Young-Laplace Simulator — Surface Tension and Capillary Action

Compute the Laplace pressure dP = 2 gamma / R for a spherical droplet, dP = gamma / R for a cylindrical column, dP = 4 gamma / R for a soap bubble, the Jurin capillary rise h and the Bond number Bo in real time from surface tension, curvature radius, contact angle and density. A droplet, capillary and soap-bubble schematic plus an R-versus-dP chart turn interface physics into something you can see.

Parameters

Surface tension gamma

mN/m

Curvature radius R

Contact angle theta

deg

Liquid density rho

kg/m^3

Defaults are water at 20 C (gamma = 72.8 mN/m rounded to 73, rho = 1000 kg/m^3, theta = 0). Gravity g = 9.81 m/s^2. R is in mm, gamma in mN/m (= 1e-3 N/m).

Results

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Sphere Laplace 2γ/R

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Cylinder Laplace γ/R

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Capillary rise h

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Bond number Bo

Droplet, capillary and soap bubble (schematic)

Left = spherical droplet (radius R, inner P_in vs outer P_out, dP = 2γ/R) / center = capillary tube (contact angle theta, rise h) / right = soap bubble (two-sided film, dP = 4γ/R)

Laplace pressure dP versus radius R (log-log)

X = R (mm, 0.005 to 10, log) / Y = dP (Pa, log) / blue = sphere 2γ/R / green = cylinder γ/R / red = soap bubble 4γ/R / yellow dot = current R

Theory & Key Formulas

The curvature of a gas-liquid interface produces a pressure jump described by the Young-Laplace equation.

Spherical droplet or bubble (radius R):

$$\Delta P = \frac{2\gamma}{R}$$

Cylindrical column (radius R, axisymmetric):

$$\Delta P = \frac{\gamma}{R}$$

Soap bubble (two interfaces, inner and outer):

$$\Delta P = \frac{4\gamma}{R}$$

Jurin's capillary rise (radius r, contact angle theta):

$$h = \frac{2\gamma\cos\theta}{\rho g r}$$

Bond number (gravity vs surface tension ratio):

$$\mathrm{Bo} = \frac{\rho g R^{2}}{\gamma}$$

Here $\gamma$ is the surface tension [N/m], $R$ the curvature radius [m], $\theta$ the contact angle, $\rho$ the liquid density [kg/m^3] and $g = 9.81$ m/s^2 the gravitational acceleration. Bo < 1 is surface-tension dominated, Bo > 1 is gravity dominated.

What the Young-Laplace Simulator does

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Smaller water droplets are nearly spherical because of surface tension, right? But how do you actually compute the inside vs outside pressure?

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That is exactly what the Young-Laplace equation gives: dP = 2 gamma / R. Surface tension acts like a stretched rubber skin, so the pressure inside a sphere of radius R exceeds the outside pressure by 2 gamma / R. Try gamma = 73 mN/m (water) and R = 0.5 mm in this tool: you should read 292 Pa. For a 0.05 mm fog droplet the value jumps to about 2920 Pa, and that is the doorway to the Kelvin effect that makes tiny droplets evaporate slower than large ones.

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The capillary picture in the middle says h = 29.8 mm. Does that mean water in a thin glass tube really climbs about 30 mm?

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It does. Jurin's law h = 2 gamma cos(theta) / (rho g r) gives about 30 mm for water (theta about 0) in a 0.5 mm glass tube. Make the tube ten times narrower (R = 0.05 mm) and h jumps to about 150 mm. That is the same physics that lets gauze and sponges suck up water; sap rising in trees uses the same capillary force in xylem vessels combined with transpiration.

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Why is the soap bubble formula 4 gamma / R, twice the droplet value?

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A soap film is bounded by two gas-liquid interfaces, an inner one and an outer one. Each one contributes 2 gamma / R, so the total inside-pressure jump is 4 gamma / R. For a 5 mm bubble with gamma about 25 mN/m that is only 20 Pa. The famous coupled-bubble experiment (two bubbles linked by a straw) is unstable for exactly this reason: the smaller bubble has higher inside pressure and pushes its air into the larger one until it deflates.

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The panel also shows Bond number 0.034. What does that mean?

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Bo = rho g R^2 / gamma is the ratio of gravity to surface tension. For water at R = 0.5 mm Bo = 0.034, deep in the surface-tension regime, so raindrops stay nearly spherical. Push R to 5 mm and Bo = 3.4: gravity dominates and raindrops flatten into a hamburger shape. The cross-over is around Bo = 1, which corresponds for water to the capillary length R about 2.7 mm. In microgravity (think the ISS) g goes to zero and Bo too, which is why a glass of water floats as a wobbly sphere.

Frequently Asked Questions

A spherical droplet has a single liquid-air interface, but a soap bubble has two: the inner and outer faces of the thin film. Each face contributes 2 gamma / R, so the net inside pressure jump is 4 gamma / R. For a 5 mm bubble with gamma about 25 mN/m the excess pressure is only about 20 Pa, which is why a strong puff bursts it instantly. Set R = 5 mm in this tool and the sphere, cylinder and soap-bubble curves line up cleanly in the 2 : 1 : 4 ratio.

Jurin's law h = 2 gamma cos(theta) / (rho g r) shows that the rise scales as 1/r. The surface tension force scales as the perimeter 2 pi r, while the column weight scales as pi r^2 h rho g, so the equilibrium gives h proportional to 1/r. Water in a 0.1 mm diameter glass capillary climbs nearly 30 cm, and the same physics is part of how plant xylem vessels lift water to the canopy. Drop R from 0.5 mm to 0.1 mm in this tool and h increases roughly five-fold.

Bo = rho g R^2 / gamma is the ratio of gravity to surface tension. Bo much smaller than 1 means surface tension dominates (small droplets stay nearly spherical), Bo much greater than 1 means gravity dominates (large puddles flatten out). The cross-over is around Bo about 1, which for water corresponds to R about 2.7 mm (the capillary length). Raindrops larger than 5 mm flatten because they cross into the gravity-dominated regime. Sweep R from 0.5 mm to 5 mm in this tool and Bo grows from 0.034 to 3.4 across that transition.

When theta exceeds 90 degrees, cos(theta) becomes negative and the capillary rise h becomes negative, meaning the meniscus is depressed below the outer level. Mercury (theta about 140 degrees) in a glass capillary depresses by about 77% of what water rises, because mercury's cohesive forces beat its adhesion to glass. This is the same physics that gives the convex meniscus on a mercury thermometer. Set theta = 130 degrees in this tool and the displayed h becomes negative, mimicking what happens on hydrophobic-coated surfaces.

Real-World Applications

Inkjet printer drop ejection design: Consumer inkjet nozzles are 20-50 microns in diameter (R = 10-25 microns). Set R = 0.025 mm and gamma = 30 mN/m (typical ink) in this tool and you get dP = 2400 Pa, the lowest pressure that needs to be exceeded to push a drop out. Real thermal inkjets generate over 100 kPa of bubble pressure, but Young-Laplace sets the floor. A 5% manufacturing variation in nozzle radius translates directly into a 5% pressure variation, which in turn affects print quality.

Plant water transport and soil physics: The fact that water rises to the canopy of a 100 m sequoia is a combination of root and xylem capillarity (radius 5-100 microns) and a strong negative pressure from leaf transpiration (over 1 MPa). With R = 0.005 mm and gamma = 72.8 mN/m this tool gives h = 2.97 m, meaning capillarity alone lifts water about 3 m. Soil water retention is governed by the same Laplace pressure across menisci between soil grains, which makes Bo a key indicator for irrigation design.

Microfluidics and MEMS devices: In a 100 micron wide microchannel the capillary pressure (about 1500 Pa) is comparable to the pump pressure, so it cannot be neglected. Lab-on-a-chip devices intentionally use hydrophobic coatings (theta > 90 degrees) to make capillary valves that block flow until a threshold pressure is exceeded. Try theta = 130 degrees and R = 0.05 mm here: h becomes about -14.8 mm, illustrating why a hydrophobic patch can stop the flow.

Bubbles and sprays: CO2 bubbles forming in carbonated drinks or fermenters experience an enormous internal pressure right after nucleation (R about 1 micron gives dP about 146 kPa), which sets the boundary between dissolution and growth. As the bubble grows the pressure drops sharply, falling to about 146 Pa for visible 1 mm bubbles. The same physics enters spray atomization size distributions and the analysis of departure-from-nucleate-boiling in reactor cooling systems.

Common Misconceptions and Pitfalls

The most common misconception is that "the internal pressure of a droplet is independent of its radius". Young-Laplace says the opposite: dP = 2 gamma / R, so the pressure scales as 1/R. A 1 mm water droplet has dP about 146 Pa, but a 1 micron fog droplet has dP about 146 kPa, a thousand-fold increase. This is the origin of the Kelvin effect: small droplets have a higher saturation vapor pressure, so in a supersaturated cloud the bigger droplets grow at the expense of the smaller ones (Ostwald ripening). Sweep R from 0.005 to 10 mm on the log-log plot in this tool and you see the slope -1 line clearly.

Next is the belief that "capillary rise is a perpetual energy source defying gravity". The Jurin rise releases surface energy and stops at the equilibrium where total system energy is minimized. To recover the column you have to pay back its potential energy, so there is no perpetual motion. The Bond number plot makes the limit clear: once Bo > 1, gravity dominates and capillary rise saturates, set by the radius and the capillary length.

Finally, people often think "the 4 gamma / R formula is unique to soap films". In fact it applies to any thin liquid film with two gas-liquid interfaces: beer foam, foamed concrete, the surfactant lining in pulmonary alveoli, and so on. Conversely, both an air droplet in liquid and a gas bubble in liquid have a single interface and obey the 2 gamma / R law equally. The three curves on the chart in this tool show that the prefactor (1, 2, 4) is just a count of how many interfaces are present.

Young-Laplace Simulator — Surface Tension and Capillary Action

What the Young-Laplace Simulator does

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

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