Structure & Transformations · Diffusion

Diffusion couples and Fick's laws

Press play in the first picture and watch atoms hop across the join. Then slide time: the counted profile follows the erf curve, and its width grows as the square root of time.

How does random hopping of atoms give Fick's law?

Walk
D = D0 exp(-Q/RT)
Diffusion distance sqrt(Dt)
Time until sqrt(Dt) = 1 µm
Time until sqrt(Dt) = 1 mm
Jumps per atom (0.25 nm each)
Total path walked
D0, Q used (Smithells data, via Callister Table 5.2)

Try it: pick Zn in Cu at 785 °C. At 1 h sqrt(Dt) is 6.4 µm. Set 4 h: it becomes 12.7 µm. Four times the time gives twice the width.

How do you get D from a profile when D changes with composition?

Matano plane x_M
Each shaded area
Slope dc/dx at c*
Integral of (x - x_M) dc
D put in at c*
D recovered at c*
Difference

Try it: set the ratio to 100. The left side, where D is high, spreads much further, but x_M stays within 0.1 µm of 0. Move c* from 0.1 to 0.9: the recovered D follows the input line within about 3 %.

Why do inert markers move in a diffusion couple?

Animation
D_A (intrinsic)
D_B (intrinsic)
Darken D~ at X_A = 0.5
Marker shift
Marker speed now
Shift / sqrt(t)
Markers move toward

Try it: set D_A / D_B to 2.5 and press play. The markers move left, into the A side, which diffuses faster, and pores appear there. At 25 h the shift is half of the shift at 100 h. Set the ratio to 1: nothing moves.

How does instrument resolution change the measured D?

Typical instruments
True D
D from fitting the blurred profile
Too high by
Need sqrt(Dt) above (for 5 %)

Try it: press SEM-EDS and set the true sqrt(Dt) to 1 µm: the fitted D is 1.13 times too high. Drop sqrt(Dt) to 0.1 µm: it is 13.5 times too high. Press APT: the error is gone (1.00 times).

What should you remember?

Width grows as sqrt(Dt)Random hops with no preferred direction give Fick's law. Four times the time only doubles the mixed zone.
D rises steeply with TD = D0 exp(-Q/RT). For Cu in Cu, going from 800 °C to 1000 °C raises D about 40 times.
Matano works for any D(c)Locate the plane of equal areas, then D(c*) comes from one slope and one area on the measured profile.
Markers show unequal fluxesIf A moves faster than B, the lattice shifts toward A and vacancies gather there as pores.
More detail: equations, assumptions and limits

Fick's first law J = -D dc/dx: atoms flow down the concentration gradient. Second law dc/dt = d/dx(D dc/dx): what flows out of a slice lowers its content.

Couple solution (constant D, two long bars joined at x = 0): c = (c1+c2)/2 - (c1-c2)/2 · erf(x / (2 sqrt(Dt))). Widget 1 uses c1 = 1 (pure A) and c2 = 0.

Random walk. Each site swaps with a random neighbour. The average of such swaps obeys the discrete diffusion equation, with D = 1/2 cell² per sweep here. For atoms, D = Γa²/6 in 3D, with Γ the jump rate and a the jump distance. The jump count shown uses a = 0.25 nm, a typical near-neighbour distance; the real distance depends on the lattice and the site.

Arrhenius data. D0 and Q are the Smithells values reproduced in Callister Table 5.2. They are fits over a limited range. C in alpha-Fe is only meaningful below 912 °C, where iron is BCC, and C in gamma-Fe only above 912 °C, where iron is FCC (a steel can hold austenite down to 727 °C). None of the pairs is used above the melting point of the host. The temperature slider is limited to sensible ranges.

Boltzmann-Matano. With c(x) depending only on x/sqrt(t), D(c*) = -(1/2t) (dx/dc)c* ∫c2c* (x - xM) dc, with xM set by equal areas. Widget 2 solves Fick's second law by finite differences (600 cells, explicit, flux-conserving) with D(c) = D(0)·rc, D(0) = 10-15 m²/s, and then runs Matano on the result. It assumes constant molar volume, so xM equals the original interface.

Darken. D~ = XBDA + XADB, marker speed v = (DA - DB) dXA/dx. Widget 3 keeps DA and DB constant, DB = 10-14 m²/s, and ignores the thermodynamic factor. Pores are drawn where the vacancy flux converges, (DA - DB) times the curvature of XA. Their count is scaled to be seen. In real couples most extra vacancies are absorbed by dislocations and boundaries.

Probe blur. An erf profile blurred by a Gaussian of sigma w is still an erf, with 4Dt replaced by 4Dt + 2w². So a fit that ignores blur gives Dfit = D + w²/(2t). Real probes are not exactly Gaussian, and APT resolution differs along and across the analysis direction.

Common questions about diffusion couples

What is a diffusion couple?

Two blocks of different composition pressed or bonded together and annealed. Atoms cross the join. Afterwards you measure composition along a line across it and read D from that profile.

What is the difference between Fick's first and second law?

The first law gives the flow of atoms from the slope of concentration. The second law says how concentration changes with time, because a slice gains what flows in minus what flows out.

How far do atoms diffuse in a given time?

About sqrt(Dt). For Zn in Cu at 785 °C, D is about 1.1 × 10-14 m²/s, so 1 hour gives about 6 µm. Use widget 1 for other metals and temperatures.

What is the Matano plane?

The position where the atoms that left one side equal the atoms that arrived on the other: the two shaded areas in widget 2 are equal. Distances in the Boltzmann-Matano formula are measured from it.

What is the Kirkendall effect?

When two kinds of atoms swap places with vacancies at different rates, more atoms cross one way than the other. Inert markers at the join move toward the faster side, and pores can form there. Smigelskas and Kirkendall showed it in 1947 with Mo wires between brass and copper.

What is the difference between intrinsic and interdiffusion coefficients?

Intrinsic DA and DB describe each species moving relative to the lattice. The interdiffusion coefficient D~ = XBDA + XADB describes the composition profile you measure. Matano gives D~; you need the marker speed as well to get DA and DB.

Why does my measured profile look wider than it should?

The probe averages over a region. If that blur is not small compared with sqrt(Dt), the fitted D comes out too high by w²/(2t). Widget 4 shows how large this is for APT and EDS.

Why does diffusion depend so strongly on temperature?

An atom must get over an energy barrier to jump, and the chance of that rises as exp(-Q/RT). A Q near 200 kJ/mol means a 100 °C rise near 900 °C gives several times faster diffusion.

Where does this connect on the site?

TTT, CCT and carburizing case depthGrain growth and Zener pinningDiffusion creep and deformation mapsAPT reconstruction and resolutionEDS quantification in the TEMSpinodal, nucleation and coarsening

Sources

Show the 12 references
  1. D. A. Porter, K. E. Easterling, M. Y. Sherif, Phase Transformations in Metals and Alloys, 3rd ed., CRC Press, 2009, Chapter 2 (Diffusion).
  2. P. Shewmon, Diffusion in Solids, 2nd ed., Springer (TMS series), 2016. doi:10.1007/978-3-319-48206-4.
  3. J. Crank, The Mathematics of Diffusion, 2nd ed., Clarendon Press, Oxford, 1975.
  4. H. Mehrer, Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes, Springer, 2007. doi:10.1007/978-3-540-71488-0.
  5. A. D. Smigelskas, E. O. Kirkendall, Zinc diffusion in alpha brass, Trans. AIME 171 (1947) 130-142.
  6. L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME 175 (1948) 184-201.
  7. C. Matano, On the relation between the diffusion-coefficients and concentrations of solid metals (the nickel-copper system), Japanese Journal of Physics 8 (1933) 109-113.
  8. L. Boltzmann, Zur Integration der Diffusionsgleichung bei variabeln Diffusionscoefficienten, Annalen der Physik 289 (1894) 959-964.
  9. W. F. Gale, T. C. Totemeier (eds.), Smithells Metals Reference Book, 8th ed., Elsevier, 2004, Chapter 13 (Diffusion in metals).
  10. W. D. Callister, D. G. Rethwisch, Materials Science and Engineering: An Introduction, Wiley, Table 5.2 (diffusion data from Smithells).
  11. D. J. Larson, T. J. Prosa, R. M. Ulfig, B. P. Geiser, T. F. Kelly, Local Electrode Atom Probe Tomography: A User's Guide, Springer, 2013. doi:10.1007/978-1-4614-8721-0.
  12. J. I. Goldstein et al., Scanning Electron Microscopy and X-Ray Microanalysis, 4th ed., Springer, 2018. doi:10.1007/978-1-4939-6676-9.
Cite this page: Tripathy, Manisha. “Diffusion Couples and Fick's Laws: Random Walk to Kirkendall.” untethered atom, 2026, https://untetheredatom.com/phase-transformations/diffusion-couples.
BibTeX 
@misc{tripathy2026diffusioncouples,
  author = {Tripathy, Manisha},
  title  = {Diffusion Couples and Fick's Laws: Random Walk to Kirkendall},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/phase-transformations/diffusion-couples}},
  note   = {Interactive web tool}
}