Elastocapillarity can control the formation and the morphology of beads-on-string structures in solid fibers

Beads-on-string patterns have been experimentally observed in solid cylinders for a wide range of material properties and structural lengths, from millimetric soft gels to nanometric hard fibers. In this work, we combine theoretical analysis and numerical tools to investigate the formation and nonlinear dynamics of such beaded structures. We show that this morphological transition is driven by elastocapillarity, i.e., a complex interplay between the effects of surface tension and bulk elasticity. Unlike buckling or wrinkling, the presence of an axial elongation is found here to favor the onset of fiber beading, in agreement with existing experimental results on electrospun fibers, hydrogels, and nerves. Our results also prove that the applied stretch can be used in fabrication techniques to control the morphology of the emerging beads-on-string patterns. Such quantitative predictions open the way for several applications, from tissue engineering to the design of stretchable electronics and the microfabrication of functionalized surfaces.

Figure 1

Bead-on-string patterns in solid fibers with different underlying physical properties: (I) rat sciatic nerve during a traction test (the scale bar is 50 $\mu \text{m}$, from Ref. [28]); (II) polymeric nanofiber under axial elongation (from Ref. [17]); (III) electrospun fiber during the extrusion process (the scale bar is 10 $\mu \text{m}$, from Ref. [30]); (IV) stretched soft cylindrical gels with radius of few mm (from Ref. [25]).

Figure 2

(a) Dispersion curves (solid black line, $\overline{k}=0.01$; dashed red line, $\overline{k}=0.1$; dotted green line, $\overline{k}=0.2$) showing the critical axial stretch ${\lambda }_z^{th}$ versus the elastocapillary ratio $L_{ec}/R_0$ at varying $\overline{k}$. The markers (black circles, $\overline{k}=0.01$; red squares, $\overline{k}=0.1$; green triangles, $\overline{k}=0.2$) indicate the thresholds computed numerically. (b) Morphological phase diagrams from the numerical simulations of two different paths, fixing $L_{ec}/R_0=7.7$ (A-B-C-D) and ${\lambda }_z=1.05$ (A-E-F-G).

Figure 3

(a) Beading amplitude $a$ and axial width $W$. (b) Numerical simulation [35] mimicking the experiments on PAN nanofibers with $R_0=300\text{nm}$ [17] giving (c) the amplitude $a$ and (d) the dimensionless width $\overline{W}=W/L_c$ for three different elastocapillary ratios. Numerical data (green triangles, $\overline{k}=0.2$ and $\frac{L_{ec}}{R_0}=5.9$; black squares, $\overline{k}=0.2$ and $\frac{L_{ec}}{R_0}=6$; red circles, $\overline{k}=0.1$ and $\frac{L_{ec}}{R_0}=5.75$) in (c) fit very well the theoretical predictions given by $\epsilon B^{st}$ with no adjustable parameters: dot-dash green line, $SSE=1.59\times {10}^{-5}$; solid black line, $SSE=3.47\times {10}^{-3}$; dashed red line, $SSE=3.10\times {10}^{-4}$ (SSE: summed square of residuals).

Figure 4

The beading amplitude $a$ versus the applied axial stretch in a loading-unloading cycle for different wave numbers $\overline{k}$ (solid black line, $\overline{k}=0.0942$; dashed red line, $\overline{k}=0.1131$; dotted green line, $\overline{k}=0.1319$). We set $R_0=0.24\mu \text{m},L_0=80\mu \text{m},\mu =20\text{kPa},\gamma =0.028\mathrm{N}/\mathrm{m}$ ($L_{ec}/R_0=5.7$).