Measurement of Capillary Forces Using Two Fibers Dynamically Withdrawn from a Liquid: Evidence for an Enhanced Cheerios Effect

We study the capillary attraction force between two fibers dynamically withdrawn from a bath. We propose an experimental method to measure this force and show that its magnitude strongly increases with the retraction speed by up to a factor of 10 compared to the static case. We show that this remarkable increase stems from the shape of the dynamical meniscus between the two fibers. We first study the dynamical meniscus around one fiber and obtain experimental and numerical scaling of its size increase with the capillary number, which is not captured by the classical Landau-Levich-Derjaguin theory. We then show that the shape of the deformed air-liquid interface around two fibers can be inferred from the linear superposition of the interface around a single fiber. These results yield an analytical expression for the capillary force which compares well with the experimental data. Our study reveals the critical role of the retraction speed to create stronger capillary interactions, with potential applications in industry or biology.

Figure 1

(a) Experimental setup. Two parallel glass fibers separated by a distance $2d$ and clamped at one end are removed from a fluid bath at a speed $V$ ($E=64\pm 1\mathrm{GPa}$). Above a critical velocity $V_c$, the fibers coalesce. (b) Length $L$ of the fibers as a function of $V_c$. Squares, $R=100\mathrm{\mu }\mathrm{m}$; circles, $R=50\mathrm{\mu }\mathrm{m}$; diamonds, $R=160\mathrm{\mu }\mathrm{m}$. Face color represents $d/R$ given by the color bar; edge color corresponds to the fluid properties; black, silicon oil V1000 ($\mu =0.96\mathrm{Pa}\mathrm{s}$); red, silicon oil V100 ($\mu =0.096\mathrm{Pa}\mathrm{s}$); gray, glycerol ($\mu =1.3\mathrm{Pa}\mathrm{s}$). $\gamma =0.021\mathrm{N}{\mathrm{m}}^{-1}$ for silicon and $0.063\mathrm{N}{\mathrm{m}}^{-1}$ for glycerol. (c) At $V=0\mathrm{m}/\mathrm{s}$, the scaling for $L=L_s$ obtained in Ref. [11] is recovered (solid curve). (d) Estimation of the dynamic capillary force $Bd/L^3$ as a function of Ca.

Figure 2

(a) Snapshots of a glass fiber ($R=50\mathrm{\mu }\mathrm{m}$) withdrawn from silicon oil ($\mu =0.96\mathrm{Pa}\mathrm{s}$) at various retraction speeds ($V=0$, 16, 128, $512\mathrm{mm}/\min$). Scale bar: 1 mm. (b) Meniscus shape as a function of the rescaled radial distance $r/{\ell }_c$ at various $V$ (mm/min). The agreement with the theoretical model proposed in Ref. [24] is good up to $V=256\mathrm{mm}/\min$ ($\mathrm{Ca}\simeq 0.2$). (c) Collapse of the dynamical meniscus profiles onto the static meniscus shape when their height are properly rescaled by $1+\zeta$. $V=2^n\mathrm{mm}/\min$ with $1\le n\le 9$. The gray shaded area indicates the region where the collapse is not satisfactory for the largest retraction speed. (d) Evolution of the scaling factor $\zeta$ as a function of Ca. ${\zeta }_{\exp }$ and ${\zeta }_{\mathrm{num}}$ correspond to the experimental and numerical profiles, respectively [25].

Figure 3

(a) Snapshots of two glass fibers ($R=100\mathrm{\mu }\mathrm{m}$) withdrawn from silicon oil ($\mu =0.96\mathrm{Pa}\mathrm{s}$) at $V=0$, 32, $256\mathrm{mm}/\min$. Scale bar: 1 mm. (b) Meniscus profiles as $V$ increases (0, 8, 32, 64, $128\mathrm{mm}/\min$). (c) Comparison between the dynamical meniscus around two fibers ($R=50\mathrm{\mu }\mathrm{m}$) separated by a distance $2d=0.88\mathrm{mm}$ and withdrawn at $V=16\mathrm{mm}/\min$ ($\mathrm{Ca}=0.012$, orange curve) and the meniscus (solid blue curve) obtained by summing the dynamical meniscus around a single fiber withdrawn at the same speed (dashed blue curve). (d) Evolution of $\mathrm{\Delta }h/h_s$ as a function of Ca. All data collapse on the same power law as the one obtained in Fig. 2: $\mathrm{\Delta }h/h_s=12.2{\mathrm{Ca}}^{4/5}$. An increase of slope is observed when $d\simeq 20R{\mathrm{Ca}}^{2/3}$. The vertical dashed lines correspond to the threshold above which a liquid film is entrained. (e) Height of the static bridge as a function of the fibers distance. The solid curve corresponds to Eq. (2b) and the dashed curve to its logarithmic approximation valid for $d/{\ell }_c\ll 1$. See Fig. 1 for the symbol and color code.

Figure 4

Evolution of $\mathrm{\Delta }F/F_s$ as a function of Ca. All data collapse on the same power law as the one obtained in Fig. 2: $\mathrm{\Delta }F/F_s=12.2{\mathrm{Ca}}^{4/5}$. An increase of slope is observed when $d\simeq 50R{\mathrm{Ca}}^{2/3}$. See Fig. 1 for the symbol and color code.