The Physics of Microdroplets

Chapter 5

Droplets Between Two Non-parallel Planes: From Tapered Planes to Wedges

5.1 Abstract

In this chapter, we focus on the behavior of a droplet or a liquid plug placed between two non-parallel plates. It is assumed that the droplet is sufficiently small that gravitational forces can be neglected (droplet Bond number smaller than 1). The case of tapered plates is first investigated. It is shown that, if the walls are perfectly smooth, e.g. there is no pinning hysteresis, a wetting droplet is not at equilibrium in such a geometry. The conventional Hauksbee’s formulation for wetting walls is presented, and a generalization for any wall wetting property (two wetting walls, two non-wetting walls, wetting and non-wetting walls) is derived. In a second part, the focus is on the Concus-Finn relations for a wedge, which governs the location of the droplet in the wedge. A generalization to different wall wetting properties is presented.

5.2 Droplet Self-motion Between Two Non-parallel Planes

It was first observed by Hauksbee [1,2] that a water plug limited by two non-parallel wetting plates – hydrophilic for a water droplet or lyophilic for any liquid – moves towards the narrow gap region. A sketch of the plug is shown in figure 5.1.

Figure 5.1 Sketch of a liquid plug between two wetting plates. The contact angle is θ < 90° and the plug is not at equilibrium: it moves in the direction of the smaller cross-section. Note that the Laplace pressures are negative in such a case.

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