Abstract <p>The formation of cuboid fibers in Ni-Al-Mo and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$[\alpha \beta ]_a[\alpha \delta ]_b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">[</mo> <mi>α</mi> <mi>β</mi> <mo stretchy="false">]</mo> </mrow> <mi>a</mi> </msub> <msub> <mrow> <mo stretchy="false">[</mo> <mi>α</mi> <mi>δ</mi> <mo stretchy="false">]</mo> </mrow> <mi>b</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> superstructures in Bi-In-Sn ternary eutectics, driven by anisotropic interfacial energies, is investigated. To explain the formed superstructures (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> solid phase arrangement mechanism), a morphology map is developed based on Directional Solidification (DS) simulation results. It is assumed that phase omission potentially occurs in locked grains, where interface anisotropy influences microstructure evolution in the following manner: Anisotropic <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \beta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> interfaces can drive the elimination of either the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> layers in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \beta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> bilayers, while <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \delta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> interface anisotropy can lead to the removal of either the <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> layers in <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \delta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> bilayers. This mechanism accounts for the formation of <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \beta \alpha \delta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> <mi>α</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> motifs in quasi-isotropic grains and the emergence of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \beta \delta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$[\alpha \beta ]_a[\alpha \delta ]_b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">[</mo> <mi>α</mi> <mi>β</mi> <mo stretchy="false">]</mo> </mrow> <mi>a</mi> </msub> <msub> <mrow> <mo stretchy="false">[</mo> <mi>α</mi> <mi>δ</mi> <mo stretchy="false">]</mo> </mrow> <mi>b</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> units in locked grains. It is additionally noted that <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \beta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> anisotropy in locked grains exerts a stronger influence on microstructural development than <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_11501_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \delta$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation> anisotropy. Subsequent rotating DS (RDS) simulations provide additional supporting evidence for the assumed formation mechanisms. The radius profile of solidified floating grains exhibits a straight spiral pattern, whereas in locked grains, it follows a tilted spiral, with the tilt angle varying proportionally to anisotropy strength. Notably, the activation of anisotropy in any interface affects neighboring interfaces, even when they are modeled isotropically.</p> Graphical abstract <p></p>

Phase-field analysis of anisotropic interfacial energy influence on microstructure evolution in directional and rotating directional solidification of ternary eutectics

期刊论文
出版模式：

开放获取
发表日期：

2025年10月13日

查看全文

Kaveh Dargahi Noubary,
Britta Nestler

摘要

Abstract

The formation of cuboid fibers in Ni-Al-Mo and $[\alpha \beta ]_a[\alpha \delta ]_b$ ${[α β]}_{a} {[α δ]}_{b}$ superstructures in Bi-In-Sn ternary eutectics, driven by anisotropic interfacial energies, is investigated. To explain the formed superstructures ( $\alpha$ $α$ , $\beta$ $β$ and $\delta$ $δ$ solid phase arrangement mechanism), a morphology map is developed based on Directional Solidification (DS) simulation results. It is assumed that phase omission potentially occurs in locked grains, where interface anisotropy influences microstructure evolution in the following manner: Anisotropic $\alpha \beta$ $α β$ interfaces can drive the elimination of either the $\alpha$ $α$ or $\beta$ $β$ layers in $\alpha \beta$ $α β$ bilayers, while $\alpha \delta$ $α δ$ interface anisotropy can lead to the removal of either the $\alpha$ $α$ or $\delta$ $δ$ layers in $\alpha \delta$ $α δ$ bilayers. This mechanism accounts for the formation of $\alpha \beta \alpha \delta$ $α β α δ$ motifs in quasi-isotropic grains and the emergence of $\alpha \beta \delta$ $α β δ$ or $[\alpha \beta ]_a[\alpha \delta ]_b$ ${[α β]}_{a} {[α δ]}_{b}$ units in locked grains. It is additionally noted that $\alpha \beta$ $α β$ anisotropy in locked grains exerts a stronger influence on microstructural development than $\alpha \delta$ $α δ$ anisotropy. Subsequent rotating DS (RDS) simulations provide additional supporting evidence for the assumed formation mechanisms. The radius profile of solidified floating grains exhibits a straight spiral pattern, whereas in locked grains, it follows a tilted spiral, with the tilt angle varying proportionally to anisotropy strength. Notably, the activation of anisotropy in any interface affects neighboring interfaces, even when they are modeled isotropically.

Graphical abstract