The ratio of the volume of an oxide to the volume of metal used to form it. A guide by which values below 1 tend to give cracked scales and values of about 1 to 2 tend to give protective ones.
Japanese name: ピリング・ベドワース比
Pilling–Bedworth ratio appears in 1 practice question across 1 topic, including High-temperature materials and creep.
In 1923, Pilling and Bedworth organized how metals oxidize by the ratio of the volume of oxide formed to the volume of metal consumed. The key is to compare equal numbers of metal atoms: the volume of 1 mol of the oxide M_aO_b is divided by the volume of the a mol of metal it contains. If the ratio is less than 1, the oxide cannot fully cover the metal surface; if it is greater than 1, the scale grows in a compressed state, squeezed sideways. Because it can be calculated from densities and molar masses alone, it has been used as a first guide for judging whether a scale will be protective.
Calculated from room-temperature densities (the values shift by 0.1 to 0.2 depending on the densities used), the PBR is about 0.57 for Na and about 0.65 for Ca, below 1, which agrees with the fact that oxidation of these metals does not easily stop. Al is about 1.3, and Ni and Cu (Cu₂O) are about 1.7, in the range that tends to give protective scales. Ti (rutile TiO₂) is about 1.8 and Cr (Cr₂O₃) about 2.0. In contrast, Nb (Nb₂O₅) is about 2.7 and W (WO₃) about 3.4, so large compressive stresses build up in the scale and cracking and spallation occur easily. W, whose oxide also volatilizes at high temperature, is hard to use in hot air as it is.
There are many cases where the PBR alone cannot judge protectiveness. Iron forms FeO with a ratio of about 1.8, but FeO formed at high temperature contains a very large number of defects through which ions diffuse rapidly, so the scale grows thick and fast. Si and Cr have rather large values of around 2, yet dense SiO₂ and Cr₂O₃ make excellent protective scales. The PBR assumes that the scale grows at its interface with the metal; when metal ions diffuse outward and oxide forms at the scale surface, stress from the volume difference does not build up much. Protectiveness is judged together with diffusion rates in the scale, adhesion and the thermal expansion mismatch on cooling.
The idea of the PBR is also useful for relating the thickness of a scale to the thickness of metal consumed. When Si is thermally oxidized to amorphous SiO₂ in semiconductor manufacturing, taking densities of 2.33 g/cm³ for Si and 2.20 g/cm³ for SiO₂ gives a ratio of about 2.27. Assuming the area stays the same and only the thickness increases, forming SiO₂ of thickness x consumes about 0.44x of Si, so about 44% of the oxide grows below the original Si surface and the rest above it. A 100 nm oxide consumes about 44 nm of Si, and this is allowed for in designing device dimensions and steps.
Answer: C. About 0.81, so the scale cannot fully cover the metal surface, cracks easily and is unlikely to be protective
PBR = (40.30 × 1.74)/(1 × 24.31 × 3.58) = 70.1/87.0 ≈ 0.81. When the PBR is less than 1, the volume of oxide is smaller than the volume of metal consumed, so the scale is under tensile stress and cracks or develops gaps, letting oxygen reach the metal easily. Around 1 to 2, the scale tends to cover the metal without gaps and to be protective, as with aluminum forming Al₂O₃ (PBR ≈ 1.29), whereas well above 2 the compressive stress becomes too large and the scale tends to spall. About 1.24 is the value with numerator and denominator swapped. The PBR is only a guide; protectiveness also depends on the rate of diffusion through the scale and its adhesion.
It is found from PBR = (M_ox ρ_M)/(n M_M ρ_ox). M_ox and ρ_ox are the molar mass and density of the oxide, M_M and ρ_M those of the metal, and n is the number of metal atoms in one formula unit of the oxide. It is a dimensionless ratio, so it can be calculated as long as the units of density and molar mass are consistent.
Because the oxide tries to swell to more than twice the volume of the original metal while being restrained sideways by the metal, so large compressive stresses build up in the scale. The stored energy grows as the scale thickens, and beyond a certain thickness the scale lifts off or cracks and spalls.
It changes slightly because the thermal expansion of the metal and oxide differs a little, but values calculated from room-temperature densities are used as a guide. What matters much more is which oxide forms; if the oxide that forms changes with temperature or oxygen partial pressure, the value for that oxide must be used.
PBR = (40.30 × 1.74)/(1 × 24.31 × 3.58) = 70.1/87.0 ≈ 0.81.
The Pilling–Bedworth ratio of a metal's oxide scale (PBR, t…Deformation mechanism map Diffusion creep Larson–Miller parameter Nickel-based superalloy Parabolic rate law Power-law creep Thermal barrier coating