A Closed-Form Interpolation Formula for the Isometric Elasticity of Finite-Segment Freely Jointed Chains

For a fixed-force Gibbs ensemble, also known as the isotensional ensemble, closed-form interpolation formula exists for the elastic free energy through the padé approximation to the inverse Langevin function; however, for the fixed-extension Helmholtz ensemble, also referred to as the isometric ensemble, no comparable interpolation formula exists at finite chain length. Using the prototypical freely jointed chain model, such a closed-form interpolation formula is derived here from the first principles. Starting from the isometric partition function, which is effectively mapped onto the Irwin-Hall distribution via the unbiased probability density functions, singularity at the finite-extensibility and stiffness at the Gaussian regime are combined together to construct the desired elastic free energy formula, from which the isometric mean force follows by differentiation. The interpolation formulas reproduce the exact isometric elastic response nearly across the entire chain length with relative errors well within acceptable margins. The treatment is then generalized by incorporating the device stiffness and it yields a meaningful mixed-ensemble result, which interpolates continuously between the two conjugate ensembles. Finally, a Monte Carlo simulation based on a two-rod closure geometry is performed and it is found that the interpolation formula for the isometric mean force agrees extremely well with the simulation data.