Intrinsic curvature of an elastic filament (${\kappa }_0(s)$)

The intrinsic signed curvature is the curvature of a stress-free filament, measured along material arc length. An elastic filament with actual curvature $\kappa$ stores bending energy $\frac{B}{2}\int (\kappa -{\kappa }_0{)}^{2}ds$, with filament bending modulus $B$. In a planar small-slope Monge representation, $\kappa \simeq {\zeta }^{\prime \prime }$ and the leading energy is quadratic in ${\zeta }^{\prime \prime }-{\kappa }_0$.

Moment-free ends require ${\zeta }^{\prime \prime }={\kappa }_0$ there, so a highly stretched finite filament can retain curvature in boundary layers of width $\ell =\sqrt{B/F}$. For constant intrinsic curvature $\kappa$, zero transverse end forces give slope ${\zeta }^{\prime }(x)=\kappa \ell \sinh [(x-L/2)/\ell ]/\cosh [L/(2\ell )]$. Direct integration yieldsIts large-force deficit is ${\kappa }^2{\ell }^3/2$, of order $F^{-3/2}$, even though the smooth bulk curvature derivative vanishes. This shows why endpoint assumptions matter in force-extension of an intrinsically curved filament.

For a planar elastic filament under positive constant extension force $F$, the quadratic force-controlled energy isThe higher-order Euler-Lagrange equation is $B{\zeta }^{\prime \prime \prime \prime }-F{\zeta }^{\prime \prime }=B{\kappa }_0^{\prime \prime }$. On a periodic interval or in a bulk Fourier transform treatment, every nonzero mode obeys the displayed transfer relation. The length deficit is $\frac{1}{2}\int ({\zeta }^{\prime }{)}^{2}dx$, so its ensemble value is determined by the power spectrum of the quenched intrinsic curvature. Boundary conditions must be specified for a finite filament.

For a stationary quenched intrinsic curvature with power spectrum $S_{\kappa }(q)$ and finite second spectral moment, the bulk deficit satisfiesThe moment is $⟨({\kappa }_0^{\prime }{)}^2⟩$ for mean-square differentiable curvature. Infinite-moment disorder and finite endpoint boundary layers can change the exponent, so this is a smooth bulk law rather than a universal law for all random polymers.

Quenched intrinsic curvature of an elastic filament is fixed during a mechanical experiment, although different filaments can have different random intrinsic shapes. Ensemble averaging is then over these shapes after mechanical minimization. It differs from averaging over time-dependent thermal bending fluctuations in a worm-like chain.