Inhibited Grain Growth Models

• Inhibited grain growth models are formulations of polycrystalline coarsening that include microstructural barriers overlooked in classical Mullins theory.
• They employ intrinsic mechanisms such as grain rotation, disrupted atomic jumps, and nonlinear attachment kinetics to capture experimental features like the 'ear' and 'tail' in grain size distributions.
• Extrinsic factors like particle pinning, solute drag, and defect chemistry are integrated to modify grain-boundary mobility, leading to diverse kinetic behaviors and microstructural stabilization.

Searching arXiv for recent and foundational papers on inhibited grain growth models and related mechanisms. An inhibited grain growth model is a formulation of polycrystalline coarsening in which grain-boundary migration is slowed, arrested, or rendered strongly nonclassical by mechanisms omitted from the standard curvature-driven description. The immediate motivation for such models is the persistent mismatch between thin-film experiments and the Mullins framework: experiments show a universal grain size distribution with an excess of small grains, termed an “ear,” and an excess of very large grains, termed a “tail,” together with much slower growth and, in some cases, stagnation, whereas the classical model predicts narrower distributions and $r \sim t^{1/2}$ coarsening (Backofen et al., 2013). Subsequent work generalized inhibition to include atomistic barriers, grain rotation, nonlinear attachment kinetics, particle pinning, solute drag, charged-defect effects, thermodynamic stabilization, and irradiation-coupled transport (Hu et al., 2018, Hussein et al., 2024).

1. Classical baseline and the anomaly that motivated inhibited-growth models

The classical reference point is the Mullins curvature-driven model, in which the normal velocity of a grain boundary is proportional to curvature,

$v_n = \mu \gamma \kappa ,$

with $\mu$ the mobility, $\gamma$ the surface tension, and $\kappa$ the local curvature. In that formulation, the average grain radius grows as $r \sim t^{1/2}$, energy dissipation occurs at the boundary only, the microstructure is represented by smooth boundaries, and atomistic features such as dislocations, lattice structure, and Peierls barriers are ignored (Backofen et al., 2013).

Thin metallic film experiments contradict this picture in several linked ways. The experimental grain size distribution peaks at $r/\langle r\rangle < 1$, whereas the Mullins model predicts a peak at $r/\langle r\rangle > 1$; experiments also display the “ear” and “tail” features and often show far slower grain growth, sometimes even stagnation (Backofen et al., 2013). A common misconception is that these discrepancies can be repaired by modest modifications of curvature-driven kinetics alone. The literature summarized here instead indicates that the discrepancy reflects missing physics at the grain-boundary, lattice, and microstructural levels.

The same papers also distinguish between universality of structure and nonuniversality of kinetics. In the thin-film context, the geometric and topological characteristics are reported to be universal, but the dynamic growth exponent is not (Backofen et al., 2013). This separation between structural universality and kinetic nonuniversality became a recurring theme in later inhibited-growth models.

2. Atomistic and mesoscale descriptions of intrinsic inhibition

Phase field crystal (PFC) models were introduced as a way to resolve the atomic scale on diffusive time scales while retaining mesoscale access to grain-network evolution. In this class of models the order parameter $\psi$, representing time-averaged atomic density, evolves according to

$$\frac{\partial\psi}{\partial t} = \nabla^2\frac{\delta\mathcal{F}[\psi; T]}{\delta\psi},$$

so that atomic-scale periodicity, dislocations, lattice rotations, and lattice-resistance effects are represented directly (Bjerre et al., 2013). In the thin-film grain-growth problem, PFC simulations recover the experimentally observed “ear” and “tail,” produce nearly log-normal grain size distributions, and yield growth exponents $v_n = \mu \gamma \kappa ,$0 in $v_n = \mu \gamma \kappa ,$1 ranging from $v_n = \mu \gamma \kappa ,$2 to $v_n = \mu \gamma \kappa ,$3, rather than the Mullins value $v_n = \mu \gamma \kappa ,$4; for some parameters and low temperatures, coarsening stagnates and the system freezes in a polycrystalline configuration (Backofen et al., 2013).

A second intrinsic mechanism identified in PFC work is grain rotation. Grain growth and stagnation depend on quenching depth relative to $v_n = \mu \gamma \kappa ,$5: near $v_n = \mu \gamma \kappa ,$6, coarsening remains continuous, whereas deep quenching produces metastable states in which the kinetic barrier for recrystallization across boundaries is too large and grain rotation with subsequent coalescence is infeasible (Bjerre et al., 2013). The mean grain area follows $v_n = \mu \gamma \kappa ,$7 at early times, with $v_n = \mu \gamma \kappa ,$8 for deep quenching and increasing toward $v_n = \mu \gamma \kappa ,$9 for shallower quenching, before saturation at late times. Small grains rotate more actively than large grains, and stagnation correlates with the suppression of rotation (Bjerre et al., 2013).

Atomistic simulation has also identified a distinct intrinsic stagnation mechanism based on disruptive atomic jumps within moving grain boundaries. In that picture, the ordered collective motion of grain-boundary atoms can be destabilized by the jump of only a few atoms, producing abrupt slowdown or full stagnation of the entire boundary. These jumps can be activated by high driving forces, high temperatures, and large model sizes, and the mechanism was proposed to explain non-Arrhenius behavior in some grain boundaries (Song et al., 2024). The paper further reports that the disruptive atoms are not distinguishable from other grain-boundary atoms by atomic energy, volume, density, local entropy, or Voronoi tessellation, which makes the events difficult to identify without the proposed displacement vector analysis (Song et al., 2024).

3. Nonlinear and unified continuum formulations

A major theoretical development was the replacement of the classically linear mobility–driving-force relation by nonlinear expressions in which boundary migration is exponentially suppressed by an attachment or step-free-energy barrier. In the unified model, the grain-boundary migration rate is derived from an atomic-jump picture with detachment and attachment contributions, and for isotropic grains reduces to

$\mu$0

where $\mu$1 is grain size, $\mu$2 measures the size ratio relative to surrounding grains, and $\mu$3 is the grain-boundary step free energy (Hu et al., 2018). High $\mu$4 or small $\mu$5 suppresses growth and produces grain growth stagnation; low $\mu$6 recovers the linear classical limit. The same framework was presented as a quantitative route to normal grain growth, abnormal grain growth, and stagnant grain growth within one equation set (Hu et al., 2018).

A closely related nonlinear capillarity-driven formulation expresses the grain volume growth rate as

$\mu$7

with the curvature difference $\mu$8, interfacial energy $\mu$9, and step free energy $\gamma$0 entering inside the exponential (Hu et al., 2017). In this model, growth accelerates when curvature differences are large or the step free energy is small, while stagnation appears when $\gamma$1 is high or the curvature difference is too small. The model was presented as a basis for interpreting normal, abnormal, and stagnant behavior without introducing ad hoc abrupt immobilization (Hu et al., 2017).

More recent work on non-Arrhenius grain growth reformulated the kinetics as

$\gamma$2

with $\gamma$3 an effective grain-boundary step energy and $\gamma$4 the curvature ratio across the boundary (Pan et al., 19 Mar 2026). That study, using SrTiO$\gamma$5 as a model system, concluded that non-Arrhenius grain growth is thermally activated and does not have a definitive characteristic temperature; instead it is controlled by the interplay between temperature-dependent factors and temperature-independent parameters such as grain size and its distribution (Pan et al., 19 Mar 2026). This differs in emphasis from the atomistic disruptive-jump interpretation of anti-thermal mobility (Song et al., 2024). A plausible implication is that “non-Arrhenius” behavior is mechanism-specific and may depend on the level of description—single-boundary atomistics versus polycrystalline growth statistics—rather than naming a single universal anomaly.

4. Extrinsic inhibition: particles, solutes, and defect chemistry

Extrinsic inhibited-growth models introduce barriers or drag terms arising from second-phase particles, segregating solutes, or charged defects. These models differ in scale and implementation, but all treat grain-boundary motion as coupled to fields or objects external to simple capillarity.

Model family	Key control variables	Reported consequence
Particle pinning in 2D polycrystals	$\gamma$6, $\gamma$7, $\gamma$8	limiting grain size and a heterogeneous mixture of large and small grains (Bignon et al., 2024)
Cellular automata with primary $\gamma$9 in FGH98	particle radius and volume fraction	experimental fit with error less than 10%; Zener $\kappa$0 minimum value $\kappa$1 at $\kappa$2 (Liua et al., 2021)
Atomistically informed solute drag and multiscale AGG models	segregation energy, diffusivity, grain-boundary character	Nb $\kappa$3 Ti $\kappa$4 Mo $\kappa$5 V $\kappa$6 Mn; AGG when 10–30% of GBs are high-mobility type (Suhane et al., 5 Mar 2025, Linda et al., 17 Oct 2025)
Defect-chemistry-informed phase field	formation-energy differences, dopant concentration, grain-boundary potentials	solute drag alone can lead to bimodal grain size distribution; larger grains tend to have lower potentials (Wang et al., 2024)

In the analytical particle-pinning model for two-dimensional polycrystals, grain boundaries are assumed not to detach from particles once the local configuration with minimal boundary length is reached unless another driving force is applied. This differs from classical force-balance approaches and leads to a stabilized microstructure containing both large and small grains rather than a single limiting size. The model predicts limiting sizes, surviving large and small grain fractions, vanishing grains, and the dependence of heterogeneity on particle surface fraction $\kappa$7, particle radius $\kappa$8, and normalized initial grain size $\kappa$9, with very good agreement against full-field level-set simulations and published phase-field results (Bignon et al., 2024).

The FGH98 nickel-based superalloy study implements inhibition in a two-dimensional cellular automata model based on thermal activation and the lowest energy principle. There the primary $r \sim t^{1/2}$0 phase, rather than carbides and borides, is identified as the dominant inhibitor during solution at 1433 K, because the primary $r \sim t^{1/2}$1 is larger and mostly located at grain boundaries. The simulations reproduce experimental grain sizes with an error less than 10% and conform to the Zener relation

$r \sim t^{1/2}$2

with the theoretically calculated coefficient $r \sim t^{1/2}$3 decreasing as particle radius increases and reaching a minimum value of $r \sim t^{1/2}$4 when the radius of primary $r \sim t^{1/2}$5 is $r \sim t^{1/2}$6 (Liua et al., 2021).

Solute-drag models derive reduced grain-boundary mobility from segregation energetics and diffusivity. In atomistically informed phase-field simulations of austenite grain growth, the solute effectiveness ranking was reported as Nb $r \sim t^{1/2}$7 Ti $r \sim t^{1/2}$8 Mo $r \sim t^{1/2}$9 V $r/\langle r\rangle < 1$0 Mn, and the anisotropy of solute drag across the studied grain-boundary types was found to play a secondary role, allowing the $r/\langle r\rangle < 1$1 boundary to serve as a representative boundary for solute-trend prediction (Suhane et al., 5 Mar 2025). A related multiscale framework for $r/\langle r\rangle < 1$2-Fe combines density functional theory, the McLean isotherm, the Cahn–Lücke–Stüwe solute-drag model, and multi-phase-field grain growth. In that study, low-energy $r/\langle r\rangle < 1$3 boundaries retain higher mobility, AGG emerges only when a minority fraction, 10–30%, of boundaries are the high-mobility type, and the distribution of grain-boundary mobility is identified as the main control parameter rather than grain-boundary energy alone (Linda et al., 17 Oct 2025).

Oxide electroceramics require an additional level of coupling because dopants and oxygen vacancies are charged species. The defect-chemistry-informed phase-field model treats grain growth, defect diffusion, and electrostatics self-consistently through Allen–Cahn, Cahn–Hilliard, and Poisson equations. Within that framework, stronger segregation energies and higher dopant concentrations increase inhibition, and simulations indicate that solute drag alone can lead to bimodal grain size distributions without any contribution from grain misorientation and other anisotropy. The same study reports that larger grains tend to have lower grain-boundary potentials than smaller grains, implying a heterogeneous potential distribution that coevolves with the microstructure (Wang et al., 2024).

5. Thermodynamic stabilization and coupled-field limits

A limiting case of inhibited grain growth is full thermodynamic stabilization, in which the driving force for coarsening vanishes because the grain-boundary free energy reaches zero. A two-dimensional kinetic Monte Carlo model of nanocrystalline alloy systems captures coupled grain-boundary migration and solute diffusion and identifies the stabilization criterion as

$r/\langle r\rangle < 1$4

In that state, reducing grain-boundary area no longer lowers the free energy, planar boundaries develop divergent capillary wave amplitudes, and a large grain can fragment into a stable ensemble of smaller grains (Hussein et al., 2024). The model also emphasizes that finite solute diffusion is not a secondary correction: if diffusion is too slow, the system may evolve toward a metastable single crystal even when the thermodynamic conditions favor a stabilized polycrystal (Hussein et al., 2024). The paper notes that this fully stabilized state has been predicted theoretically and by simulations but is yet to be confirmed experimentally (Hussein et al., 2024).

Coupled-field models can also delimit the range over which inhibition is effective by introducing size-dependent auxiliary driving forces. In nanocrystalline UO$r/\langle r\rangle < 1$5, a phase-field grain-growth model coupled to heat conduction with stochastic thermal spikes was used to describe irradiation-assisted grain growth. The heat spikes enhance grain-boundary mobility locally, but only when a spike overlaps a grain boundary; the overlap fraction is estimated as

$r/\langle r\rangle < 1$6

As grains coarsen, the overlap probability falls, and the study concludes that irradiation-assisted grain growth is only significant with average grain sizes less than 35 nm and can be neglected for typical UO$r/\langle r\rangle < 1$7 fuel pellets (Muntaha et al., 12 Dec 2025). Although this model concerns enhancement rather than inhibition, it shows that coupled transport processes can make grain-growth kinetics self-limiting through geometry alone.

6. Microstructural signatures, universality, and open controversies

Across the inhibited-growth literature, the most persistent observable signature is a departure from the Mullins distribution toward either a broad, nearly log-normal distribution with “ear” and “tail” features or an explicitly bimodal distribution containing coexisting small and large grains. In the thin-film PFC work, the distribution is self-similar, small grains predominantly have 3 or 4 sides, large grains have more than 6, and the geometric and topological characteristics are universal even though the dynamic growth exponent is not (Backofen et al., 2013). In particle-pinned, solute-drag, and charged-defect models, heterogeneity and bimodality arise because growth inhibition prevents the disappearance of some grains while allowing a subset of grains to grow or persist abnormally (Bignon et al., 2024, Wang et al., 2024).

One recurring misconception is that inhibition must be extrinsic, for example by Zener pinning or solute drag. The surveyed work shows otherwise. Intrinsic inhibition appears in PFC models through lattice-scale resistance, deep-quench metastability, and suppressed grain rotation (Bjerre et al., 2013), and in atomistic simulations through disruptive jumps that arrest collective boundary migration even in pure materials (Song et al., 2024). Conversely, several extrinsic models show that the relevant control variable is not simply the average grain-boundary energy but the distribution of mobility, defect chemistry, or particle interception statistics (Linda et al., 17 Oct 2025, Bignon et al., 2024).

A second controversy concerns non-Arrhenius behavior. One line of work attributes it to disruptive atomic jumps and even anti-thermal mobility at the grain-boundary level (Song et al., 2024). Another argues that non-Arrhenius grain growth in SrTiO$r/\langle r\rangle < 1$8 is thermally activated, lacks a definitive characteristic temperature, and emerges from the interplay between temperature-dependent step-energy effects and temperature-independent grain size and size distribution (Pan et al., 19 Mar 2026). These positions are not formally identical, and the discrepancy remains an active point of interpretation.

Taken together, the inhibited grain growth model has evolved from a critique of curvature-driven coarsening into a broad family of theories that connect microstructural arrest to atomistic barriers, nonlinear attachment kinetics, pinning, segregation, space charge, and free-energy cancellation. This suggests that inhibition is not a single mechanism but a regime of grain-growth dynamics in which the effective mobility or effective driving force is strongly conditioned by microstructure, chemistry, and scale.