Spreading law on a completely wettable spherical substrate: The energy balance approach

The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The nonequilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect. Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet. The time evolution of the contact angle $\theta$ of a droplet obeys a power law $\theta \sim t^{-\alpha }$ with the exponent $\alpha$, which is different from that derived from Tanner's law on a flat substrate. Furthermore, the line tension must be positive to promote complete wetting on a spherical substrate, while it must be negative on a flat substrate.

Figure 1

A spherical liquid droplet on a convex spherical substrate. The centers of the droplet with radius $r$ and that of the spherical substrate with radius $R$ are separated by a distance $C$. The radius of the contact line is denoted by $R_{\mathrm{L}}$ and the dynamic contact angle is denoted by $\theta$. Note that the three-phase contact line passes through the equator when half of the central angle becomes $\varphi ={90}^{\circ }$ and the contact line moves from the upper hemisphere to the lower hemisphere on the sphere. A small distortion of a spherical cap induces the displacement of radius $\delta r_{\mathrm{n}}$ and that of contact line $\delta R_{\mathrm{L}}$.

Figure 2

(a) A spreading droplet on a spherical substrate. The three-phase contact line shrinks towards the south pole of the substrate to realize the complete wetting state. (b) A spherical droplet on a flat substrate. In this case, the three-phase contact line expands toward infinity to realize the complete wetting state. Therefore, the line tension on a spherical substrate plays the opposite role to that on a flat substrate.