Simultaneous observation of asymmetric speed-dependent capillary force hysteresis and slow relaxation of a suddenly stopped moving contact line

We report 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 liquid-air interface. By using eight different liquid interfaces with varying solid-liquid molecular interactions, we find a universal behavior of the asymmetric speed dependence of CFH and CL relaxation. A unified model based on force-assisted barrier crossing is used to connect the mesoscopic measurements of CFH and CL relaxation with the energy barrier height $E_b$ and size $\lambda$ associated with the surface defects. The experiment demonstrates that the CL pinning (relaxation) and depinning dynamics are closely related and can be described by a common microscopic framework.

Figure 1

(a) Sketch of the AFM-based capillary force apparatus at a liquid-air interface. (b) A scanning electron microscope image of the actual hanging glass fiber of diameter $d\simeq 2.2\mu \mathrm{m}$ and length $l\simeq 150\mu \mathrm{m}$ (partially adopted from Ref. [23]).

Figure 2

Sketch of the linear-cubic potential $U\left(x\right)$ with three different characteristic forces: $f_T$ (top black curve), $f_T<f_2<f_c$ (middle green curve), and $f_c$ (bottom red curve). The three curves are displaced vertically for clarity. The intrinsic barrier height $E_b$ is shown for $\delta f=f_T$.

Figure 3

3D plot of the normalized rupture force $\delta f_1/f_c$ as a function of the normalized loading rate $\dot{f}/\dot{f_T}$ and intrinsic energy barrier $E_b/k_{B}T$, given in Eq. (5). The grid used on the surface plot shows increments of ${log}_{10}(\dot{f}/\dot{f_T})=0.5$ and $E_b/k_{B}T=5$, respectively. The red arrow indicates the variation of $\delta f_1/f_c$ with increasing $\dot{f}/\dot{f_T}$ for a fixed barrier $E_b/k_{B}T=10$.

Figure 4

(a) AFM topographic image of the top surface of a FTS-coated glass fiber in air. Typical height contrast is $\pm 0.4$ nm. (b) Sketch of the FTS monolayer grafted on a glass fiber surface with defects of size $\lambda$ (side view) (partially adopted from Ref. [23]).

Figure 5

Variations of the measured capillary force $f$ as a function of traveling distance $s$ when the FTS-coated glass fiber is pushed downward [advancing $(\to )$, A to C, red] and is pulled upward [receding $(\leftarrow )$, D to F, black]. The measurement is made at a water-air interface with the fiber speed $u=10 \mu \mathrm{m}$/s.

Figure 6

Measured speed dependence of the hysteresis loop of the capillary force $f$ and corresponding contact angle ${\theta }_i$, when the FTS-coated glass fiber is pushed downward and is pulled upward through four different liquid-air interfaces: (a) water-air, (b) octanol-air, (c) decane-air, and (d) FC77-air. The measurements are made with fiber speed $u=2\mu \text{m/s}$ (black), $5 \mu \text{m/s}$ (red), $20 \mu \text{m/s}$ (green), and $50 \mu \text{m/s}$ (blue), respectively. The black long arrow points the direction of increasing $u$.

Figure 7

(a) Replot of the same data shown in Fig. 5 as a function of time $t$ when the fiber is pushed downward (advancing, A to C, red) and is pulled upward (receding, D to F, black). Curves C to D (green) and F to G (blue) show the relaxation of the measured $f\left(t\right)$ when the (moving) fiber is suddenly stopped at C and F, respectively, for 10 s. (b) Measured relaxation of the capillary force $f\left(t\right)$ when a MCL with speed $u=20\mu \text{m/s}$ is suddenly stopped at $t=0$. The measurements are made at the octanol-air interface in the advancing (red line) and receding (black line) directions.

Figure 8

(a) Measured depinning force $\mathrm{\Delta }{F}_r$ for the receding CL as a function of fiber speed $u$ for eight liquid samples: water (black circles), decane (black squares), octane (red circles), dodecane (green up triangles), hexadecane (blue diamonds), octanol (blue left triangles), hexanol (red right triangles), and FC77 (orange down triangles). The three solid lines show, respectively, the fits to Eq. (8) for water (black line), octanol (blue line), and hexanol (red line). (b) Magnified plots of the measured $\mathrm{\Delta }{F}_r$ for alkanes: decane (black squares), octane (red circles), dodecane (green triangles), and hexadecane (blue diamonds). The error bars show the standard deviation of the measurements. The solid lines are the fits to Eq. (8).

Figure 9

Measured depinning force $\mathrm{\Delta }{F}_a$ for the advancing CL as a function of fiber speed $u$ for three liquid samples: water (black circles), octanol (blue left triangles), and hexanol (red right triangles). The error bars show the standard deviation of the measurements. The solid lines are the fits to Eq. (8).

Figure 10

Relaxation of the measured capillary force $f\left(t\right)$ and corresponding contact angle ${\theta }_i\left(t\right)$ as a function of time $t$ after a MCL is suddenly stopped at $t=0$. The measurements are made for the decane-air interface with different initial speeds: $u=1 \mu \mathrm{m}$/s (black), $2 \mu \mathrm{m}$/s (red), $5 \mu \mathrm{m}$/s (green), and $20 \mu \mathrm{m}$/s (blue). The upper four curves are obtained for the advancing CL and the lower four curves are obtained for the receding CL. The black long arrow points the direction of increasing $u$. The solid lines show the fits to Eq. (11).

Figure 11

Normalized mean rupture force per defect $\delta f_1/f_c$ as a function of the normalized pulling rate ${\dot{f}}_1/\dot{f_T}$ for different liquid samples. The solid symbols are obtained in the receding direction, and the open symbols are obtained in the advancing direction. The solid lines are given by Eq. (8) for $E_b=5$, 10, 15, 25, 50, and $150 k_{B}T$, respectively. The entire “phase diagram” may be divided into three regions (see text for more details): (I) strong pinning region (upper shaded area), (II) speed-dependent intermediate region (middle unshaded area), and (III) weak pinning region (lower shaded area).

Figure 12

Measured hysteresis force ${\left(f_h\right)}_0$ as a function of the receding critical force ${\left(f_c\right)}_r$ for eight liquid samples: water (black circle), decane (black square), octane (red circle), dodecane (green up triangle), hexadecane (blue diamond), octanol (blue left triangle), hexanol (red right triangle), and FC77 (orange down triangle). The error bars show the standard deviation of the force measurements. The solid line is a linear fit to the data points (excluding the blue diamond): ${\left(f_h\right)}_0=b{\left(f_c\right)}_r$ with $b=23.1/\mu \text{m}$.