State and Rate Dependent Contact Line Dynamics over an Aging Soft Surface

We report direct atomic-force-microscope measurements of capillary force hysteresis (CFH) of a circular contact line (CL) formed on a long glass fiber, which is coated with a thin layer of soft polymer film and intersects a water-air interface. The measured CFH shows a distinct overshoot for the depinning of a static CL, and the overshoot amplitude grows logarithmically with both the hold time $\tau$ and fiber speed $V$. A unified model based on the slow growth of a wetting ridge and force-assisted barrier crossing is developed to explain the observed time (or state) and speed (or rate) dependent CL depinning dynamics over an aging soft surface. The experimental findings have important implications to a common class of problems involving depinning dynamics in a defect or roughness landscape, such as friction of solid interfaces.

Figure 1

Sketch of the AFM-based capillary force apparatus at a liquid-air interface. The “long needle” AFM involves a vertical glass fiber of diameter $d$ in the range $1–3\mu \mathrm{m}$ and length $150–300\mu \mathrm{m}$, which is glued on to the front end of a rectangular cantilever beam. The surface of the glass fiber is coated with a thin layer of soft polymer film (PNIPAM). The inset indicates a slow growing wetting ridge (dashed lines) of height $h(\tau )$ formed beneath the three-phase CL. The solid line indicates the wetting ridge with a tangential displacement $\epsilon$ and corresponding tilted angle $\delta {\varphi }_0$ resulting from the CL pulling when the fiber moves downward.

Figure 2

Variations of the measured capillary force $f$ and corresponding contact angle $\theta$ [see Eq. (1)] when the glass fiber is first pushed downward (advancing $\to$) and then is pulled upward (receding $\leftarrow$) through a water-air interface. The measurements are made at a fixed fiber speed $V=10\mu \mathrm{m}/\mathrm{s}$ and sample temperature $T=26°\mathrm{C}$ for different hold times: $\tau =1$ (black), 10 (red), 30 (green), and 100 s (blue).

Figure 3

Variations of the measured $f$ and corresponding $\theta$ as a function of traveling distance $s$ for different fiber speeds: $V=3$ (black), 10 (red), 30 (green), and $100\mu \mathrm{m}/\mathrm{s}$ (blue). The measurements are made at a fixed hold time $\tau =1\mathrm{s}$ and sample temperature $T=32°\mathrm{C}$.

Figure 4

Measured overshoot amplitude $\mathrm{\Delta }{F}_{\mathrm{os}}(\tau ,V)$ as a function of hold time $\tau$ at a fixed fiber speed $V=10\mu \mathrm{m}/\mathrm{s}$. The measurements are made at two sample temperatures: $T=26°\mathrm{C}$ (red triangles) and $T=32°\mathrm{C}$ (black squares). The error bars show the standard deviation of the measurements. The solid lines are the fits of Eq. (2) to the data points with $A(26°\mathrm{C})=3.99\pm 0.25\mathrm{mN}/\mathrm{m}$, ${\tau }_0(26°\mathrm{C})=3.4\pm 0.5\mathrm{s}$, and $A(32°\mathrm{C})=2.97\pm 0.20\mathrm{mN}/\mathrm{m}$, ${\tau }_0(32°\mathrm{C})=0.05\pm 0.02\mathrm{s}$.

Figure 5

Comparison between the measured $\mathrm{\Delta }{F}_{\mathrm{os}}(\tau ,V)$ for a static CL (black symbols) and $\mathrm{\Delta }{F}_a(V)$ for a moving CL (red symbols) both as a function of fiber speed $V$. The measurements are made at a fixed hold time $\tau =1\mathrm{s}$ and at temperatures $T=26°\mathrm{C}$ (triangles) and $32°\mathrm{C}$ (squares). The error bars show the standard deviation of the measurements. The black lines are the fits of Eq. (2) to the data points with $B(26°\mathrm{C})=3.71\pm 0.50\mathrm{mN}/\mathrm{m}$, $V_0(26°\mathrm{C})=19.6\pm 5.0\mu \mathrm{m}/\mathrm{s}$ and $B(32°\mathrm{C})=2.63\pm 0.40\mathrm{mN}/\mathrm{m}$, $V_0(32°\mathrm{C})=1.25\pm 0.40\mu \mathrm{m}/\mathrm{s}$. The red lines are fits of Eq. (3) to the data points with $(B-A)(26°\mathrm{C})=-0.26\pm 0.02\mathrm{mN}/\mathrm{m}$ and $(B-A)(32°\mathrm{C})=-0.35\pm 0.02\mathrm{mN}/\mathrm{m}$.