Thermalized buckling of extensible, semiflexible polymers

TL;DR

The paper shows that finite-temperature fluctuations dramatically alter the Euler buckling of extensible semiflexible polymers in the isometric ensemble by softening the effective elasticity and delaying buckling. A Ginzburg-like length $\ell_{\rm th}$ separates perturbative and strongly fluctuating regimes, while a one-loop RG reveals a nontrivial thermal fixed point with nonclassical exponents, and Monte Carlo simulations confirm the qualitative predictions of increased buckling threshold with system size. The work demonstrates Fisher-renormalized ensemble inequivalence between isometric and isotensional conditions and provides a framework for predicting observable nonclassical scaling in biological and nanotechnological polymers such as microtubules, actin filaments, and carbon nanotubes.

Abstract

The Euler buckling of rods is a long-studied mechanical instability, and it remains relevant to this day, as the constituent components in many biological and physical systems are linear polymers, such as microtubules or carbon nanotubes. At finite temperature, if a polymer is shorter than its persistence length, the polymer is semiflexible, and its elasticity remains rod-like. But polymers can also stretch due to their finite extensibility, which can couple to energetically cheap bending deformations in nonlinear ways when a load is applied to the system. We show how the interplay between thermal fluctuations and nonlinear elasticity dramatically modifies the Euler buckling instability for compressed semiflexible polymers in a fixed strain ensemble. We identify a Ginzburg-like length scale beyond which thermally excited undulations lead to a softened Young's modulus, while the polymer nevertheless remains semiflexible. Both perturbative calculations and numerical Monte Carlo simulations suggest a qualitative change in several scaling properties of the buckling transition. The critical compressional strain for thermal buckling now increases with system size, in contrast to athermal buckling, where it decreases with system size. Renormalization group calculations confirm this picture, and also show that thermal buckling is controlled by a new fixed point with different critical exponents compared to classical Euler buckling.

Figures (15)

• Figure 1: (a) Modeling a polymer in an isometric ensemble. The polymer has a zero temperature rest length $L_0$, and a fixed compressive strain $\epsilon$ is imposed by fixing the endpoints in place a distance $L=L_0(1-\epsilon)$ apart. Examples of physically relevant systems: (b) Fluorescence microscropy image of bovine pulmonary artery endothelial cells - microtubules (green), actin filaments (red), nuclei (blue). Source: example image from ImageJ (public domain) https://imagej.net/ij/images/ (c) Self-assembled colloidal chain from Ref. stuij2019 (scale bar: 3 $\mu$m) (d) Carbon nanotube from Ref. fakhri2009 (scale bar: 2 $\mu$m)
• Figure 2: Diagrammatic notation for the bare propagator and quartic interactions in the transverse deformation field $\mathbf{h}$. The second, hyperlocal contribution to the quartic interactions is present even for $f=0$.
• Figure 3: One loop corrections to the vertex and the propagator at the critical buckling transition $f_c\to 0$
• Figure 4: Momentum shell RG to one loop. The red lines indicate fast modes in the momentum shell to be integrated out. The first line shows the corrections to the propagator, the second line shows the corrections to the $y$ vertex, and the third line shows the corrections to the $g$ vertex.
• Figure 5: One loop RG flows within an invariant plane $g=\frac{d_c}{4-d_c}y$ for codimension $d_c=2$. This plane is an attractive subspace within the three dimensional parameter space of RG flows. There are two fixed points: I corresponds to classical Euler buckling, while II corresponds to thermal buckling in an isometric ensemble. The incoming part of the green separatrix connecting I and II marks the boundary between the buckled (orange) and unbuckled (purple) phases. Since the phase boundary curves rightward away from the vertical axis, the critical buckling compression is increased for finite temperature.