The Washburn Equation in Mercury Porosimetry

The formula

• d — diameter of the cylindrical pore (pore entrance) being filled
• γ — surface tension of mercury, ≈ 0.485 N/m (485 mN/m) at room temperature
• θ — contact angle between mercury and the solid, usually assumed 130° (some labs use 140°)
• P — applied pressure (absolute pressure on the mercury)

Mercury does not wet most solids (θ > 90°, so cos θ < 0). The minus sign makes the diameter positive: pressure must be applied to push mercury into a pore, and the narrower the pore, the higher the pressure. The relation is a force balance between the applied pressure acting on the pore cross-section (π d²/4 · P) and the surface-tension force around the pore perimeter (−π d γ cos θ).

Edward W. Washburn proposed the method in 1921 (Proc. Natl. Acad. Sci. USA 7, 115). Ritter and Drake built the first practical mercury porosimeter in 1945. Today the procedure is standardised in ISO 15901-1 and ASTM D4284.

Handy unit forms

With the conventional γ = 485 mN/m and θ = 130° (cos θ = −0.6428), the constant 4γ|cos θ| = 1.247 N/m and the equation reduces to:

With θ = 140° (cos θ = −0.7660) the constants become 1.486 / P[MPa] and 215.5 / P[psia].

Pressure to pore diameter table

Values for γ = 485 mN/m and θ = 130°. Pore-size classes follow IUPAC: micropores < 2 nm, mesopores 2–50 nm, macropores > 50 nm.

The table shows why mercury porosimeters have two stages. A low-pressure port (vacuum up to roughly 30–50 psia) covers pores from several hundred micrometres down to a few micrometres. A high-pressure chamber (up to 33,000 or 60,000 psia depending on model) covers the rest down to about 3 nm.

Worked examples

1. Pressure to pore size (SI units)

Mercury enters a sample at P = 50 MPa. With γ = 0.485 N/m and θ = 130°:

2. Pore size to pressure

What pressure is needed to fill 10 nm pores? Rearranging, P = −4γ cos θ / d = 1.247 / (10 × 10⁻⁹) Pa = 1.25 × 10⁸ Pa, or about 125 MPa (≈ 18,100 psia).

3. Using psia directly

An intrusion step at 2,500 psia corresponds to d = 180.9 / 2500 = 0.072 μm = 72 nm.

You can check any value with the Washburn calculator.

Sensitivity to the contact angle

The diameter scales with |cos θ|, so the assumed angle matters more than any other input:

The true contact angle depends on the solid (oxides, carbons and polymers differ) and on whether mercury is advancing or receding. Most reports use 130° or 140° by convention; what matters is that the value is stated, so that results from different laboratories can be compared.

Assumptions and their consequences

• Cylindrical pores. For parallel-sided slits of width w, the balance becomes w = −2γ cos θ / P, i.e. half the cylinder-equivalent diameter at the same pressure.
• Pore entrances, not bodies. Mercury fills a cavity only when it can pass the narrowest throat leading to it, so the measured size is the entrance size. This is the ink-bottle effect.
• Rigid sample. At high pressure some materials compress or collapse; the apparent intrusion is then deformation, not pore filling.
• Constant γ and θ. Both are treated as fixed over the whole pressure range, which is an approximation, especially in pores of a few nanometres.

Converting mercury capillary pressure to a reservoir fluid system

In petrophysics, mercury injection capillary pressure (MICP) curves are rescaled to another fluid pair by the ratio of γ cos θ:

For example, with air–mercury values of 485 mN/m and 140° (γ|cos θ| ≈ 372 mN/m) and an air–brine system of 72 mN/m and 0°, the factor is 72 / 372 ≈ 0.19. The input values are conventions that vary between laboratories, so they should be quoted with the converted curve.

Not to be confused with: the Washburn capillary-flow equation

Washburn also published the equation for spontaneous imbibition of a wetting liquid into a capillary (the Lucas–Washburn equation):

Here L is penetration length after time t, r the capillary radius and η the liquid viscosity. It is used for wicking of paper, textiles and powders, not for pressure-driven mercury intrusion.

Related guides

• Mercury intrusion porosimetry — full method guide
• Reading a mercury porosimetry report
• The ink-bottle effect
• Bubble point and mean flow pore size (the same Young–Laplace physics with a wetting liquid)

Frequently asked questions

What is the Washburn equation in mercury porosimetry?

It is d = −4γcosθ/P, which gives the diameter d of the cylindrical pore that mercury enters at pressure P. γ is the surface tension of mercury (about 0.485 N/m) and θ its contact angle with the solid (commonly 130° to 140°). Because θ is above 90°, cosθ is negative and d comes out positive.

What pore size does 60,000 psia correspond to?

With γ = 485 mN/m and θ = 130°, d(μm) = 180.9 / P(psia), so 60,000 psia (414 MPa) corresponds to about 3.0 nm. That is the practical lower limit of commercial mercury porosimeters.

Why do different labs report different pore sizes for the same sample?

The calculated diameter is proportional to γ·|cosθ|. Changing the assumed contact angle from 130° to 140° makes every diameter about 19 % larger. Always report the γ and θ used.

Is the Washburn equation in porosimetry the same as the Washburn capillary-rise equation?

No. Washburn published both in 1921. The porosimetry relation is a pressure balance, d = −4γcosθ/P. The capillary-flow (Lucas–Washburn) equation, L² = γ r t cosθ / (2η), describes how far a wetting liquid penetrates a capillary in time t.