The relation x² = k_p t, in which the square of the oxide scale thickness increases in proportion to time. It holds when diffusion of ions through a dense scale controls the growth rate.
Japanese name: 酸化の放物線則
Parabolic rate law appears in 1 practice question across 1 topic, including High-temperature materials and creep.
When diffusion through the scale controls growth, the flow of matter through the scale is inversely proportional to its thickness x, so the growth rate is dx/dt ∝ 1/x, which integrates to x² = k_p t. In 1933, Wagner related k_p, through the rate at which ions and electrons move via point defects in the scale, to the defect concentration and diffusion coefficients of the oxide and to the difference in oxygen partial pressure across the scale. This theory also explains why small additions of elements that change the defect concentration of the oxide alter k_p. k_p increases with temperature in the Arrhenius form k_p = k₀exp(−Q/RT).
In actual tests the mass gain of a specimen is often measured continuously with a thermobalance, and with the mass gain per unit area Δm/A the law is written (Δm/A)² = k_p t. The unit of k_p is then g²/(cm⁴·s) or similar, and its value differs from a k_p in m²/s written with thickness, so the definition must be checked when comparing literature values. The mass gained is that of the oxygen taken into the scale. For Al₂O₃, for example, oxygen makes up about 47% of the mass, so with a density of 3.95 g/cm³, a gain of 0.1 mg/cm² corresponds to a scale about 0.54 µm thick.
The parabolic law is a sign that a dense, adherent scale controls growth. If the scale is cracked or porous, a linear law results, proceeding at a constant rate regardless of thickness, and thin films near room temperature show a logarithmic law in which growth almost stops at a few nm. If the scale volatilizes at high temperature, growth by diffusion and loss by volatilization combine in a paralinear law; the thickness approaches a constant value, but metal continues to be lost at a constant rate. Volatilization of Cr₂O₃ as CrO₃ is a typical example. Breakaway oxidation, in which the scale cracks partway through and the rate suddenly increases, also needs attention.
For the thermal oxidation of Si in semiconductors, Deal and Grove in 1965 expressed the relation between oxide thickness x and time t as x² + Ax = B(t + τ). It is a model in which diffusion of oxygen through the SiO₂ film and reaction at the interface with Si act in series. While the film is thin, the interface reaction is the slow step and the thickness grows nearly in proportion to time; as it thickens, diffusion takes control and growth approaches the parabolic law x² ≈ Bt. The parabolic rate constant B is larger in water vapor than in dry oxygen, so thick films are grown quickly in steam. Very thin films grown in dry oxygen, however, are known to grow faster than this model predicts.
Answer: B. About 6.0 µm
Under the parabolic law the thickness is proportional to the square root of time, so x = 2.0 × √(900/100) = 2.0 × 3 = 6.0 µm. As the scale thickens, the distance that ions and electrons must travel through it grows and growth slows, giving this form (Wagner's theory). About 18 µm is calculated with the linear law, in which thickness is proportional to time; the linear law is seen when the scale is porous or cracked and not protective, or when a reaction at an interface controls the rate. About 4.2 µm is calculated with the cubic law, in which thickness is proportional to the cube root of time. The smaller the parabolic rate constant k_p of a scale, the more protective it is, and this serves as a guide in choosing alloys.
Written with thickness, x² = k_p t, it is in m²/s or cm²/s; written with mass gain, (Δm/A)² = k_p t, it is in g²/(cm⁴·s), mg²/(cm⁴·h) or similar. Some definitions include a factor of 2, as in x² = 2k_p t, so the form of the equation and the units should be checked when using literature values.
Plot the square of the thickness or mass gain against time and see whether it is a straight line, or see whether the slope on log-log axes is 0.5. Other processes often mix in at short times, so that part is excluded when judging. A slope near 1 suggests a linear law, and about 0.33 a cubic law.
Both reflect the same property: the distance of a process driven by diffusion is proportional to the square root of time. Because diffusion through the scale controls growth, k_p is related to the diffusion coefficients in the scale and the difference in oxygen partial pressure, and like a diffusion coefficient it increases with temperature in Arrhenius form.
Under the parabolic law the thickness is proportional to the square root of time, so x = 2.0 × √(900/100) = 2.0 × 3 = 6.0 µm.
When an alloy was oxidized in air at high temperature, it f…Deformation mechanism map Diffusion creep Larson–Miller parameter Nickel-based superalloy Pilling–Bedworth ratio Power-law creep Thermal barrier coating