Spreading law of non-Newtonian power-law liquids on a spherical substrate by energy balance approach

TL;DR

This paper theoretically investigates the spreading behavior of non-Newtonian power-law liquids on spherical substrates, deriving a new power-law relation for contact angle evolution and exploring late-stage line-tension effects.

Contribution

It introduces a novel energy balance model for non-Newtonian droplet spreading on spherical surfaces, deriving a unique spreading law different from classical Tanner's law.

Findings

Contact angle follows a power law $ heta o t^{-eta}$ with a new exponent.

Spreading dynamics differ from flat substrate cases for non-Newtonian liquids.

Line-tension effects dominate in late-stage spreading on spherical substrates.

Abstract

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids, both shear-thinning and shear-thinning liquids, that completely wet a spherical substrate is theoretically investigated in the capillary-controlled spreading regime. The crater-shaped droplet model with the wedge-shaped meniscus near the three-phase contact line is used to calculate the viscous dissipation near the contact line. Then the energy balance approach is adopted to derive the equation which governs the evolution of the contact line. The time evolution of the dynamic contact angle $\theta$ of a droplet obeys a power law $\theta \sim t^{-\alpha}$ with the spreading exponent $\alpha$, which is different from Tanner's law for Newtonian liquids and those for non-Newtonian liquids on a flat substrate. Furthermore, the line-tension dominated spreading, which could be realized on a spherical substrate for late-stage of spreading when the contact angle becomes low and the curvature of the contact line becomes large, is also investigated.