Nonlinear multimode buckling dynamics examined with semiflexible paramagnetic filaments

We present the contractile buckling dynamics of superparamagnetic filaments using experimental, theoretical, and simulation approaches. Under the influence of an orthogonal magnetic field, flexible magnetic filaments exhibit higher-order buckling dynamics that can be identified as occurring in three stages: initiation, development, and decay. Unlike initiation and decay stages where the balance between magnetic interactions and elastic forces is dominant, in the development stage, the influence of hydrodynamic drag results in transient buckling dynamics that is nonlinear along the filament contour. The inhomogeneous temporal evolution of the buckling wavelength is analyzed and the contractions under various conditions are compared.

Figure 1

(a) A schematic of the electromagnet microscopy setup used to image filament buckling. (b) The change of induced dipoles within the superparamagnetic filaments when the external field is switched to perpendicular direction. (c) A zoom-in image of the local structure of a buckling superparamagnetic filament under orthogonal magnetic field. Scale bar, 10 μm.

Figure 2

A time series of snapshots of three superparamagnetic filament buckles in the experiment. The shortest filament realigns to the direction of the magnetic field 2.5 s after the buckling initiation (c); the filament with the medium length buckles into a quasistable hairpin shape 5.5 s from the direction switch of the external field (d); and it takes 11.5 s for the longest filament to fold into a 14-curve shape (e). Persistence length of the filaments $L_p=1.20\mathrm{mm}$. The buckling field strength $B=77$ Gauss. Scale bar, 100 μm.

Figure 3

Plot of (a) filament shape evolution captured in experiments (left) and numerical simulations (right), and (b) magnetic potential and elastic bending energy evolution in a buckled filament when an orthogonal magnetic field is applied, based on both experiment and numerical simulation results. Persistence length of the filament $L_p=1.33\mathrm{mm}$. The buckling field strength $B=43\mathrm{G}$. Scale bar, 24 μm. Filament length $L=155\mu \mathrm{m}$. The numerical simulation result of energy evolution is calculated based on 100 runs. The shaded area indicates the error.

Figure 4

The development stage of simulated contractile buckling dynamics of a paramagnetic colloidal filament with filament length $L=246\mu \mathrm{m}$, persistence length $L_p=1.33\mathrm{mm}$. The buckling field strength is 43 G. (a,b) Color maps showing the relationship between (a) transverse displacement from the initial aligned configuration and (b) tangent angle, along the filament and normalized contour length $s$ (by filament contour length) as well as time $t$. (c) Contraction speed $v_{\parallel }\left(s,t\right)$ in the longitudinal direction along the filament backbone at different times. Results based on 100 runs. The shaded area indicates the error. (d) Log-log colormap of normalized Fourier mode amplitude of the filament shape (by the maxium mode amplitude) with the normalized wave number $\tilde{k}$ and time $t$.

Figure 5

(a) Snapshots of the simulated filament conformation during the orthogonal magnetic field induced buckling process without contraction. The buckling process has $\mathrm{Mn}=1.47\times {10}^5$, the persistence length of the filament $L_p=1.33\mathrm{mm}$, and the buckling field strength $B=77\mathrm{G}$. (b) The rescaled slope-slope correlation function of buckling shapes in (a) collapse onto a single curve. (c,d) The temporal evolution of (c) normalized amplitude $w/w_0$ and (d) normalized wavelength $\lambda /{\lambda }_0$ of the buckling filament. Simulation system is restricted to only transverse displacement by coupling the dynamics of the free ends together. Experimental results from the system of the same buckling condition with the same Mn but the ends of the filament remain free.

Figure 6

Normalized longitudinal displacement along the filament backbone at (main plot) 10% end-to-end contraction ratio and (inset) four different end-to-end contraction ratios of 10%, 20%, 30%, and 40%, with a system of different Mn. The normalized longitudinal displacement is defined as the displacement of a small segment on the filament at a particular time from the initial straight configuration normalized by the length of the filament, and the normalized contour length from center is defined as the distance between the segment and the center of the filament normalized by the length of the filament. Both experiment (solid points) and numerical simulation (hollow points) results of higher-order buckling ($\mathrm{Mn}:2.7\times {10}^3-1.7\times {10}^5$, or approximately 8–45 buckling modes) are plotted out. The numerical simulation result is an average of 100 runs with the error bars indicating the variance. The first $\left(\mathrm{Mn}=16\right)$ and second $\left(\mathrm{Mn}=63\right)$ mode buckling results are also included in the main plot with two example buckling shapes for each case next to the result curves and the shaded area (light blue for $\mathrm{Mn}=16$ and light pink for $\mathrm{Mn}=63$) indicating the error. Example buckling conformations of 10%, 20%, 30%, and 40% end-to-end contraction ratios of a $\mathrm{Mn}=1.7\times {10}^4$ system is shown at the top left corner of the inset.