Filament formation via the instability of a stretching viscous sheet: Physical mechanism, linear theory, and fiber applications

Many intriguing patterns are driven by fluid instabilities, even down to micrometer and nanometer scales, with numerous technological and industrial applications. Here, we propose a novel physical mechanism for filament formation in a stretched viscous sheet by combining stretching and van der Waals forces. We show that a transverse instability is enhanced while the longitudinal instability is suppressed, resulting in filamentation. We find a close agreement between the experimental data and the predictions of a linear-stability analysis. These results offer a pathway to achieving sophisticated microstructuring–nanostructuring for functional devices in either a single fiber or in integrated fabrics for large-scale textiles.

Figure 1

Physical mechanism. (a) Sketch of thermal drawing. A cylindrical preform, composed by a thin glass sheet in a polymer matrix, is heated into viscous fluid and stretched into fibers, resulting in the axial extension and transverse shrinkage. For the typical geometrical parameters, the radii of interfaces $2R_1\sim 2R_2\sim R_3\sim 1\mathrm{cm}$ at the inlet, while the thickness $H_0=R_2-R_1<10\mu \mathrm{m}$ and the stretching length $L>10\mathrm{cm}$. Due to $H_0\ll R_1$, the curvature effect of cylindrical structures can be neglected. (b) Sketch of three-layer liquid sheets. A stretching sheet (viscosity ${\mu }_2$, an initial width $2{\ell }_0=2\pi R_1$) sandwiched by liquid with viscosity ${\mu }_1$. An initial velocity $w_0$, and the final velocity $w_f$ out of the stretching region. (c) Sketch for the mechanism of AISVS, where the transverse direction is along $x$ while the longitudinal direction is along $z$.

Figure 2

The attenuation rates of thickness $S_H=H_0/H$ (dotted line) and width $S_{\ell }={\ell }_0/\ell$ (dashed line) along the sheet and their approximate scale factor $S$ (solid line) in the “fiber limit” case at the final scaling factor $S_f=10$ and the ratio $a=\pi R_1/L=0.15$.

Figure 3

Instability of a sandwiched sheet due to van der Waals force. (a) Growth rate $\mathrm{\Omega }$ versus wave number $K$ ($\alpha =100,\eta =1$); (b) Maximum growth rate ${\mathrm{\Omega }}_m$ (solid line) and its corresponding wave number $K_m$ (dashed line) dependent on dimensionless ratio $\alpha$ of van der Waals force to interfacial tension ($\eta =1$).

Figure 4

Anisotropic instability arising from stretching. (a) Maximum total growth rate ${\mathrm{\Phi }}_{\parallel }^m$, ${\mathrm{\Phi }}_{\perp }^m$, and (b) their ratio ${\mathrm{\Phi }}_{\perp }^m/{\mathrm{\Phi }}_{\parallel }^m$, as a function of $S_f$ (${\alpha }_0={10}^{-3},\eta =1,w_0{\mu }_1/\gamma =1,z_p=0,L/H_0={10}^3$).

Figure 5

Anisotropic instability associated with the viscosity ratio (${\alpha }_0={10}^{-2},S_f=10,w_0{\mu }_1/\gamma =1,z_p=0,L/H_0={10}^3$).

Figure 6

Fiber applications. (a), (b) Theoretical predictions agreeing with experimental data. (a) ${\mathrm{\Phi }}_{\perp }^m$ for $\mathrm{PES}/{\mathrm{As}}_2{\mathrm{Se}}_3/\mathrm{PES}$ and PSU/Se/PSU, the dashed line for ${\mathrm{\Phi }}_{\mathrm{cri}}=2$. (b) For PSU/Se/PSU, the predicted wavelength agreeing with the measured wavelength (Deng et al 2008: [11]). (c)–(e) SEM imaging for evolution of instability and breakup for five final targeted film thickness of 300, 170, 90, 40, and 10 nm, respectively. (f) SEM imaging for a ribbon-fiber cross-section for a final targeted 10-nm-thick film, and its inset presenting the circular shapes. Dark color for PSU, bright color for Se, and arrow indicating the decreased targeted thickness.