Capillary Action

Definition

Capillary action (also called capillarity or wicking) is the movement of a liquid through a narrow channel — a thin tube, a crack, a fibre network, or a porous solid — under the action of interfacial forces alone. There is no external pump. The liquid moves because raising or lowering the interface reduces the total surface energy of the system, and it stops when that energetic gain is balanced by the weight of the raised (or lowered) column.

The whole effect rests on two molecular attractions:

• Cohesion — the attraction of the liquid's molecules for one another. Cohesion is what produces surface tension γ, an energy per unit area (J/m²) equivalently expressed as a force per unit length (N/m).
• Adhesion — the attraction between the liquid's molecules and the atoms of the solid wall.

The outcome is summarised by the contact angle θ, the angle the liquid surface makes with the solid measured through the liquid. Adhesion winning gives θ < 90° (the liquid wets the wall and climbs); cohesion winning gives θ > 90° (the liquid is non-wetting and is pushed down).

Jurin's law — the governing equation

For a liquid in a narrow, vertical, circular tube of internal radius r, the equilibrium rise is:

h = 2·γ·cos(θ) / (ρ·g·r)

Every symbol, with SI units:

Symbol	Meaning	Units
h	Equilibrium height the liquid rises (or, if negative, falls) relative to the outside level	m
γ (gamma)	Surface tension of the liquid–air interface	N/m (= J/m²)
θ (theta)	Contact angle between liquid and wall, measured through the liquid	rad or °
ρ (rho)	Density of the liquid	kg/m³
g	Gravitational acceleration (9.81 m/s² at Earth's surface)	m/s²
r	Internal radius of the tube (for two parallel plates a gap d apart, replace r with d)	m

The physical derivation is a force balance. The wetted perimeter of the meniscus is 2πr, and surface tension pulls upward along it at angle θ, giving a total upward force of 2πr·γ·cos(θ). This supports a liquid column of weight ρ·g·(πr²·h). Equating the two:

2πr·γ·cos(θ) = ρ·g·π·r²·h   ⟹   h = 2γcos(θ)/(ρgr)

Equivalently, in the Young–Laplace picture, the curved meniscus (radius of curvature R = r/cos θ for a hemispherical cap) creates a pressure jump ΔP = 2γcos(θ)/r across the interface. That reduced pressure just under the meniscus draws liquid up until the hydrostatic head ρgh restores balance — the same result by a different route.

Why narrower tubes lift higher

The dominant feature of Jurin's law is h ∝ 1/r. Height and radius are inversely proportional, so a tube ten times narrower lifts the liquid ten times higher. This is the counter-intuitive heart of capillarity: the tiniest channels are the strongest pumps.

Plugging pure water on clean glass into Jurin's law (γ = 0.0728 N/m, θ ≈ 0° so cos θ = 1, ρ = 1000 kg/m³, g = 9.81 m/s²) gives a compact rule:

h ≈ 1.48 × 10⁻⁵ / r   (metres, with r in metres)

Tube radius r	Water rise h	Where you meet it
1 mm (1×10⁻³ m)	≈ 15 mm	A thin drinking straw
0.1 mm (1×10⁻⁴ m)	≈ 15 cm	Fine glass capillary
10 µm (1×10⁻⁵ m)	≈ 1.5 m	Paper towel / cloth pores
1 µm (1×10⁻⁶ m)	≈ 15 m	Fine soil pores, wood cell walls
0.1 µm (1×10⁻⁷ m)	≈ 150 m	Clay pores (idealised)

These are ceiling values — real porous media rarely reach the ideal height because pores are tortuous, not perfectly cylindrical, and evaporation, air pockets, and pore-size variation intervene. Still, the 1/r law is why fine-grained materials wick water so much further than coarse ones.

The capillary length: where surface tension wins

Capillarity only matters when the tube or pore is small enough that surface tension can overpower gravity. The natural crossover scale is the capillary length:

λ_c = √(γ / (ρ·g))

For water at 20°C, λ_c = √(0.0728 / (1000 × 9.81)) ≈ 2.7 mm. Below this scale, surface tension dominates: drops stay spherical, menisci curve sharply, and capillary rise is large. Above it, gravity flattens interfaces and puddles spread. This is why a raindrop a few millimetres across is nearly spherical while a spilled cup of water lies flat. In Jurin's law, the rise h and radius r combine so that h·r = 2λ_c²cos θ (≈ 2λ_c² for a wetting liquid) — the product of height and radius is fixed by the capillary length.

How it works, step by step

1. Contact. Dip a clean glass tube into water. Water molecules near the wall are attracted to the silica surface (hydrogen bonding to surface –OH groups). Adhesion outcompetes cohesion, so the water spreads up the wall — the contact angle is near 0°.
2. Meniscus forms. The liquid surface curves concave-up, hugging the wall higher than the centre. Surface tension, acting tangent to this curved film, has a net upward vertical component along the wetted perimeter.
3. Pressure drops. By the Young–Laplace relation, the concave meniscus lowers the pressure just beneath it by ΔP = 2γcos(θ)/r. The higher reservoir pressure below now pushes liquid up the tube.
4. Column rises. Liquid climbs until the extra weight ρgh of the raised column exactly cancels the pressure deficit. That equilibrium is Jurin's height.
5. Balance. At steady state the upward pull of surface tension around the rim equals the downward weight of the suspended column. Nothing flows; the meniscus sits fixed.

For a non-wetting liquid such as mercury, step 1 reverses: cohesion beats adhesion, θ ≈ 140°, the meniscus bulges convex, cos θ is negative, and h comes out negative — the liquid is depressed below the reservoir.

Contact angles and capillary behaviour

Liquid / surface	Contact angle θ	Behaviour
Pure water on clean glass	≈ 0°	Strong rise (fully wetting)
Water on typical soda-lime glass	≈ 20–30°	Rise, slightly reduced
Water on cellulose (paper, cotton)	≈ 0–30°	Wicks readily
Water on paraffin wax / PTFE	≈ 108–110°	Beads up; slight depression
Water on a superhydrophobic leaf	> 150°	Rolls off (lotus effect)
Mercury on glass	≈ 140°	Depression (non-wetting)
Ethanol on glass	≈ 0°	Rise (low γ ≈ 0.022 N/m)

Worked example: how high does sap climb?

Consider water rising in a single xylem conduit of radius r = 20 µm = 2×10⁻⁵ m, treating the wall as fully wetting (θ ≈ 0°):

h = 2 × 0.0728 × cos(0°) / (1000 × 9.81 × 2×10⁻⁵)
  = 0.1456 / 0.1962
  ≈ 0.74 m

So pure capillary rise in a 20 µm vessel lifts water less than a metre — nowhere near the 100 m height of a redwood. This is the key lesson: capillary action alone does not lift sap to the treetops. Trees mainly use the cohesion–tension mechanism, where evaporation from leaf pores (transpiration) puts the water column under tension and cohesion holds it together, pulling the whole thread up. Capillarity sets the meniscus geometry in the fine cell-wall pores and helps refill embolised vessels, but the heavy lifting is done by transpirational tension, with pressures reaching roughly −1.5 to −2 MPa (about −15 to −20 atm) at the top of a tall tree.

JavaScript — Jurin's law calculations

const g = 9.81;            // m/s^2
const GAMMA_WATER = 0.0728; // N/m at 20 C
const RHO_WATER = 1000;    // kg/m^3

// Jurin's law: equilibrium capillary rise (m). theta in radians.
function jurinRise(gamma, theta, rho, r) {
  return (2 * gamma * Math.cos(theta)) / (rho * g * r);
}

// Water on clean glass (theta = 0) in a 0.5 mm radius tube
const r1 = 0.5e-3;
console.log(`r = 0.5 mm: h = ${(jurinRise(GAMMA_WATER, 0, RHO_WATER, r1) * 1000).toFixed(1)} mm`); // ~29.7 mm

// The 1/r scaling: shrink radius 10x, rise grows 10x
const r2 = 0.05e-3;
console.log(`r = 0.05 mm: h = ${(jurinRise(GAMMA_WATER, 0, RHO_WATER, r2) * 1000).toFixed(0)} mm`); // ~297 mm

// Mercury: non-wetting, theta ~ 140 deg, denser
const GAMMA_HG = 0.485, RHO_HG = 13534, THETA_HG = 140 * Math.PI / 180;
const hHg = jurinRise(GAMMA_HG, THETA_HG, RHO_HG, r1);
console.log(`Mercury r = 0.5 mm: h = ${(hHg * 1000).toFixed(2)} mm (negative = depression)`); // ~ -11.2 mm

// Capillary length: scale where surface tension balances gravity
function capillaryLength(gamma, rho) {
  return Math.sqrt(gamma / (rho * g));
}
console.log(`Water capillary length: ${(capillaryLength(GAMMA_WATER, RHO_WATER) * 1000).toFixed(2)} mm`); // ~2.72 mm

// Young-Laplace pressure jump across a hemispherical meniscus (Pa)
function laplacePressure(gamma, theta, r) {
  return (2 * gamma * Math.cos(theta)) / r;
}
console.log(`Suction under meniscus (r = 10 um): ${laplacePressure(GAMMA_WATER, 0, 10e-6).toFixed(0)} Pa`); // ~14560 Pa

Where capillary action shows up

• Plants. Fine cell-wall pores set the meniscus geometry that lets transpirational tension pull sap up xylem; capillarity keeps the water threads continuous.
• Soil and hydrology. The capillary fringe above the water table is water held in pore space against gravity; it controls how deep roots can drink and how quickly ground dries.
• Paper, fabric, and wicks. Paper towels, sponges, candle wicks, and fountain-pen nibs all move liquid by wicking through fibre pores.
• Chromatography and lab assays. Thin-layer chromatography and lateral-flow tests (like pregnancy and rapid antigen strips) rely on capillary flow to carry samples across a matrix.
• Microfluidics. Lab-on-a-chip devices use engineered channels to pump nanolitre volumes with no moving parts.
• Building materials. Rising damp in masonry is capillary transport of groundwater up through porous brick and mortar.
• Spacecraft. In microgravity, propellant management devices use capillarity to feed liquid fuel to engine inlets where gravity cannot settle it.

Common misconceptions

• "Capillary action lifts water to the top of tall trees." No — a 20 µm vessel gives under 1 m of pure capillary rise. Transpiration-driven cohesion–tension does the lifting; capillarity handles the fine-pore menisci.
• "Surface tension pushes the water up." Surface tension acts along the interface; it is the resulting reduced pressure under the curved meniscus (Young–Laplace) that lets atmospheric pressure below drive the liquid up. Framing it as a direct upward "push" from the water surface is loose.
• "Wider tubes lift water higher because they hold more." The opposite: h ∝ 1/r, so narrower tubes lift higher. Wider tubes hold more volume but reach a much lower equilibrium height.
• "All liquids rise in capillaries." Only wetting liquids (θ < 90°) rise. Non-wetting ones (θ > 90°, like mercury on glass) are depressed — cos θ is negative and h is negative.
• "Contact angle is a property of the liquid alone." It depends on the liquid, the solid, and the surrounding gas — plus surface cleanliness and roughness. Greasing clean glass can turn a 0° water contact angle into a non-wetting one.
• "Jurin's law works for any tube size." It assumes a narrow tube (r ≲ λ_c) so the meniscus is nearly spherical and hydrostatic. In wide tubes the meniscus flattens, gravity distorts it, and the simple 2γcosθ/(ρgr) form breaks down.