Young’s Equation at the Nanoscale

In 1805, Thomas Young was the first to propose an equation to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s equation.

Figure 1

Schematic illustration of the tangential forces exerted on a partially immersed rod.

Figure 2

Front and top views showing our molecular-dynamics simulations of a cylindrical rod (red atoms) dipping into a cylindrical liquid bath (green molecules) held in place within a frozen liquid annulus. The configuration shows the equilibrium configuration for $C_{S-L}=1.0$.

Figure 3

Contact angle versus time for five different liquid-solid interactions (from top to bottom, $C_{S-L}=0.4$, 0.6, 0.8, 0.9, and 1.0 with, respectively, ${\theta }^0=133.0°\pm 3.3°$, $108.8°\pm 3.3°$, $85.7°\pm 3.4°$, $70.0°\pm 3.5°$, and $51.0°\pm 3.5°$). Each point is the average of 50 successive values, and the error bars are the standard deviations.

Figure 4

Tangential force distribution at equilibrium along the nanofiber obtained from the simulations at two solid-liquid couplings: $C_{S-L}=0.8$ (triangles) and 1.0 (squares). The inset shows an enlarged view of the force distribution at the contact line. The error bars are the standard deviations on the average values of the forces calculated for each layer of the fiber.

Figure 5

The total force (a) at the bottom of the fiber and (b) across the three-phase zone for $C_{S-L}=0.4$ to 1.05, plotted as a function of the equilibrium contact angle. The solid lines through the data represent (a) ${\gamma }_{LV}$ and (b) $-{\gamma }_{LV}(1+\cos {\theta }^0)$. (c) The intermediate plot compares the sum of the measured forces at both locations with that predicted by Eq. (3): $-{\gamma }_{LV}\cos {\theta }^0$ (because we measure the forces in the $z$ direction, attractive forces between the liquid and the solid have a minus sign).