The microscopic origin of droplet line tension

Abstract

The size dependence of the equilibrium droplet contact angle is governed by line tension. In this work, we identify a contribution to line tension arising from gravitational effects and pressure-induced changes in volume-fraction-dependent interfacial tensions within an adsorption layer. This mechanism addresses a multiscale problem of line tension in droplets ranging from nanometric to millimetric sizes that change sign and span several orders of magnitude, in agreement with experimental and simulation results. The sign of the apparent line tension is controlled by surface wettability, the initial volume fraction in the adsorption layer, and the droplet size, which also strongly influences its magnitude. Our results provide a unified physical interpretation of the experimentally observed variability in both the sign and magnitude of line tensions.

Figures (4)

• Figure 1: (A) Schematic cross-sectional representation of the liquid droplet (L, blue) on a solid substrate (S, gray) in a gas phase (G, white), illustrating the contact angle $\theta$, the base radius $r$, the cap radius $R$, and the interfacial tensions $\sigma$, $\gamma _G$, and $\gamma _L$. The colored layer represents the adsorption layer on the solid, which modifies the interfacial tensions. The wetted area, the local droplet penetration area and the exterior solid-gas interfacial area are represented by $S_1$, $S_2$, and $S_3$, respectively. (B) Enlarged view at the triple line and the adsorption layer of thickness $\ell$ showing the liquid volume fraction $\phi_{L1}$ and $\phi_{L2}$ dependent interfacial tensions $\gamma(\phi_{L1})$ (green) and $\gamma(\phi_{L2})$ (pink) in the vicinity of the droplet, which are affected by the liquid pressure $p_L$ and the gas pressure $p_G$, respectively. Outside the penetration area, the interfacial tension $\gamma_{G0}$ is considered to be constant (gray).
• Figure 2: Line tension $\tau$ [N] as a function of the base radius $r$ [m] with $\theta_0=40^\circ$, $\Delta\rho=\qty{998}{kg\per m^{3}}$, $g=\qty{9.81}{m\per s^2}$, $\sigma=\qty{0.055}{N\per m}$, and $\ell=\qty{5}{nm}$, in comparison with previous experiments zhaoResolvingApparentLine2019mugeleCapillarityNanoscaleAFM2002gaydosDependenceContactAngles1987aduncanCorrelationLineTension1995wallaceLineTensionSessile1988amunzSizeDependenceShape2014zhangInterfacialOilDroplets2008bergImpactNegativeLine2010heimMeasurementLineTension2013checcoNonlinearDependenceContact2003seemannPolystyreneNanodroplets2001amirfazliLineTensionMeasurements1998drelicheffectSolidSurface1994werderWaterCarbonInteractionUse2003DussaudLineTension1996AmirfazliDetermination2003 and simulations wlochNewForcefieldWater2017kanducGeneralizedLineTension2018zhangCriticalAssessmentLine2018scocchiWettingContactlineEffects2011khalkhaliWettingNanoscaleMolecular2017ShintakuMeasuring2024, where Young's contact angle, $\theta_0$, is indicated by the color of each point.
• Figure 3: (A) Energy landscape $E(\phi_{L1},\phi_{L2})$ with local energy minima marked by crosses with $S_2=0.1S_1$. (B) Line tension as a function of the base radius: single pressure effect (dashed lines), single gravity effect (dotted lines), combined effect (solid lines; micro--Cassie--Baxter state: pink, micro--Wenzel: blue). (C) Apparent line tension as a function of the base radius for different $\theta_0$ (different materials) at (i) micro--Cassie--Baxter state and (ii) micro--Wenzel state.
• Figure 4: Apparent line tension at the micro--Cassie--Baxter and the micro--Wenzel state as a function of the apparent contact angle $\theta$ in comparison with experimental measurements zhaoResolvingApparentLine2019pompeLineTensionBehavior2002.