Structure & Transformations · Grain boundary migration
Grain boundary migration: what moves a boundary and how fast
A grain boundary is the wall where two crystals in a metal meet; push one and watch it move.
What pushes a grain boundary to move?
Speed = mobility × push (v = M·P). Pick the push: curvature, or a clean grain eating a deformed one.
Try it: drag R from 10 to 1 µm: the push rises tenfold and the grain vanishes in 0.17 s, not 17 s.
Does a grain boundary always move faster when it is hotter?
No: in computer simulations of 388 nickel boundaries, about 1 in 5 kept the same speed or slowed down when hotter.
I rises, II flat or falls, III mixed (14%), IV unclear (9%)
Try it: press Slows down and drag the temperature up: the boundary moves slower.
How does a grain boundary move by shear?
A small step on the boundary (a disconnection) glides along it: the boundary rises by h and the top grain slides sideways by b.
Try it: at 36.9° switch to the <110> mode: β flips from +0.67 to −1.00 and the top grain shifts the other way.
What should you remember?
Speed = mobility × push
v = M·P. Halving a grain's radius doubles its curvature push.
Hotter is not always faster
Of 388 simulated Ni boundaries, about 57% sped up when hotter; about 20% stayed flat or slowed.
Many boundaries move by shear
A gliding disconnection lifts the boundary by h and slides the grain by b; β = b/h.
More detail: the equations, the numbers and their limits
Speed and curvature
The boundary speed is v = M·P, with M the mobility and P the driving pressure (Suo, lecture notes). For curvature, P = γK, where K is the sum of the two principal curvatures, K = 1/R₁ + 1/R₂. A sphere has K = 2/R, so P = 2γ/R; the page uses this. A long cylinder, or a circle in a 2D model, has K = 1/R, so P = γ/R.
Shrink time
For one isolated round grain with fixed M and γ, dR/dt = −2Mγ/R, so it vanishes after t = R₀²/(4Mγ). For a cylinder or 2D circle the same steps give t = R₀²/(2Mγ), twice as long. This is a derivation from the two lines above, not a grain growth law for a real polycrystal.
Stored energy
Florez, Alvarado and Bernacki write the stored energy as E = ½μb²ρ, with μ the shear modulus and b the Burgers vector length. The push on a boundary is the energy difference, P = τΔρ, with τ = ½μb². Raabe gives dislocation densities of 10¹⁰ to 10¹¹ m⁻² annealed, 10¹² to 10¹³ m⁻² after modest deformation and up to 10¹⁶ m⁻² after very heavy deformation, and stored energies of about 1 to 10 MPa.
The numbers used
Parameters given by Florez and co-workers as representative of 304L stainless steel: M₀ = 1.56 × 10¹¹ mm⁴/(J·s), Q = 2.8 × 10⁵ J/mol, T = 1353 K, γ = 0.6 J/m², τ = 1.28331 × 10⁻⁹ J/m. With M = M₀ exp(−Q/RT) this gives M = 2.41 × 10⁻¹² m⁴/(J·s). Here R = 8.314 J/(mol·K) is the gas constant, because Q is per mole; with Q per atom the same law uses Boltzmann's constant kB. The "about 1 J/m²" button is Raabe's value for a new high-angle boundary. The 388 Ni boundaries of widget 2 show that mobility differs from one boundary to another.
Mobility against temperature
Mobility is often fit to M = M₀ exp(−Q/kBT). Homer and co-workers sorted the simulated mobilities of 388 Ni boundaries into four classes: thermally activated (about 57%), independent of temperature or falling with temperature (about 20%), mixed trends over different temperature ranges (about 14%) and unclassifiable or immobile (about 9%). The curve shapes in widget 2 are schematic, with no numbers on the axes.
Why can mobility fall?
Olmsted, Foiles and Holm described one boundary whose motion looked like barrier-free motion with damping that grows with temperature. Chen, Han and Srolovitz model migration by the formation and glide of disconnections; when several disconnection modes with different activation energies compete, mobility can rise, fall, stay flat or show peaks.
Coupling factor
Cahn, Mishin and Suzuki write the grain translation rate as v∥ = βvn + vs, where vn is the boundary speed normal to itself and vs is sliding. For [001] symmetric tilt boundaries their geometric model gives β = 2 tan(θ/2) for the <100> mode; Trautt, Adland, Karma and Mishin write the <110> mode as β = 2 tan(θ/2 − 45°), the same as −2 tan(45° − θ/2), with β = v∥/vn. In simulations the <100> mode was seen at small tilt angles and the <110> mode at large ones. The picture draws positive β as a shift to the right; which way counts as positive is a choice of axes. In the copper simulations of Cahn, Mishin and Suzuki β switched branch near 35°, and at high temperature high-angle boundaries changed from coupling to sliding.
Disconnections
Thomas and co-workers give β = b/h for one disconnection. A boundary can have a series of disconnection modes (b, h), so the β you see is an average over the modes that operate.
Questions people ask
What is grain boundary mobility?
Mobility M is the boundary speed per unit driving pressure, from v = M·P. Its unit is m⁴/(J·s), which is the same as m/s per Pa. It is often fit to an Arrhenius law, M = M₀ exp(−Q/kBT), but simulations show many boundaries that do not follow it.
What is antithermal grain boundary mobility?
It is mobility that falls as temperature rises. In the Homer and co-workers survey of 388 simulated Ni boundaries, it sits in a class of about 20% together with mobility that does not change with temperature. One proposed picture is barrier-free motion with damping that grows with temperature.
What is shear-coupled grain boundary motion?
The boundary moves normal to itself and, at the same time, one grain slides past the other. Cahn, Mishin and Suzuki showed this in copper simulations, and showed that a shear stress on the boundary can drive it. The ratio of grain translation to boundary motion is the coupling factor β.
What is a disconnection?
A disconnection is a line defect that lives in a grain boundary. It is both a step, of height h, and a dislocation, with Burgers vector b. When it glides along the boundary, the boundary moves by h and the grains shift by b.
Is boundary speed always proportional to curvature?
Classical grain growth models assume v = Mγκ. Thomas and co-workers showed in Ni simulations that this misses effects such as grain rotation, stress build-up and boundaries that stop. Boundaries can keep moving by switching between disconnection modes.
On this site: Interfaces and the coincidence site lattice · Grain growth and Zener pinning · Sliding-induced microstructure · Dislocations and Burgers vectors
References
Show the 10 references
- J. W. Cahn, Y. Mishin and A. Suzuki, Coupling grain boundary motion to shear deformation, Acta Materialia 54, 4953 to 4975 (2006). doi:10.1016/j.actamat.2006.08.004
- Z. T. Trautt, A. Adland, A. Karma and Y. Mishin, Coupled motion of asymmetrical tilt grain boundaries: molecular dynamics and phase field crystal simulations, arXiv:1206.3549 (2012).
- S. L. Thomas, K. Chen, J. Han, P. K. Purohit and D. J. Srolovitz, Reconciling grain growth and shear-coupled grain boundary migration, Nature Communications 8, 1764 (2017). doi:10.1038/s41467-017-01889-3
- J. Han, S. L. Thomas and D. J. Srolovitz, Grain-boundary kinetics: a unified approach, Progress in Materials Science 98, 386 to 476 (2018). doi:10.1016/j.pmatsci.2018.05.004
- E. R. Homer, E. A. Holm, S. M. Foiles and D. L. Olmsted, Trends in grain boundary mobility: survey of motion mechanisms, JOM 66, 114 to 120 (2014). doi:10.1007/s11837-013-0801-2
- D. L. Olmsted, S. M. Foiles and E. A. Holm, Grain boundary interface roughening transition and its effect on grain boundary mobility for non-faceting boundaries, Sandia report SAND2007-0507J. osti.gov/servlets/purl/1427002
- K. Chen, J. Han and D. J. Srolovitz, On the temperature dependence of grain boundary mobility, arXiv:2003.01191 (2020).
- S. Florez, K. Alvarado and M. Bernacki, A new front-tracking Lagrangian model for the modeling of dynamic and post-dynamic recrystallization, Modelling and Simulation in Materials Science and Engineering (2021), doi:10.1088/1361-651X/abd837; preprint arXiv:2009.08368: 304L parameters and E = ½μb²ρ.
- D. Raabe, Recovery and recrystallization: phenomena, physics, models, simulation, in Physical Metallurgy, 5th edition, Elsevier (2014), pages 2291 to 2397.
- Z. Suo, Lecture 10: grain growth, course notes, Evolving Small Structures: v = mp and p = γK.
BibTeX
@misc{tripathy2026gbmigration,
author = {Tripathy, Manisha},
title = {Grain Boundary Migration Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/grain-boundary-migration}},
note = {Interactive web tool}
}