The coefficient D expressing how readily atoms diffuse, in units of m²/s. It depends on temperature in the Arrhenius form D = D₀exp(−Q/RT).
Japanese name: 拡散係数
Diffusion coefficient appears in 6 practice questions across 1 topic, including Crystal defects and diffusion.
The diffusion coefficient D is the proportionality constant in Fick's first law, J = −D(dC/dx), that links the concentration gradient to the flow of atoms (the flux). Its unit is m²/s (cm²/s is also common), and the larger it is, the faster atoms spread. A rough distance over which atoms spread in time t is √(Dt). For example, with D = 10⁻¹¹ m²/s, atoms spread about 0.2 mm in one hour (3600 s). It is the basic quantity for estimating carburized case depths, dopant diffusion depths and the growth of reaction layers in solder joints.
The diffusion coefficient depends exponentially on temperature, D = D₀exp(−Q/RT). Q is the activation energy needed for an atom to jump to a neighboring site, and D₀ is a constant that includes the vibration frequency and jump distance. Plotting ln D against 1/T (an Arrhenius plot) gives a straight line, and Q is obtained from its slope −Q/R. The larger Q, the stronger the effect of temperature, and a difference of a few tens of degrees can change the diffusion rate several times over. In calculations, it is essential to convert temperatures to absolute temperature.
Small atoms such as carbon, nitrogen and hydrogen in iron move by interstitial diffusion, jumping between interstitial sites, so their diffusion coefficients are large. Substitutional atoms diffuse by the vacancy mechanism, moving only when a vacancy arrives next to them; the activation energy also includes the vacancy formation energy, so diffusion is far slower. In addition, diffusion along grain boundaries, surfaces and dislocations, where the atomic arrangement is disordered, has a lower activation energy and can dominate over lattice diffusion at low temperatures. This is why grain boundary diffusion stands out in polycrystalline thin films.
When different atoms diffuse into each other, as when two metals are bonded, the diffusion coefficient of each kind of atom (the intrinsic diffusion coefficient) must be distinguished from the interdiffusion coefficient that describes the overall change in concentration. The difference between them shows up as marker movement and void formation through the Kirkendall effect. In practice, the diffusion coefficient at the treatment temperature is calculated from literature values of D₀ and Q, and the concentration profile is predicted with the error-function solution. Values differ between data sources, so it is safer to compare several before use.
Answer: C. About 2.4×10⁻⁹ kg/(m²·s)
At steady state, Fick's first law J = −D(dC/dx) applies. The concentration gradient is (0.8 − 1.2)/(5.0×10⁻³ m) = −80 kg/m⁴, so J = 3.0×10⁻¹¹ × 80 = 2.4×10⁻⁹ kg/(m²·s), flowing from the high-concentration side to the low-concentration side. About 2.4×10⁻¹² is a unit-conversion error from dividing by the thickness left as 5.0 in mm, and about 7.2×10⁻⁹ uses the concentration on one side, 1.2 kg/m³, instead of the concentration difference. It is the concentration difference that drives the flow, so getting the gradient right matters. The same approach is used in hydrogen permeation tests and in evaluating gas separation membranes.
Answer: A. About 8.3×10⁻¹² m²/s
Convert the temperature to absolute temperature: T = 927 + 273 = 1200 K. Q/RT = 148000/(8.314 × 1200) ≈ 14.83, so D = 2.3×10⁻⁵ × exp(−14.83) ≈ 2.3×10⁻⁵ × 3.6×10⁻⁷ ≈ 8.3×10⁻¹² m²/s. About 1.1×10⁻¹³ is the mistake of substituting the temperature as 927 in °C. About 2.3×10⁻⁵ is the mistake of leaving Q as 148 in kJ, which makes the exponent nearly zero and gives almost the same value as D₀. Because D changes exponentially with temperature, raising the carburizing temperature by just a few tens of degrees can shorten the treatment time considerably.
In J = −D(dC/dx), the flux J has units of (amount)/(m²·s) and the concentration gradient (amount)/m⁴, so D has units of m²/s. This is also consistent with √(Dt) being a length.
It depends on the activation energy and the temperature. Solve ln 2 = (Q/R)(1/T₁ − 1/T₂); for example, with Q = 150 kJ/mol and T₁ = 1200 K, raising the temperature by about 58 K roughly doubles the diffusion coefficient.
A rough diffusion distance in time t is √(Dt) (or 2√(Dt), depending on the definition). The distance is proportional to the square root of time, so doubling the depth takes four times as long.
Paths with lower activation energy lose less of their diffusion coefficient as the temperature falls, so the relative contribution of grain boundary diffusion grows at lower temperatures.
Which statement about diffusion paths in a polycrystalline …