Capillary Condensation in Nanopores

Pore & Adsorptive

Pore radius r_p 3.0 nm

Adsorptive gas

Relative Pressure

P / P₀ 0.30

Readouts

Kelvin radius r_k

— nm

Film thickness t

— nm

Pore state

Empty (vapor)

Pore filled

Isotherm Trace

Adsorption (P rising) Desorption (P falling)

How it works

Below a pore's condensation pressure, gas molecules only build up a thin liquid film on the pore wall. The Kelvin equation gives the pore (meniscus) radius r_k in equilibrium with relative pressure P/P₀:

ln(P/P0) = -k·γ·Vm / (r_k·R·T)

k = 1  cylindrical meniscus (filling / adsorption branch)
k = 2  hemispherical meniscus (emptying / desorption branch)
γ = liquid surface tension, Vm = molar volume, T = temperature

Because filling and emptying involve menisci of different curvature, the two branches trigger at different pressures for the same pore radius — this is the physical origin of the adsorption/desorption hysteresis loop seen in Type IV isotherms of real mesoporous solids (silica gels, MCM-41, activated alumina).

Rising P — a film of thickness t(P) coats the wall until the shrunken core radius r_c = r_p − t satisfies the k=1 equation; the pore then snaps to fully liquid-filled (capillary condensation).
Falling P — the filled pore stays condensed until the k=2 equation is satisfied at the full pore radius r_p; it then empties back to a thin film.
Pore radius slider — smaller pores condense/evaporate at lower P/P₀; this radius-dependence is exactly what BJH analysis inverts to recover a pore-size distribution from a measured isotherm.
Gas selector — N₂ at 77 K and Ar at 87 K are the two adsorptives actually used in commercial porosimeters; they shift the loop because γ, V_m and T differ.

Film thickness here uses a simplified monotonic statistical-thickness approximation (t grows smoothly with P/P₀, capped below r_p) rather than a specific literature t-curve fit — it is illustrative of the real multilayer-adsorption trend, not a metrology-grade fit.

Empty

Adsorbed film Condensed liquid

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