Solid State
The Kirkendall Effect: Unequal Diffusion and Void Formation
In 1947, Ernest Kirkendall and his student Alice Smigelskas sealed a brass bar wrapped in 127-micrometer molybdenum wires inside a block of copper and baked it at 785 °C for up to 56 days. When they measured the wires afterward, the two marker planes had crept 0.1 mm closer together. Inert wires cannot move on their own, so something inside the metal was moving them: zinc atoms were leaving the brass faster than copper atoms were entering it.
The Kirkendall effect is the shift of an interface (or of inert markers embedded in it) that occurs when two species in a diffusion couple move at unequal rates. Because atomic motion in substitutional alloys proceeds by the vacancy mechanism, unequal fluxes produce a net flux of vacancies in the opposite direction. Those vacancies must be created on one side and destroyed on the other; when they instead supersaturate and coalesce, they nucleate microscopic voids known as Kirkendall porosity.
- TypeSolid-state diffusion / interface phenomenon
- IntroducedSmigelskas & Kirkendall, 1947; theory by Darken, 1948
- Governing equationv = (D_A - D_B)·(∂C_A/∂x)/C (marker/Kirkendall velocity)
- Classic systemCu / alpha-brass (Cu-Zn); D_Zn > D_Cu
- Applies toSubstitutional alloys, solder joints, coatings, hollow nanocrystals
- Measured byInert markers (Mo, W, ThO2, oxide films) tracked by microscopy
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What the Kirkendall Effect Is and Where It Shows Up
The Kirkendall effect is the observation that in a diffusion couple — two materials joined and heated so their atoms interpenetrate — the two chemical species generally do not move at the same rate. When the flux of one atom (say Zn) outpaces the flux of the other (Cu), the crystal lattice itself must drift to compensate. Inert markers embedded at the original interface ride along with the lattice and appear to move relative to the ends of the sample.
This matters far beyond the original brass experiment. It governs:
- Solder and microelectronics: voids at Cu/Sn joints and the 'purple plague' in Au/Al wire bonds are Kirkendall failures.
- Protective coatings and cladding: unequal diffusion undermines the bond line.
- Nanomaterials: the same imbalance is deliberately exploited to grow hollow oxide, sulfide, and selenide nanocrystals.
The key insight is that in substitutional alloys atoms move by swapping places with vacancies, not by direct exchange — so unequal atomic fluxes necessarily create an equal-and-opposite vacancy flux.
The Mechanism and the Darken Derivation, Step by Step
Step 1 — Two intrinsic fluxes. Each species obeys Fick's first law in the lattice frame: J_A = -D_A·(∂C_A/∂x) and J_B = -D_B·(∂C_B/∂x), where D_A and D_B are the intrinsic diffusion coefficients and may differ.
Step 2 — Conservation demands a vacancy flux. Since C_A + C_B = C is roughly constant, ∂C_A/∂x = -∂C_B/∂x. If D_A > D_B the atom fluxes do not cancel; the imbalance is a net vacancy flux J_V = -(J_A + J_B) moving opposite to the fast species.
Step 3 — The lattice drifts. Vacancies are created on one side and annihilated on the other (at dislocations, grain boundaries, surfaces). This makes the lattice plane move with a marker (Kirkendall) velocity:
v = (D_A − D_B)·(∂C_A/∂x) / C
Step 4 — Darken (1948) combined drift and diffusion into a single laboratory-frame interdiffusion coefficient:
D̃ = X_A·D_B + X_B·D_A
where X_A, X_B are mole fractions. Where vacancy sinks are too few to keep up, vacancies supersaturate and condense into voids — the porosity Kirkendall porosity.
Key Quantities and a Worked Example
Define every symbol: D_A, D_B are intrinsic diffusivities (m²/s); D̃ is the interdiffusion coefficient; X_A, X_B mole fractions; C the total molar concentration (mol/m³); v the marker velocity (m/s); ∂C_A/∂x the concentration gradient (mol/m⁴).
Characteristic diffusion length scales as x ≈ √(D̃·t). At 785 °C, interdiffusion in Cu-Zn has D̃ on the order of 1 × 10⁻¹³ m²/s.
Worked example: For t = 56 days ≈ 4.8 × 10⁶ s, x ≈ √(1 × 10⁻¹³ × 4.8 × 10⁶) ≈ √(4.8 × 10⁻⁷) ≈ 6.9 × 10⁻⁴ m ≈ 0.7 mm — matching the depth over which composition changed in the original couple, and consistent with a marker shift of roughly 0.05-0.1 mm.
- Marker shift Δx ∝ √t (parabolic), the fingerprint of a diffusion-controlled process.
- Activation energy for vacancy diffusion in Cu-Zn is ~180-200 kJ/mol, so D̃ roughly doubles per 30-40 °C near 785 °C via the Arrhenius law D̃ = D₀·exp(−Q/RT).
How It Is Measured and Used in Practice
Marker experiments. The canonical measurement embeds chemically inert, insoluble markers at the interface — Kirkendall used 127 µm molybdenum wires because Mo is essentially insoluble in brass. Alternatives include tungsten, ThO₂ particles, or a thin oxide film. After annealing, the separation of the two marker planes is measured by optical or electron microscopy; plotting shift versus √t confirms the parabolic law and yields v.
Extracting coefficients. Combining the measured marker velocity with the Boltzmann–Matano analysis of the concentration profile lets you solve simultaneously for D_A, D_B, and D̃ — this is the standard route to individual intrinsic diffusivities.
Engineering use. Reliability engineers model Kirkendall void growth to predict lifetimes of solder joints and wire bonds; diffusion barriers (e.g., Ni between Cu and Sn) are inserted precisely to suppress it. In nanochemistry the effect is a synthetic tool: Yin and Alivisatos (Science, 2004) reacted Co nanocrystals with O, S, or Se and let the fast outward Co flux hollow the particles into oxide/chalcogenide nanoshells.
How It Differs From Its Close Cousins
The Kirkendall effect is easy to confuse with related diffusion phenomena. The distinctions:
- Self-diffusion / tracer diffusion: movement of atoms in a chemically uniform crystal (no gradient). There is no marker motion and no void formation — it is the baseline the Kirkendall effect departs from.
- Interdiffusion (D̃): the net mixing described by a single coefficient in the lab frame. The Kirkendall effect is what reveals that this single D̃ actually hides two unequal intrinsic coefficients.
- Frenkel effect / nanoscale Kirkendall: the same vacancy-condensation mechanism, but the resulting single central void in a nanoparticle is sometimes called the Frenkel void.
- Void swelling / creep: vacancy phenomena driven by irradiation or stress, not by a chemical concentration gradient.
The defining signatures of a true Kirkendall process are: (1) a chemical gradient, (2) unequal intrinsic diffusivities, (3) marker drift ∝ √t, and (4) an accompanying vacancy flux that can nucleate porosity.
Exceptions, Significance, and Famous Cases
Historical significance: Kirkendall's result was initially rejected. Robert Mehl, a leading metallurgist and editor, refused to accept a mechanism implying atoms and vacancies rather than direct place-exchange, and the paper was contested for years. Vindication came when Darken's 1948 analysis showed the data demanded the vacancy mechanism — a turning point that established vacancies as the carriers of substitutional diffusion and effectively founded modern diffusion theory.
Exceptions and limits:
- In interstitial systems (C, N, H in iron) the small atom moves without vacancies, so the classic marker effect is muted.
- Abundant vacancy sinks (dislocations, incoherent boundaries) can annihilate vacancies fast enough that no voids form, only a marker shift.
- Volume changes and lattice-parameter mismatch add drift terms beyond the simple Darken velocity.
Famous cases: the Au/Al 'purple plague' that plagued early semiconductor packaging; Cu₃Sn/Cu₆Sn₅ voiding in lead-free solder; and the deliberate synthesis of hollow Co₃O₄, CoS, and Pt–CoO yolk-shell nanoreactors — turning a century-old failure mode into designer nanostructures.
| Diffusion couple | Faster species | Slower species | Consequence |
|---|---|---|---|
| Cu / alpha-brass (785 °C) | Zn (D ~ 3-4 × faster) | Cu | Markers shift toward Cu; ~0.05-0.1 mm over weeks |
| Cu / Ni (1000 °C) | Cu (D_Cu > D_Ni) | Ni | Markers shift toward Ni; classic Kirkendall porosity on Cu side |
| Au / Al (thin film, 200-300 °C) | Al into Au | Au | Al-rich phases + 'purple plague' voids in wire bonds |
| Cu / Sn solder (150 °C) | Cu | Sn | Cu6Sn5 / Cu3Sn IMC growth + voids at joint |
| Co nanocrystal + O/S (nanoscale) | Co (outward) | O or S (inward) | Hollow oxide/sulfide nanoshell forms |
Frequently asked questions
Why do the markers move if they are inert?
The markers themselves don't diffuse — they are pinned to the crystal lattice. When zinc leaves the brass faster than copper enters, the lattice must shed planes on the fast-diffusing side and add them on the other, so the whole lattice (and the embedded markers) drifts. The markers are simply witnesses to lattice motion driven by an unequal atomic flux.
What is the difference between intrinsic and interdiffusion coefficients?
Intrinsic coefficients D_A and D_B describe each species' motion relative to the moving lattice and are generally unequal — that inequality is the Kirkendall effect. The interdiffusion coefficient D̃ = X_A·D_B + X_B·D_A (Darken's equation) describes the net mixing in the fixed laboratory frame and is what you measure from a concentration profile alone.
Why does the Kirkendall effect create voids?
The unequal atomic fluxes generate a net flux of vacancies toward the fast-diffusing side. Those vacancies must be destroyed at sinks like dislocations or boundaries. If they are created faster than the sinks can absorb them, they supersaturate and condense into microscopic pores — Kirkendall porosity — which weaken solder joints and bonds.
How is the Kirkendall shift related to time?
The marker displacement grows as the square root of annealing time, Δx ∝ √t, the hallmark of a diffusion-controlled process. This parabolic dependence lets researchers confirm the mechanism and extract the marker velocity v and coefficients by plotting shift against √t.
What was actually measured in the 1947 experiment?
Smigelskas and Kirkendall placed 127 µm molybdenum wires on a brass bar, electroplated copper over it, and annealed at 785 °C for up to 56 days. They tracked the separation of the two wire planes and found it decreased with √t, proving zinc diffused out of the brass faster than copper diffused in.
How is the effect used to make hollow nanoparticles?
At the nanoscale (Yin and Alivisatos, 2004), a solid metal nanocrystal such as cobalt is reacted with oxygen, sulfur, or selenium. The metal diffuses outward faster than the nonmetal diffuses inward, so vacancies pile up in the core and coalesce into a central void, leaving a hollow oxide or chalcogenide shell — a designer nanoreactor built from a former failure mode.