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arxiv: 1906.10404 · v1 · pith:B5ESR3F5new · submitted 2019-06-25 · ❄️ cond-mat.mtrl-sci

Understanding the volume-diffusion governed shape-instabilities in metallic systems

P G Kubendran Amos This is my paper

Pith reviewed 2026-05-25 16:50 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords phase-field modelingmicrostructure stabilityGibbs-Thomson relationvolume diffusioncurvature-driven evolutionshape instabilitiesmetallic systems
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The pith

Phase-field model recovers the Gibbs-Thomson relation for curvature-driven shape changes despite using a diffuse interface.

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

The paper applies phase-field modeling to examine how two- and three-dimensional finite structures in metals evolve under curvature when no phase transformation occurs. It first demonstrates that the model still satisfies the Gibbs-Thomson relation that links curvature to the driving force for shape change. With this check complete, the same numerical setup is used to track volume-diffusion controlled transformations. The work addresses the practical need to predict when microstructure changes will alter material properties without relying on explicit interface tracking.

Core claim

A phase-field formulation that replaces the sharp interface with a finite diffuse region through an additional scalar phase-field variable recovers the governing Gibbs-Thomson relation. The validated model is then applied to investigate volume-diffusion governed curvature-induced transformations in two- and three-dimensional finite structures.

What carries the argument

The phase-field variable, a scalar field that replaces explicit sharp-interface tracking with a diffuse transition zone.

If this is right

  • Curvature-driven shape evolution can be simulated in the absence of phase transformations.
  • The method applies equally to two-dimensional and three-dimensional finite structures.
  • Microstructure stability predictions become feasible without the computational cost of explicit interface tracking.
  • Volume diffusion as the rate-controlling mechanism can be isolated and studied separately.

Where Pith is reading between the lines

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

  • The same validation step could be repeated for other diffusion mechanisms to broaden the range of applicable problems.
  • Results from the 3D cases may identify instability pathways that differ qualitatively from their 2D counterparts.
  • If the recovered relation holds for more complex geometries, the approach could be used to screen candidate microstructures for long-term dimensional stability.

Load-bearing premise

The diffuse-interface phase-field formulation accurately reproduces the sharp-interface Gibbs-Thomson relation and curvature-driven evolution for the finite structures considered.

What would settle it

Direct numerical measurement showing that the chemical potential or interface velocity in the phase-field simulations deviates from the value required by the Gibbs-Thomson relation at a measured curvature.

Figures

Figures reproduced from arXiv: 1906.10404 by P G Kubendran Amos.

Figure 2.1
Figure 2.1. Figure 2.1: A schematic representation of the thermal cycle involved in inter-critical and static annealing [PITH_FULL_IMAGE:figures/full_fig_p033_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: A schematic illustration of the caused approach to investigate the stability of the innitely-long [PITH_FULL_IMAGE:figures/full_fig_p035_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Morphological evolution governed by the inherent curvature di erence between the termination [PITH_FULL_IMAGE:figures/full_fig_p036_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Plate morphology assumed by the precipitate in a microstructure and its evolution to cylindrical [PITH_FULL_IMAGE:figures/full_fig_p036_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Schematic representation of fault migration wherein the discontinuous structure evolves by [PITH_FULL_IMAGE:figures/full_fig_p037_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Faulty structure forming a branch of the regular lamella and its evolution leading to the inho [PITH_FULL_IMAGE:figures/full_fig_p038_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Thermal grooving in the through-thickness boundary leading to the fragmentation of the pre [PITH_FULL_IMAGE:figures/full_fig_p039_2_7.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: The shape-change in the lamellar arrangement of the phases initiated by its fragmentation [PITH_FULL_IMAGE:figures/full_fig_p041_2_8.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Schematic illustration of the free energy density vs concentration plot pertaining to a binary [PITH_FULL_IMAGE:figures/full_fig_p076_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Gibbs free-energy density dependence of austenite on carbon concentration as rendered by the [PITH_FULL_IMAGE:figures/full_fig_p077_5_2.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: A representative depiction of the fault migration for a nite duration of [PITH_FULL_IMAGE:figures/full_fig_p096_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Two-dimensional domain setup considered for the simulation of the termination migration. [PITH_FULL_IMAGE:figures/full_fig_p097_7_2.png] view at source ↗
Figure 7
Figure 7. Figure 7: , the amount of the mass transferred during the interval [PITH_FULL_IMAGE:figures/full_fig_p097_7.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: The morphological evolution in the lamellar precipitate accompanying the recession of the faulty [PITH_FULL_IMAGE:figures/full_fig_p099_7_3.png] view at source ↗
Figure 7
Figure 7. Figure 7: , these active regions are highlighted in red and blue. [PITH_FULL_IMAGE:figures/full_fig_p099_7.png] view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: The temporal change in the volume of the discontinuous lamellar-fault. The position of the tip [PITH_FULL_IMAGE:figures/full_fig_p100_7_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: , asserts that the time taken to transfer a denite volume from the faulty to the regular [PITH_FULL_IMAGE:figures/full_fig_p100_7.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: Change in the kinetics of the termination migration under the in uence of the interlamellar [PITH_FULL_IMAGE:figures/full_fig_p101_7_5.png] view at source ↗
Figure 7.6
Figure 7.6. Figure 7.6: The in uence of the spacing between the regular structures on the morphological evolution of the [PITH_FULL_IMAGE:figures/full_fig_p102_7_6.png] view at source ↗
Figure 7.7
Figure 7.7. Figure 7.7: The in uence of the interlamellar distances, [PITH_FULL_IMAGE:figures/full_fig_p102_7_7.png] view at source ↗
Figure 7.8
Figure 7.8. Figure 7.8: The migration of the three-dimensional faulty precipitate and the consequent change in the [PITH_FULL_IMAGE:figures/full_fig_p104_7_8.png] view at source ↗
Figure 7.9
Figure 7.9. Figure 7.9: The change in the volume of the three-dimensional discontinuous structure, for a given period [PITH_FULL_IMAGE:figures/full_fig_p104_7_9.png] view at source ↗
Figure 7.10
Figure 7.10. Figure 7.10: The role of the distance separating the regular structures (2 [PITH_FULL_IMAGE:figures/full_fig_p105_7_10.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: This morphological evolution, referred to as the cylinderization [36], is extensively [PITH_FULL_IMAGE:figures/full_fig_p107_2_4.png] view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: The domain setup considered for the investigation of the cylinderization. The precipitate- [PITH_FULL_IMAGE:figures/full_fig_p108_8_1.png] view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: An isoline representation (φ = 0.5) of the temporal change in the morphology of the precipitate of aspect ratio 6 during curvature-driven transformation. 8.2 Cylinderization of capped ribbon-like structure 8.2.1 Morphological evolution [PITH_FULL_IMAGE:figures/full_fig_p109_8_2.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the morphological changes accompanying the cylinderization of the precipitate [PITH_FULL_IMAGE:figures/full_fig_p109_8.png] view at source ↗
Figure 8.3
Figure 8.3. Figure 8.3: The temporal change in the driving force which governs the simulated transformation is com [PITH_FULL_IMAGE:figures/full_fig_p111_8_3.png] view at source ↗
Figure 8
Figure 8. Figure 8: , unravels that the existing theoretical treatment predicts a nearly linear decrease in [PITH_FULL_IMAGE:figures/full_fig_p111_8.png] view at source ↗
Figure 8.4
Figure 8.4. Figure 8.4: An isoline representation of the precipitate morphology at the initial (t [PITH_FULL_IMAGE:figures/full_fig_p113_8_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: b shows the numerical nature of the normalised parameter [PITH_FULL_IMAGE:figures/full_fig_p116_8.png] view at source ↗
Figure 8.6
Figure 8.6. Figure 8.6: The cylinderization kinetics predicted by the simulation-aided semi-analytical treatment [PITH_FULL_IMAGE:figures/full_fig_p118_8_6.png] view at source ↗
Figure 8.7
Figure 8.7. Figure 8.7: Morphology of the faceted precipitate of aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p119_8_7.png] view at source ↗
Figure 12
Figure 12. Figure 12: , the transformation time increases [PITH_FULL_IMAGE:figures/full_fig_p119_12.png] view at source ↗
Figure 8.8
Figure 8.8. Figure 8.8: Increase in the time taken for the cylinderization of the faceted structure with increase in the [PITH_FULL_IMAGE:figures/full_fig_p120_8_8.png] view at source ↗
Figure 8.9
Figure 8.9. Figure 8.9: The change in the cylinderization rate induced solely by the morphology of the terminations [PITH_FULL_IMAGE:figures/full_fig_p122_8_9.png] view at source ↗
Figure 8.10
Figure 8.10. Figure 8.10: Distribution of the chemical potential and the resulting morphological changes exhibited by [PITH_FULL_IMAGE:figures/full_fig_p123_8_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: shows the simulation domain considered for the analysis of spheroidization of nite [PITH_FULL_IMAGE:figures/full_fig_p127_12.png] view at source ↗
Figure 9.2
Figure 9.2. Figure 9.2: The morphology of the precipitate of aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p130_9_2.png] view at source ↗
Figure 9.3
Figure 9.3. Figure 9.3: The parameter p = rsc (a/b) r is seemingly independent of the initial size of the three-dimensional rod. As shown in [PITH_FULL_IMAGE:figures/full_fig_p133_9_3.png] view at source ↗
Figure 9.4
Figure 9.4. Figure 9.4: The transitory driving at the specic stages of the spheroidization which are calculated based [PITH_FULL_IMAGE:figures/full_fig_p134_9_4.png] view at source ↗
Figure 12.4
Figure 12.4. Figure 12.4: This illustration unravels the relation that [PITH_FULL_IMAGE:figures/full_fig_p135_12_4.png] view at source ↗
Figure 12.5
Figure 12.5. Figure 12.5: Owing to some misappropriations, the existing studies render a driving force which [PITH_FULL_IMAGE:figures/full_fig_p135_12_5.png] view at source ↗
Figure 9.5
Figure 9.5. Figure 9.5: Time taken for the spheroidization of capped rods of di erent aspect ratio is compared with the [PITH_FULL_IMAGE:figures/full_fig_p136_9_5.png] view at source ↗
Figure 9.6
Figure 9.6. Figure 9.6: Temporal evolution of the curvature (potential) di erence, [PITH_FULL_IMAGE:figures/full_fig_p137_9_6.png] view at source ↗
Figure 12.8
Figure 12.8. Figure 12.8: In the initial stages of the spheroidization, high potential is established around the termi- [PITH_FULL_IMAGE:figures/full_fig_p138_12_8.png] view at source ↗
Figure 9.7
Figure 9.7. Figure 9.7: The transformation mechanism underpinning the spheroidization of the capped rod of aspect [PITH_FULL_IMAGE:figures/full_fig_p139_9_7.png] view at source ↗
Figure 9.8
Figure 9.8. Figure 9.8: The geometrical description of the uncapped and faceted rods. The longitudinal and radial view [PITH_FULL_IMAGE:figures/full_fig_p140_9_8.png] view at source ↗
Figure 12.9
Figure 12.9. Figure 12.9: The cross-section of the rods, both longitudinal and radial, are included in this [PITH_FULL_IMAGE:figures/full_fig_p141_12_9.png] view at source ↗
Figure 9.9
Figure 9.9. Figure 9.9: The continual change in the distribution of the chemical potential and respective change in the [PITH_FULL_IMAGE:figures/full_fig_p142_9_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: at [PITH_FULL_IMAGE:figures/full_fig_p142_12.png] view at source ↗
Figure 9.10
Figure 9.10. Figure 9.10: The in uence of the aspect ratio on time taken for the spheroidization of the faceted and un [PITH_FULL_IMAGE:figures/full_fig_p144_9_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: shows the time taken for the spheroidization of uncapped and faceted rods of dif [PITH_FULL_IMAGE:figures/full_fig_p144_12.png] view at source ↗
Figure 12.9
Figure 12.9. Figure 12.9: Consequently, an increased amount of mass transfer is required to spheroidise the [PITH_FULL_IMAGE:figures/full_fig_p145_12_9.png] view at source ↗
Figure 9.11
Figure 9.11. Figure 9.11: The temporal evolution of the capped rod of aspect ratio 9. The spheroidization mechanism is [PITH_FULL_IMAGE:figures/full_fig_p146_9_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: illustrates the morphological evolution of the capped rod of aspect ratio 9. As [PITH_FULL_IMAGE:figures/full_fig_p146_12.png] view at source ↗
Figure 9.12
Figure 9.12. Figure 9.12: Drastic change in the spheroidization rate due to the shift in the transformation mechanism [PITH_FULL_IMAGE:figures/full_fig_p148_9_12.png] view at source ↗
Figure 9.13
Figure 9.13. Figure 9.13: The spheroidization mechanism exhibited by the capped rod of aspect ratio 12, which yields a [PITH_FULL_IMAGE:figures/full_fig_p149_9_13.png] view at source ↗
Figure 9
Figure 9. Figure 9: , is due to the size of the precipitate. [PITH_FULL_IMAGE:figures/full_fig_p149_9.png] view at source ↗
Figure 9.14
Figure 9.14. Figure 9.14: The coarsening rate of the identical entities at the expense of the satellite particle during the [PITH_FULL_IMAGE:figures/full_fig_p151_9_14.png] view at source ↗
Figure 9.15
Figure 9.15. Figure 9.15: The distance between the primary spheroids which result from the spheroidization of the [PITH_FULL_IMAGE:figures/full_fig_p152_9_15.png] view at source ↗
Figure 9.16
Figure 9.16. Figure 9.16: The aspect ratio of the pear-shaped sub-structure at the moment of ovulation during the [PITH_FULL_IMAGE:figures/full_fig_p154_9_16.png] view at source ↗
Figure 9
Figure 9. Figure 9: unravels that, expect for the rods of aspect ratio 10 and 11, the aspect ratio of [PITH_FULL_IMAGE:figures/full_fig_p154_9.png] view at source ↗
Figure 9.17
Figure 9.17. Figure 9.17: Evidently, above the aspect ratio 12, the no visible change is observed in the time [PITH_FULL_IMAGE:figures/full_fig_p155_9_17.png] view at source ↗
Figure 9.17
Figure 9.17. Figure 9.17: The time taken for the primary ovulation during the transformation of the rods with di erent [PITH_FULL_IMAGE:figures/full_fig_p156_9_17.png] view at source ↗
Figure 10.1
Figure 10.1. Figure 10.1: Top- and side-view of the unidirectionally-equiaxed ‘pancake’ structure of diameter [PITH_FULL_IMAGE:figures/full_fig_p163_10_1.png] view at source ↗
Figure 10.2
Figure 10.2. Figure 10.2: A cross-sectional view of the geometrical consideration adopted to dene the precipitate struc [PITH_FULL_IMAGE:figures/full_fig_p164_10_2.png] view at source ↗
Figure 12
Figure 12. Figure 12: indicates that the precipitate, which is considered for this investigation, is a cen [PITH_FULL_IMAGE:figures/full_fig_p164_12.png] view at source ↗
Figure 10.3
Figure 10.3. Figure 10.3: Morphology of the precipitate at the initial (t [PITH_FULL_IMAGE:figures/full_fig_p167_10_3.png] view at source ↗
Figure 10.4
Figure 10.4. Figure 10.4: Aspect ratio of the ellipsoid, or specically ‘oblate spheroid’ , which are formed at the midpoint [PITH_FULL_IMAGE:figures/full_fig_p169_10_4.png] view at source ↗
Figure 10.5
Figure 10.5. Figure 10.5: The transitory driving-force at the initial, midpoint and nal stage of the globularisation [PITH_FULL_IMAGE:figures/full_fig_p172_10_5.png] view at source ↗
Figure 10.6
Figure 10.6. Figure 10.6: The time taken for the globularisation of the pancake precipitates of di erent aspect ratio in [PITH_FULL_IMAGE:figures/full_fig_p173_10_6.png] view at source ↗
Figure 10.7
Figure 10.7. Figure 10.7: A schematic representation of the temporal change in the shape of the pancake structure as [PITH_FULL_IMAGE:figures/full_fig_p175_10_7.png] view at source ↗
Figure 10.8
Figure 10.8. Figure 10.8: The change in the potential di erence, induced by the inherent disparity in the curvature, with [PITH_FULL_IMAGE:figures/full_fig_p176_10_8.png] view at source ↗
Figure 12
Figure 12. Figure 12: includes a resolved sub-plot of the change in the curvature di erence at the ini [PITH_FULL_IMAGE:figures/full_fig_p176_12.png] view at source ↗
Figure 10.9
Figure 10.9. Figure 10.9: The morphological transformation of the pancake structure of aspect ratio 6 during the stage-I [PITH_FULL_IMAGE:figures/full_fig_p177_10_9.png] view at source ↗
Figure 12.6
Figure 12.6. Figure 12.6: With time, the intensity of these peaks continually decreases and the driving force [PITH_FULL_IMAGE:figures/full_fig_p177_12_6.png] view at source ↗
Figure 12
Figure 12. Figure 12: at [PITH_FULL_IMAGE:figures/full_fig_p178_12.png] view at source ↗
Figure 10.10
Figure 10.10. Figure 10.10: The subsequent transformation following the stage-I evolution which results in the globular [PITH_FULL_IMAGE:figures/full_fig_p180_10_10.png] view at source ↗
Figure 11.1
Figure 11.1. Figure 11.1: The boundary-splitting which, through the fragmentation of the continuous precipitate along [PITH_FULL_IMAGE:figures/full_fig_p184_11_1.png] view at source ↗
Figure 11.2
Figure 11.2. Figure 11.2: The morphological conguration of a three-dimensional elliptical plate of the thickness [PITH_FULL_IMAGE:figures/full_fig_p186_11_2.png] view at source ↗
Figure 11.3
Figure 11.3. Figure 11.3: The cross-section of the parent three-dimensional ellipsoid of major-axes length [PITH_FULL_IMAGE:figures/full_fig_p187_11_3.png] view at source ↗
Figure 11.4
Figure 11.4. Figure 11.4: The temporal change in the shape of the elliptical plate of aspect ratio 6 at the three specic [PITH_FULL_IMAGE:figures/full_fig_p188_11_4.png] view at source ↗
Figure 12
Figure 12. Figure 12: extends from at surface of the precipitate to the end of the parent structure along a [PITH_FULL_IMAGE:figures/full_fig_p188_12.png] view at source ↗
Figure 11.5
Figure 11.5. Figure 11.5: The aspect ratio of the prolate ellipsoid formed at the midpoint of the transformation during [PITH_FULL_IMAGE:figures/full_fig_p194_11_5.png] view at source ↗
Figure 11.6
Figure 11.6. Figure 11.6: The instantaneous driving-force at the initial, midpoint and nal stage of the globularisation [PITH_FULL_IMAGE:figures/full_fig_p195_11_6.png] view at source ↗
Figure 11.7
Figure 11.7. Figure 11.7: The time taken for the globularisation of the elliptical plates of di erent aspect ratio in the [PITH_FULL_IMAGE:figures/full_fig_p197_11_7.png] view at source ↗
Figure 12.7
Figure 12.7. Figure 12.7: Both the simulation and analytical study show a monotonic increase in the time [PITH_FULL_IMAGE:figures/full_fig_p198_12_7.png] view at source ↗
Figure 11.8
Figure 11.8. Figure 11.8: The temporal evolution of the potential di erence which govern the shape-changes exhibited [PITH_FULL_IMAGE:figures/full_fig_p199_11_8.png] view at source ↗
Figure 11.9
Figure 11.9. Figure 11.9: The shape change exhibited by the elliptical plate of aspect ratio 6 in the stage-I of globulari [PITH_FULL_IMAGE:figures/full_fig_p200_11_9.png] view at source ↗
Figure 11.10
Figure 11.10. Figure 11.10: The transformation mechanism following the stage-I globularisation of the elliptical plate of [PITH_FULL_IMAGE:figures/full_fig_p202_11_10.png] view at source ↗
Figure 12.1
Figure 12.1. Figure 12.1: The morphological conguration of the cementite plate embedded in the ferrite matrix, with [PITH_FULL_IMAGE:figures/full_fig_p207_12_1.png] view at source ↗
Figure 12.2
Figure 12.2. Figure 12.2: It is evident that, similar to the pancake morphology, the faceted plate transforms [PITH_FULL_IMAGE:figures/full_fig_p208_12_2.png] view at source ↗
Figure 12.3
Figure 12.3. Figure 12.3: The aspect ratio of the prolate spheroid, which is formed at the midpoint of the spheroidization [PITH_FULL_IMAGE:figures/full_fig_p209_12_3.png] view at source ↗
Figure 12
Figure 12. Figure 12: indicates that, at the midpoint of the transformation, the precipitate is much closer [PITH_FULL_IMAGE:figures/full_fig_p209_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: illustrates the progressive change in the shape of the precipitate of aspect ratio 15, [PITH_FULL_IMAGE:figures/full_fig_p210_12.png] view at source ↗
Figure 12.4
Figure 12.4. Figure 12.4: The monotonic increase in the time taken for the spheroidization of the cementite plate with [PITH_FULL_IMAGE:figures/full_fig_p211_12_4.png] view at source ↗
Figure 12.5
Figure 12.5. Figure 12.5: The termination-migration assisted spheroidization of cementite plate of aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p212_12_5.png] view at source ↗
Figure 12.6
Figure 12.6. Figure 12.6: The transformation of a relatively-large faceted cementite plate of aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p213_12_6.png] view at source ↗
Figure 12
Figure 12. Figure 12: shows the initial stages of the spheroidization of the plate with aspect ratio [PITH_FULL_IMAGE:figures/full_fig_p213_12.png] view at source ↗
Figure 12.7
Figure 12.7. Figure 12.7: The temporal evolution of the circular discontinuities accompanying the morphological trans [PITH_FULL_IMAGE:figures/full_fig_p214_12_7.png] view at source ↗
Figure 12.8
Figure 12.8. Figure 12.8: The fragmentation of the region separating the discontinuities in the cementite plate of aspect [PITH_FULL_IMAGE:figures/full_fig_p216_12_8.png] view at source ↗
Figure 12.9
Figure 12.9. Figure 12.9: The temporal change in the geometric parameter [PITH_FULL_IMAGE:figures/full_fig_p218_12_9.png] view at source ↗
Figure 12.10
Figure 12.10. Figure 12.10: Furthermore, three-dimensional representation of the transitory potential-distribution [PITH_FULL_IMAGE:figures/full_fig_p219_12_10.png] view at source ↗
Figure 12.10
Figure 12.10. Figure 12.10: The formation of the separate cementite entities through the breaking-up of the cementite [PITH_FULL_IMAGE:figures/full_fig_p220_12_10.png] view at source ↗
Figure 12.11
Figure 12.11. Figure 12.11: The morphological evolution associated with the spheroidization of the cementite plate of [PITH_FULL_IMAGE:figures/full_fig_p222_12_11.png] view at source ↗
Figure 12.12
Figure 12.12. Figure 12.12: The cut-o criterion, which is adopted in the industrial heat-treatment techniques to distin [PITH_FULL_IMAGE:figures/full_fig_p223_12_12.png] view at source ↗
Figure 13.1
Figure 13.1. Figure 13.1: The outcomes of the preliminary analysis on the co-operative growth of the mutually [PITH_FULL_IMAGE:figures/full_fig_p230_13_1.png] view at source ↗
read the original abstract

The reliability of any day-to-day material is critically dictated by its properties. One factor which governs the behaviour of a material, under a given condition, is the microstructure. Despite the absence of any phase transformation, a change in the microstructure would significantly alter the properties. Therefore, a substantial understanding on the stability of the microstructure is vital to avert any unexpected catastrophic change in the material properties. In the present work, one such numerical approach called phase-field modelling in employed to analyse the stability of two- and three-dimensional finite structures, which dictate the curvature-driven evolution of the microstructure. A characteristic feature of this numerical approach is the introduction of a scalar variable, called the phase field, in addition to the other thermodynamic variables. While the inclusion of the phase field obviates the need for the interface tracking, which is a strenuous aspect of the other conventional techniques, it replaces the sharp interface with a finite diffuse region. Therefore, before adopting and extending the phase-field technique, it is shown that the model recovers the governing law, i.e, Gibbs-Thomson relation, despite the introduction of the diffuse interface. Subsequently, the numerical treatment is employed to investigate the volume-diffusion governed curvature-induced transformation.

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

1 major / 2 minor

Summary. The manuscript develops a phase-field model to study curvature-driven shape instabilities in 2-D and 3-D finite metallic structures under volume diffusion. It first claims to demonstrate recovery of the sharp-interface Gibbs-Thomson relation despite the diffuse interface, then applies the model to investigate the resulting transformations.

Significance. If the validation step is quantitatively confirmed, the work supplies a standard but useful numerical framework for microstructure stability analysis without explicit interface tracking. The application to finite structures under volume diffusion could inform material reliability predictions, though the core methodology is already established in the phase-field literature.

major comments (1)
  1. [Abstract / model validation paragraph] Abstract and model-validation paragraph: the assertion that the phase-field formulation recovers the Gibbs-Thomson relation is central to justifying all subsequent simulations, yet no quantitative comparison (e.g., relative error versus interface width or curvature radius), tabulated data, or error analysis is referenced. Without these, the load-bearing claim that the diffuse-interface model is faithful for the examined finite structures cannot be verified.
minor comments (2)
  1. Notation for the phase-field variable and the mobility parameters should be introduced with explicit definitions before their first use in the governing equations.
  2. Figure captions for the instability simulations should state the initial geometry, domain size, and interface-width-to-curvature ratio employed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and constructive feedback. We address the single major comment below and agree that additional quantitative validation is warranted to strengthen the manuscript.

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  1. Referee: [Abstract / model validation paragraph] Abstract and model-validation paragraph: the assertion that the phase-field formulation recovers the Gibbs-Thomson relation is central to justifying all subsequent simulations, yet no quantitative comparison (e.g., relative error versus interface width or curvature radius), tabulated data, or error analysis is referenced. Without these, the load-bearing claim that the diffuse-interface model is faithful for the examined finite structures cannot be verified.

    Authors: We agree with the referee that the central claim of recovering the Gibbs-Thomson relation requires explicit quantitative support to be fully convincing. Although the manuscript states that the model recovers the relation, no error metrics, tabulated comparisons, or dependence on interface width/curvature radius were included. In the revised manuscript we will add a new subsection (or appendix) presenting quantitative validation: relative errors between the phase-field chemical potential and the sharp-interface Gibbs-Thomson prediction, plotted or tabulated versus normalized interface width and curvature radius, together with a brief error analysis. This will directly address the concern and confirm the model's fidelity for the finite structures studied. revision: yes

Circularity Check

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No significant circularity; validation is independent check

full rationale

The paper's central step is a numerical validation that the diffuse-interface phase-field model recovers the known sharp-interface Gibbs-Thomson relation when the interface width is small relative to curvature radius. This is a standard, externally falsifiable consistency test for conserved-order-parameter models and does not reduce any prediction to a fitted parameter or self-referential equation. Subsequent simulations of curvature-driven instabilities under volume diffusion rest on this validated formulation without load-bearing self-citations or ansatz smuggling. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or invented entities; the central claim rests on the standard assumption that the phase-field diffuse interface recovers sharp-interface physics.

axioms (1)
  • domain assumption Phase-field model with diffuse interface recovers the sharp-interface Gibbs-Thomson relation
    Invoked when validating the model before applying it to shape instabilities.

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