eduKateSG · How Science Works
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A drop can remain on a slope because its contact line resists rearrangement. Different contact angles around the footprint allow a capillary retention force to balance the downhill component of its weight.
Contents
Understand the mechanism · Chapters 1–3
1. The drop changes shape before it moves
Work the calculation · Chapters 4–5
Test the explanation · Chapters 6–8
6. History and surface condition can change the answer
A water drop placed on a gently tilted surface may stay put. Increase the tilt and it stretches downhill before eventually moving. Gravity was acting all along. What changed was the balance between the downhill load and the surface’s capacity to retain the drop.
The boundary where liquid, solid and gas meet is the contact line. Tiny variations in texture or chemistry can hinder its motion. A drop can respond by changing shape while parts of this boundary remain pinned.
This guide focuses on retention and the onset of sliding. For leaf texture and dirt removal, use the existing lotus-leaf guide. A rounded bead and a drop that moves easily are distinct observations.
The contact angle is measured through the liquid where its surface meets the solid. On an ideal, smooth, chemically uniform surface at equilibrium, interfacial tensions specify an ideal angle. A real surface can support a range of apparent angles before the contact line changes position.
When liquid is slowly added, the angle can rise while the footprint remains pinned, then the boundary advances. During slow withdrawal, the angle can fall before the boundary recedes. Advancing and receding measurements depend on a stated procedure.
The difference between these limiting behaviours is contact-angle hysteresis. It records path dependence rather than a contradiction in the measurement. Tilt experiments also create front-to-back angle differences, but their angles must be interpreted with the observed contact-line motion.
An original inclined-drop study examined hysteresis cycles, free-surface shape and footprint changes. The footprint is part of the data, not merely a convenient background for measuring two angles.
The downhill component of weight is mg sin α, where α is the surface tilt from horizontal. A commonly used retention relation has the form F ≈ kγw(cos θR − cos θA). Here γ is liquid–gas surface tension, w is a relevant footprint width, and k represents geometric details.
The relation is useful only when the angle definitions, width and geometry are consistent. It does not mean all contact-line forces act at just two points. A full treatment accounts for the distribution around the perimeter.
Stern and colleagues’ Furmidge-equation analysis investigates how the angular distribution affects the geometric prefactor. Setting k = 1 is a modelling choice for an example, not a universal constant supplied by nature.
Before calculating, distinguish static retention at impending motion from resistance after sliding begins. A threshold force does not by itself determine a moving drop’s speed.
Original teaching example. Assign γ = 0.072 N/m, w = 0.0030 m, θR = 60°, θA = 100° and k = 1. The cosine difference is 0.5000 − (−0.17365) = 0.67365.
The model retention force is approximately 0.0001455 N, or 145.5 micronewtons. For a drop of mass 30 milligrams, mg = 294.3 micronewtons. Equating downhill load and retention gives sin α ≈ 145.5/294.3 = 0.4944, so α ≈ 29.6°.
This is a predicted threshold for the assigned parameters. It is not an experimental water-on-glass value. Surface condition, drop shape and the definitions used for the angles and width can shift the result.
At 10° the downhill force is about 51.1 micronewtons; at 20° it is 100.7; at 30° it is 147.2. Comparing these loads with 145.5 makes the threshold visible without imagining that gravity suddenly switches on.
| Tilt | Downhill weight component | Compared with 145.5 micronewtons |
|---|---|---|
| 10° | 51.1 micronewtons | Below model retention |
| 20° | 100.7 micronewtons | Below model retention |
| 30° | 147.2 micronewtons | Above model retention |
A ten-degree difference is not the same capillary contribution at every pair of angles. With the same γ, w and k, angles of 60° and 70° give a cosine difference about 0.1580. Angles of 10° and 20° give about 0.0451.
The corresponding force scales in our example are about 34.1 and 9.75 micronewtons. The identical ten-degree angular separation produces different values because the force relation uses cosines.
This does not establish that either pair will occur on a particular material. It exposes why a reported hysteresis measured only in degrees may not be sufficient for a mechanical comparison.
Likewise, a very high apparent contact angle does not guarantee weak retention. Ask for advancing and receding behaviour, footprint dimensions and a sliding or retention measurement appropriate to the intended use.
Two equal-volume drops on the same nominal material can encounter different defects or contamination. Their footprints may pin differently. Depositing, adding, withdrawing and tilting can also lead to different intermediate states.
A useful study repeats measurements at multiple locations and records the preparation and timing. Evaporation can reduce volume during a slow experiment, making the load and geometry drift while the tilt changes.
Do not explain every pinned line as a visible scratch. Chemically heterogeneous patches and small-scale roughness can both matter. The observed threshold identifies retention; additional evidence is needed to identify the particular microscopic obstacles.
The original modelling work on inclined sliding drops explicitly separates stages of motion. A model for one stage should not silently become a claim about every stage.
Measure drop volume, tilt, footprint shape and side-view angles under the same preparation protocol. Record when each part of the contact line first moves. A video can distinguish smooth sliding from intermittent depinning and repinning that a single final photograph conceals.
For an original comparison, imagine surface A gives an initial angle of 120° but needs 40° tilt to move the drop. Surface B gives 100° but motion starts at 15°. The first surface produces the rounder initial bead; the second releases it more easily in this specified test.
Neither observation alone proves better self-cleaning, lower contamination or long-term durability. Those require their own measurements. Matching the metric to the claim is more useful than ranking surfaces by whichever number looks most impressive.
Fit a retention model using independently measured inputs where possible. Choosing k separately for every observation can reproduce the data while removing much of the model’s predictive power. Hold some conditions aside to test the fitted relation.
Why can a drop remain still although gravity pulls downhill? The forces can balance. The liquid shape and contact-line configuration support a resisting capillary contribution.
If the simple equation gives sin α greater than one, what follows? There is no predicted gravity-driven threshold below 90° with those fixed parameters. It does not prove permanent attachment: geometry and wetting states can change, and other failure routes exist.
Can doubling drop mass be analysed by doubling weight alone? Only as a deliberately constrained thought experiment. A real larger drop generally changes its width, shape and angles too.
For the transport framework, continue to Fluid Dynamics. For a different consequence of a pinned footprint, read the coffee-ring mechanism.