Surface tension is a key thermophysical property governing the phase stability and interfacial behavior of liquid metals, yet its accurate measurement at high temperatures remains experimentally challenging. In this study, a refined semi-empirical model is developed to predict the surface tension of liquid metals using only thermodynamic and structural parameters, without requiring experimental density–temperature data. A strong linear relationship between the surface tension at the melting point (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\sigma }_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>) and the extrapolated value at 0 K (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\sigma }_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>) enables an empirical estimation of the critical temperature (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">${T}_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>) from the melting temperature (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">${T}_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation>), expressed as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${T}_{c}\approx 5.124\, {T}_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>≈</mo> <mn>5.124</mn> <mspace width="0.166667em" /> <msub> <mi>T</mi> <mi>m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. The proportionality between <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${T}_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${T}_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> is thermodynamically justified by cohesive energy considerations. Furthermore, a universal correlation between <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d\sigma /dT$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>σ</mi> <mo stretchy="false">/</mo> <mi>d</mi> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$-{\sigma }_{m}/({T}_{c}-{T}_{m})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>σ</mi> <mi>m</mi> </msub> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>T</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is established, leading to a practical expression d<InlineEquation ID="IEq10"> <EquationSource Format="TEX">${\sigma }_{m}/dT=-0.358\,{\sigma }_{m}/{T}_{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>m</mi> </msub> <mo stretchy="false">/</mo> <mi>d</mi> <mi>T</mi> <mo>=</mo> <mo>-</mo> <mn>0.358</mn> <mspace width="0.166667em" /> <msub> <mi>σ</mi> <mi>m</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>T</mi> <mi>m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. The model yields a consistent atomic packing fraction (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\eta \approx 0.459$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>≈</mo> <mn>0.459</mn> </mrow> </math></EquationSource> </InlineEquation>), in good agreement with theoretical dense-liquid values. Using these correlations, surface tensions of 26 additional liquid metals were successfully predicted, expanding the dataset to 68 elements. The proposed framework provides an efficient and physically meaningful approach for estimating surface tension and its temperature dependence in metallic systems, with implications for alloy design and high-temperature process modeling. Graphical Abstract

Predicting Surface Tension of Liquid Metals

期刊论文
发表日期：

2026年3月30日

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Yoongu Kang,
In-Ho Jung

摘要

Surface tension is a key thermophysical property governing the phase stability and interfacial behavior of liquid metals, yet its accurate measurement at high temperatures remains experimentally challenging. In this study, a refined semi-empirical model is developed to predict the surface tension of liquid metals using only thermodynamic and structural parameters, without requiring experimental density–temperature data. A strong linear relationship between the surface tension at the melting point ( ${\sigma }_{m}$ $σ_{m}$ ) and the extrapolated value at 0 K ( ${\sigma }_{0}$ $σ_{0}$ ) enables an empirical estimation of the critical temperature ( ${T}_{c}$ $T_{c}$ ) from the melting temperature ( ${T}_{m}$ $T_{m}$ ), expressed as ${T}_{c}\approx 5.124\, {T}_{m}$ $T_{c} \approx 5.124 T_{m}$ . The proportionality between ${T}_{c}$ $T_{c}$ and ${T}_{m}$ $T_{m}$ is thermodynamically justified by cohesive energy considerations. Furthermore, a universal correlation between $d\sigma /dT$ $d σ / d T$ and $-{\sigma }_{m}/({T}_{c}-{T}_{m})$ $- σ_{m} / (T_{c} - T_{m})$ is established, leading to a practical expression d ${\sigma }_{m}/dT=-0.358\,{\sigma }_{m}/{T}_{m}$ $σ_{m} / d T = - 0.358 σ_{m} / T_{m}$ . The model yields a consistent atomic packing fraction ( $\eta \approx 0.459$ $η \approx 0.459$ ), in good agreement with theoretical dense-liquid values. Using these correlations, surface tensions of 26 additional liquid metals were successfully predicted, expanding the dataset to 68 elements. The proposed framework provides an efficient and physically meaningful approach for estimating surface tension and its temperature dependence in metallic systems, with implications for alloy design and high-temperature process modeling.

Graphical Abstract