Arrhenius Diffusion Coefficient

Because T appears inside an exponential. The fraction of atoms whose thermal energy exceeds the barrier Q is given by the Boltzmann factor exp(−Q/(k_BT)), and this fraction changes rapidly with T. For a typical metallic activation energy Q ≈ 150 kJ/mol, the Boltzmann factor at 900 K is about 1.8×10⁻⁹ while at 1000 K it is 1.4×10⁻⁸ — nearly a factor of 8 change for a 100 K rise. A 200 K rise to 1100 K pushes it to 7.5×10⁻⁸, another factor of 5. So in the range 900–1100 K, D changes by a factor of 40. This exponential sensitivity is what makes diffusion-controlled processes so sharply temperature-dependent.

D₀ is the pre-exponential factor, also called the frequency factor. It has units of diffusivity (m²/s) and represents the value D would take in the limit T → ∞ if the Arrhenius law held all the way. Physically it combines several factors: the vibrational attempt frequency of the diffusing atom (typically 10¹³ Hz), the square of the jump distance (a few tenths of a nanometre), a geometric factor that counts the number of equivalent jump directions, and an entropy factor exp(ΔS_m/R) that accounts for the change in vibrational entropy when the atom moves through the saddle point. For interstitial solutes in metals D₀ is typically 10⁻⁷–10⁻⁵ m²/s; for substitutional self-diffusion it is 10⁻⁵–10⁻⁴ m²/s; for oxygen in silicon it reaches 10⁻¹ m²/s because of a large entropy term.

Take the natural logarithm of both sides of the Arrhenius equation: ln D = ln D₀ − Q/(R·T). Plot ln D on the y-axis against 1/T on the x-axis. The data points should fall on a straight line; fit a line and read off the slope m and intercept b. Then Q = −m·R and D₀ = exp(b). It is best practice to use at least four or five temperatures spanning a wide range — at least 200 K — because the slope estimate is sensitive to how much the x-range varies. If the plot curves, the mechanism is changing across the temperature range and a single Arrhenius fit is not appropriate. This calculator's Arrhenius plot reproduces the linearised form in real time.

In interstitial diffusion, small atoms — H, C, N, O — sit in the gaps between the host lattice atoms and hop directly from one interstitial site to the next. The barrier they must overcome is the migration energy alone, typically 10–40 kJ/mol for H and 80–150 kJ/mol for C and N in steel. In substitutional diffusion, an atom occupies a regular lattice site and can only move if a vacancy is next to it — the atom swaps places with the vacancy. The total barrier is Q = Q_formation + Q_migration, typically 200–280 kJ/mol for metallic self-diffusion. The extra formation energy makes substitutional diffusion orders of magnitude slower than interstitial diffusion at the same temperature. The two mechanisms also differ in their concentration dependence: interstitial D is nearly independent of composition, while substitutional D depends on the vacancy concentration, which is itself temperature-dependent.

The characteristic diffusion distance over a time t is x ≈ √(D·t). This is the fundamental result of the diffusion equation, obtained by solving Fick's second law for a point source or step profile. It says that diffusion distance grows as the square root of both D and t — doubling the time increases the distance by only a factor of √2, and quadrupling the time doubles it. Combined with the exponential temperature dependence of D, this gives the two basic levers of a diffusion process. This calculator's Advanced panel reports the diffusion length at 1 s, 1 hour, and 24 hours for the current D, so you can quickly estimate how far a process will penetrate in a realistic time. For example, carburising steel at 950 °C for 4 hours gives a case depth of about 1 mm.

The simple Arrhenius form assumes a single, temperature-independent activation energy and a single diffusion mechanism. It breaks down when: (1) two or more mechanisms operate simultaneously, giving a curved Arrhenius plot; (2) the material undergoes a phase transformation within the temperature range, so the lattice and hence D₀ and Q change; (3) the temperature approaches the melting point, where the lattice becomes anharmonic and the activation energy itself becomes temperature-dependent; (4) diffusion is anisotropic, giving different D along different crystallographic directions; or (5) diffusion is enhanced by short-circuit paths — grain boundaries and dislocations — which have much lower Q and dominate at low temperatures. In all of these cases a modified or multi-mechanism model is needed.