Exact solution for the force-extension relation of a semiflexible polymer under compression

Exact solutions for the elastic and thermodynamic properties for the wormlike chain model are elaborated in terms of Mathieu functions. The smearing of the classical Euler buckling instability for clamped polymers is analyzed for the force-extension relation. Interestingly, at strong compression forces the thermal fluctuations lead to larger elongations than for the elastic rod. The susceptibility defined as the derivative of the force-extension relation displays a prominent maximum at a force that approaches the critical Euler buckling force as the persistence length is increased. We also evaluate the excess entropy and heat capacity induced by the compression and find that they vary nonmonotonically with the load. These findings are corroborated by pseudo-Brownian simulations.

Figure 1

Typical time evolution in simulation of a clamped (top) and a half-clamped (bottom) polymer starting from a straight configuration. The simulation is for a rather stiff polymer, ${\ell }_{\text{p}}=10L$, subject to a strong compression force $\left|F\right|=2F_c$.

Figure 2

End-to-end distance in the direction of the applied force (top) and susceptibility (bottom) of polymers with different persistence lengths and boundary conditions, clamped (a) and (d), half-clamped (b) and (e), and free ends (c) and (f). The black dashed lines indicate the linear response and the gray dashed line corresponds to the classical Euler buckling, $F_c$ is the classical Euler buckling force $F_c={\pi }^2\kappa /{\left(\gamma L\right)}^2$. In (c) and (f) we use $\gamma =2$ in $F_c$ to normalize the forces. The forces $|F_{\text{max}}|$ are extracted from the maxima of the susceptibility. In (d) the dashed line (ap. 10) represents the approximate susceptibility derived from the solution in Ref. [44]. Theory and pseudodynamics simulations are shown with lines and dots, respectively.

Figure 3

Comparison of our analytic results to the approximate solution (dashed lines) of Ref. [44] for different persistence lengths. $|F_{\text{max}}|$ denotes the force at maximal susceptibility from the analytic theory and $|F_{\text{ap}}^c|$ the critical force for the onset of buckling, taken from Ref. [44]. The black dashed line indicates the linear response, and the gray dashed line corresponds to the classical Euler buckling.

Figure 4

End-to-end distance in the direction of the force for pulling of clamped and half-clamped (inset) polymers with different persistence lengths. We use the critical buckling force $F_c$ to normalize the forces. The black-dashed lines indicate the weakly-bending approximation.

Figure 5

Excess entropy $\mathrm{\Delta }S$ and excess heat capacity $\mathrm{\Delta }{C}_F$ (inset) for (a) clamped and (b) free polymers as a function of the force $\left|F\right|/F_c$. Temperature enters via ${\ell }_{\text{p}}/L=2\kappa /L{k}_{\text{B}}T=1.4,1.7,2.0,2.2,2.5$.

Figure 6

(a) Even Mathieu functions ${\text{ce}}_{2n}(q,0)$ and (b) corresponding eigenvalues $a_{2n}\left(q\right)$ with deformation parameter $q=2\left|f\right|{\ell }_{\text{p}}$, reduced force $f=F/k_{\text{B}}T$, and persistence length ${\ell }_{\text{p}}=10L$. Here, $F_c={\pi }^2\kappa /L^2$ denotes the critical Euler buckling force of a clamped polymer.

Figure 7

Number of modes $n$ required to achieve an accuracy of ${|\langle X\rangle }_{n+1}-{\langle X\rangle }_n|/L<{10}^{-8}$ with respect to the force $F$ for a clamped (a), half-clamped (b), and free polymer (c) of different persistence lengths ${\ell }_{\text{p}}$. Here, $F_c={\pi }^2\kappa /{\left(\gamma L\right)}^2$ denotes the critical Euler buckling force. In (c) we use $\gamma =2$ in $F_c$ to normalize the forces.