Force-extension formula for the worm-like chain model from a variational principle

Stiff polymers, such as single-stranded DNA, unstructured RNA and cellulose, are all basically extremely long rods with relatively short repeating monomers. The simplest model for describing such stiff polymers is called the freely jointed chain model, which treats a molecule as a chain of perfectly rigid subunits of orientationally independent statistical segments, joined together by perfectly flexible hinges. A more realistic model that incorporates the entropic elasticity of a molecule, called the worm-like chain model, has been proposed by assuming that each monomer resists the bending force. Some force-extension formulae for the worm-like chain model have been previously found in terms of interpolation and numerical solutions resulting from statistical mechanics. In this paper, however, we adopt a variational principle to seek the minimum energy configuration of a stretched molecule by incorporating all the possible orientations of each monomer under thermal equilibrium, i.e., constant temperature. We determine a force-extension formula for the worm-like chain model analytically. We find that our formula suggests new terms such as the free energy and the cut-off force of a molecule, which define a clear transition from the entropic regime to the enthalpic regime and the fracture of the molecule, respectively. In addition, we predict two possible phase changes for a stretched molecule, i.e., from a super-helix to a soliton and then from a soliton to a vertical twisted line. We show theoretically that a molecule must undergo at least one phase change before it is fully stretched into its total contour length. This new formula is used to fit recent experimental data and shows a good agreement with some current literature that uses a statistical approach. Finally, an instability analysis is adopted to investigate the sensitivity of the new formula subject to small changes in temperature.