Buckling a Semiflexible Polymer Chain under Compression

Abstract

Instability and structural transitions arise in many important problems involving dynamics at molecular length scales. Buckling of an elastic rod under a compressive load offers a useful general picture of such a transition. However, the existing theoretical description of buckling is applicable in the load response of macroscopic structures, only when fluctuations can be neglected, whereas membranes, polymer brushes, filaments, and macromolecular chains undergo considerable Brownian fluctuations. We analyze here the buckling of a fluctuating semiflexible polymer experiencing a compressive load. Previous works rely on approximations to the polymer statistics, resulting in a range of predictions for the buckling transition that disagree on whether fluctuations elevate or depress the critical buckling force. In contrast, our theory exploits exact results for the statistical behavior of the worm-like chain model yielding unambiguous predictions about the buckling conditions and nature of the buckling transition. We find that a fluctuating polymer under compressive load requires a larger force to buckle than an elastic rod in the absence of fluctuations. The nature of the buckling transition exhibits a marked change from being distinctly second order in the absence of fluctuations to being a more gradual, compliant transition in the presence of fluctuations. We analyze the thermodynamic contributions throughout the buckling transition to demonstrate that the chain entropy favors the extended state over the buckled state, providing a thermodynamic justification of the elevated buckling force.

Keywords: elasticity; fluctuations; semiflexible polymers.

Figure 1

Free energy F as a function of end-to-end separation over contour length $R/L$ for a semiflexible chain of length $L/\left(2l_p\right)=0.25$ at four different applied forces $f/f_{\mathrm{E}}$. Solid curves show the total free energy F, and the dashed lines indicate the force energy $E_{\mathrm{f}}$. Monte Carlo snapshots of the chain conformations at the free-energy minima are marked by the circles labeled A, B, C, and D.

Figure 2

The left side shows the minimum free-energy end extension $R_{\min }$ versus external force $f/f_{\mathrm{E}}$ for values of $L/\left(2l_p\right)$ ranging from $L/\left(2l_p\right)=0.25$ (blue curve) to $L/\left(2l_p\right)=2$ (red curve). For $L/\left(2l_p\right)=0.25$, the top plot includes pre-buckling (marked A and B) and post-buckling (marked C and D) conformations calculated via Monte Carlo simulation at the same force values as in Figure 1. The right side shows a surface plot of $R_{\min }$ ($R_{\min }/L=1$ in red to $R_{\min }/L=0$ in blue as indicated by the colorbar) versus $f/f_{\mathrm{E}}$ and $L/\left(2l_p\right)$. The vertical lines indicate the $L/\left(2l_p\right)$-value slices that are plotted in the top plot. The black curve on the bottom surface plot gives the critical buckling force made dimensionless by the zero-temperature critical force $f_{\mathrm{c}}/f_{\mathrm{E}}$ versus $L/\left(2l_p\right)$.

Figure 3

Thermodynamic contributions to the free energy F for a worm-like chain versus the end-to-end distance R. The free energy F (black dots) is determined using our exact analytical treatment (Equation (4)). The average polymer energy $\langle E_{\mathrm{poly}}\rangle -E_0$ (blue dots) is found from discretized Monte Carlo simulations, and the entropy S (red dots) is calculated from $S=(\langle E_{\mathrm{poly}}\rangle -E_0-F)/T$. The dashed curve is the zero-temperature bending energy, and the four Monte Carlo conformations span the range of end retractions.

Figure A1

The left plot shows the average polymer energy $\langle E_{\mathrm{poly}}\rangle$ versus the end extension from Monte Carlo simulations for a chain of length $L=l_p/2$ and number of beads $N_b=11$ (blue), $N_b=31$ (purple), and $N_b=51$ (red). The right plot demonstrates that subtracting the references’ energy $E_0=\frac{5}{2}{k}_{\mathrm{B}}T(N_b-4)$ from each average polymer energy $\langle E_{\mathrm{poly}}\rangle$ gives an average polymer energy that is invariant to the discretization and reflects the deformation energy of the fluctuating polymer chain under compression.