Microscopic Origin of Capillary Force Balance at Contact Line

We investigate the underlying mechanism of capillary force balance at the contact line. In particular, we offer a novel approach to describe and quantify the capillary force on the liquid in coexistence with its vapor phase, which is crucial in wetting and spreading dynamics. Its relation with the interface tension is elucidated. The proposed model is verified by our molecular dynamics simulations over a wide contact angle range. Differences in capillary forces are observed in evaporating droplets on homogeneous and decorated surfaces. Our findings not only provide a theoretical insight into capillary forces at the contact line, but also validate Young’s equation based on a mechanical interpretation.

Figure 1

(a) Schematic representation of Young’s equation. (b) Sketch of force balance at the contact line. The dark region indicates the liquid corner that is subject to capillary forces. (c) Distribution of tangential force on solid. The contour shows the forces by liquid on each solid atom. The profile gives the sum of every column and manifests a width of $12\sigma$. All the results are normalized by the corresponding peak value. The lattice constant of substrate is $0.8\sigma$ and $\sigma$ is the characteristic size of particles. (d) Determination of the size of liquid corner. When the normalized tangential force ${\tilde{\tau }}_{S\parallel }$ exhibits a saturation, the liquid corner size (radius of the sector, denoted by $R$ in the inset) is considered large enough to describe the contact line.

Figure 2

(a) Validation of Young’s equation in MD simulations. (b) Tangential force exerted on the contact line versus $\theta$. Black dots were calculated directly from MD simulations and red dots were obtained using Eq. (5). The solid line is fitted by $3{\mathrm{\Omega }}_{LS}/2$, while the dashed line is fitted using ${\gamma }_{\mathrm{LV}}(1+\cos \theta )$. The inset shows the profile of tangential forces on solid by liquid. The cohesive liquid-liquid interactions pull the contact line towards the interior of the droplet. Thus, the tangential forces on the solid from both sides are opposite to each other. (c) Normal force exerted on the contact line versus $\theta$. The solid line is fitted using ${\gamma }_{\mathrm{LV}}\sin \theta$. The inset provides the profile of normal forces on solid by liquid.

Figure 3

(a) Overlap of solid-liquid and liquid-vapor interfaces. Layering structures of liquid and density profiles were plotted for both of the interfaces. The insets illustrate the influence of interface overlap on liquid structure. (b) Relation between ${\tau }_{S\parallel }$ and $\cos \theta$. The dashed line gives the linear fitting of the calculated ${\tau }_{S\parallel }$ as $\theta >30°$. The dark region denotes the interface overlap when $\theta \le 30°$. The inset interprets the physical meaning of ${\mathrm{\Omega }}_{\mathrm{SL}}$.

Figure 4

Evaporation of nanodroplets on solid surfaces with different local interactions at the contact line (CL). (a) Droplet base diameter as a function of simulation time. The insets show the MD snapshots of the two cases at $t=0$, 5, and 9.5 ns, respectively. (b) Evaluation of the instantaneous tangential capillary forces during evaporation. The inset shows the tangential force profiles at $t=0$ and 9.5 ns. The results presented are normalized by their initial values before evaporation.