Diffusion Coefficient and Diffusion Length Calculator
Find the diffusion coefficient at a given temperature from the pre-exponential factor D0 and activation energy Q, and estimate from the heating time how deep atoms diffuse. The defaults are carbon in austenite (γ-iron) (D0 = 2.3 × 10⁻⁵ m²/s, Q = 148 kJ/mol), carburized at 900 °C for 4 hours.
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Concentration ratio at depth x, (Cx − C0)/(Cs − C0)
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D0 and Q depend on the combination of material and diffusing element. The concentration ratio is calculated with the error-function solution, valid when the surface concentration Cs is held constant and the specimen is thick enough to be treated as semi-infinite. Gas constant R = 8.314 J/(mol·K).
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
Temperature dependence of the diffusion coefficient (Arrhenius equation)
Atoms in a solid move in discrete jumps via vacancies or interstitial sites. Each jump must overcome an activation energy Q, so the diffusion coefficient varies exponentially with temperature as D = D0·exp(−Q/RT). For the defaults (carbon in γ-iron, 900 °C = 1173.15 K), D = 2.3 × 10⁻⁵ × exp(−148000/(8.314 × 1173.15)) ≈ 5.91 × 10⁻¹² m²/s. Raising the temperature by just 100 °C increases D several times, which is why temperature control matters in heat treatment. Taking logarithms of both sides, ln D = ln D0 − Q/(RT) is a straight line against 1/T, and its slope gives −Q/R.
The approximate diffusion length √(Dt)
The distance atoms move by diffusion is proportional not to time but to the square root of time. The estimate x ≈ √(Dt) is widely used; for the default 4 hours (14,400 s), √(5.91 × 10⁻¹² × 14400) ≈ 0.292 mm. To double the diffusion depth at the same temperature, the time must be quadrupled. Depending on the definition, the coefficient may be given differently, as in 2√(Dt), so use the same definition when comparing.
Carburizing and the error-function solution
When the surface of a steel is held in an atmosphere of carbon concentration Cs for carburizing, the concentration profile in a thick specimen of initial concentration C0 is given by the solution of Fick's second law, (Cx − C0)/(Cs − C0) = 1 − erf(x/(2√(Dt))). At the default depth of 0.5 mm, x/(2√(Dt)) ≈ 0.857, and the concentration ratio is about 22.6 %. With C0 = 0.20 wt% and Cs = 1.00 wt%, for example, the carbon concentration at 0.5 mm is about 0.38 wt%. erf(z) is a function that is 0 at z = 0 and nearly 1 for z above 2.
Typical values of D0 and Q
Diffusion is faster for small atoms that move interstitially and slower for substitutional atoms. For carbon in γ-iron (interstitial diffusion), D0 ≈ 2.3 × 10⁻⁵ m²/s and Q ≈ 148 kJ/mol; for carbon in α-iron, D0 ≈ 1.1 × 10⁻⁶ m²/s and Q ≈ 87 kJ/mol. Self-diffusion of iron (in γ-iron, by the vacancy mechanism), by contrast, has a large Q of around 280 kJ/mol and is many orders of magnitude slower at the same temperature. Values vary between sources, so when calculating, enter the values from your source as given.
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.
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.
Taking the ratio of the expressions at the two temperatures eliminates D₀ and gives ln(D₂/D₁) = (Q/R)(1/T₁ − 1/T₂). D₂/D₁ = 40, ln 40 ≈ 3.689 and 1/800 − 1/1000 = 2.5×10⁻⁴ K⁻¹, so Q = 8.314 × 3.689 / (2.5×10⁻⁴) ≈ 1.23×10⁵ J/mol, about 123 kJ/mol. About 53 kJ/mol uses the common logarithm (log₁₀ 40 ≈ 1.602) instead of the natural logarithm, and about 245 kJ/mol is double the value, from confusing the expression with the form exp(−Q/2RT). In practice, ln D is plotted against 1/T and Q is obtained from the slope −Q/R of the straight line.
FAQ
What is the unit of the diffusion coefficient?
In SI units it is m²/s. Literature values are often given in cm²/s; convert with 1 cm²/s = 10⁻⁴ m²/s. Enter D0 in this calculator in m²/s.
Why is the diffusion length proportional to the square root of time?
Atoms repeat jumps in random directions (a random walk), so the average distance traveled is proportional to the square root of the number of jumps. The number of jumps is proportional to time, so the distance is proportional to √t.
What if I know the activation energy in eV?
1 eV per atom is about 96.49 kJ/mol; for 1.5 eV, for example, enter about 145 kJ/mol. When calculating per atom, use Boltzmann's constant k = 8.617 × 10⁻⁵ eV/K instead of the gas constant R, with D = D0·exp(−Q/kT).