Find the mechanism before the formula
A vapour can condense inside a wetting pore before the surrounding bulk reaches saturation. Curved liquid–vapour interfaces shift chemical equilibrium, producing the Kelvin equation. The same compact formula powers pore-size analysis—and fails when adsorbed films, metastability, molecular scale or critical fluctuations become comparable with the pore.
We will connect curvature to chemical potential, calculate relative humidity for two pore radii, transfer the result across contact angle and adsorbed-film corrections, and separate equilibrium capillary condensation from hysteresis and near-critical finite-size crossover.
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Choose a reading route
Start with the question closest to the decision you need to make. Every route returns to the same worked evidence and transfer checks.
| Route | Chapters | Use it when |
|---|---|---|
| 1 | 1–3 | You need curvature, wetting and the Kelvin equation. |
| 2 | 4–5 | You want worked ten- and five-nanometre calculations. |
| 3 | 6–8 | You need contact-angle, film and hysteresis transfer checks. |
| 4 | 9–10 | You must handle critical crossover and model limits. |
| 5 | 11–12 | You want FAQs and a reproducible capillary checklist. |
Chapter contents
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Build the model
Test the mechanism
1. Curvature shifts pressure across an interface
The Young–Laplace relation connects the pressure jump across a liquid–vapour interface to surface tension γ and mean curvature. A concave meniscus in a wetting pore permits liquid equilibrium with vapour below the flat-interface saturation pressure.
The sign depends on curvature and convention. State whether radius refers to the meniscus, the physical pore or an effective pore after subtracting an adsorbed film. These radii are not automatically equal.
- Define pore geometry.
- Report temperature and surface tension.
- State liquid molar volume and contact angle.
- Separate adsorption film thickness from core radius.
2. The Kelvin equation balances chemical potentials
For an ideal vapour and an incompressible liquid in a cylindrical meniscus model, ln(p/psat)=−2γVmcosθ/(rRT). Here p/psat is relative vapour pressure, Vm liquid molar volume, r an effective curvature radius and θ the contact angle.
The equation predicts earlier condensation for smaller wetting pores because the exponent becomes more negative. Its geometry factor changes for slit pores and other curvatures, so the formula must follow the meniscus model rather than the shape of a convenient plot.
3. Condensation pressure is not a direct ruler without a model
An adsorption isotherm records uptake versus relative pressure. A step or loop can reflect capillary condensation, but converting it to pore radius requires surface tension, molar volume, contact angle, film thickness, pore geometry and an equilibrium branch.
The Kelvin equation is macroscopic. At a few molecular diameters, layering, density oscillations and solid–fluid potentials can dominate. A precise numerical radius from an inappropriate continuum model is not accurate evidence.
| Input | Physical role | Frequent shortcut |
|---|---|---|
| γ | cost of liquid–vapour area | bulk value used near criticality |
| Vm | liquid chemical-potential scale | treated constant over wide T |
| θ | wall wetting preference | assumed zero |
| r | meniscus curvature radius | equated to physical pore radius |
| p/psat | vapour undersaturation | taken from a metastable branch |
4. Worked example: water in a ten-nanometre pore
Use water at 298 K with γ=0.072 N/m, Vm=18×10−6 m³/mol, θ=0°, r=10 nm, R=8.314 J mol⁻¹ K⁻¹. The exponent is −2(0.072)(18×10⁻⁶)/[(10×10⁻⁹)(8.314)(298)]=−0.1046.
Therefore p/psat=exp(−0.1046)=0.901. In this idealised model, condensation occurs near 90 percent relative humidity, before bulk saturation. The result is a curvature-equilibrium estimate, not a prediction of the full adsorption loop.
5. Transfer check: halve the effective radius
With r=5 nm and all other inputs unchanged, the exponent doubles to −0.2092. The relative pressure becomes exp(−0.2092)=0.811. Smaller curvature radius produces condensation at lower vapour pressure.
The shift is nonlinear in relative pressure but linear in ln(p/psat) versus 1/r. This is why Kelvin plots can look simple while uncertainty in a one-nanometre film thickness produces a large radius error in small pores.
6. Contact angle changes the curvature driving term
If θ=60°, cosθ=0.5. For the ten-nanometre example the exponent becomes −0.0523 and p/psat=0.949. Weaker wetting moves condensation closer to saturation.
A contact angle measured on a flat macroscopic surface may not transfer into a chemically heterogeneous nanopore. Treat θ as a model parameter with uncertainty, and test whether adsorption and desorption branches imply physically consistent values.
7. Adsorbed films reduce the open core before condensation
A wetting film forms on the wall before the pore core condenses. If the physical radius is 10 nm and the film is 1 nm thick, an approximate meniscus radius is 9 nm. The Kelvin exponent increases in magnitude by 10/9, shifting the predicted pressure.
Methods such as Barrett–Joyner–Halenda combine a thickness model with Kelvin inversion. Their output depends on both pieces. Report the film model and do not label a derived distribution as geometry measured independently of adsorption physics.
8. Hysteresis records metastability and network effects
Adsorption and desorption can follow different paths because nucleation barriers, pore blocking, ink-bottle geometry and cavitation delay equilibration. A loop is not proof that one branch is the unique equilibrium Kelvin transition.
Use independent structural evidence and scan-rate tests. In connected porous media, an observed emptying pressure may be controlled by a narrow neck or collective network event rather than the body radius that stores most fluid.
9. Near criticality, surface tension and density contrast vanish
As a fluid approaches its critical point, γ and the liquid–vapour density difference decrease while the correlation length ξ grows. The classical Kelvin balance changes, and when ξ becomes comparable with pore width L, finite-size scaling and critical adsorption replace a sharp bulk-interface picture.
A capillary critical point can terminate the first-order condensation line. Pseudocritical shifts and rounding depend on L, surface fields and universality class. Extrapolating room-temperature Kelvin parameters into this regime misses the crossover mechanism.
10. Limits and misconceptions
- The Kelvin radius is a meniscus-curvature radius, not automatically the physical pore radius.
- Contact angle and adsorbed-film thickness can dominate small-pore inference.
- Adsorption–desorption hysteresis is not a single equilibrium curve.
- Continuum capillarity weakens at molecular dimensions.
- Bulk γ and density values fail close to criticality.
- Pore networks can make a local radius inversion nonunique.
The equation remains valuable when its scale and geometry match the experiment. Its limits should narrow the claim, not be hidden behind extra decimal places.
11. Frequently asked questions
Why does condensation happen below saturation?
A concave meniscus lowers the liquid chemical potential relative to a flat interface. Curvature allows equilibrium with an undersaturated vapour.
Which branch should be used for pore size?
There is no universal answer. It depends on geometry, metastability and the chosen analysis model. Report the branch and validate against independent structure.
Does the Kelvin equation apply to slit pores?
The chemical-potential logic applies, but the curvature factor and effective width differ. Use the equation derived for the actual meniscus geometry.
12. A reproducible capillary-condensation checklist
| Stage | Evidence to retain |
|---|---|
| Fluid | T, γ, Vm and saturation pressure |
| Surface | contact angle and chemical heterogeneity |
| Geometry | pore shape and effective curvature radius |
| Film | thickness model before condensation |
| Path | adsorption, desorption and rate dependence |
| Limits | molecular scale and ξ/L critical crossover |
The gold standard is to calculate the relative pressure, repeat it for a changed radius or contact angle and show how the film and path alter inference. Capillary condensation becomes useful evidence when curvature, wetting and metastability are kept separate.
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Authoritative reading
- Evans, Stewart and Wilding, order of condensation in capillary grooves
- Thommes and colleagues, historical aspects and limits of capillary condensation
- Gelb, Gubbins, Radhakrishnan and Sliwinska-Bartkowiak, phase separation in confined systems
- Evans, Marconi and Tarazona, capillary condensation and adsorption in cylindrical pores