Formula used
Capillary pressure is the pressure jump across a curved interface caused by surface tension. The general Laplace relation is: ΔP = γ(1/R1 + 1/R2), where γ is surface tension and R1, R2 are the principal radii of curvature.
- Spherical droplet / single interface: ΔP = 2γ/R
- Soap bubble (two interfaces): ΔP = 4γ/R
- Cylindrical capillary tube with contact angle: ΔP = 2γ cos(θ) / r
Units: γ in N/m and radii in meters produce ΔP in pascals (Pa).
How to use this calculator
- Select a geometry model that matches your interface shape.
- Enter surface tension γ and choose its unit.
- Provide the required radius values and units for that model.
- For capillary tubes, enter the contact angle θ.
- Press
Calculateto view pressure in multiple units. - Use the download buttons to export results as CSV or PDF.
Example data table
| Case | Model | γ (N/m) | Radius / Radii | θ | ΔP (Pa) |
|---|---|---|---|---|---|
| 1 | Cylindrical capillary | 0.072 | r = 0.5 mm | 0° | 288 |
| 2 | Spherical droplet | 0.072 | R = 1.0 mm | — | 144 |
| 3 | Soap bubble | 0.030 | R = 2.0 mm | — | 60 |
| 4 | General curvature | 0.050 | R1 = 2.0 mm, R2 = 4.0 mm | — | 37.5 |
Example values are illustrative and rounded.
Capillary pressure in real systems
Capillary pressure is the pressure difference created by a curved liquid–gas or liquid–liquid interface. In practice, it governs wetting in pores, droplet stability, and meniscus-driven transport. Because ΔP scales with γ and inversely with radius, small geometries generate surprisingly large pressures. This calculator helps you quantify that effect across common interface models.
1) Why Laplace pressure matters
Laplace pressure links geometry to mechanics: a tighter curvature produces a larger pressure jump. In porous media, this pressure controls which pores fill first and how fluids redistribute after drainage. In microfluidics, it sets the force needed to push droplets through channels and constrictions.
2) Typical surface tension values
Surface tension depends strongly on composition and temperature. Water–air is often near 0.072 N/m around room temperature, while many oils range from roughly 0.020–0.035 N/m. Surfactants can lower γ substantially, reducing ΔP and making interfaces easier to deform.
3) Radius scale drives the magnitude
The inverse-radius scaling is the main “lever.” With γ = 0.072 N/m and R = 1 mm, ΔP ≈ 144 Pa. If the radius shrinks to 10 μm, ΔP rises to about 14.4 kPa. At 100 nm, the same γ would imply pressures on the order of 1.44 MPa.
4) Droplet vs bubble models
A spherical droplet has a single interface, so ΔP = 2γ/R. A soap bubble has two interfaces (inner and outer films), doubling the pressure jump to ΔP = 4γ/R. This distinction is important when comparing foam films to ordinary droplets.
5) Contact angle effects in capillaries
For a cylindrical capillary, the wetting condition enters through cos(θ). When θ < 90°, cos(θ) is positive and the tube can pull liquid inward. When θ > 90°, cos(θ) is negative and the meniscus resists invasion, increasing the required driving pressure.
6) General curvature for complex menisci
Many interfaces are not perfectly spherical. The two-principal-radius form, ΔP = γ(1/R1 + 1/R2), captures saddle-like or elongated shapes. If one radius is very large, its curvature contribution becomes small, approaching a cylindrical limit.
7) Common application ranges
In laboratory capillaries (r ~ 0.1–1 mm), ΔP typically falls in the tens to hundreds of pascals. In porous rocks and membranes (pore radii ~ 0.1–10 μm), ΔP commonly reaches kilopascals to megapascals. These magnitudes influence filtration, ink penetration, and evaporation fronts.
8) Practical tips for accurate inputs
Use consistent geometry: radius should reflect the actual curvature of the meniscus, not merely a container size. For θ, confirm whether your measurement is through the liquid phase. If using surfactants, note that γ can change with concentration and time, especially under flow.
FAQs
1) What does a positive ΔP mean?
A positive ΔP indicates the pressure is higher on the concave side of the interface, given the chosen sign convention. The direction depends on how you define R1 and R2.
2) Which model should I choose?
Use the capillary tube model for wetting in cylindrical pores with a contact angle. Use the droplet model for a single spherical interface. Use the bubble model for soap films. Use general curvature for non-spherical menisci.
3) Why does a bubble use 4γ/R?
A soap bubble has two liquid–air interfaces, one on each side of the thin film. Each interface contributes 2γ/R, so the total pressure jump across the film becomes 4γ/R.
4) Can ΔP be negative?
Yes. In the capillary formula, cos(θ) becomes negative for θ > 90°, which flips the sign of ΔP. In the general formula, negative radii can represent opposite curvature directions.
5) How sensitive is ΔP to radius errors?
Very sensitive. Because ΔP scales as 1/R, a 10% uncertainty in radius typically leads to about a 10% uncertainty in pressure. Small radius measurements need careful microscopy or calibration.
6) What units should I use for γ?
Any supported unit is fine. The calculator converts mN/m and dyn/cm into N/m internally, then reports ΔP in multiple pressure units. Ensure your γ value matches the selected unit.
7) Does temperature affect the result?
Yes. Surface tension generally decreases with temperature for many liquids. If you need higher accuracy, use γ measured at your operating temperature and consider composition changes due to evaporation or surfactants.