💧 Capillary Action

The surface-tension force lifting the liquid scales with the tube perimeter (2πr) — a linear dimension — whilst the weight of liquid resisting that lift scales with the cross-sectional area (πr²) — a quadratic dimension. Dividing force by weight gives h ∝ (r/r²) = 1/r. This fundamental geometric argument means that for a given liquid and surface, halving the tube radius doubles the rise height. Below approximately 1 μm radius, however, the assumptions of continuum mechanics break down and molecular-scale effects dominate — relevant to water transport in biological nanopores such as aquaporins.

In tall trees such as Sequoias (height up to 116 m), water is transported from roots to leaves through xylem vessels — hollow dead cells with diameters typically 20–200 μm. Jurin's law alone predicts capillary rise of only about 1.5 m for a 100 μm vessel, far short of 116 m. The actual mechanism is the cohesion–tension theory: transpiration at leaves pulls the water column under tension (negative pressure, as low as −5 MPa in conifers) whilst the narrow xylem vessels resist collapse due to thickened lignified walls. Capillary action provides the initial "wick" but tension from above drives the bulk flow.

The contact angle θ is the angle between the liquid–gas interface and the solid surface, measured through the liquid phase, at the three-phase contact line. It is defined by Young's equation: cos θ = (γ_SG − γ_SL)/γ_LG, where γ_SG, γ_SL, and γ_LG are the solid–gas, solid–liquid, and liquid–gas interfacial energies respectively. Contact angles <90° indicate wetting (hydrophilic for water); angles >90° indicate non-wetting (hydrophobic). It is measured with a goniometer by photographing a sessile drop on the surface and fitting the profile shape. Superhydrophobic surfaces (lotus leaves, Gore-Tex) achieve θ > 150° through micro/nanoscale surface roughness.

Lateral flow assays (LFA) — the technology behind home pregnancy tests, COVID-19 antigen tests, and many point-of-care diagnostics — exploit capillary action in nitrocellulose membranes to transport analyte-containing fluid along a defined path without any external pumps. The sample is applied to an absorbent pad, which draws fluid by capillarity into a conjugate pad containing antibody-labelled gold or latex nanoparticles. These flow by capillary action along a nitrocellulose strip to test and control lines where capture antibodies create the visible coloured bands. The entire assay runs in 5–15 minutes driven entirely by Jurin-type capillary forces.

In microgravity (g → 0), Jurin's law predicts h → ∞: liquid would completely fill any capillary tube without limit. This has been demonstrated on the International Space Station — water placed near a capillary structure immediately climbs and floods it. In spacecraft water management, this creates both challenges (water wicks into unwanted crevices) and opportunities (passive wicking systems can distribute coolant or propellant without pumps). The "gravity" button in this simulator showing Moon (g = 1.62 m/s²) vs Earth (9.81 m/s²) illustrates that the Moon's lower gravity increases capillary rise by a factor of 9.81/1.62 ≈ 6.

Surfactant molecules (surface-active agents, such as the soap molecule sodium stearate) have a hydrophilic head group and a long hydrophobic tail. They preferentially adsorb at the water–air interface, tail pointing outward, replacing high-energy water–water interactions at the surface with lower-energy water–surfactant interactions, reducing γ from ~72 mN/m to ~25–40 mN/m. This is why soapy water spreads better across surfaces and penetrates fabrics more effectively for cleaning — lower surface tension reduces capillary resistance and lowers the contact angle on hydrophobic fibres, improving wetting.

Paper chromatography uses capillary action through cellulose fibres of the paper to carry a solvent front (the mobile phase) up the stationary paper. Different compounds travel different distances because they partition between the stationary phase (water adsorbed on cellulose) and the mobile phase (solvent) according to their polarity. The retention factor Rf = distance of compound / distance of solvent front characterises each compound's relative affinity. The capillary rise rate follows Washburn's equation: x² = (rγ cosθ/2η)t, where η is the solvent viscosity — the distance the front travels grows as √t rather than linearly with time.

The Young–Laplace equation ΔP = γ(1/R₁ + 1/R₂) gives the pressure jump across a curved liquid interface, where R₁ and R₂ are the principal radii of curvature. For a hemispherical meniscus in a tube of radius r, ΔP = 2γ/r — a higher pressure on the concave (gas) side. This pressure difference drives the capillary rise: the liquid inside the tube is at lower pressure than the reservoir (because the concave meniscus creates suction), and atmospheric pressure on the reservoir surface pushes liquid up until the hydrostatic pressure ρgh balances the Laplace pressure, recovering Jurin's law h = 2γ cosθ/(ρgr).