Formula used
The pressure jump across a curved interface is described by the Young–Laplace relation: ΔP = γ(1/R1 + 1/R2). Here, γ is surface tension and R1, R2 are principal radii of curvature. Special cases follow directly:
- Spherical droplet: ΔP = 2γ/r (one interface).
- Soap bubble: ΔP = 4γ/r (two interfaces).
- Cylinder: ΔP = γ/r (one non‑zero curvature).
Sign depends on your curvature convention and which side is considered inside.
How to use this calculator
- Select the model that matches your interface shape.
- Enter surface tension and choose its unit.
- Provide the radius r, or radii R1 and R2 for the general case.
- Choose the curvature sign if you need a negative result.
- Click Calculate to view ΔP above the form.
- Use Download CSV or Download PDF for reports.
Example data table
| Case | γ (mN/m) | Radius (mm) | Model | ΔP (Pa) |
|---|---|---|---|---|
| Water droplet | 72 | 1 | ΔP = 2γ/r | 144 |
| Soap bubble | 30 | 0.5 | ΔP = 4γ/r | 240 |
| Cylindrical meniscus | 50 | 0.2 | ΔP = γ/r | 250 |
Values are illustrative; real systems vary with temperature and impurities.
Practical guide to surface-tension pressure
1) Why curvature creates pressure
Any curved liquid interface stores surface energy. To balance that energy, a pressure difference forms between the two sides of the interface. This pressure jump is central to capillarity, foams, emulsions, and droplet transport. Smaller radii mean larger curvature, so micro-scale features produce surprisingly large pressures.
2) Typical surface-tension values
Surface tension depends on fluid, temperature, and contamination. At room conditions, water is often near 72 mN/m, many light oils are roughly 25–35 mN/m, and alcohol–water mixtures can be much lower. Surfactants can reduce tension by factors of two or more, strongly lowering the predicted pressure jump.
3) Droplets versus bubbles
A spherical droplet has one interface, giving a factor of 2 in the pressure expression. A soap bubble has two interfaces, so the factor becomes 4. For the same radius and tension, bubbles therefore require about twice the pressure difference compared with droplets, which matters in foam stability and film drainage.
4) Order-of-magnitude examples
Consider γ = 72 mN/m and r = 1 mm. The droplet pressure jump is about 144 Pa. If r decreases to 100 µm, the jump rises to roughly 1440 Pa. At r = 10 µm, it reaches about 14.4 kPa, comparable to many laboratory pressure drops, explaining why tiny channels can pin or propel drops.
5) General radii for real interfaces
Real menisci are rarely perfect spheres. The general model uses two principal radii, R1 and R2, which may differ in magnitude and sign. A cylindrical surface is a special case where one principal radius is finite and the other is effectively infinite, leaving a single curvature contribution. This calculator supports these cases directly.
6) Sign conventions and interpretation
The sign of ΔP depends on which side you label as “inside” and whether curvature is convex or concave. For a convex droplet surface, internal pressure is typically higher than external. When curvature reverses, the computed pressure jump should switch sign. Use the sign selector to match your convention consistently.
7) Unit discipline and reporting
Surface tension is commonly reported in mN/m, while radii may be in mm or µm. Mixing units is the most frequent source of error. This tool converts all inputs to SI internally, then reports ΔP in Pa along with kPa, bar, atm, and mmHg for quick comparison with gauges and reference tables.
8) Where these results are used
Capillary pressure estimates inform droplet generation in microfluidics, wetting and wicking in porous media, inkjet and coating processes, bubble rise and breakup, and stability limits for emulsions and foams. In design work, pairing ΔP with flow resistance helps predict whether a meniscus will move, remain pinned, or fracture into satellites.
FAQs
1) What does ΔP represent here?
ΔP is the pressure difference across the interface caused by curvature and surface tension. It is not the hydrostatic head from gravity unless you add that separately.
2) Why is the bubble formula different from the droplet formula?
A bubble has two interfaces, inner and outer films. Each contributes the same curvature effect, so the total pressure jump doubles compared with a single-interface droplet.
3) Which radius should I use for a meniscus in a tube?
Use the principal radii of the meniscus. If you approximate a spherical cap, a single effective radius can be used. For more accuracy, estimate R1 and R2 from geometry or imaging.
4) What units are most reliable for lab work?
Use N/m for surface tension and meters for radii whenever possible. If you enter mN/m and mm or µm, double-check magnitudes and look at the Pa and kPa outputs for sanity checks.
5) Can ΔP be negative?
Yes. If your curvature convention defines the interface as concave relative to the “inside,” the computed pressure jump reverses sign. Choose the sign option that matches your definitions.
6) Does temperature matter?
Yes. Surface tension usually decreases with temperature. A few degrees can change γ by a noticeable amount in sensitive systems, so use a γ value measured at your operating temperature.
7) Are surfactants important for accuracy?
Very. Surfactants can lower γ dramatically and may vary with concentration and time. If a detergent or additive is present, use an appropriate γ value rather than pure-fluid tables.