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Sandbox Physics
Crystal defects and microstructure

FR39 · An interactive course

Crystal defects and microstructure: dislocations, grain growth and dendrites

Sandbox Physics ·

A metal bends because rows of atoms slip one at a time, it coarsens because grain boundaries straighten, and a casting freezes into branches because solute has to get out of the way. This course follows those three processes from the defect or interface that drives each one to the equation that predicts it, and checks every equation against a simulation you can run.

Start with the Burgers vector Open an experiment
345678910−30+3number of sidesarea rate
Mean area rate of grains with nn sides, in units of LkLk, pooled over three seeded 160160 by 160160 polycrystals (dots, sample counts from 1717 to 37313731). The dashed line is the von Neumann–Mullins law (π/3)(n−6)(\pi/3)(n-6). Five-sided grains sit on it; six-sided grains grow slightly, and grains with more sides grow 3%3\% to 6%6\% faster than the law.

From one extra half-plane of atoms to a published phase-field model

The first chapter defines a dislocation by the loop that fails to close around it and writes down its stress field. The second turns that stress into a force and shows why like dislocations stack into walls. The third finds the stress that tears a dipole apart. The fourth moves to grain boundaries, which travel toward their centre of curvature, and the fifth counts sides to explain why a polycrystal coarsens. The sixth explains why a phase-field model of a freezing alloy traps too much solute unless it adds an extra current, and the last chapter reads the paper that introduced that current and reproduces its figures.

You need vectors and derivatives, the idea of stress and strain, and the diffusion equation. Phase fields, Euler’s formula and the thin-interface limit are introduced when they are first used. FR39

  1. 01 · L1

    The Burgers vector and the stress field of an edge dislocation

    Define a dislocation by the loop that fails to close around it, write down its elastic field, and check it against a numerically computed Burgers circuit.

  2. 02 · L2

    The Peach–Koehler force makes like dislocations stack into walls

    Turn stress into a force on a dislocation, let dislocations glide in each other’s fields, and see why like signs stack and opposite signs settle at 45 degrees.

  3. 03 · L2

    Dislocation dipoles and the stress that tears them apart

    A pair of opposite dislocations on nearby planes resists shear up to a passing stress that falls as one over their spacing. Derive it and test it.

  4. 04 · L2

    Grain boundaries move by curvature, so a round grain shrinks at a steady rate

    A curved grain boundary moves toward its centre of curvature, so a round grain loses area at a steady rate. Derive the law from a phase field and test it.

  5. 05 · L3

    The von Neumann–Mullins law says grains with fewer than six sides shrink

    In a two-dimensional polycrystal, a grain’s area rate depends only on how many sides it has. Derive the law, count sides with Euler, and measure both.

  6. 06 · L3

    Solute trapping in alloy phase-field models and the anti-trapping current

    A growing alloy crystal must shed solute. A diffuse interface wide enough to compute traps too much of it; an extra current removes the error.

  7. 07 · L4

    Reading Karma 2001 on alloy phase fields and reproducing its dendrite figures

    A guide to the four-page paper that introduced the anti-trapping current: where its claims sit, how it tests convergence, and which figure values we reproduce.

Experiments along the way

01Shear a crystal and pull a dipole apartLoad edge dislocations with a cyclic shear, watch like ones stack into a wall, and find the stress that tears an opposite pair apart.02Anneal a polycrystalLet grain boundaries move by curvature, time a shrinking round grain, and measure how each grain’s area rate depends on its number of sides.03Grow an alloy dendriteGrow a four-armed crystal from a supersaturated melt, then switch off the anti-trapping current and see how much extra solute the solid keeps.

Every model in these experiments is checked against a closed-form result or a published figure, and the route benchmark records each comparison. The Volterra field, the passing stress and the von Neumann–Mullins law are drawn as separate reference lines; nothing on screen is a micrograph.

Where these defects are measured

These Labs are not part of the course. They show how real specimens are loaded and imaged, which is where dislocations and boundaries are seen in practice.

Stress–strain material testerPull a bar until it yields. Yield is where dislocation glide, the subject of chapters 1 to 3, takes over from elastic stretching.In-situ TEM: stimulus or beam?Load, heat or bias a specimen inside an electron microscope and ask how much of the response the imaging beam causes. The Lab tracks markers; it does not model dislocations.Electron holographyRecover the phase an electron picks up crossing a specimen and separate its magnetic part. It shows how microscopes measure fields inside a sample; it does not model defects.

Sources and what we recomputed

  1. A. Kauffmann and D. Schliephake, Plasticity, chapter 4c: Elastic theory of dislocations, lecture slides, Karlsruhe Institute of Technology, version 24-05-02 (2024)

    Version 24-05-02, all slides. Slides 9 and 10: displacement and stress field of a straight edge dislocation with line along zz and Burgers vector along xx, after Hirth and Lothe. Slide 11: maps of the stress components. Slide 13: interactions follow from superposing the fields. Slide 15: the field is cut off near the core, at about 1.5 b1.5\,b for an edge dislocation. Slide 23: the glide force per unit length is bτb\tau.

  2. X. Niu, Y. Zhu, S. Dai and Y. Xiang, A continuum model for distributions of dislocations incorporating short-range interactions, arXiv:1707.05161v2 (2018)

    Version 2, Section 2, the discrete dislocation model. Eq. (2.1): the Peach–Koehler force f=(σ⋅b)×τ\mathbf f=(\boldsymbol\sigma\cdot\mathbf b)\times\boldsymbol\tau, with glide component bσxyb\sigma_{xy} and climb component −bσxx-b\sigma_{xx}. Eqs. (2.2)–(2.3): the isotropic stress field of an edge dislocation. Eqs. (2.4)–(2.7): forces summed over all other dislocations plus the applied stress. Eq. (2.8): the mobility law vg=mgfgv_{\mathrm g}=m_{\mathrm g}f_{\mathrm g}. Section 4 discusses dipolar walls; its continuum results are not used here.

  3. A. Kazaryan, Y. Wang, S. A. Dregia and B. R. Patton, Generalized phase field model for computer simulation of grain growth in anisotropic systems, arXiv:cond-mat/9912044v1 (1999)

    Version 1, full text. Eq. (1): one Allen–Cahn equation per order parameter. Eq. (2): the free energy with gradient terms. Eq. (5): the bulk free-energy density, used with a1=a2=1a_1=a_2=1 and a3=2a_3=2. Eq. (6) and Fig. 3: an isolated grain shrinks at a constant area rate. The closing paragraphs and Fig. 5: the mean grain area grows linearly in time. This course uses only the isotropic case.

  4. J. K. Mason, E. A. Lazar, R. D. MacPherson and D. J. Srolovitz, Geometric and topological properties of the canonical grain growth microstructure, arXiv:1507.03379v1 (2015)

    Version 1. Section II, Eq. (1): the von Neumann–Mullins relation dA/dt=−2πMγ(1−n/6)dA/dt=-2\pi M\gamma(1-n/6), exact for isotropic two-dimensional grain growth. Section III A 1 and Fig. 4(a): the distribution of edges per grain, and Euler’s theorem fixing the mean number of edges at six.

  5. A. Karma, Phase-field formulation for quantitative modeling of alloy solidification, arXiv:cond-mat/0103289v1 (2001)

    Version 1, full text. Eqs. (16)–(18) and (22): the isothermal dilute-alloy model with the anti-trapping current. Eqs. (19)–(20): the equilibrium profile and the choice of q(ϕ)q(\phi) that removes surface diffusion. The parameter paragraph: Δx=0.4\Delta x=0.4, Δt=0.008\Delta t=0.008, k=0.15k=0.15, Ω=0.55\Omega=0.55, ϵ4=0.02\epsilon_4=0.02, a1=0.8839a_1=0.8839, a2=0.6267a_2=0.6267, a seed of radius 22 d022\,d_0. Fig. 1: tip velocity against time. Fig. 2: solute in the solid along the dendrite axis.

  6. B. Echebarria, R. Folch, A. Karma and M. Plapp, Quantitative phase-field model of alloy solidification, arXiv:cond-mat/0404164v2 (2004)

    Version 2. Section I B, Eq. (2): with a mesoscopic interface thickness, solute trapping, surface diffusion and interface stretching all grow in proportion to WW. Section I D: the anti-trapping current as a non-variational fix. Section I E, Eqs. (7)–(13): the full model, the current of Eq. (10) with a=1/(22)a=1/(2\sqrt2), and the relations d0=a1W/λd_0=a_1W/\lambda and β\beta of Eq. (13). Section II A, Eqs. (17)–(21): the sharp-interface problem for isothermal growth, with the Stefan condition and the Gibbs–Thomson relation. Section IV gives the thin-interface analysis; it was skimmed, not rederived.

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