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arxiv: 2604.16740 · v1 · submitted 2026-04-17 · ❄️ cond-mat.mtrl-sci

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

Jing Tang , Runlu Yan , Donglan Zhang , Ronald Schnitzer , Lorenz Romaner , Marlene Kapp , Marco Salvalaglio , Oliver Renk This is my paper

Pith reviewed 2026-05-10 07:42 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords grain growthfree surfacegrain size gradientnickel polycrystalshear couplingelastic relaxationmicrostructureannealing
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The pith

Free surfaces cause smaller grains near the exterior than deeper inside in annealed nickel because they relax elastic stresses from shear-coupled boundary motion.

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

The paper studies grain growth in high-purity bulk nickel polycrystals during annealing, focusing on how a free surface changes the process. It finds that grain size increases steadily from the surface inward over depths of five to ten grain layers, well beyond where thermal grooves act. The authors trace this gradient to elastic relaxation at the free surface, which modifies the stress fields created when grain boundaries migrate with shear coupling and thereby changes the driving forces for further growth. This shows that surface effects reach farther into the material than curvature-driven models alone would predict. The evidence comes from comparing cross-section measurements with in-plane observations on specimens of different thicknesses.

Core claim

In high-purity nickel polycrystals, annealing produces an intrinsic grain-size gradient in which grain diameter increases gradually from the free surface toward the interior. This grading extends five to ten grain layers deep, where thermal-groove effects are negligible. The gradient forms because elastic relaxation at the free surface alters the internal stress fields that accompany shear-coupled grain-boundary migration, thereby slowing growth near the surface relative to the bulk.

What carries the argument

Elastic relaxation at the free surface that modifies internal stress fields generated by shear-coupled grain boundary migration.

If this is right

  • Grain-size distributions in annealed polycrystals vary systematically with depth from any free surface.
  • Models of grain growth near surfaces must include elastic stress relaxation in addition to curvature flow.
  • Thinner specimens experience stronger surface-induced gradients than thicker ones.
  • Microstructure evolution at depths of several grain diameters can still be influenced by the presence of a free surface.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Surface conditions could be engineered to tailor grain-size gradients in thin-sheet or near-surface regions of components.
  • Similar depth-dependent gradients may appear in other metals where shear-coupled migration contributes to grain growth.
  • Grain-growth simulations that omit elastic boundary conditions at free surfaces will under-predict interior grain sizes.

Load-bearing premise

The grain-size gradient is produced by elastic relaxation of shear-coupling stresses rather than by specimen-preparation artifacts, thermal history differences, or other unmeasured surface influences.

What would settle it

Bonding a free surface to a rigid substrate to suppress elastic relaxation and then annealing the specimen; absence of the grain-size gradient in that case would falsify the proposed mechanism.

Figures

Figures reproduced from arXiv: 2604.16740 by Donglan Zhang, Jing Tang, Lorenz Romaner, Marco Salvalaglio, Marlene Kapp, Oliver Renk, Ronald Schnitzer, Runlu Yan.

Figure 1
Figure 1. Figure 1: Schematic illustration of the sample extraction from the HPT disc and the two different set of specimens investigated in this work. Dimensions are not to scale [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: To separate grooving effects, and assess the microstructural evolution at different depths from the surface, specimens were sequentially characterized using EBSD. Layers were removed by electropolishing (thin specimens) and a combination of grinding and electropolishing in case of thick specimens. Dimensions are not to scale. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of an infinite straight free surface on elastic fields. (a) Dislocation with b = (√ 2/2, √ 2/2) illustrated by stress field components with (three panels on the right) and without (three panels on the left) a free surface with nˆ = yˆ at y = 5. (b) Internal resolved shear stress (RSS) generated by disconnections (Eq. (4)) over a circular grain boundary changing the orientation (α) of references e (k… view at source ↗
Figure 4
Figure 4. Figure 4: Color IPF maps and corresponding grain size distribution curves obtained on specimen set i. (a) Schematic depticting the analyzed sample regions; (b) Detailed EBSD IPF color maps from the regions in (a); (c) Cumulative grain size distribution curves based on the data set displayed in (b); Considering that about 1500 grains were analyzed for each depth, there is a clear tendency for larger grains in the spe… view at source ↗
Figure 5
Figure 5. Figure 5: Representative color IPF maps taken at different depths from the surface of a 1 mm thick sample (a); of a 40 µm thick specimen (b) and of two ultra-thin lamellae taken at the very surface and after removing additional 4 µm thickness by FIB polishing (c). The Z direction and associated color code refers to the direction perpendicular to the specimen surface. thickness) at 13 µm and 500 µm depth emerged. Mor… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Representative grain size distribution curves of various 1 mm and 40 µm thick samples obtained at different representative depths from the surface. Each distribution contains grain size data from EBSD scans covering about 30,000 grains. #2, #3, etc. refers to the individual specimen number, respectively; (b) Plot of all the grain size distributions of one 1 mm thick and one 40 µm thick specimen at diff… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the fraction of small grains (<3 µm size) surviving the heat treatment in the specimen center (at approximately half thickness) for different sample thicknesses. Grain growth is clearly subdued as specimen size decreases. The scale bars for the two maps within a particular dotted-line frame are the same. Z direction and related color code refer to the direction perpendicular to the specimen s… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Illustration of grain growth by mean-curvature flow with (blue) and without (red) surface pinning. (b) L2 norm of the difference between the fields corresponding to the networks with and without pinning, as shown in panel (a), plotted as a function of the distance from the surface D, normalized by the average grain size ⟨R⟩. The solid blue line denotes the average over different times (every ∆t = 250) … view at source ↗
read the original 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 $\mu$m, and 10 $\mu$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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper experimentally investigates grain growth in high-purity polycrystalline nickel, reporting an intrinsic grain-size gradient near free surfaces that extends 5–10 grain layers into the interior. Using cross-sectional analysis of 1 mm thick specimens and in-plane observations on samples of three thicknesses (1 mm, 40 μm, 10 μm), the authors conclude that elastic relaxation at the free surface modifies internal stress fields arising from shear-coupled grain-boundary migration, thereby producing the observed gradient beyond the range of thermal-groove effects.

Significance. If the mechanistic attribution is substantiated, the result would strengthen the case for incorporating elastic and shear-coupling contributions into grain-growth models, particularly for near-surface and thin-film regimes where free-surface relaxation is pronounced. The work supplies direct microstructural evidence that challenges purely curvature-driven descriptions and could guide processing strategies for controlling grain-size distributions in polycrystalline materials.

major comments (3)
  1. [Abstract] Abstract: The central claim that the gradient 'is significantly affected by elastic relaxation at the free surface, which alters the internal stress fields generated by shear-coupled grain boundary migration' is not supported by any quantitative elastic calculations, finite-element stress maps, or in-situ strain measurements within the manuscript; the thickness and cross-section comparisons demonstrate differences but do not isolate or quantify the proposed stress-field mechanism.
  2. [Methods] Experimental methods (specimen preparation): The preparation protocols for the 40 μm and 10 μm foils are not described in sufficient detail to exclude confounding effects such as residual stresses introduced during thinning, surface damage, or altered initial texture; without these controls, the thickness-dependent grain-size trends cannot be unambiguously attributed to free-surface elastic relaxation.
  3. [Results] Results (cross-section and grain-size data): The reported gradient extending 5–10 grain layers lacks accompanying statistical tables, error bars on grain-size measurements, EBSD acquisition parameters, or explicit depth calibration; without these, it is impossible to assess whether the gradual increase is statistically robust or influenced by sampling bias or post-processing choices.
minor comments (2)
  1. [Abstract] Abstract contains a grammatical error: 'elastic effects (intrinsic stress) arised' should read 'arise'.
  2. Notation for specimen thicknesses is inconsistent between text (1 mm, 40 μm, 10 μm) and any accompanying figures; uniform use of SI units and explicit labeling of in-plane versus cross-section views would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We have carefully considered each comment and revised the manuscript accordingly to strengthen the presentation of our experimental findings and their interpretation.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the gradient 'is significantly affected by elastic relaxation at the free surface, which alters the internal stress fields generated by shear-coupled grain boundary migration' is not supported by any quantitative elastic calculations, finite-element stress maps, or in-situ strain measurements within the manuscript; the thickness and cross-section comparisons demonstrate differences but do not isolate or quantify the proposed stress-field mechanism.

    Authors: We agree that the manuscript lacks direct quantitative elastic calculations or finite-element modeling to explicitly map the stress fields. Our interpretation relies on the comparative experimental data: the grain-size gradient persists to depths where thermal grooving is negligible, and thinner specimens show enhanced effects consistent with surface relaxation influencing shear-coupled migration. We will revise the abstract and discussion to more cautiously phrase the claim as 'consistent with' elastic relaxation effects, and add references to theoretical works on shear coupling and elastic interactions to provide indirect support. Full quantitative modeling is planned for future work but is outside the current experimental focus. revision: partial

  2. Referee: [Methods] Experimental methods (specimen preparation): The preparation protocols for the 40 μm and 10 μm foils are not described in sufficient detail to exclude confounding effects such as residual stresses introduced during thinning, surface damage, or altered initial texture; without these controls, the thickness-dependent grain-size trends cannot be unambiguously attributed to free-surface elastic relaxation.

    Authors: We will expand the Methods section to provide more detailed preparation protocols for the 40 μm and 10 μm foils, including steps taken to minimize residual stresses, surface damage, and ensure consistent initial texture. Specific controls such as pre- and post-annealing characterization will be described to demonstrate that the thickness-dependent trends are due to free-surface elastic relaxation. revision: yes

  3. Referee: [Results] Results (cross-section and grain-size data): The reported gradient extending 5–10 grain layers lacks accompanying statistical tables, error bars on grain-size measurements, EBSD acquisition parameters, or explicit depth calibration; without these, it is impossible to assess whether the gradual increase is statistically robust or influenced by sampling bias or post-processing choices.

    Authors: We will add statistical tables with mean grain sizes, standard deviations, and number of measurements at different depths, include error bars in the relevant figures, specify EBSD acquisition parameters and depth calibration procedures in the Methods section, and provide additional analysis to confirm the statistical robustness of the observed gradient. revision: yes

Circularity Check

0 steps flagged

Experimental observation of grain-size gradient; attribution interpretive but non-circular

full rationale

The paper reports direct microstructural measurements on cross-sections and specimens of different thicknesses, identifying a grain-size gradient extending 5-10 grain layers. This is an empirical finding from image analysis rather than a derivation or model that reduces to its own fitted inputs. The mechanistic link to elastic relaxation of shear-coupling stresses is presented as an interpretation based on geometry comparisons, without equations that define the gradient in terms of itself or predictions that are statistically forced by parameter fitting within the paper. No self-citation load-bearing steps, uniqueness theorems, or ansatzes smuggled via prior work are required for the central claim. The result is self-contained as an observation with supporting discussion.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The claim rests on experimental observation of microstructure evolution plus an interpretive link to elastic relaxation; no explicit free parameters, ad-hoc axioms, or new entities are stated in the abstract.

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Reference graph

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