A self-contained, Fourier-matched derivation is presented that extends the classical Jackson–Hunt lamellar spacing-selection law from binary eutectic solidification to multicomponent solid-state eutectoid transformations in steels. The resulting selection law, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V\lambda _*^2 = B/A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mmultiscripts> <mi>λ</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mn>2</mn> </mmultiscripts> <mo>=</mo> <mi>B</mi> <mo stretchy="false">/</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, retains the elegant structure of the classical result while embedding full multicomponent thermodynamic content. The selection constant B/A depends explicitly on the substitutional composition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{x}_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> through the conditional solvus-slope difference <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g_\Delta (\varvec{x}_\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mi mathvariant="normal">Δ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mi>∞</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the carbon chemical diffusivity <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D_{\text{C}}^\gamma (\varvec{x}_\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>D</mi> <mrow> <mtext>C</mtext> </mrow> <mi>γ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mi>∞</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and the capillarity parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {G}}(\varvec{x}_\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mi>∞</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, thereby providing a direct, closed-form link between alloy chemistry and pearlite microstructural refinement. The derivation exploits a dual Péclet-number ordering—carbon fast, substitutionals frozen at the lamellar scale—to reduce the multicomponent diffusion problem to a single-species Fourier solution, while the parametric influence of every substitutional alloying element is preserved through conditional phase-equilibrium quantities. Capillarity (Gibbs–Thomson) undercooling is treated through trijunction force balance and circular-arc curvature, interface mobility (kinetic) undercooling is incorporated as an additive contribution, and a framework for boundary-diffusion corrections that interpolates smoothly between the volume-diffusion (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$V\lambda ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>λ</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>) and boundary-diffusion (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$V\lambda ^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>λ</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>) limits is presented. Comparison with four independent experimental datasets spanning 0 to 3.5 wt pct Mn in Fe–C and Fe–C–Mn steels demonstrates that the theory captures approximately Mn-induced growth rate retardation without adjustable parameters, correctly ranks the alloy systems, and identifies the transition to partitioning pearlite as the natural model boundary. A Robin boundary condition extension is outlined that provides a clear path toward quantitative treatment of the partitioning regime.

Lamellar Spacing Selection in Multicomponent Eutectoid Transformations: Extension of the Jackson–Hunt Framework to Fe–C–X Steels

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发表日期：

2026年7月7日

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P. G. Kubendran Amos

摘要

A self-contained, Fourier-matched derivation is presented that extends the classical Jackson–Hunt lamellar spacing-selection law from binary eutectic solidification to multicomponent solid-state eutectoid transformations in steels. The resulting selection law, $V\lambda _*^2 = B/A$ $V λ_{*}^{2} = B / A$ , retains the elegant structure of the classical result while embedding full multicomponent thermodynamic content. The selection constant B/A depends explicitly on the substitutional composition $\varvec{x}_\infty $ $x_{\infty}$ through the conditional solvus-slope difference $g_\Delta (\varvec{x}_\infty )$ $g_{Δ} (x_{\infty})$ , the carbon chemical diffusivity $D_{\text{C}}^\gamma (\varvec{x}_\infty )$ $D_{C}^{γ} (x_{\infty})$ , and the capillarity parameter ${\mathcal {G}}(\varvec{x}_\infty )$ $G (x_{\infty})$ , thereby providing a direct, closed-form link between alloy chemistry and pearlite microstructural refinement. The derivation exploits a dual Péclet-number ordering—carbon fast, substitutionals frozen at the lamellar scale—to reduce the multicomponent diffusion problem to a single-species Fourier solution, while the parametric influence of every substitutional alloying element is preserved through conditional phase-equilibrium quantities. Capillarity (Gibbs–Thomson) undercooling is treated through trijunction force balance and circular-arc curvature, interface mobility (kinetic) undercooling is incorporated as an additive contribution, and a framework for boundary-diffusion corrections that interpolates smoothly between the volume-diffusion ( $V\lambda ^2$ $V λ^{2}$ ) and boundary-diffusion ( $V\lambda ^3$ $V λ^{3}$ ) limits is presented. Comparison with four independent experimental datasets spanning 0 to 3.5 wt pct Mn in Fe–C and Fe–C–Mn steels demonstrates that the theory captures approximately Mn-induced growth rate retardation without adjustable parameters, correctly ranks the alloy systems, and identifies the transition to partitioning pearlite as the natural model boundary. A Robin boundary condition extension is outlined that provides a clear path toward quantitative treatment of the partitioning regime.