Quick answer Capillary rise follows Jurin's law h = 2γ·cos θ/(ρ·g·r) (γ = surface tension, θ = contact angle, r = tube radius), so narrower tubes lift liquid higher. The Laplace pressure jump is ΔP = 2γ/r and the Bond number Bo = ρ·g·r²/γ. Mercury, at θ ≈ 140°, has negative cos θ and is depressed instead.
Calculate capillary rise, Laplace pressure and Bond number from surface tension, tube radius and contact angle in real time, and watch the droplet contact-angle shape update interactively on a Canvas diagram.
Parameters
Quick Settings
Viz
Hydrophilic (θ = 20°)
Capillary rise height
—
Laplace pressure difference
—
Bond number Bo
—
Pull-up force
—
1
0.0
Rise Height h [mm]
20°
Contact Angle θ
0
Laplace Pressure Δp [Pa]
0
Bond number Bo
Left: The Liquid Surface in the Capillary Rises to Its Equilibrium Height hRises/falls to =2γcosθ/(ρgr). Wetting liquids form a concave meniscus; non-wetting liquids (mercury) form a convex meniscus. Right: a droplet on a solid surface changes shape at contact angle θ (spreads at θ<90° and becomes spherical at θ>90°). A narrower tube produces a greater rise.
Professor, why does the water surface curve upward at the rim of a glass—what's that "meniscus"? It's fascinating how water rises on its own.
🎓
That's because of "adhesion" between water and glass. Water molecules are attracted enough to form hydrogen bonds with silanol groups (-SiOH) on the glass surface. This adhesion is stronger than the "cohesion" between water molecules, so water spreads along the wall. As a result, the contact angle becomes small (about 20° for water-glass), and the meniscus curves upward.
🙋
Then why does the mercury in a thermometer dip downward inside the tube?
🎓
For mercury, the metallic bonds between mercury atoms (cohesion) are much stronger than the adhesion to glass. So the contact angle becomes about 140°, and it "tries to pull away from the glass wall." The meniscus curves downward, and when you calculate with Jurin's equation $h = 2\gamma\cos(140°)/(\rho g r)$, cos is negative, so h is negative—meaning capillary depression. You can check it right away by selecting "Mercury" from the presets.
🙋
Is it true that the narrower the tube, the higher water rises? Can this explain how plants suck up water?
🎓
Yes, Jurin's equation shows h is inversely proportional to r. For water (γ=72.8 mN/m, θ=20°): r=1 mm gives h≈14 mm, r=0.1 mm gives h≈140 mm. But capillary action alone can't deliver water to the top of a 100 m tall tree. The main mechanism in plants is the "transpiration-tension mechanism," where water columns are pulled up by evaporation from leaves. Capillary action plays a supporting role.
🙋
What's the "lotus effect" on lotus leaves? How is it different from water-repellent coatings?
🎓
The surface of a lotus leaf has a dual structure: micrometer-sized wax bumps plus nanoscale fine structures on top. Droplets can only touch the tips of these bumps, so the actual contact area is only a few percent of the apparent area. As a result, the contact angle exceeds 160°, and droplets roll off at tilt angles of just 3–5°. Ordinary water-repellent coatings (fluorine-based) give contact angles of about 120–140°, which can't match the lotus effect. Modern Gore-Tex and car glass coatings artificially replicate this nano-dual structure.
🙋
Is a smaller Bond number always better? Where is it used industrially?
🎓
It's not about good or bad—it's an indicator of which physics dominates. Bo ≪ 1 (surface tension dominated) is critical for inkjet printer droplet control (10–100 μm), microfluidic devices, and photoresist coating on semiconductor wafers. Bo ≫ 1 (gravity dominated) matters for large tank liquid behavior and wave calculations. The capillary length $l_c = \sqrt{\gamma/(\rho g)}$ is the boundary—about 2.7 mm for water. Below this scale, surface tension cannot be ignored.
Frequently Asked Questions
What is surface tension?
Molecules at the liquid surface have fewer neighbors than those inside, so they carry extra energy. The liquid minimizes its surface area to reduce energy, which is observed as surface tension. It is expressed as force per unit length (N/m) or energy per unit area (J/m²). For water at 20°C: 72.8 mN/m; ethanol: 22.3 mN/m; mercury: 485 mN/m.
Why does capillary action occur?
When the adhesion between solid and liquid is greater than the liquid's cohesion (contact angle θ < 90°), the liquid tends to spread along the tube wall. This creates a Laplace pressure ΔP = 2γcosθ/r, which pushes the liquid upward. Eventually, it balances with gravity at height $h = 2\gamma\cos\theta/(\rho g r)$ (Jurin's equation).
Why does mercury fall in a capillary tube?
The contact angle between mercury and glass is about 140°, so cos(140°) ≈ −0.766 < 0, making the rise height in Jurin's equation negative (depression). This is called capillary depression. The downward-curving meniscus in a mercury thermometer is an example. Because mercury has low affinity for glass containers, special care is needed when handling liquid metals.
How does water-repellent coating (lotus effect) work?
Reducing surface energy with fluorine compounds gives contact angles of about 120–140°. Adding a micro+nanoscale dual structure (lotus effect) reduces the actual contact area to just a few percent, achieving contact angles over 160° and roll-off at tilt angles below 3°. Applications: anti-fouling glass, water-repellent textiles (Gore-Tex), car body coatings, and semiconductor photoresist coating control.
What is the Bond number?
The Bond number $Bo = \rho g r^2 / \gamma$ is a dimensionless number indicating the relative importance of gravity versus surface tension. Using the capillary length $l_c = \sqrt{\gamma/(\rho g)}$ (water: about 2.7 mm), it can also be expressed as $Bo = (r/l_c)^2$. Bo ≪ 1 means surface tension dominates (spherical droplets, microfluidics); Bo ≫ 1 means gravity dominates (large tanks, waves). Inkjet printers (10–100 μm droplets) have Bo ≈ 10⁻³ to 10⁻⁵, strongly surface tension dominated.
Select a fluid from the dropdown (water, ethanol, mercury, or liquid metal) or enter surface tension (N/m) in the vGamma field.
Input the capillary tube radius in millimeters using vRho; set contact angle θ in degrees via vTheta (0° = perfectly wetting, 180° = non-wetting).
Click Calculate to compute capillary rise height (h = 2γcosθ/ρgr), Laplace pressure (ΔP = 2γcosθ/r), and Bond number (Bo = ρgr²/γ) to assess gravity versus surface tension dominance.
Worked Example
Water at 20°C in a 0.5 mm radius glass capillary: γ = 0.073 N/m, ρ = 998 kg/m³, θ = 0° (cosθ = 1). Capillary rise h = (2 × 0.073 × 1)/(998 × 9.81 × 0.0005) = 29.8 mm. Laplace pressure ΔP = 2 × 0.073/0.0005 = 292 Pa. Bond number Bo = (998 × 9.81 × 0.00000025)/0.073 = 0.034, confirming surface tension dominates. Mercury in the same tube with θ = 140° (cosθ = −0.766): capillary depression = −22.3 mm; ΔP = −306 Pa (pressure jump into droplet).
Practical Notes
Contact angle strongly affects wetting: water on clean glass ≈ 0°, silicone oil on hydrophobic surfaces ≈ 110°. Measure on your actual substrate; manufacturers publish advancing/receding hysteresis values.
Bond number < 1 means capillarity wins (micropatterned surfaces, MEMS); Bo > 1 favors gravity (large pools, industrial reactors). Transition occurs around r = 2.7 mm for water.
Laplace pressure drives spontaneous curvature in emulsions and foam. Glycerol (γ = 0.064 N/m) vs water shows minimal difference; molten solder (γ ≈ 0.87 N/m) creates much higher interfacial stress in electronics assembly.
Temperature reduces surface tension ~0.15% per °C for most liquids; water drops from 0.0728 N/m at 20°C to 0.0652 N/m at 60°C, reducing capillary rise by 10%.