<p><InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{10\bar{1}1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>10</mn> <mover accent="true"> <mrow> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> twins are known to play a pivotal role in the deformation and fracture of hcp materials under <i>c</i>-axis compression. In this paper, the nucleation and migration of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{10\bar{1}1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>10</mn> <mover accent="true"> <mrow> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> twins are investigated through the integration of atomistic simulations and theoretical calculations. The atomistic simulations reveal extensive nucleation of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{10\bar{1}1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>10</mn> <mover accent="true"> <mrow> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> twins, occurring either directly at the free surface or via a bcc intermediate state in bulk. The theoretical calculations identify the surface-nucleation as the low-shear mode with conjugate twinning plane <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_2=\{10\bar{1}3\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>10</mn> <mover accent="true"> <mrow> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In contrast, the bulk-nucleation is determined as the high-shear mode with irrational twinning shear and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> plane. Our analyses pinpoint that their fundamental distinction lies in whether the twinning shear encompasses an <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\langle a \rangle\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>a</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> component along the common zone axis. During subsequent twin growth, the high-shear mode activates sequential <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(b_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> dislocations, while the low-shear mode involves concurrent activation of two stacked <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(b_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> dislocations, collectively forming a <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(b_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> dislocation. This paper provides valuable insights into the critical distinctions and the complex interplay between the two <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10853_2025_10844_Article_IEq12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{10\bar{1}1\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>10</mn> <mover accent="true"> <mrow> <mn>1</mn> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> modes in hcp materials.</p> Graphical abstract <p></p>

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The nucleation and migration of \(\{10\bar{1}1\}\) twins in hcp materials

  • Jamie Ombogo,
  • Eduardo Vitral,
  • Amir Zahiri,
  • Lei Cao

摘要

\(\{10\bar{1}1\}\) { 10 1 ¯ 1 } twins are known to play a pivotal role in the deformation and fracture of hcp materials under c-axis compression. In this paper, the nucleation and migration of \(\{10\bar{1}1\}\) { 10 1 ¯ 1 } twins are investigated through the integration of atomistic simulations and theoretical calculations. The atomistic simulations reveal extensive nucleation of \(\{10\bar{1}1\}\) { 10 1 ¯ 1 } twins, occurring either directly at the free surface or via a bcc intermediate state in bulk. The theoretical calculations identify the surface-nucleation as the low-shear mode with conjugate twinning plane \(K_2=\{10\bar{1}3\}\) K 2 = { 10 1 ¯ 3 } . In contrast, the bulk-nucleation is determined as the high-shear mode with irrational twinning shear and \(K_2\) K 2 plane. Our analyses pinpoint that their fundamental distinction lies in whether the twinning shear encompasses an \(\langle a \rangle\) a component along the common zone axis. During subsequent twin growth, the high-shear mode activates sequential \(b_2\) b 2 dislocations, while the low-shear mode involves concurrent activation of two stacked \(b_2\) b 2 dislocations, collectively forming a \(b_4\) b 4 dislocation. This paper provides valuable insights into the critical distinctions and the complex interplay between the two \(\{10\bar{1}1\}\) { 10 1 ¯ 1 } modes in hcp materials.

Graphical abstract