Asymmetric and Speed-Dependent Capillary Force Hysteresis and Relaxation of a Suddenly Stopped Moving Contact Line

We report on direct atomic-force-microscope measurements of capillary force hysteresis (CFH) and relaxation of a circular moving contact line (CL) formed on a long micron-sized hydrophobic fiber intersecting a water-air interface. The measured CFH and CL relaxation show a strong asymmetric speed dependence in the advancing and receding directions. A unified model based on force-assisted barrier crossing is utilized to find the underlying energy barrier $E_b$ and size $\lambda$ associated with the defects on the fiber surface. The experiment demonstrates that the pinning (relaxation) and depinning dynamics of the CL can be described by a common microscopic framework, and the advancing and receding CLs are influenced by two different sets of relatively wetting and nonwetting defects on the fiber surface.

Figure 1

(a) Sketch of the AFM-based capillary force apparatus. (b) A scanning electron microscope image of the actual hanging glass fiber of diameter $d\simeq 2.2\mu \mathrm{m}$ and length $165\mu \mathrm{m}$. (c) Sketch of the FTS coating on a glass fiber surface with defects of size $\lambda$ (side view).

Figure 2

Variations of the measured $f$ and the corresponding contact angle ${\theta }_i$ [see Eq. (1)] when the glass fiber is pushed downward (advancing $\to$, solid line) and is pulled upward (receding $\leftarrow$, dashed line) through a water-air interface. The measurements are made at fiber speed $u=1$ (black lines), 3.5 (red lines), 10 (green lines), and $35\mu \mathrm{m}/\mathrm{s}$ (blue lines), respectively. For clearance, each hysteresis loop is coded with a different color. The black long arrow indicates the direction of increasing $u$.

Figure 3

Relaxation of the measured capillary force $f(t)$ as a function of time $t$ for a moving fiber through a water-air interface in the advancing (red line) and receding (black line) directions with an initial speed $u=20\mu \mathrm{m}/\mathrm{s}$ at $t=0$. The corresponding value of ${\theta }_i(t)$ [see Eq. (1)] is marked on the right vertical axis. The inset shows the relaxation of $f(t)$ in the receding direction with different initial speeds at $t=0$: $u=1$ (black), 2 (red), 5 (blue), 10 (orange), and $20\mu \mathrm{m}/\mathrm{s}$ (green). The black long arrow points in the direction of increasing $u$. The solid lines show the fits to Eq. (5).

Figure 4

Measured rupture force $\mathrm{\Delta }F$ as a function of fiber speed $u$ in the advancing (red squares) and receding (black circles) directions. The error bars show the standard deviation of the measurements. The solid lines are the fits to Eq. (4) with $(E_b{)}_a=14.6k_{B}T$ and ${\lambda }_a=1.30\mathrm{nm}$ in the advancing direction and $(E_b{)}_r=16.8k_{B}T$ and ${\lambda }_r=0.73\mathrm{nm}$ in the receding direction.