E6 Atomic Diffusion

Topic

The temperature of this metal determines—following the same Arrhenius relationship that governs the equilibrium vacancy concentration and atomic diffusion coefficient within the metal's lattice in another model of this set—the diffusion coefficient of atoms within that lattice; indeed, it is these very vacancies that allow a neighboring atom to jump from its own lattice site to an adjacent empty one, and the higher the metal's temperature, the more vacancies are available and the more thermal energy each atom possesses to jump into them. Consequently, a higher metal temperature results in a higher diffusion coefficient, in accordance with this same Arrhenius relationship.

This diffusion coefficient governs—according to Fick's second law—how the concentration of a solute within the metal evolves over space and time in regions where the concentration is non-uniform; Fick's second law describes how the solute concentration at any point in the metal changes over time in proportion to the curvature of the concentration profile at that point, progressively smoothing out concentration differences until uniformity is reached. A higher diffusion coefficient causes this smoothing of the concentration profile to proceed more rapidly, in accordance with Fick's second law.

In the specific case of semi-infinite diffusion—where the metal is placed in contact at a single surface with a solute source maintained at a constant concentration—Fick's second law is solved using the error function; this function describes, for any given point at a certain depth below the surface and at any moment in time, what fraction of the original concentration difference has diffused to that point. A longer elapsed time or a higher diffusion coefficient results—according to the error function—in the solute diffusing to a greater depth within the metal over that same period. For this reason, this model—along with the diffusion coefficient and Fick's second law, with which it was introduced—governs the kinetics of four other models in this set: quenching and tempering, precipitation, recrystallization, and sintering. None of these processes could proceed at any specific rate without this same diffusion coefficient as a foundation. Furthermore, the fact that this diffusion coefficient is the very parameter used by another model in the set to describe the metal's diffusion-controlled creep—and by yet another to describe how dissolved hydrogen reaches a crack tip—establishes this model as the physical basis upon which those two models already relied.

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