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

## TL;DR

This paper investigates the spreading dynamics of a droplet on a spherical substrate using thermodynamic and energy balance methods, revealing unique power-law behavior and the influence of line tension.

## Contribution

It introduces a novel energy balance approach to model droplet spreading on spherical substrates, accounting for line tension effects and deriving a new contact angle evolution law.

## Key findings

- Contact angle follows a power law with a different exponent than Tanner's law.
- Line tension must be positive for complete wetting on a sphere.
- The spreading behavior differs from flat substrates due to curvature effects.

## Abstract

The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The non-equilibrium 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.