Capillary Condensation and Pore Analysis

Fill a coffee mug's worth of silica gel with nitrogen gas at 77 K, and something strange happens. Below its normal boiling point, nitrogen condenses to liquid inside the tiny pores long before the bulk gas would ever condense in the open. This is capillary condensation, and it is the workhorse phenomenon behind every commercial pore-size-distribution measurement. Catalysts, membranes, pharmaceutical excipients, and battery electrodes all live or die by their pore architecture. A catalyst with the right surface area but the wrong pore size will strangle diffusion and waste its expensive active sites. Engineers need a routine, quantitative way to convert a gas adsorption isotherm into a distribution of pore sizes, and that is exactly what capillary condensation theory provides. This article builds on the Kelvin equation and the BET isotherm you have already met. You will see why adsorption and desorption trace different paths on the same isotherm. Then you will learn how that hysteresis loop shape encodes pore geometry. Finally, you will work through the BJH-style calculation that turns raw pressure-volume data into a pore size distribution curve.

Key takeaways

Capillary condensation is the filling of narrow pores with liquid adsorbate at pressures below the bulk saturation vapor pressure $p_0$ .
It occurs because a concave meniscus lowers the equilibrium vapor pressure, as described by the Kelvin equation.
Adsorption and desorption branches diverge into a hysteresis loop whenever pores are wide enough (typically above ~2 nm) and have restrictive necks or ink-bottle shapes.
The IUPAC H1–H4 hysteresis loop shapes are a diagnostic fingerprint of pore geometry: cylindrical, ink-bottle, slit, or interconnected networks.
The Barrett–Joyner–Halenda (BJH) method combines the Kelvin equation with a statistical multilayer thickness correction to extract a pore size distribution from one isotherm branch.
Key idea: Capillary condensation shifts the condensation pressure below $p_0$ by an amount set by pore curvature, and inverting that shift branch-by-branch across the isotherm reconstructs the full pore size distribution.

The core insight

Because a curved liquid meniscus inside a narrow pore has a lower equilibrium vapor pressure than a flat surface, gas condenses to liquid in progressively larger pores as relative pressure $p/p_0$ rises. Reading the isotherm as a sequence of pore-filling events, each governed by the Kelvin equation and corrected for the pre-existing adsorbed film, converts a single pressure-volume curve into a quantitative distribution of pore radii.

The physical picture

Consider a rigid, mesoporous solid — silica, alumina, or a porous carbon — exposed to nitrogen vapor at 77 K in a static volumetric adsorption apparatus. At very low relative pressure $p/p_0$ , gas molecules physisorb as a submonolayer, then a full monolayer, then successive multilayers, exactly as BET theory describes. This is ordinary multilayer adsorption on an essentially flat internal wall. As $p/p_0$ climbs into the range 0.3–0.95, something new happens inside pores narrower than about 50 nm (the IUPAC mesopore range, 2–50 nm). The adsorbed multilayer film thickens until the residual open channel becomes so narrow that its liquid-vapor interface is forced into a strongly curved, concave meniscus. A concave meniscus has a lower vapor pressure than a flat liquid surface, by the Kelvin relation. So at a pressure still below $p_0$ , the vapor becomes supersaturated with respect to that curved meniscus, and the entire remaining pore volume snaps to liquid — capillary condensation. Wider pores fill at higher $p/p_0$ ; narrower pores fill at lower $p/p_0$ . Scanning the whole isotherm and identifying at which pressures the sharp condensation steps occur therefore maps directly onto a distribution of pore sizes. The catch is that filling and emptying are not always symmetric. During adsorption, condensation in a pore can be delayed until a stable meniscus can form — often requiring the pore to be nearly full, or governed by metastability that postpones nucleation. During desorption, the liquid empties by retreat of a meniscus that may be pinned at a narrow neck (an "ink-bottle" pore) or may empty cooperatively across a connected network. The result is that the isotherm traces one path on the way up and a different, lower-pressure path on the way down. This adsorption-desorption mismatch is the hysteresis loop, and its shape is diagnostic of pore shape and connectivity.

Figure

Capillary condensation inside a cylindrical mesopore, showing the curved meniscus and adsorbed film

The core relations

\ln\frac{p}{p_0} = -\frac{2\gamma V_{\mathrm m}\cos\theta}{r_{\mathrm k}\,RT}

where: $\gamma$ is the liquid surface tension, $V_{\mathrm m}$ its molar volume, $\theta$ the contact angle (0° for full wetting), $r_{\mathrm k}$ the Kelvin radius of curvature of the meniscus, $R$ the gas constant, $T$ the temperature.

r_{\mathrm p} = r_{\mathrm k} + t(p/p_0)

where: $r_{\mathrm p}$ is the true (physical) pore radius, $t(p/p_0)$ the statistical thickness of the pre-adsorbed multilayer film at that relative pressure (from a reference $t$ -curve such as the Halsey or Harkins–Jura equation).

t = 0.354\left[\frac{-5.00}{\ln(p/p_0)}\right]^{1/3}\ \text{nm}

where: this is the Halsey equation for nitrogen at 77 K, giving statistical film thickness $t$ in nanometers as a function of $p/p_0$ .

\frac{\Delta V_{\mathrm p}}{\Delta r_{\mathrm p}} \approx \frac{1}{\Delta r_{\mathrm p}}\left[\Delta V_{\mathrm{liq}} - \bar r_{\mathrm p}^{\,2}\sum_j \frac{\Delta t_j\, A_{\mathrm{cyl},j}}{\bar r_{\mathrm k,j}^{\,2}}\right]

where: $\Delta V_{\mathrm{liq}}$ is the volume of liquid desorbed in one pressure step, and the summation subtracts the volume released purely from thinning of the film already lining previously emptied larger pores (the BJH thickness-correction term).

Worked examples

Derivation AThe Kelvin equation and the BJH pore-filling ladder

Goal. Starting from the equilibrium condition between a curved liquid meniscus and its vapor, derive the Kelvin equation, then show how it is adapted into the BJH algorithm that converts a desorption isotherm into a pore size distribution.

Step 1 — mechanical equilibrium across a curved interface. A meniscus of mean radius of curvature $r_{\mathrm k}$ supports a pressure difference given by the Young–Laplace equation,

p_{\mathrm{liq}} - p_{\mathrm{vap}} = -\frac{2\gamma\cos\theta}{r_{\mathrm k}}

for a concave (wetting) meniscus in a cylindrical pore, the minus sign showing the liquid pressure is reduced below the vapor pressure by curvature.

Step 2 — chemical potential balance. At equilibrium, the chemical potential of the liquid under this reduced pressure must equal that of the vapor at pressure $p$ . Using $\mathrm d\mu_{\mathrm{liq}} = V_{\mathrm m}\,\mathrm dp_{\mathrm{liq}}$ and $\mathrm d\mu_{\mathrm{vap}} = RT\,\mathrm d\ln p$ , integrating from the flat-surface reference state (pressure $p_0$ , liquid pressure equal to $p_0$ ) to the curved state gives

RT\ln\frac{p}{p_0} = V_{\mathrm m}(p_{\mathrm{liq}} - p_0) = -\frac{2\gamma V_{\mathrm m}\cos\theta}{r_{\mathrm k}}

Step 3 — the Kelvin equation. Rearranging Step 2 directly yields

\ln\frac{p}{p_0} = -\frac{2\gamma V_{\mathrm m}\cos\theta}{r_{\mathrm k}RT}

For full wetting ( $\theta = 0$ ), this simplifies to $\ln(p/p_0) = -2\gamma V_{\mathrm m}/(r_{\mathrm k}RT)$ , showing smaller $r_{\mathrm k}$ (more curvature) forces condensation at lower relative pressure.

Step 4 — separating curvature from film. The Kelvin radius $r_{\mathrm k}$ describes only the curvature of the liquid meniscus, not the physical pore. Before condensation, a statistical adsorbed film of thickness $t(p/p_0)$ already lines the pore wall (BET-like multilayer growth). So the true pore radius is the sum,

r_{\mathrm p} = r_{\mathrm k} + t(p/p_0)

This is essential: ignoring $t$ systematically underestimates every pore radius, sometimes by 30–50% for small mesopores.

Step 5 — the BJH ladder. Starting from the highest relative pressure (largest pores, all filled), the desorption isotherm is stepped downward in small pressure decrements. At each step, some volume of liquid $\Delta V_{\mathrm{liq}}$ desorbs. Part of this volume comes from complete emptying of the new, smaller class of pores that were just able to hold a meniscus stable at the previous step's pressure; another part comes merely from thinning of the residual adsorbed film still coating the walls of all the larger pores that emptied at earlier (higher-pressure) steps. Subtracting this film-thinning contribution, weighted by the ratio of pore cross-section to core cross-section $(\bar r_{\mathrm p}/\bar r_{\mathrm k})^2$ , isolates the volume genuinely due to newly emptied pores of that specific size class.

Step 6 — assembling the distribution. Repeating Step 5 down the entire desorption branch, converting each surviving $\Delta V_{\mathrm p}$ into a differential $\mathrm dV_{\mathrm p}/\mathrm dr_{\mathrm p}$ against the corresponding $r_{\mathrm p}$ from Step 4, builds up the full pore size distribution curve.

\boxed{\ \frac{\mathrm dV_{\mathrm p}}{\mathrm dr_{\mathrm p}}\ \text{vs.}\ r_{\mathrm p}\ \text{obtained pointwise from the desorption isotherm via Steps 1–6 (the BJH method)}\ }

Worked examples

Example 1 — Foundational: finding the Kelvin radius for a single condensation step. Nitrogen at 77 K is used to probe a mesoporous alumina. At $p/p_0 = 0.60$ a sharp condensation step is observed. Nitrogen's properties at 77 K: $\gamma = 8.85\times10^{-3}\ \mathrm{N\,m^{-1}}$ , $V_{\mathrm m} = 34.7\times10^{-6}\ \mathrm{m^3\,mol^{-1}}$ , and assume complete wetting, $\theta = 0°$ so $\cos\theta = 1$ . Step 1: rearrange the Kelvin equation for $r_{\mathrm k}$ ,

r_{\mathrm k} = -\frac{2\gamma V_{\mathrm m}}{RT\ln(p/p_0)}

Step 2: compute the numerator,

2\gamma V_{\mathrm m} = 2 \times 8.85\times10^{-3} \times 34.7\times10^{-6} = 6.142\times10^{-7}\ \mathrm{N\,m^2\,mol^{-1}}

Step 3: compute $RT$ at 77 K,

RT = 8.314 \times 77 = 640.2\ \mathrm{J\,mol^{-1}}

Step 4: compute $\ln(0.60) = -0.5108$ . Step 5: combine,

r_{\mathrm k} = -\frac{6.142\times10^{-7}}{640.2 \times (-0.5108)} = \frac{6.142\times10^{-7}}{327.1} = 1.878\times10^{-9}\ \mathrm m = 1.88\ \mathrm{nm}

Step 6: add the film thickness at $p/p_0 = 0.60$ using the Halsey relation,

t = 0.354\left[\frac{-5.00}{\ln(0.60)}\right]^{1/3} = 0.354\left[\frac{5.00}{0.5108}\right]^{1/3} = 0.354\,(9.789)^{1/3} = 0.354 \times 2.135 = 0.756\ \mathrm{nm}

Step 7: the true pore radius is

r_{\mathrm p} = r_{\mathrm k} + t = 1.88 + 0.76 = 2.64\ \mathrm{nm}, \quad \text{diameter} \approx 5.3\ \mathrm{nm}

Sanity check: this falls squarely in the mesopore range (2–50 nm), consistent with alumina catalysts support pores, and is significantly larger than the bare Kelvin radius — showing why the film correction is not optional.

Example 2 — Applied: estimating pore volume in a distribution bin from a desorption step. A desorption isotherm on a mesoporous silica (SBA-15 type) shows that between $p/p_0 = 0.52$ and $p/p_0 = 0.50$ , the volume of nitrogen desorbed (converted to liquid volume at STP density) is $\Delta V_{\mathrm{liq}} = 0.048\ \mathrm{cm^3\,g^{-1}}$ . Assume that in this narrow window essentially all of this volume corresponds to newly emptying pores (the film-thinning correction on other, larger pores has already been subtracted and is negligible for this illustrative bin). Estimate the pore radius at the midpoint of the step and the contribution to $\mathrm dV_{\mathrm p}/\mathrm dr_{\mathrm p}$ . Step 1: take the midpoint relative pressure, $p/p_0 = 0.51$ , $\ln(0.51) = -0.6733$ . Step 2: Kelvin radius,

r_{\mathrm k} = \frac{6.142\times10^{-7}}{640.2 \times 0.6733} = \frac{6.142\times10^{-7}}{431.0} = 1.425\times10^{-9}\ \mathrm m = 1.43\ \mathrm{nm}

Step 3: film thickness via Halsey,

t = 0.354\left[\frac{5.00}{0.6733}\right]^{1/3} = 0.354\,(7.426)^{1/3} = 0.354 \times 1.949 = 0.690\ \mathrm{nm}

Step 4: true pore radius,

r_{\mathrm p} = 1.43 + 0.69 = 2.12\ \mathrm{nm}

Step 5: at the adjacent, higher pressure boundary $p/p_0 = 0.52$ ( $\ln = -0.6539$ ), a parallel calculation gives $r_{\mathrm k} = 1.467$ nm, $t = 0.700$ nm, $r_{\mathrm p} = 2.17$ nm. So the bin width is

\Delta r_{\mathrm p} = 2.17 - 2.12 = 0.05\ \mathrm{nm}

Step 6: the differential pore volume distribution for this bin is

\frac{\Delta V_{\mathrm p}}{\Delta r_{\mathrm p}} = \frac{0.048\ \mathrm{cm^3\,g^{-1}}}{0.05\ \mathrm{nm}} = 0.96\ \mathrm{cm^3\,g^{-1}\,nm^{-1}}

Sanity check: this places a peak in the pore size distribution near 4.2 nm diameter, consistent with the known ~5–8 nm main channel pores of SBA-15 materials once the neighboring bins (which would show the true peak) are included; a single narrow bin like this only samples one slice of a full distribution, which is why in practice the full desorption branch is processed step by step across the whole pressure range.

Limiting cases & common traps

Situation	What happens	Common trap
Micropores below ~2 nm	Kelvin equation breaks down; pore filling becomes governed by enhanced adsorption potential, not simple meniscus curvature	Applying BJH to micropores gives meaningless "pore sizes"; use density functional theory (NLDFT/QSDFT) instead
Macropores above ~50 nm	Capillary condensation pressure shift becomes negligible; pores fill only near $p/p_0 \to 1$	Mercury intrusion porosimetry is the appropriate technique here, not gas sorption
Ink-bottle pores (narrow neck, wide body)	Desorption is delayed until the neck's own Kelvin pressure is reached, even though the body could empty earlier; produces a steep, near-vertical desorption branch	Using the desorption branch naively assigns the wrong (too small) size to the body; adsorption branch or independent methods needed to deconvolve neck vs. body
Cavitation-controlled desorption	For necks below ~4–5 nm (for N2 at 77 K), the liquid may empty by spontaneous cavitation rather than meniscus retreat through the neck	The sharp "cavitation step" near $p/p_0 \approx 0.42$ is instrument-independent and should not be mistaken for a real pore size feature
Non-wetting or high contact angle systems	$\cos\theta < 1$ shifts every calculated radius	Assuming $\theta = 0$ for all adsorbates (e.g., some organics on carbons) introduces systematic error

The chemist behind it

The Kelvin equation itself dates to Lord Kelvin's 1871 work on vapor pressure over curved surfaces, but its application to real porous solids required a practical algorithm. That came in 1951, when Elliott Barrett, Leslie Joyner, and Paul Halenda, working at the Bureau of Mines and later at Esso (Standard Oil), published the stepwise desorption-branch calculation now universally known as the BJH method. Their original paper analyzed nitrogen isotherms on silica-alumina cracking catalysts — materials central to the petroleum industry at the time — and the method they devised remains, seven decades later, the default pore-size algorithm built into essentially every commercial gas sorption analyzer.

Origins · 1 minElliott P. Barrett, Leslie G. Joyner, Paul P. Halenda and how this was discovered

Elliott P. Barrett, Leslie G. Joyner, Paul P. Halenda

active mid-20th century · American physical chemists whose 1951 desorption-isotherm algorithm turned Kelvin-equation curvature effects into a routine industrial pore-size measurement

Ask the tutor: How does the NLDFT method improve on BJH for narrow mesopores and micropores?

How is the Kelvin equation related to the BET isotherm covered earlier?

BET theory describes the buildup of an adsorbed multilayer film on an open, essentially flat surface, valid roughly for $p/p_0$ between 0.05 and 0.35. The Kelvin equation takes over once that film grows thick enough, in a confined pore, that the remaining open channel develops a curved meniscus. BJH analysis actually reuses the BET-derived $t$ -curve (statistical film thickness versus $p/p_0$ ) as an input, so the two theories are complementary stages of the same isotherm rather than competing models.

Why do adsorption and desorption branches not overlap — what causes hysteresis physically?

On adsorption, the vapor must nucleate a stable meniscus inside a pore, which can require metastable supersaturation beyond the equilibrium Kelvin pressure, delaying condensation to higher $p/p_0$ . On desorption, an already-formed meniscus retreats under genuinely reversible thermodynamic control, or is blocked by a narrower neck elsewhere in a connected pore network. Because these two processes probe different physical events, they trace different pressures for the same pore, opening a loop.

Which branch of the isotherm should be used for pore size analysis, adsorption or desorption?

IUPAC recommends the desorption branch for simple, well-connected mesopore networks because it more closely reflects true thermodynamic equilibrium during meniscus retreat. However, for ink-bottle or highly interconnected networks, desorption reflects the neck size rather than the true pore body size. In such cases, the adsorption branch (analyzed with an adsorption-specific Kelvin relation) or independent techniques like electron tomography give a more reliable body-size estimate.

Can the BJH method be applied to any adsorbate gas, not just nitrogen?

In principle yes; argon at 87 K is increasingly preferred because its interaction with oxide surfaces is simpler and it avoids nitrogen's quadrupole-induced anomalies in micropore-sized channels. The same Kelvin and $t$ -curve framework applies, but $\gamma$ , $V_{\mathrm m}$ , and the statistical thickness equation must all be re-parametrized for the chosen adsorbate and temperature.

Does the BJH method give the true, absolute pore size, or only a relative ranking?

BJH pore sizes carry systematic uncertainty, typically 20–30% underestimation for pores below about 7–10 nm, because the classical Kelvin equation assumes bulk liquid properties that break down at small curvature. Modern NLDFT and QSDFT methods, calibrated against molecular simulations, are more accurate for narrow mesopores. BJH remains extremely useful for comparing pore sizes within a series of related materials, even where its absolute values are approximate.

Capillary Condensation and Pore Analysis

The physical picture

The core relations

Worked examples

Worked examples

Limiting cases & common traps

The chemist behind it

Before this