MCEN90014 · Materials Engineering
Diffusion & Kinetics of Phase Transformation
Diffusion & Kinetics of Phase Transformation is the Week-6 rate-controlling half of MCEN90014 Materials Engineering at the University of Melbourne: the phase diagram tells you what forms, and diffusion tells you how fast. This chapter covers the two Fick laws, the error-function solution for non-steady diffusion, the Arrhenius temperature law for the diffusion coefficient, the mean-diffusion-distance estimate, and the interstitial-versus-vacancy mechanisms — the toolkit behind carburizing and case-hardening. It is one of the recurring quantitative topics on the exam's mix of phase diagrams, mechanical properties, strengthening, failure and materials thermodynamics.
What this chapter covers
- 01Distinguish interstitial diffusion (small C, N, H atoms hopping through voids) from vacancy/substitutional diffusion, and explain why interstitial is faster
- 02State Fick's 1st law J = -D dC/dx with correct units and the down-gradient sign convention
- 03Use the linear steady-state shortcut J = D(C1 - C2)/dx to solve for flux, gradient, concentration or thickness
- 04Recognise when a problem is steady-state (Fick 1) versus non-steady (Fick 2) and pick the right law
- 05Apply Fick's 2nd law and its semi-infinite error-function solution (Cx - C0)/(Cs - C0) = 1 - erf[x/(2 sqrt(Dt))]
- 06Read an error-function table and invert it to find z, then solve for a diffusion time or depth
- 07Compute the diffusion coefficient from the Arrhenius law D = D0 exp(-Qd/RT) with T in kelvin and Qd in J/mol
- 08Estimate how far atoms travel using the mean diffusion distance x = sqrt(Dt) and its square-root-of-time scaling
- 09Set up a carburizing / case-hardening calculation: temperature fixes D, then solve for the time to a target case depth
Diffusion coefficient and penetration depth via Arrhenius
- +1Convert the temperature to kelvin: T = 900 + 273 = 1173 K. Arrhenius always uses absolute temperature.
- +1Form the exponent: Qd/(RT) = 148000 / (8.314 x 1173) = 148000 / 9752 = 15.18.
- +1Diffusion coefficient: D = D0 exp(-15.18) = 2.3 x 10^-5 x 2.57 x 10^-7 = 5.9 x 10^-12 m^2/s.
- +1For how far atoms travel, use the mean diffusion distance x = sqrt(Dt) with t = 1 hour = 3600 s.
- +1Compute the product Dt = 5.9 x 10^-12 x 3600 = 2.12 x 10^-8 m^2.
- +1Take the square root: x = sqrt(2.12 x 10^-8) = 1.46 x 10^-4 m, i.e. about 0.15 mm.
Key terms
- Diffusion flux (J)
- The amount of a species crossing unit area per unit time, in kg/m^2/s or mol/m^2/s. At steady state it is constant everywhere and given by Fick's 1st law, J = -D dC/dx.
- Fick's 1st law
- The steady-state law J = -D dC/dx: flux is proportional to the concentration gradient, with the minus sign showing atoms flow from high to low concentration. Valid only when the profile does not change with time.
- Fick's 2nd law
- The non-steady law dC/dt = D d2C/dx2, describing how a concentration profile evolves in time. For a semi-infinite solid with a fixed surface concentration its solution is the error function.
- Error-function solution
- (Cx - C0)/(Cs - C0) = 1 - erf[x/(2 sqrt(Dt))], where Cs is the held surface concentration, C0 the initial bulk value, and Cx the value at depth x after time t. The single argument z = x/(2 sqrt(Dt)) sets the whole profile.
- Diffusion coefficient (D)
- The proportionality constant in Fick's laws, in m^2/s, measuring how mobile a species is. It rises steeply with temperature via the Arrhenius law D = D0 exp(-Qd/RT).
- Activation energy (Qd)
- The energy barrier an atom must overcome to make a diffusive jump, in J/mol. Interstitial atoms have a lower Qd (and larger D) than substitutional atoms, so they diffuse faster.
- Mean diffusion distance
- The characteristic penetration depth after time t, x = sqrt(Dt). It captures the key scaling that diffusion depth grows with the square root of time.
- Carburizing / case-hardening
- A surface heat treatment that diffuses carbon into a low-carbon steel at temperature (in the FCC gamma field) to create a hard, wear-resistant case over a tough core. Its design equation is the error-function solution.
Diffusion & Kinetics of Phase Transformation FAQ
How do I know whether to use Fick's 1st or 2nd law?
Ask whether the concentration profile is changing with time. If both faces are held at fixed concentrations so the flux is constant (for example a membrane), the profile is steady and you use Fick's 1st law, J = -D dC/dx. If the concentration at a point is still evolving - a part being carburized, a diffusion couple homogenising - the problem is non-steady and you need Fick's 2nd law and its error-function solution. Picking the wrong law is the biggest structural error on a diffusion question.
Why does interstitial carbon diffuse so much faster than substitutional alloying atoms?
A small interstitial atom sits in the voids between host atoms and is surrounded by empty space, so it can jump without waiting for anything and climbs a low energy barrier. A substitutional atom occupies a lattice site and can only move when a neighbouring site happens to be vacant, which is rare, so its activation energy is higher. Because Qd sits in the exponent of D = D0 exp(-Qd/RT), that higher barrier makes substitutional diffusion orders of magnitude slower at the same temperature.
Can AI help me with diffusion and kinetics in MCEN90014?
Yes, within the University of Melbourne's academic-integrity rules. An AI tutor like Sia can explain the difference between Fick's 1st and 2nd laws step by step, walk through how the Arrhenius law fixes the diffusion coefficient, and check your reasoning on a practice carburizing or mean-diffusion-distance question. It cannot sit your invigilated final exam or guarantee a mark, and MCEN90014's assessed work must be your own; always confirm on Canvas which tools are permitted.
Studying with AI? Sia — free AI mechanical engineering tutor works through MCEN90014 step by step.
Exam move
Treat this chapter as a decision tree followed by careful arithmetic. First decide steady versus non-steady: held-fixed faces mean Fick's 1st law and the linear shortcut J = D(C1 - C2)/dx, while an evolving profile means Fick's 2nd law and the error-function solution. For any error-function problem the routine never changes: evaluate (Cx - C0)/(Cs - C0), set it equal to 1 - erf(z), look up z in the table, then solve z = x/(2 sqrt(Dt)) for the unknown time or depth - keeping the factor of two and squaring sqrt(Dt) back to Dt. Get the diffusion coefficient from D = D0 exp(-Qd/RT) with temperature in kelvin and the activation energy in J/mol, and use x = sqrt(Dt) for quick penetration estimates. Because the final exam is worth 50%, is a hurdle you must pass, and provides a formula sheet, practise substituting cleanly and carrying units on every line so these become reliable, self-contained marks. Confirm the exam duration and format on Canvas.
