Intrinsic grain-size gradients upon grain growth near a free surface

Abstract: Grain growth fundamentally shapes the microstructure of crystalline materials upon annealing, affecting their overall mechanical and functional properties. Recently, it has been rationalized that grain growth in polycrystals does not result solely from weighted curvature flow, but elastic effects (intrinsic stress) arised from shear coupling also need to be taken into account. We characterize and examine the effect of free surfaces on grain growth kinetics of high-purity, bulk polycrystalline nickel. By analyzing the microstructural evolution on cross sections of 1 mm thick specimens from the surface to the interior, as well as through in-plane investigations on specimens with varying thickness (1 mm, 40 $μ$m, and 10 $μ$m), an intrinsic grain-size gradient was identified, characterized by a gradual increase in grain size towards the interior. Interestingly, this grading was not restricted to the very surface but continued to depths of five to ten layers of grains, where effects from thermal grooves are considered negligible. We demonstrate that this behavior is significantly affected by elastic relaxation at the free surface, which alters the internal stress fields generated by shear-coupled grain boundary migration. These findings emphasize the relevance of free surfaces to the microstructural evolution of polycrystal.

• The paper demonstrates that free surfaces create intrinsic grain-size gradients in nickel, with grains growing from about 5 μm near the surface to 8 μm in the bulk and differences persisting to roughly 100 μm depth.
• Experiments using depth profiling and specimens from 10 μm to 1 mm thick show that thin samples retain substantially more small grains—about 50% more in 40 μm samples and 90% more in 10 μm lamellae than in bulk-like samples.
• The paper’s disconnection-based phase-field model attributes the extended gradient primarily to surface-induced elastic relaxation of shear-coupling stresses, while thermal grooving affects only approximately the first grain layer.

Motivation and context

Grain growth in polycrystalline solids has traditionally been described as mean-curvature flow, $v = M\kappa\gamma$, with grain boundary (GB) mobility as the only kinetic parameter. This capillarity-only picture has been increasingly challenged by 4D X-ray diffraction studies: analysis of roughly 50,000 GB segments in nickel found that GB velocity is weakly, if not at all, correlated with curvature (2604.16740), and similar results were reported for iron and strontium titanate. A key mechanism reconciling these observations is shear-coupled GB migration mediated by disconnection flow, which generates internal stresses during thermally driven growth and modifies GB kinetics through a Peach–Koehler-type force.

Against this background, the paper addresses a question that is usually neglected for bulk specimens: how do free surfaces affect grain growth? While thermal grooving at surface-intersecting GBs is well known to retard growth in thin films, its influence is assumed to be confined to the first one or two grain layers. The authors show experimentally and theoretically that free surfaces alter grain growth kinetics over much larger depths—five to ten grain layers—through elastic relaxation of the internal stresses generated by shear-coupled migration.

Experimental approach

The study used severely plastically deformed high-purity nickel (99.99%) processed by quasi-constrained high-pressure torsion (HPT) to obtain a homogeneous ultrafine-grained structure (~200 nm), which coarsens to ~5 μm upon annealing at moderate temperatures. Specimens of three thicknesses—1 mm, ~40 μm, and ~10 μm—were annealed at 400 °C for 1 h in vacuum, a temperature range where shear-coupled migration was previously observed in nickel with comparable grain sizes.

Two complementary strategies isolated depth-dependent effects:

• Cross-sectional depth profiling: EBSD characterization at successive depths from the surface (0, 13, 40, up to 500 μm) on 1 mm thick specimens, using sequential grinding/electropolishing to remove material between scans.
• Thickness variation: comparison of mid-thickness microstructures across specimens spanning two orders of magnitude in thickness, with six ~40 μm specimens prepared to control for volume effects. Surface layers affected by thermal grooving were removed by electropolishing or FIB polishing after annealing before EBSD acquisition.

Large-area EBSD scans (~30,000 grains per scan, 872 μm field of view, 500 nm step size) provided statistically robust grain size distributions (GSDs).

Experimental findings

The central observation is an intrinsic grain-size gradient: average grain size increases progressively from the free surface toward the specimen interior. The GSD peak shifts from about 5 μm near the surface to about 8 μm in the bulk, and the gradient saturates only beyond ~100 μm depth; differences remain clearly detectable down to depths of ~40 μm, corresponding to five to ten grain layers. Island-like small grains embedded in grown grains are abundant directly at the surface—consistent with grooving drag—but largely absent already at 13 μm depth, while the GSD still differs significantly from the bulk there.

The thickness-series experiments reinforce this conclusion. At half-thickness (where grooving effects are safely excluded), the fraction of small grains (<3 μm) surviving annealing increases strongly as specimen thickness decreases: it is about 50% higher in the 40 μm specimens and almost 90% higher in the 10 μm lamellae compared to the 1 mm reference. Notably, no difference exists between thin and thick samples at the very surface or at 13 μm depth—the divergence appears only in the interior—confirming that the trend reflects depth-dependent growth kinetics rather than sample-to-sample scatter. The implication is that specimen dimensions influence measured grain growth even when several grains span the thickness, well above the semi-2D limit where grooving stagnation is classically invoked.

Continuum modeling

To rationalize these observations, the authors extend a disconnection-based continuum model of GB migration (2604.16740) to include an infinitely extended free surface. The equation of motion combines capillarity ($\Gamma\kappa$) with the coupling between resolved shear stress (RSS) and the shear-coupling factor $\boldsymbol{\beta}$:

$$\partial_t \mathbf{x} = \mathbf{M}(\Gamma\kappa + \boldsymbol{\tau}\cdot\boldsymbol{\beta})\hat{\mathbf{n}}$$

The internal stress field is computed by superposing analytic elastic fields of dislocations, using Head's solution for edge dislocations near a stress-free surface: the bulk field plus an image dislocation term plus correction terms enforcing the zero-traction boundary condition. The essential physics is that the long-range dislocation-like elastic field of disconnections becomes effectively short-ranged in the presence of a free surface, with decay scaling with distance from the surface—so the surface modifies internal stresses even for GBs that do not terminate at it.

Simulations address two scenarios:

• Surface pinning (grooving): multi-phase-field simulations of curvature flow with frozen boundary layers show that pinning effects vanish beyond a depth of roughly 1.25 average grain radii. Grooving alone therefore cannot explain gradients extending over five to ten grain layers.
• Isolated circular grain near a free surface: single-grain phase-field simulations varying the grain-surface distance $d$ and the relative orientation $\alpha$ of the shear direction reveal non-trivial behavior. For $\alpha = 0$ and $\alpha = \pi/2$, evolution is slightly faster or slower than in bulk, respectively; for intermediate orientations (e.g., $\alpha = \pi/4$), the RSS changes sign and magnitude near the surface, attracting the GB and slowing shrinkage. Across configurations, deceleration dominates: the slowest case takes double the bulk annihilation time, while the fastest case reduces it by only 30%. This asymmetry is consistent with the experimental observation that most—but not all—near-surface grains grow more slowly.

The relative importance of shear coupling versus capillarity is expected to increase with grain size, so the effect does not vanish as growth proceeds, short of regime changes such as plastic relaxation.

Discussion

The combination of experiments and simulations supports a specific mechanistic assignment. Thermal grooves, typically 50–100 nm deep, can explain growth stagnation in films with one to a few grains across the thickness and the island grains at the very surface, but their influence decays within about one grain layer. The observed gradient over 40 μm instead arises from free-surface-induced modification of internal stresses generated by shear-coupled migration. This also aligns with prior proposals that stress buildup from shear coupling, rather than grooving alone, causes growth stagnation in thin films—an interpretation supported by observations of stagnation even under protective capping layers.

A practically significant consequence concerns 3D X-ray diffraction studies of grain growth, which typically use platelet or needle specimens with minimum dimensions of several hundred micrometers. For typical grain sizes of tens of micrometers, such specimens contain fewer than ten grain layers across their thinnest dimension and thus fall squarely within the regime where free-surface effects are pronounced. Kinetics extracted from such experiments—and comparisons with bulk 2D sections—must therefore be interpreted with care.

Limitations and open questions

The modeling relies on simplifying assumptions that the authors state explicitly: a single active disconnection mode per reference orientation, isotropic interface properties, an idealized straight infinite free surface, and neglect of surface grooving in the single-grain dynamics (acknowledged to shift quantitative values but not trends). Real grains experience different shear-coupling behavior at each delimiting GB, and a fully self-consistent description of polycrystalline growth with realistic free-surface morphologies remains beyond current modeling capabilities. Experimentally, the prediction that free surfaces broaden the grain size distribution (since shear orientation can either accelerate or retard shrinkage) would require highly accurate statistics to confirm, and the role of multiple free surfaces or engineered surface conditions is unexplored. Whether the effect generalizes beyond nickel and the specific annealing temperature also remains to be established.

Conclusion

This work demonstrates that free surfaces produce intrinsic grain-size gradients extending five to ten grain layers into bulk polycrystalline nickel—far deeper than thermal grooving can account for. Phase-field simulations based on disconnection-mediated GB migration attribute this to elastic relaxation at the zero-traction surface, which reshapes the long-range internal stress fields generated by shear coupling and predominantly retards, though occasionally accelerates, local growth. The results establish free-surface elasticity as a relevant factor in microstructural evolution whenever the grain-layer count is limited, with direct implications for thin-film processing and for the design and interpretation of 3D grain growth experiments.