<p>Melting of metallic nanostructures such as nanoparticles and nanowires has been extensively studied particularly using atomistic simulations and thermodynamic relations for melting temperature. In addition, continuum modeling due to its capabilities has recently received attention for modeling of melting at nanoscale. In this work, melting of copper nanowires is investigated using a phase field model as a continuum model. Axisymmetric model is used to effectively solve the combined Ginzburg–Landau and elasticity equations in order to capture the melting process. The obtained melting temperature shows a nonlinear reduction as the radius decreases and stands between two known thermodynamic relations. The premelting temperature is found slightly below the melting temperature for the radii of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="161_2025_1418_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$R\ge 3\text{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>≥</mo> <mn>3</mn> <mtext>nm</mtext> </mrow> </math></EquationSource> </InlineEquation> while is significantly lower than it for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="161_2025_1418_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$R<3\text{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo><</mo> <mn>3</mn> <mtext>nm</mtext> </mrow> </math></EquationSource> </InlineEquation>. The obtained melting temperature also shows a nonlinear reduction as the length decreases. Also, the obtained MT is averagely 3.5% larger than the melting temperature from the thermodynamic relation for the lengths of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="161_2025_1418_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$L<80\text{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo><</mo> <mn>80</mn> <mtext>nm</mtext> </mrow> </math></EquationSource> </InlineEquation>; and this difference reduces for lower radii. The melting mechanism differs for radii smaller than the solid-melt interface width where the interface propagates only along the nanowire length and not radially. Having studied different thermodynamic driving forces of melting, the transformation strain driving force is found the dominant mechanical term while thermal strain practically shows no impact on the melting temperature for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="161_2025_1418_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$R>3\text{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>></mo> <mn>3</mn> <mtext>nm</mtext> </mrow> </math></EquationSource> </InlineEquation>. The obtained melting temperature well matches the start temperature of melting from the existing molecular dynamics simulations for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="161_2025_1418_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$R\ge 2\text{nm}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>≥</mo> <mn>2</mn> <mtext>nm</mtext> </mrow> </math></EquationSource> </InlineEquation>.</p>

Melting of copper nanowires: phase field simulations and comparison with existing analytical and molecular dynamics results

期刊论文
发表日期：

2025年10月16日

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Mahdi Javanbakht,
Amir Mohammad Mostajeran,
Emilio Barchiesi

摘要

Melting of metallic nanostructures such as nanoparticles and nanowires has been extensively studied particularly using atomistic simulations and thermodynamic relations for melting temperature. In addition, continuum modeling due to its capabilities has recently received attention for modeling of melting at nanoscale. In this work, melting of copper nanowires is investigated using a phase field model as a continuum model. Axisymmetric model is used to effectively solve the combined Ginzburg–Landau and elasticity equations in order to capture the melting process. The obtained melting temperature shows a nonlinear reduction as the radius decreases and stands between two known thermodynamic relations. The premelting temperature is found slightly below the melting temperature for the radii of $R\ge 3\text{nm}$ $R \geq 3 nm$ while is significantly lower than it for $R<3\text{nm}$ $R < 3 nm$ . The obtained melting temperature also shows a nonlinear reduction as the length decreases. Also, the obtained MT is averagely 3.5% larger than the melting temperature from the thermodynamic relation for the lengths of $L<80\text{nm}$ $L < 80 nm$ ; and this difference reduces for lower radii. The melting mechanism differs for radii smaller than the solid-melt interface width where the interface propagates only along the nanowire length and not radially. Having studied different thermodynamic driving forces of melting, the transformation strain driving force is found the dominant mechanical term while thermal strain practically shows no impact on the melting temperature for $R>3\text{nm}$ $R > 3 nm$ . The obtained melting temperature well matches the start temperature of melting from the existing molecular dynamics simulations for $R\ge 2\text{nm}$ $R \geq 2 nm$ .