Stretching of buckled filaments by thermal fluctuations

Krzysztof Baczynski, Reinhard Lipowsky, and Jan Kierfeld

Phys. Rev. E 76, 061914 – Published 21 December 2007

Abstract

We study the buckling instability of filaments or elastic rods in two spatial dimensions in the presence of thermal fluctuations. We present an analytical solution based on a renormalizationlike procedure where we integrate out short wavelength fluctuations in order to obtain an effective theory governing the buckling instability. We calculate the resulting shift of the critical force by fluctuation effects and the average projected filament length parallel to the force direction as a function of the applied force and of the contour length of the filament. We find that, in the buckled state, thermal fluctuations lead to an increase in the mean projected length of the filament in the force direction. As a function of the contour length, the mean projected length exhibits a cusp at the buckling instability, which becomes rounded by thermal fluctuations. Our analytic results are confirmed by Monte Carlo simulations.

Received 6 August 2007

DOI:https://doi.org/10.1103/PhysRevE.76.061914

Authors & Affiliations

Krzysztof Baczynski, Reinhard Lipowsky, and Jan Kierfeld

Max Planck Institute of Colloids and Interfaces, Science Park Golm, D-14424 Potsdam, Germany

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Vol. 76, Iss. 6 — December 2007

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Images

Figure 1
Thermally fluctuating filament under a compressive force $F$ for (a) free and (b) clamped boundary conditions at both ends. For absolute values $| F |$ of the force larger than the critical buckling force $F_{c}$ the filament is buckled; for forces smaller than $F_{c}$ it remains unbuckled. The filament has contour length $L$ , $t (s)$ is the unit tangent vector, and $ϕ (s)$ is the corresponding tangent angle at arc length $s$ . $L_{∥}$ is the projected length in the force direction.Reuse & Permissions
Figure 2
Reduced projected length $L_{∥} / L$ as a function of the reduced force $F / F_{c, 0}$ . For $F < F_{c, 0}$ , the filament is straight with $L_{∥} = L$ and $1 - L_{∥} / L = 0$ . The buckled solution appears for $F > F_{c, 0}$ . The solid curve is obtained numerically from relations (7, 8) by a parametric plot using the buckling angle $ϕ^{*}$ as the curve parameter. The dashed line is the linear approximation (12). For $F / F_{c, 0} > 2.183$ , $L_{∥}$ becomes negative.Reuse & Permissions
Figure 3
Reduced projected length $L_{∥} / L_{c, 0}$ as a function of the reduced contour length $L / L_{c, 0}$ . For $L < L_{c, 0}$ , the filament is straight with $L_{∥} = L$ which corresponds to the left part of the diagram with $L / L_{c, 0} < 1$ . The buckled solution appears for $L > L_{c, 0}$ . The solid curve is obtained numerically by a parametric plot using the buckling angle $ϕ^{*}$ as the curve parameter in Eqs. (7, 8). The dashed line is the linear approximation (12). For $L / L_{c, 0} > 1.478$ , $L_{∥}$ becomes negative.Reuse & Permissions
Figure 4
(Color) Reduced projected length $⟨ L_{∥} ⟩ / L$ as a function of the reduced force $F / F_{c, 0}$ for $L_{p} / L = 100$ (red, $○$ ), 10 (green, $▵$ ), 2 (light blue, $▿$ ), and 1 (blue, $◻$ ) corresponding to $t ≃ 10^{- 3}$ , $10^{- 2}$ , $5 \times 10^{- 2}$ , and $10^{- 1}$ . The solid curves show the analytic result (41). (a) Comparison with Monte Carlo simulation data for two clamped ends using the full model (2). The dashed line corresponds to the zero temperature solution from Fig. 2. (b) Comparison with Monte Carlo simulation data for two clamped ends using the fourth order approximation of Eq. (14). The analytical zero temperature solution $1 - L_{∥} / L = 1 - {\bar{F}}^{- 2}$ is included as a dashed line.Reuse & Permissions
Figure 5
(Color) Reduced projected length $⟨ L_{∥} ⟩ / L_{c, 0}$ as a function of the reduced contour length $L / L_{c, 0}$ for $L_{p} / L = 100$ (red, $○$ ), 10 (green, $▵$ ), 2 (light blue, $▿$ ), and 1 (blue, $◻$ ) corresponding to $t ≃ 10^{- 3}$ , $10^{- 2}$ , $5 \times 10^{- 2}$ , and $10^{- 1}$ . The solid curves show the analytic result. (a) Comparison with Monte Carlo simulation data for two clamped ends using the full model (2). The analytical zero temperature solution from Fig. 3 is shown as a dashed line. (b) Comparison with Monte Carlo simulation data for two clamped ends using the fourth order approximation of Eq. (14). The analytical zero temperature solution $1 - L_{∥} / L = 1 - {\bar{L}}^{- 4}$ is included as a dashed line.Reuse & Permissions

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