Minus sign indicates the diffusion is down the concentration gradient Negative
Fick’s first law
Mathematics of Diffusion (4)
• Nonsteady-stDiffusion in Solids
Diffusion - Introduction
• A phenomenon of material transport by atomic migration The mass transfer in macroscopic level is implemented by the motion of atoms in microscopic level • Self-diffusion and interdiffusion (or impurity diffusion) • Topics: mechanisms of diffusion, mathematics of diffusion, effects of temperature and diffusing species on the rate of diffusion, and diffusion of vacancy-solute complexes
Position of interstitial atom after diffusion
The activation energy is the migration energy of the interstitial atom.
Mathematics of Diffusion (1)
• Steady-state diffusion
– Time-dependent process, the rate of mass transfer is expressed as a diffusion flux (J)
M J At
In differential form
Mass transferred through a crosssectional area Diffusion time Area across which the diffusion occurs
dC J D dx
For this case, the unit of C is in mass per unit volume, e.g. g/m3
where D is the diffusion coefficient (m2/s), showing the rate of diffusion
The diffusion flux at a particular point varies with time.
Mathematics of Diffusion (5)
The diffusion equation is represented by
C C (D ) t x x
e.g. the diffusion of atoms of a gas through a metal plate
concentration gradient = dC/dx
dC/dx
J
Mathematics of Diffusion (3)
For steady-state diffusion, the diffusion flux is proportional to the concentration gradient The mathematics of steady-state diffusion in one dimension is given by
For self-diffusion, the activation energy is vacancy formation energy + vacancy migration energy.
Diffusion – Mechanisms (3)
• Interstitial diffusion In interstitial solid solutions, the diffusion of interstitial solute atoms is the migration of the atoms from interstitial site to interstitial site
Demonstration of diffusion
Before heat treatment
After heat treatment
Diffusion – Mechanisms (1)
Two mechanisms:
• Vacancy diffusion • Interstitial diffusion
1 dM J A dt
J = Mass transferred through a unit area per unit time (g/m2 s))
Mathematics of Diffusion (2)
Concentration profile does not change with time – steady-state diffusion
Fick’s second law C is a function of x and t If D is independent of the composition, the above equation changes to
C 2C D 2 t x
Unit area cross-section C = mass per unit volume (concentration)
Diffusion – Mechanisms (2)
• Vacancy diffusion In substitutional solid solutions, the diffusion (both self-diffusion and interdiffusion) must involve vacancies