Remarks on Flory theory of a self-avoiding chain under cylindrical confinement

Despite its limitations, mainly due to its simplicity, Flory theory has been extended to many other important cases, e.g., linear chains with stiffness and polymers of various topology in a confined space. Surprisingly, the severe limitations of the applicability of the Flory-type free energy for cylindrical confinement have not been well noticed. In this note, we present a simple "renormalized" form of Flory energy for 1D system, from which one can obtain the following three quantities consistently: the equilibrium end-to-end distance of the chain, the confinement free energy, and the effective "Hookian" spring constant of the chain. Our result has practical implications for many experimental studies concerning DNA molecules in nano-/micro-channels.