Fick’s Laws of Diffusion in Metals: First Law, Second Law and Applications

Diffusion is the thermally activated migration of atoms through a crystal lattice, and it is the rate-controlling mechanism behind carburizing, nitriding, homogenisation, sintering, and countless other metallurgical processes. This article develops Fick’s first and second laws from first principles, explains the Arrhenius temperature dependence of the diffusion coefficient, and applies the error-function solution to practical case-hardening problems.

Fick’s First Law: Steady-State Diffusion

Fick’s first law states that the diffusive flux of atoms is proportional to the local concentration gradient and acts down that gradient, from regions of high concentration toward regions of low concentration.

J = -D · (dC/dx)

where:
  J = diffusive flux (atoms or mass per unit area per unit time)
  D = diffusion coefficient (m²/s or cm²/s)
  dC/dx = concentration gradient (concentration per unit distance)
  (negative sign: flux runs opposite to the gradient, i.e. down-hill in concentration)

When Fick’s First Law Applies

Fick’s first law is strictly valid only under steady-state conditions, meaning the concentration profile C(x) does not change with time anywhere in the system. This condition is closely approximated in thin-membrane permeation problems, such as hydrogen diffusing through a steel wall of constant thickness under a constant upstream partial pressure once an initial transient has decayed, or in diffusion couples analysed far from their transient early stage. Most heat-treating processes of interest to metallurgists, however, are inherently transient and require Fick’s second law instead.

Diffusion Mechanisms and the Arrhenius Temperature Dependence

Vacancy (Substitutional) Diffusion

Substitutional atoms diffuse by exchanging position with an adjacent vacant lattice site. Because equilibrium vacancy concentration itself rises exponentially with temperature, and because the atom must additionally possess enough thermal energy to squeeze past its neighbours during the jump, substitutional diffusion coefficients carry a comparatively high activation energy and are markedly slower than interstitial diffusion at a given temperature.

Interstitial Diffusion

Small solute atoms such as carbon, nitrogen, boron, and hydrogen occupy interstitial sites, principally octahedral interstices in iron, and can jump directly to an adjacent interstitial site without requiring a vacancy. This removes one major kinetic barrier, so interstitial diffusion coefficients are typically several orders of magnitude larger than substitutional coefficients, which is precisely why carburizing (interstitial carbon diffusion) can build a useful case in hours, while homogenising a substitutionally alloyed casting can require many hours at high temperature to achieve comparable penetration.

where D0 is a pre-exponential (frequency) factor characteristic of the diffusing species and lattice, Q is the activation energy for diffusion (J/mol), R is the universal gas constant (8.314 J/mol·K), and T is absolute temperature (K). Because Q sits in the exponent, diffusion rate is exquisitely sensitive to processing temperature: raising a carburizing temperature by only 50°C can more than double the diffusion coefficient. This exponential relationship also governs the composition-dependent transformation kinetics discussed in the iron-carbon phase diagram and the diffusional transformations covered under the eutectoid reaction.

Values are representative literature averages for illustrative and educational use; production process design should reference certified data for the specific alloy grade and temperature range in service.

Fick’s Second Law: Non-Steady-State Diffusion

Combining Fick’s first law with a mass balance on a thin differential slab, requiring that any net flux imbalance across the slab must accumulate as a local concentration change with time, yields Fick’s second law:

This partial differential equation, not the simpler first law, is the correct starting point for essentially all case-hardening, homogenisation, and decarburization calculations, because the concentration profile in these processes evolves continuously with time.

Error Function Solution for a Semi-Infinite Solid

For the practically important case of a semi-infinite solid initially at uniform concentration C0, with its surface held at a constant concentration Cs from time zero onward (the boundary condition of carburizing and nitriding furnaces), Fick’s second law has a closed-form solution in terms of the Gaussian error function:

where Cx is the concentration at depth x and time t. Because the argument of the error function is x divided by 2√(Dt), any fixed target concentration Cx corresponds to a fixed value of x/2√(Dt); therefore, the depth at which that concentration is reached scales with √(Dt), i.e. with the square root of time at fixed temperature. This is the origin of the well-known heat-treater’s rule of thumb that doubling case depth requires roughly quadrupling treatment time.

Grain Boundaries, Dislocations and Short-Circuit Diffusion Paths

The Arrhenius relation above describes lattice (volume) diffusion through the bulk crystal, but atoms also diffuse faster along grain boundaries, free surfaces, and dislocation cores, collectively called short-circuit diffusion paths, because the locally disordered atomic packing lowers the effective activation energy. Grain boundary diffusion coefficients can exceed lattice diffusion coefficients by three to six orders of magnitude at a given temperature, though the effect on overall mass transport depends on the fraction of the diffusion path actually occupied by boundaries or dislocations, so grain boundary diffusion becomes progressively more important, relative to lattice diffusion, as temperature decreases and as grain size decreases. This is one reason fine-grained microstructures (see the grain boundaries guide) can show accelerated diffusion-controlled degradation phenomena at temperatures where lattice diffusion alone would be negligible.

Industrial Applications and Significance

Carburizing and nitriding are the most direct industrial applications of the error-function solution developed above: carburizing builds a hard, wear-resistant, compressively stressed case by diffusing carbon into austenite followed by quenching and tempering, while nitriding diffuses nitrogen directly into ferrite at lower temperature to form hard alloy nitrides with minimal distortion. Homogenisation heat treatments rely on the same second-law framework, but in reverse, using extended high-temperature soaks to flatten dendritic microsegregation left over from casting solidification through long-range substitutional diffusion. In welding metallurgy, diffusion-controlled processes govern hydrogen redistribution and the risk of hydrogen-induced cracking in the heat-affected zone (see also HAZ microstructure), where post-weld heat treatment or controlled cooling is specified explicitly to allow diffusible hydrogen time to escape before it can concentrate at susceptible microstructural sites. Diffusion bonding, sintering of powder metallurgy compacts, and the decarburization that must be controlled during annealing and normalising of high-carbon and spring steels are further processes governed directly by the same first- and second-law framework.

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