, the authors obtain an array-factor expression for the polar order and argue that the effective angular increment must shrink as the number of monomers increases. However, the self-consistent bond-angle scaling generated by the fully coupled positional and orientational dynamics is not analytically determined.

Resolving this problem would provide a first-principles explanation of the large-chain limit, including whether the total angular span saturates and whether the polar order approaches a nonzero asymptotic value rather than eventually decreasing.

References

A full analysis of this would require the self-consistent scaling of $\alpha*(N)$ itself, obtained here only via the externally-fixed-$\alpha$ reduction (Sec.~\ref{sec:nmer_det}), not solved for; a first-principles derivation of $\Delta\theta(N)$ is left to future work.

— Mechanics and statistics of a solvable model of an autophoretic colloidal chain  (2608.28041 - Subramaniam et al., 28 Aug 2026) in Section 3.2, “Polar order and stability for generic N-mers” (following Fig. 3)