Pilling-Bedworth Ratio: Why Some Oxide Layers Protect and Others Flake

What the Pilling-Bedworth ratio is and where it applies

When a metal reacts with oxygen it grows a solid oxide on its surface. Whether that oxide is a shield or a liability depends on how much space it takes up relative to the metal it replaced. The Pilling-Bedworth ratio (R_PB) captures exactly this: it is the volume of oxide formed divided by the volume of metal consumed.

• If the oxide is smaller than the metal it came from (R_PB < 1), it cannot fully cover the surface - it grows under tension, cracks, and leaves bare metal exposed.
• If the oxide is slightly larger (1 < R_PB < 2), it forms a continuous, compressively-stressed, adherent film that seals the surface and slows further reaction - passivation.
• If the oxide is much larger (R_PB > 2), the compressive stress becomes so severe that the film buckles, cracks, and spalls off, again exposing fresh metal.

The criterion governs dry, high-temperature oxidation - jet-engine blades, furnace components, boiler tubes, and the native passive films that make aluminium and stainless steel corrosion-resistant. It is a screening tool, not a full theory.

Deriving the ratio step by step

Consider the generic oxidation reaction a M + (b/2) O2 -> M_a O_b. One mole of oxide M_a O_b contains n = a metal atoms. We compare the volume of that oxide to the volume of the n moles of metal it consumed.

The molar volume of any solid is V = M / rho, where M is molar mass and rho is density. So:

• Volume of oxide formed = M_oxide / rho_oxide (per mole of oxide)
• Volume of metal consumed = n * (M_metal / rho_metal), because n metal atoms went into that mole of oxide

Dividing gives the governing equation:

R_PB = (M_oxide * rho_metal) / (n * M_metal * rho_oxide)

where M_oxide is the molar mass of the oxide, M_metal the atomic mass of the metal, rho_metal and rho_oxide their densities, and n the number of metal atoms per formula unit of oxide. Every term is a tabulated constant, so R_PB is computed from a handbook in seconds - no experiment required. The n term is the step students most often drop: for Al2O3, n = 2, not 1.

A worked example and characteristic numbers

Take aluminium forming Al2O3. The constants: M_Al = 26.98 g/mol, rho_Al = 2.70 g/cm3; M(Al2O3) = 101.96 g/mol, rho(Al2O3) = 3.95 g/cm3; and n = 2 aluminium atoms per formula unit.

R_PB = (101.96 * 2.70) / (2 * 26.98 * 3.95) = 275.3 / 213.1 = 1.29

That lands squarely in the protective 1-2 window, which is why aluminium self-passivates with a ~2-4 nm native oxide that regrows instantly when scratched.

Contrast magnesium forming MgO: M_Mg = 24.31, rho_Mg = 1.74; M(MgO) = 40.30, rho(MgO) = 3.58; n = 1.

R_PB = (40.30 * 1.74) / (1 * 24.31 * 3.58) = 70.1 / 87.0 = 0.81

Below 1 - the oxide cannot cover the metal, so magnesium keeps oxidizing. Characteristic values span from K (0.47) and Na (0.57) at the porous low end, through the protective band (Al 1.28, Ni 1.65, Fe/FeO 1.70, Ti 1.73), up to the spalling high end (Nb 2.69, Ta 2.47, W 3.30).

How it is calculated and used in practice

Because R_PB depends only on molar masses and densities, engineers compute it directly from crystallographic and handbook data - no gravimetry or microscopy needed for a first estimate. The workflow is:

• Identify the stable oxide that actually forms at the service temperature (e.g. Cr2O3 on stainless steel, Al2O3 on FeCrAl alloys).
• Look up M and rho for metal and oxide; count n metal atoms per formula unit.
• Plug into R_PB and read off the regime.

In alloy design this is why chromium (R_PB = 2.07) and aluminium (1.28) are the classic protective-scale formers deliberately added to superalloys and stainless steels: they build slow-growing Cr2O3 or Al2O3 barriers. Actual oxidation kinetics are then measured by thermogravimetric analysis (TGA), tracking mass gain versus time; a truly protective scale follows parabolic kinetics (mass gain proportional to sqrt(t)), whereas a non-protective one grows linearly. R_PB tells you which to expect before you run the furnace.

How it compares to related concepts

R_PB is a geometric criterion; it is often confused with thermodynamic and kinetic ones that answer different questions.

• Gibbs free energy / Ellingham diagrams tell you whether an oxide will form (is oxidation spontaneous?). Magnesium's MgO is enormously stable (very negative delta-G), yet R_PB = 0.81 says the film still won't protect. Stability and protectiveness are independent.
• Parabolic rate constant (k_p) from Wagner's oxidation theory tells you how fast a protective scale thickens by ionic diffusion. R_PB only predicts whether a protective scale can exist at all.
• Passivation potential in aqueous electrochemistry is the wet-corrosion analogue; R_PB is the dry, high-temperature cousin.

Think of it as a hierarchy: Ellingham says an oxide is thermodynamically wanted, R_PB says its geometry allows a coherent film, and Wagner kinetics say how slowly that film then grows. All three must line up for real protection.

Exceptions, limits, and why it still matters

Pilling and Bedworth's 1923 rule is famous precisely because it is a superb first approximation that nonetheless fails in instructive cases:

• Iron has R_PB = 1.7 (protective on paper) yet rusts badly - because above 570 C it forms a three-layer FeO/Fe3O4/Fe2O3 scale, and the porous, fast-diffusing wustite (FeO) undermines the barrier. R_PB ignores multi-layer scales and defect chemistry.
• Silicon (R_PB = 2.15) sits above 2 yet forms an exceptionally protective amorphous SiO2 - the rule's upper bound assumes brittle crystalline oxides and misjudges glassy, plastically-relaxing ones.
• Adhesion, ductility, thermal-expansion mismatch, volatility (e.g. volatile MoO3, WO3 at red heat), and epitaxial growth stress all matter and are absent from the formula.

Despite these caveats, R_PB remains a cornerstone taught in every corrosion course and the reason we understand, in one number, why aluminium foil survives the oven, magnesium ribbon burns to powder, and tungsten filaments must run in vacuum.

Pilling-Bedworth ratios for common metals and the behaviour they predict

Metal	Oxide	R_PB	Predicted behaviour
Potassium	K2O	0.47	R_PB < 1: film too small, tensile, porous - no protection
Magnesium	MgO	0.81	R_PB < 1: non-protective, oxide cannot cover the metal
Aluminium	Al2O3	1.28	1 < R_PB < 2: dense, adherent, passivating
Iron	FeO	1.70	1 < R_PB < 2: protective in ideal case (in practice multi-layer)
Chromium	Cr2O3	2.07	R_PB ~ 2: borderline; largely protective in alloys
Tungsten	WO3	3.30	R_PB > 2: high compressive stress, spalls and flakes