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REVIEW 3 major objections 6 minor 59 references

A unified variational model for grain boundary dynamics incorporating microscopic structure

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives all major grain boundary motions as mobility limits of one constrained variational principle.

desk verdict A genuinely unifying variational framework for grain boundary dynamics, but the exactness of the Frank-Bilby constraint rests on an unquantified timescale separation. read the letter →

arxiv 2411.15747 v1 pith:6FK4MZ4K submitted 2024-11-24 cond-mat.mtrl-sci

Luchan Zhang , Xiaoxue Qin , Yang Xiang This is my paper · ORCID
classification cond-mat.mtrl-sci MSC 74A50
keywords grainboundarydynamicsdislocationsdisconnectionsFrank-BilbyequationsOnsagerprinciplevariationalframeworkshearcouplingrotation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the many distinct modes of grain boundary motion—curvature-driven migration, sliding, shear coupling, and grain rotation—are not separate theories but limiting cases of one variational principle. The principle minimizes the Onsager dissipation plus the rate of energy change, subject to dynamic Frank-Bilby equations that force the moving misorientation, velocity, and line-defect reaction rates to remain compatible with the microscopic dislocation or disconnection structure. In the low-angle case the minimizer reproduces the known coupling and sliding models in the slow- and fast-reaction mobility limits, and the same construction extends to high-angle boundaries whose structure is carried by disconnections with both Burgers-vector and step character. A sympathetic reader would care because a single constrained minimization offers a common mathematical language for a family of models that were previously derived piecemeal from discrete line-defect dynamics.

What carries the argument

The load-bearing object is the dynamic Frank-Bilby constraint $h=0$ (Eq. (2.6)): during any admissible motion, the misorientation rate, the velocity gradient along the boundary, and the line-defect reaction rate must combine so that the defect content stays consistent with the misorientation angle on the timescale where the constraint is assumed to hold. The variational mechanism is the Onsager dissipation $Q=\frac{1}{2}\int_\Gamma\left(v^2/M_n+(B_t^0)^2/M_B+\dot{\theta}^2/M_\theta\right)ds$ together with the energy release rate $\dot{E}$, minimized over $v$, $B_t^0$, and $\dot{\theta}$ subject to $h=0$; Lagrange multipliers convert the constrained minimization into the ODE system (3.14)-(3.18). For high-angle boundaries the analogous constraints (6.6)-(6.8) encode the step character of disconnections and produce the coupling factor $k=\beta_1+2\sin(\theta/2)$ in the grain-rotation relation $-\dot{\theta} y n_2=k v\cdot n$.

What would settle it

Run a discrete dislocation dynamics or atomistic simulation of an elliptical low-angle boundary and evaluate the residual of Eq. (2.6) from the instantaneous velocity, reaction rate, and misorientation rate; if the residual is comparable in size to the individual terms instead of negligible, the constraint is not instantaneous. For a driven planar high-angle boundary with a nonzero reference-plane dislocation density, measure the tangential relative velocity as a function of normal velocity and misorientation angle: the predicted coupling factor $k=\beta_1+2\sin(\theta/2)$ would be ruled out if the measured slope departs from this expression over a range of $\theta$.

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Extended reading notes

Core claim

The central discovery is a variational structure for grain boundary dynamics whose constraints are the dynamic Frank-Bilby equations. Taking a time derivative of the static Frank-Bilby relation $B=-2\sin(\theta/2)n$ gives $h=\dot{\theta}\cos(\theta/2)n-2\sin(\theta/2)\frac{d}{ds}(v\times\hat{z})+B_t^0=0$, and this $h=0$ is imposed exactly inside a minimization of $Q+\dot{E}$ over the boundary velocity $v$, the line-defect reaction rate $B_t^0$, and the misorientation rate $\dot{\theta}$. In the low-angle setting, the slow-reaction limit reproduces the shape-preserving coupling/sliding dynamics of the earlier dislocation-based continuum models, while the fast-reaction limit returns the classical curvature-stiffness velocity and the misorientation relaxation $\dot{\theta}=-M_\theta\partial\gamma/\partial\theta$. For high-angle boundaries the model uses a unified Frank-Bilby structure $B=(\beta_1 n_1,-2\sin(\theta/2)n_2)$ combining disconnection and reference-plane dislocation contents, and the resulting coupling factor for grain rotation is $k=\beta_1+2\sin(\theta/2)$, which reduces to the disconnection-only factor when the reference-plane contribution is neglected.

Load-bearing premise

The load-bearing premise is that the boundary velocity, the line-defect reaction rate, and the misorientation rate adjust on a timescale much shorter than the boundary and defect-density evolution, so the dynamic Frank-Bilby equations can be imposed as exact instantaneous constraints; in the high-angle case the rotation constraint is admitted to hold only in an averaged sense.

Editorial extensions

If this is right

  • In the slow-dislocation-reaction limit, the model gives shape-preserving inward radial shrinkage with increasing misorientation, reproducing the established coupling and sliding continuum dynamics for low-angle boundaries.
  • In the fast-dislocation-reaction limit, the boundary velocity reduces to the classical curvature-stiffness law $v=M_n(\gamma+\gamma'')\kappa n$ and the misorientation evolves by $\dot{\theta}=-M_\theta\partial\gamma/\partial\theta$, so motion by mean curvature and sliding are recovered.
  • For an infinite planar tilt boundary under constant shear with fixed misorientation, the model yields velocity $M_n\tau b\hat{x}$ and the shear-coupling factor $2\tan(\theta/2)$, matching the classical coupling theory.
  • For a high-angle boundary described by disconnections, normal motion induces a tangential relative velocity with coupling factor $k=\beta_1+2\sin(\theta/2)$, and grain rotation follows Eq. (6.29) with separate contributions from disconnection flow and dislocation reactions.
  • The variational form gives a mathematically tractable basis for analyzing the resulting PDE models and for building efficient numerical methods for grain boundary networks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the instantaneous-constraint premise holds, the same minimization template should extend to semicoherent hetero-interfaces by replacing the Frank-Bilby right-hand side with $-2\sin(\theta/2)n+\Sigma T$, as the paper sketches; the open question is which mobility regimes survive the lattice-mismatch term.
  • The two-speed mobility structure (fast disconnection nucleation, slow vertical dislocation motion) suggests a measurable anisotropy: for a fixed driving force, a boundary with larger $\beta_1$ should rotate faster than one whose coupling comes mainly from $2\sin(\theta/2)$.
  • One can test the variational principle's predictive content by comparing its predicted aspect-ratio evolution for shrinking elliptical grains with atomistic simulations in the slow, median, and fast reaction regimes, since the model already matches those regimes qualitatively.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a variational continuum model for two-dimensional grain-boundary dynamics that incorporates the underlying dislocation or disconnection microstructure. Starting from the static Frank-Bilby relation B = -2 sin(theta/2) n, the authors differentiate (2.1) and use the kinematic relations (2.3)-(2.5) to derive the dynamic Frank-Bilby constraint h = 0 in Eq. (2.6). They then formulate an Onsager-principle constrained minimization, Eqs. (3.1)-(3.2), with a quadratic dissipation function (3.3) and energy dissipation rate (3.5). The Euler-Lagrange equations lead to the closed rate equations (3.14)-(3.21). In Sec. 4, slow and fast dislocation-reaction limits are shown to reduce to coupling motion and to sliding/mean-curvature motion, respectively, and numerical simulations of an elliptical boundary are reported for three mobility regimes. Section 5 adds stress and synthetic-force contributions, and Sec. 6 generalizes the framework to high-angle boundaries using a unified Frank-Bilby equation (6.3), dynamic constraints (6.6)-(6.8), and a grain-rotation formula (6.29) with coupling factor k = beta1 + 2 sin(theta/2). The paper concludes with a discussion of extensions to hetero-interfaces.

Significance. If the underlying assumptions hold, the framework provides a single variational umbrella for mean-curvature motion, sliding, shear coupling, and disconnection-mediated grain rotation, and it gives a tractable starting point for numerical methods and analysis. The analytic derivations are self-contained: the dynamic Frank-Bilby equations are obtained by differentiating the static equations, and the target models are not used to construct the variational principle, so the argument is not circular. The reduction to classical motion-by-curvature in the fast-reaction limit and the emergence of the Cahn-Taylor coupling factor beta = 2 tan(theta/2) in Secs. 4 and 6 are concrete successes, and the high-angle coupling factor k = beta1 + 2 sin(theta/2) is a falsifiable prediction. The paper is not parameter-free, however: it relies on five phenomenological mobilities, and its predictive content is conditional on instantaneous Frank-Bilby relaxation and, in Sec. 6, on an average constraint. The numerical validation is qualitative and is primarily a consistency check against the authors' prior continuum models rather than against atomistic or experimental data.

major comments (3)
  1. [§3, Eq. (3.2)] The constrained minimization is built on the assumption that the dynamic Frank-Bilby equation h = 0 holds instantaneously, yet the only justification is the statement after (3.2) that the velocity, reaction rate, and rotation are adjusted on a timescale much shorter than the evolution of the boundary and dislocation density. No estimate of this relaxation timescale is given, and in Sec. 4.3 the model is applied to the comparable regime Mn/MB ~ l^2/B^2 (case (ii)), where the separation of timescales cannot be assumed. Since the rate equations (3.14)-(3.18) are derived from the constrained minimization, the validity of the model in this regime is not established; please provide estimates of the relaxation time in terms of the mobilities and geometric length scales, or restrict the claimed validity to the separated regimes and present case (ii) as a heuristic interpolation.
  2. [§6, Eqs. (6.8), (6.12)-(6.14)] For high-angle boundaries, the paper derives the pointwise relation -theta_dot y(s) n2 = k v·n in (6.12), then states that this relation "should hold in some average sense since instant equilibrium is not required" and replaces it by the integral constraint (6.8). The constrained minimization (6.13)-(6.14) then imposes (6.8) as an exact equality, and the rotation formula (6.29) depends directly on the Lagrange multiplier lambda3 determined by this constraint. The manuscript does not quantify the error incurred by replacing the pointwise relation by its average, nor does it show that local violations do not bias (6.29); without such an estimate, the high-angle rotation predictions are not fully justified. A similar issue arises in Eq. (6.3), where the equilibrium Frank-Bilby density B2 = -2 sin(theta/2) n2 of the reference plane is maintained while the disconnection density B1 evolves; the paper does not justify this persistence.
  3. [§6, after Eq. (6.25)] The closed-form high-angle evolution (6.26)-(6.32) is derived under the new assumptions Mn2/MB1 << L^2 and Mtheta << MB2, introduced immediately before the solution. These inequalities are not derived or related to the physical mobility parameters, and they are needed to eliminate the Lagrange multipliers lambda1 and lambda2 and to express v0 and theta_dot as in (6.28)-(6.31). Without them, the system (6.18)-(6.22) is not solved. Please state the range of material parameters for which these assumptions hold, or give the general solution without these reductions; this is necessary for the claimed unification at high angles.
minor comments (6)
  1. [§3, Eq. (3.21)] The functions pn and p* used in the solution formula lambda = (1/MB) theta_dot cos(theta/2) pn + (1/MB) p* are not defined before they are used; please define them in the main text as the periodic solutions of the ODE in Appendix A with f = n and f = -2 sin(theta/2) d/ds(v* × zhat) + B0*_t, respectively.
  2. [§2, Eqs. (2.7)-(2.8)] The definition S = integral_0^s B0_t(w) dw depends on the choice of the arclength origin; the paper should state that the origin is fixed and that condition (2.11) makes S well defined as a periodic function, or discuss the gauge dependence of the tangential-velocity formula.
  3. [Appendix A] The solution formula (A.3)-(A.4) uses both l and L for the perimeter, and the exponents in D1 and D2 are written with inconsistent notation; please use a single symbol for the perimeter and verify the periodic Green's function formula by substitution into (A.1) and (A.2).
  4. [§4.1, Eq. (4.1)] The constant CB is described as ensuring that Eq. (2.11) holds, but the displayed formula does not show how CB is constructed from the integral condition; please spell out the formula for CB.
  5. [§6, Eq. (6.35)] The self-stress formula is written for small-slope graphs h(x), and the general curved-boundary expression is stated immediately afterward without derivation; please specify the principal-value regularization and the range of h_x for which the small-slope form is valid.
  6. [Throughout] There are several typographical issues: "The F rank-Bilby equations" before Eq. (2.1) has a stray space, reference [57] contains "grian boundaries", and the notation for B0_t and the vector lambda is sometimes inconsistent. A careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dynamic Frank-Bilby constraints are derived from static Frank-Bilby and kinematics, and the recovered models are not used as premises.

full rationale

The paper's claimed derivation chain is self-contained. The static Frank-Bilby equation (2.1) is taken from classical theory; taking its time derivative and using the kinematic evolution laws (2.3)-(2.5) yields the dynamic Frank-Bilby constraint (2.6) as a mathematical consequence (Sec. 2). The variational principle (3.1)-(3.2) is an Onsager minimization with the dissipation function (3.3) and energy dissipation rate (3.5) as independent physical inputs, not as definitions of the target motions; the Euler-Lagrange equations (3.7)-(3.9) and the reduced rate equations (3.14)-(3.18) follow by standard constrained calculus. The recovery in Sec. 4 of classical mean-curvature motion, Cahn-Taylor coupling, sliding models, and the authors' earlier continuum models is a consistency check performed after the derivation, not a premise of it. The high-angle extension derives the unified Frank-Bilby structure (6.3) geometrically from the reference-plane Frank-Bilby density and the disconnection density, and the dynamic constraints (6.6)-(6.8) are derived from (6.3) plus plastic-flow kinematics; the coupling factor k = beta1 + 2 sin(theta/2) emerges from these equations rather than being imposed to match [36]. No fitted parameter is renamed as a prediction: the mobilities Mn, MB, Mtheta are free Onsager coefficients, and no uniqueness theorem is imported from the authors' prior work. The paper does contain physically fragile assumptions: the instantaneous Frank-Bilby timescale separation stated after Eq. (3.2), and the admission after Eq. (6.12) that Eq. (6.8) 'should hold in some average sense since instant equilibrium is not required'. These are unproved modeling assumptions or acknowledged limitations, not circular reductions of the output to the input. Accordingly, the core derivation does not reduce to its own inputs.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The framework introduces no new particles, fields, or conserved quantities. All degrees of freedom, dislocation density, disconnection density, misorientation, and mobilities, are standard in the cited literature.

free parameters (4)
  • Grain boundary mobility M_n = M_n = M_d b/B in simulations
    Introduced as a constitutive mobility in the dissipation function (3.3); the specific form and values are chosen for simulations in Sec. 4.3, not derived.
  • Dislocation reaction mobility M_B = M_B = K_B M_d B/b, with K_B set so M_B values are 2.86e-7, 2.86e-5, and 8.58e-5 M_d B/b in cases (i)-(iii)
    Free mobility controlling reaction rate; values are hand-picked to realize slow, median, and fast reaction regimes in Sec. 4.3.
  • Grain rotation mobility M_theta = M_theta = 2.34e-3 M_d/b
    Introduced in the dissipation function (3.3); the value is chosen from atomistic-informed estimates in Sec. 4.3.
  • High angle mobilities M_n1, M_n2, M_B1, M_B2 = Not specified numerically
    Appear in Eq. (6.15) and are assumed to satisfy Mn2/MB1 << L^2 and M_theta << M_B2; no constitutive values are given.
assumptions (7)
  • domain assumption Grain boundary energy density is gamma(B) = gamma0 B(A0 - log B)
    Taken from Read-Shockley type formulas in [20,41,56,54], used throughout the derivations, e.g., Eq. (2.2).
  • domain assumption Static Frank-Bilby equation B = -2 sin(theta/2)n holds at every instant for low angle boundaries
    Assumed in Eq. (2.1) and used to compute B from the boundary profile after solving, Sec. 3.
  • ad hoc to paper Dynamic Frank-Bilby constraint h=0 holds exactly at all times
    Stated after Eq. (3.2): velocity, reaction rate, and rotation are adjusted to satisfy Frank-Bilby on a timescale much shorter than the evolution. This is the key timescale-separation assumption.
  • domain assumption Onsager principle with quadratic dissipation function Q in Eq. (3.3) governs the dynamics
    Used to define the constrained minimization (3.1)-(3.2); this is a linear response assumption.
  • domain assumption High angle reference plane keeps equilibrium Frank-Bilby dislocation density B2 = -2 sin(theta/2)n2
    Eqs. (6.1)-(6.3); needed to define the misorientation-dependent energy and the rotation formula.
  • ad hoc to paper High angle constraints hd in Eqs. (6.6)-(6.8), including the average constraint (6.8), are valid
    Derived in Sec. 6, but Eq. (6.8) is imposed only in an average sense because instant equilibrium is not required; this weakens the constraint.
  • ad hoc to paper Fast disconnection nucleation assumptions Mn2/MB1 << L^2 and M_theta << M_B2
    Stated before Eq. (6.26); needed to obtain the simplified velocity and rotation formulas.

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Cite this review

Pith. "Pith review of A unified variational model for grain boundary dynamics incorporating microscopic structure." pith.science (2026). https://pith.science/paper/6FK4MZ4K

@misc{pith2026241115747,
  author       = {Pith},
  title        = {Pith review of: A unified variational model for grain boundary dynamics incorporating microscopic structure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6FK4MZ4K}},
  note         = {Machine review of arXiv:2411.15747}
}
read the original abstract

Recent experiments, atomistic simulations, and theoretical predictions have identified various new types of grain boundary motions that are controlled by the dynamics of underlying microstructure of line defects (dislocations or disconnections), to which the classical motion by mean curvature model does not apply. Different continuum models have been developed by upscaling from discrete line defect dynamics models under different settings (dislocations or disconnections, low angle grain boundaries or high angle grain boundaries, etc.), to account for the specific detailed natures of the microscopic dynamics mechanisms, and these continuum models are not in the variational form. In this paper, we propose a unified variational framework to account for all the underlying line defect mechanisms for the dynamics of both low and high angle grain boundaries and the associated grain rotations. The variational formulation is based on the developed constraints of the dynamic Frank-Bilby equations that govern the microscopic line defect structures. The proposed variational framework is able to recover the available models for different motions under different conditions. The unified variational framework is more efficient to describe the collective behaviors of grain boundary networks at larger length scales. It also provides a mathematically tractable basis for rigorous analysis of these partial differential equation models and for the development of efficient numerical methods.

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [55]

    Zhang, J

    L.C. Zhang, J. Han, Y. Xiang, and D.J. Srolovitz. Equation of motion for a grain boundary. Phys. Rev. Lett., 119:246101, 2017

    work page 2017

  2. [36]

    C. Qiu, M. Salvalaglio, D.J. Srolovitz, and J. Han. Disconnection flow–mediated grain rotation. Proceedings of the National Academy of Sciences, 121:e2310302121, 2024

    work page 2024

  3. [1]

    M. F. Ashby. Boundary defects, and atomistic aspects of boundary sliding and diffusional creep. Surface Sci., 31:498–542, 1972

    work page 1972

  4. [2]

    B. A. Bilby. Bristol conference report on defects in crystalline materials. Phys. Soc., London, page 123, 1955

    work page 1955

  5. [3]

    J.W. Cahn, Y. Mishin, and A. Suzuki. Coupling grain boundary motion to shear deformation. Acta Mater., 54:4953–4975, 2006. This manuscript is for review purposes only. A UNIFIED V ARIATIONAL MODEL FOR GRAIN BOUNDARY DYNAMICS 23

    work page 2006

  6. [4]

    Cahn and J.E

    J.W. Cahn and J.E. Taylor. A unified approach to motion of grain boundaries, relative tan- gential translation along grain boundaries, and grain rotation. Acta Mater., 52:4887–4898, 2004

    work page 2004

  7. [5]

    Chen and W

    L.Q. Chen and W. Yang. Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics. Phys. Rev. B, 50:15752–15756, 1994

    work page 1994

  8. [6]

    Chen, H.L

    Y.B. Chen, H.L. Deng, Q. Zhu, H.F. Zhou, and J.W. Wang. Direct observation of disconnection- mediated grain rotation. Scripta Mater., 252:116279, 2024

    work page 2024

Show all 59 references
  1. [7]

    Y.B. Chen, J. Han, H.L. Deng, G. Cao, Z. Zhang, Q. Zhu, H.F. Zhou, D.J. Srolovitz, and J.W. Wang. Revealing grain boundary kinetics in three-dimensional space. Acta Mater., 268:119717, 2024

    2024

  2. [8]

    S. B. Dai, B. Li, and J. F. Lu. Convergence of phase-field free energy and boundary force for molecular solvation. Arch. Ration. Mech. Anal., 227:105–147, 2018

    2018

  3. [9]

    Du and X

    Q. Du and X. B. Feng. The phase field method for geometric moving interfaces and their numerical approximations. Handbook of Numerical Analysis, 21:425–508, 2020

    2020

  4. [10]

    Elsey, S

    M. Elsey, S. Esedoglu, and P. Smereka. Diffusion generated motion for grain growth in two and three dimensions. J. Comput. Phys., 228:8015–8033, 2009

    2009

  5. [11]

    Epshteyn, C

    Y. Epshteyn, C. Liu, and M. Mizuno. Large time asymptotic behavior of grain boundaries motion with dynamic lattice misorientations and with triple junctions drag. Commun. Math. Sci., 19:1403–1428, 2021

    2021

  6. [12]

    Epshteyn, C

    Y. Epshteyn, C. Liu, and M. Mizuno. Motion of grain boundaries with dynamic lattice misori- entations and with triple junctions drag. SIAM J. Math. Anal., 53:3072–3097, 2021

    2021

  7. [13]

    Esedoglu

    S. Esedoglu. Grain size distribution under simultaneous grain boundary migration and grain rotation in two dimensions. Comput. Mater. Sci., 121:209–216, 2016

    2016

  8. [14]

    F. C. Frank. The resultant content of dislocations in an arbitrary intercrystalline boundary. pages 150–154. Office of Naval Research, Pittsburgh, 1950

    1950

  9. [15]

    Gong, H.Y

    M.Y. Gong, H.Y. Ma, K.M. Yang, Y. Liu, J.-F. Nie, and J. Wang. Symmetric or asymmetric glide resistance to twinning disconnection. npj Comput. Mater., 8:168, 2022

    2022

  10. [16]

    J. Han, D. J. Srolovitz, and M. Salvalaglio. Disconnection-mediated migration of interfaces in microstructures: I. continuum model. Acta Mater., 227:117178, 2022

    2022

  11. [17]

    Han, S.L

    J. Han, S.L. Thomas, and D.J. Srolovitz. Grain-boundary kinetics: A unified approach. Prog. Mater. Sci., 98:386–476, 2018

    2018

  12. [18]

    C. Herring. Surface tension as a motivation for sintering. In W. E. Kingston, editor, The Physics of Powder Metallurgy, pages 143–179. McGraw-Hill, New York, 1951

    1951

  13. [19]

    Hirth, G

    J.P. Hirth, G. Hirth, and J. Wang. Disclinations and disconnections in minerals and metals. Proc. Natl. Acad. Sci., 117:196–204, 2020

    2020

  14. [20]

    Hirth and J

    J.P. Hirth and J. Lothe. Theory of Dislocations. Wiley, New York, 2nd edition, 1982

    1982

  15. [21]

    Hirth and R.C

    J.P. Hirth and R.C. Pond. Steps, dislocations and disconnections as interface defects relating to structure and phase transformations. Acta Mater., 44:4749–4763, 1996

    1996

  16. [22]

    Kazaryan, Y

    A. Kazaryan, Y. Wang, S.A. Dregia, and B.R. Patton. Generalized phase-field model for computer simulation of grain growth in anisotropic systems. Phys. Rev. B, 61:14275–14278, 2000

    2000

  17. [23]

    Kinderlehrer and C

    D. Kinderlehrer and C. Liu. Evolution of grain boundaries. Math. Models Methods Appl. Sci., 11:713–729, 2001

    2001

  18. [24]

    Kinderlehrer, I

    D. Kinderlehrer, I. Livshits, and S. Taasan. A variational approach to modeling and simulation of grain growth. SIAM J. Sci. Comput., 28:1694–1715, 2006

    2006

  19. [25]

    A. H. King and D. A. Smith. The effects on grain-boundary processes of the steps in the boundary plane associated with the cores of grain-boundary dislocations. Acta Cryst., A36:335–343, 1980

    1980

  20. [26]

    Kobayashi, J.A

    R. Kobayashi, J.A. Warren, and W.C. Carter. A continuum model of grain boundaries. Phys. D, 140:141–150, 2000

    2000

  21. [27]

    Lazar, R.D

    E.A. Lazar, R.D. MacPherson, and D.J. Srolovitz. A more accurate two-dimensional grain growth algorithm. Acta Mater., 58:364–372, 2010

    2010

  22. [28]

    C.H. Li, E.H. Edwards, J. Washburn, and E.R. Parker. Stress-induced movement of crystal boundaries. Acta Metall., 1:223–229, 1953

    1953

  23. [29]

    J.C.M. Li. Possibility of subgrain rotation during recrystallization. J. Appl. Phys., 33:2958– 2965, 1962

    1962

  24. [30]

    McReynolds, K.A

    K. McReynolds, K.A. Wu, and P. Voorhees. Grain growth and grain translation in crystals. Acta Mater., 120:264–272, 2016

    2016

  25. [31]

    Onsager principle as a tool for approximation

    M.Doi. Onsager principle as a tool for approximation. Chin.Phys.B, 24:020505, 2015

    2015

  26. [32]

    Molodov, V.A

    D.A. Molodov, V.A. Ivanov, and G. Gottstein. Low angle tilt boundary migration coupled to shear deformation. Acta Mater., 55:1843–1848, 2007. This manuscript is for review purposes only. 24 LUCHAN ZHANG, XIAOXUE QIN AND YANG XIANG

    2007

  27. [33]

    W. Mullins. Two-dimensional motion of idealized grain boundaries. J. Appl. Phys., 27:900–904, 1956

    1956

  28. [34]

    L. Onsager. Reciprocal relations in irreversible processes. Phys. Rev., 37:405, 1931

    1931

  29. [35]

    X. X. Qin, L. C. Zhang, and Y. Xiang. A three-dimensional continuum simulation method for grain boundary motion incorporating dislocation structure. J. Sci. Comput., 90:3, 2022

    2022

  30. [37]

    Rajabzadeh, M

    A. Rajabzadeh, M. Legros, N. Combe, F. Mompiou, and D.A. Molodov. Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration. Philos. Mag., 93(10-12):1299–1316, 2013

    2013

  31. [38]

    Salvalaglio, R

    M. Salvalaglio, R. Backofen, K.R. Elder, and A. Voigt. Defects at grain boundaries: A coarse- grained, three-dimensional description by the amplitude expansion of the phase-field crystal model. Phys. Rev. Mater., 2:053804, 2018

    2018

  32. [39]

    P.G. Shewmon. In H. Margolin, editor, Recrystallization, grain growth and textures, pages 165–199. American Society of Metals, Metals Park, 1966

    1966

  33. [40]

    Srinivasan and J.W

    S.G. Srinivasan and J.W. Cahn. Challenging some free-energy reduction criteria for grain growth. In S. Ankem, C. S. Pande, I. Ovid’ko, and S. Ranganathan, editors, Science and Technology of Interfaces, pages 3–14. TMS, Seattle, 2002

    2002

  34. [41]

    Sutton and R.W

    A.P. Sutton and R.W. Balluffi. Interfaces in Crystalline Materials. Clarendon, 1995

    1995

  35. [42]

    Taylor and J.W

    J.E. Taylor and J.W. Cahn. Shape accommodation of a rotating embedded crystal via a new variational formulation. Interfaces and Free Boundaries, 9:493–512, 2007

    2007

  36. [43]

    S. L. Thomas, C.Z. Wei, J. Han, Y. Xiang, and D. J. Srolovitz. Disconnection description of triple junction motion. Proc. Natl. Acad. Sci., 116:8756–8765, 2019

    2019

  37. [44]

    Thomas, K.T

    S.L. Thomas, K.T. Chen, J. Han, P.K. Purohit, and D.J. Srolovitz. Reconciling grain growth and shear-coupled grain boundary migration. Nature Commun., 8:1764, 2017

    2017

  38. [45]

    Y. Tian, X. Gong, M. Xu, C. Qiu, Y. Han, Y. Bi, L. Estrada, E. Boltynjuk, H. Hahn, J. Han, D.J. Srolovitz, and X.Q. Pan. Grain rotation mechanisms in nanocrystalline materials: Multiscale observations in pt thin films. Science, 386:49–54, 2024

    2024

  39. [46]

    Trautt and Y

    Z.T. Trautt and Y. Mishin. Grain boundary migration and grain rotation studied by molecular dynamics. Acta Mater., 60:2407–2424, 2012

    2012

  40. [47]

    Upmanyu, D

    M. Upmanyu, D. Srolovitz, A. Lobkovsky, J. Warren, and W. Carter. Simultaneous grain boundary migration and grain rotation. Acta Mater, 54:1707–1719, 2006

    2006

  41. [48]

    L.H. Wang, Y. Zhang, Z. Zeng, H. Zhou, J. He, P. Liu, M.W. Chen, J. Han, D.J. Srolovitz, J. Teng, Y.Z. Guo, G. Yang, D.L. Kong, E. Ma, Y.L. Hu, B.C. Yin, X.X. Huang, Z. Zhang, T. Zhu, and X.D. Han. Tracking the sliding of grain boundaries at the atomic scale. Science, 375:1261...

    2022

  42. [49]

    Wei, S.L

    C.Z. Wei, S.L. Thomas, J. Han, D.J. Srolovitz, and Y. Xiang. A continuum multi-disconnection- mode model for grain boundary motion. J. Mech. Phys. Solids, 133:103731, 2019

    2019

  43. [50]

    Wei, L.C

    C.Z. Wei, L.C. Zhang, J. Han, D.J. Srolovitz, and Y. Xiang. Grain boundary triple junction dynamics: a continuum dislocation model. SIAM J. Appl. Math., 80:1101–1122, 2020

    2020

  44. [51]

    Wu and P.W

    K.A. Wu and P.W. Voorhees. Phase field crystal simulations of nanocrystalline grain growth in two dimensions. Acta Mater., 60:407–419, 2012

    2012

  45. [52]

    Yamanaka, K

    A. Yamanaka, K. McReynold, and P.W. Voorhees. Phase field crystal simulation of grain boundary motion, grain rotation and dislocation reactions in a bcc bicrystal. Acta Mater., 133:160–171, 2017

    2017

  46. [53]

    L. C. Zhang, J. Han, D. J. Srolovtiz, and Y. Xiang. Equation of motion for grain boundaries in polycrystals. npj Comput. Mater., 7:64, 2021

    2021

  47. [54]

    Zhang, Y.J

    L.C. Zhang, Y.J. Gu, and Y. Xiang. Energy of low angle grain boundaries based on continuum dislocation structure. Acta Mater., 126:11–24, 2017

    2017

  48. [56]

    Zhang and Y

    L.C. Zhang and Y. Xiang. Motion of grain boundaries incorporating dislocation structure. J. Mech. Phys. Solids, 117:157–178, 2018

    2018

  49. [57]

    Zhang and Y

    L.C. Zhang and Y. Xiang. A new formulation of coupling and sliding motions of grian bound- aries based on dislocation structure. SIAM J. Appl. Math., 80:2365–2387, 2020

    2020

  50. [58]

    Q. Zhu, G. Cao, J.W. Wang, C. Deng, J.X. Li, Z. Zhang, and S.X. Mao. In situ atomistic observation of disconnection-mediated grain boundary migration.Nature Communications, 10:156, 2019

    2019

  51. [59]

    Zhu and Y

    X.H. Zhu and Y. Xiang. Continuum framework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries. J. Mech. Phys. Solids, 69:175–194, 2014. This manuscript is for review purposes only

    2014

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.