Stochastic buckling of self-assembled colloidal structures

The vast majority of soft and biological materials, gels, and tissues are made from micrometer-size slender structures such as biofilaments and colloidal and molecular chains, which are believed to crucially control their mechanics. These constituents show intriguing extreme mechanics, mechanical instabilities, and plasticity, which, besides attracting significant theoretical attention, have not been studied experimentally and as such remain poorly understood. Here we investigate, by experiments, simulations, and theory, the mechanical instabilities of a slender self-assembled colloidal structure, observing a form of stochastic buckling where thermal fluctuations and associated entropic force effects are amplified in the vicinity of a buckling instability. We fully characterize how the persistence length and plasticity control the stochastic buckling transition, leading to intriguing higher-order buckling modes. These results elucidate the interplay of geometrical, thermal, and plastic interactions in the nonlinear mechanics of thermal self-assembled structures, crucial to the mechanical response and function of fiber-based soft and biological materials, as well as the rational design of micro- and nanoscale architectures.

Figure 1

Buckling of a colloidal chain. (a) Sketch of the colloidal chain compressed by optical tweezers (red dots). (b) Bright-field microscope images of the chain at $u=-0.45, -0.25$, 0.2, 0.6, 0.8, and 1.0 $\mu \mathrm{m}$, respectively from left to right, under a compressive displacement. White bars indicate the position of the laser trap that is slowly displaced and red bars the position of the static trap. The scale bar is $3\mu \mathrm{m}$. The color code demarcates regions of the straight and elastically and plastically buckled chains, bounded by $u_c$ and $u_p$, respectively. (c) Overlay of three reconstructed chains, one corresponding to the still shown in (b), one taken 1.5 s earlier (light gray) and one 1.5 s later (dark gray). (See Supplemental video 1 [38] for microscopy video.)

Figure 2

Elastic buckling regime. The bending force and first Fourier mode are from experiments (gray dots), simulations (blue dots), and the continuum model (purple line). (a) Compressive force $F$ exerted by the tweezers on the chain versus displacement $u$. Note that all experimental data points are depicted, while the simulation data have been averaged over fixed $u$. (b) Amplitude of the first Fourier mode $M_1$ of the particle deflections. Only positive mode amplitudes corresponding to positive deflection of the chain are shown. The inset shows the same quantity squared. (c) Entropic force from simulations (see the text).

Figure 3

Schematic of the model system used for the simulations and the analytical model.

Figure 4

Fluctuations close to buckling. (a) Variance ${\sigma }_{M_1}^2$ of $M_1$ above the mode of its distribution versus the compressive displacement $u$. The inset shows a log-log plot of ${\sigma }_{M_1}^2$ versus $|u-u_c|/u$. (b) Correlation time ${\tau }_{M_1}$ of $M_1$. The inset shows a log-log plot of ${\tau }_{M_1}$ versus $|u-u_c|/u$. Experimental, numerical, and continuum model data are represented by gray triangles, blue triangles, and purple lines, respectively. Simulation and continuum model values are divided by a factor of 3 to fit on the same axis. Also shown are the MD simulations for decreasing chain persistence length showing (c) the normalized first Fourier amplitude $M_1$, (d) the variance, and (e) the entropic force $F_e$ versus normalized $u$ shifted by the zero-temperature buckling compression $u_c^0={\pi }^{2}B/S{L}_0^2$. Blue, orange, green, and red curves correspond, respectively, to 1, $1/10, 1/100$, and 0 times the experimental bending rigidity. The variance was calculated over a limited time window of $1\mathrm{s}$. (See Supplemental video 2 [38] for an animation of the simulation data.)

Figure 5

Transition to low persistence length. (a) Scaling collapse of the first mode amplitude $M_1$ versus normalized $u-u_c$, with $u_c$ the buckling displacement. (See Supplemental video 3 [38] for an animation of the simulation data.) (b) Extracted normalized buckling point $u_c$ as a function of flexibility. Two scaling regimes are evident: exponent $-1$ associated with the chain compressibility for low flexibility (high bending stiffness) and a regime with exponent $+1$ associated with the loss of bending rigidity. (See Supplemental video 4 [38] for microscopy video.)

Figure 6

Plastic buckling. (a) Inverse participation ratio of the experimental chain (gray) and of 50 independent MD simulation runs (blue shading) as a function of continuously increasing compression. The simulations are performed with elastic parameters as in Fig. 2 and ${\theta }_c=0.21\mathrm{rad}$. Vertical lines and colors distinguish regimes of the straight and elastically and plastically buckled chains and indicate two plastic slippage events. Reconstructed snapshots show the experimental chain for $u=0$, 0.5, 1, and $1.5\mu \mathrm{m}$, with snapped bonds highlighted in red. (b) The IPR (black), $M_1$ (blue), and $M_2$ (yellow) versus time of a different compression experiment. Here the chain was shorter ($N=7$) and the trap was moved stepwise, by $\delta u=0.1\mu \mathrm{m}$ every $60\mathrm{s}$.

Figure 7

MD simulation effect on $F_e$ and ${\sigma }^2$ of three different routes that increase the flexibility of the system: left column, increasing $T$; middle column, $B$; and right column $N$. Here $F_e$ is calculated by subtracting the nonthermal $F$ [Eq. (A9)] from the measured $F_{\mathrm{sim}}, {\sigma }_{\mathrm{cut}}^2$ is the variance of the $M_1$ over a time window of around $1\mathrm{s}$, and ${\sigma }_{>\mathrm{mode}}^2$ are fluctuations above the mode of the distribution as defined in the main text. All simulations were run using the same protocol as described above for the elastic simulation.