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Modulation of heat flux and thermal stress at the double interface by nano-coating thickness

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Abstract

Nano-coatings hold the key to unlocking the vital potential of three-phase nanocomposites by urgently addressing thermal stress concentration. Consequently, the characteristics of double-interfacial heat flux and thermal stress have garnered considerable attention. In this paper, we introduce a set of interfacial conditions aimed at modeling double-interfacial phonon scattering and double-interfacial elasticity in three-phase nanocomposites. We derive closed-form solutions for the thermo-elastic field surrounding a circular nano-inclusion containing the nano-coatings. Through control over the nano-coating thickness and their properties, including thermal conductivity and interfacial phonon scattering coefficient, we can achieve a “neutral inclusions and coatings” temperature field, promoting uniform heat transfer. However, it is important to note that this approach leads to a noticeable increase in thermal stresses at the interfaces and the emergence of stress discontinuities due to the double-interfacial phonon scattering effect. Furthermore, the impact of coating thickness, coating thermal conductivity, and coating thermal expansion coefficient on the maximum interfacial thermal stress exhibits significant variability. Crucially, our findings reveal that adjusting the coating parameters can result in a substantial reduction in the overall thermal stress field’s maximum value. This discovery carries significant implications for the design of nanocomposites and holds great promise for various applications.

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Acknowledgements

H. J. Huang appreciates the support of the National Natural Science Foundation of China (Grant No. 12002119); K. Song appreciates the support of the National Natural Science Foundation of China (Grant No. 12102126).

Author information

Authors and Affiliations

  1. School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, People’s Republic of China

    Haojie Huang

  2. School of Mechanical and Power Engineering, Nanjing Tech University, 30 Puzhu South Road, Nanjing, Jiangsu, People’s Republic of China

    Kun Song

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  1. Haojie Huang
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Contributions

HH contributed to the conceptualization; HH and KS were involved in the methodology; HH contributed to writing—original draft preparation; KS assisted in writing—review and editing; HH and KS acquired the funding; KS contributed to the supervision.

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Correspondence to Kun Song.

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Appendix

Appendix

$$\begin{aligned} A_{-1}= & {} \overline{A_{1} } R_{1}^{2} (R_{1}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} -R_{2}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} +R_{1}^{3} k_{1} k_{3} k_{4} \gamma _{2} -R_{1}^{3} k_{3}^{2} k_{4} \gamma _{2} \nonumber \\{} & {} + R_{1}^{2} R_{2} k_{1} k_{3}^{2} \gamma _{1} +R_{1}^{2} R_{2} k_{1} k_{3} k_{4} \gamma _{1} +R_{1} R_{2}^{2} k_{1} k_{3} k_{4} \gamma _{2} +R_{1} R_{2}^{2} k_{3}^{2} k_{4} \gamma _{2} -R_{2}^{3} k_{1} k_{3}^{2} \gamma _{1} \nonumber \\{} & {} + R_{2}^{3} k_{1} k_{3} k_{4} \gamma _{1} {+ }R_{1}^{3} R_{2} k_{1} k_{3} +\, R_{1}^{3} R_{2} k_{1} k_{4} -R_{1}^{3} R_{2} k_{3}^{2} -R_{1}^{3} R_{2} k_{3} k_{4} +R_{1} R_{2}^{3} k_{1} k_{3} \nonumber \\{} & {} - R_{1} R_{2}^{3} k_{1} k_{4} +R_{1} R_{2}^{3} k_{3}^{2} -R_{1} R_{2}^{3} k_{3} k_{4} )/(R_{1}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} -R_{2}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} \nonumber \\{} & {} + R_{1}^{3} k_{1} k_{3} k_{4} \gamma _{2} +R_{1}^{3} k_{3}^{2} k_{4} \gamma _{2} +R_{1}^{2} R_{2} k_{1} k_{3}^{2} \gamma _{1} \text{+ }R_{1}^{2} R_{2} k_{1} k_{3} k_{4} \gamma _{1} +R_{1} R_{2}^{2} k_{1} k_{3} k_{4} \gamma _{2} \nonumber \\{} & {} - R_{1} R_{2}^{2} k_{3}^{2} k_{4} \gamma _{2} -R_{2}^{3} k_{1} k_{3}^{2} \gamma _{1} +R_{2}^{3} k_{1} k_{3} k_{4} \gamma _{1} \text{+ } R_{1}^{3} R_{2} k_{1} k_{3} +R_{1}^{3} R_{2} k_{1} k_{4} +R_{1}^{3} R_{2} k_{3}^{2} \nonumber \\{} & {} + R_{1}^{3} R_{2} k_{3} k_{4} +R_{1} R_{2}^{3} k_{1} k_{3} -R_{1} R_{2}^{3} k_{1} k_{4} -R_{1} R_{2}^{3} k_{3}^{2} +R_{1} R_{2}^{3} k_{3} k_{4} ) \end{aligned}$$
(93)
$$\begin{aligned} B_{-1}= & {} 2\overline{A_{1} } R_{1}^{3} R_{2}^{2} k_{1} (k_{3} k_{4} \gamma _{2} +R_{2} k_{3} -R_{2} k_{4} )/(R_{1}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} -R_{2}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} \nonumber \\{} & {} + R_{1}^{3} k_{1} k_{3} k_{4} \gamma _{2} +R_{1}^{3} k_{3}^{2} k_{4} \gamma _{2} +R_{1}^{2} R_{2} k_{1} k_{3}^{2} \gamma _{1} +R_{1}^{2} R_{2} k_{1} k_{3} k_{4} \gamma _{1} +R_{1} R_{2}^{2} k_{1} k_{3} k_{4} \gamma _{2} \nonumber \\{} & {} - R_{1} R_{2}^{2} k_{3}^{2} k_{4} \gamma _{2} -R_{2}^{3} k_{1} k_{3}^{2} \gamma _{1} +R_{2}^{3} k_{1} k_{3} k_{4} \gamma _{1} {+ }R_{1}^{3} R_{2} k_{1} k_{3} +R_{1}^{3} R_{2} k_{1} k_{4} +R_{1}^{3} R_{2} k_{3}^{2} \nonumber \\{} & {} + R_{1}^{3} R_{2} k_{3} k_{4} +R_{1} R_{2}^{3} k_{1} k_{3} -R_{1} R_{2}^{3} k_{1} k_{4} -R_{1} R_{2}^{3} k_{3}^{2} +R_{1} R_{2}^{3} k_{3} k_{4} ) \end{aligned}$$
(94)
$$\begin{aligned} C_{1}= & {} 2A_{1} R_{1}^{3} k_{1} (k_{3} k_{4} \gamma _{2} +R_{2} k_{3} +R_{2} k_{4} )/(R_{1}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} -R_{2}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} \nonumber \\{} & {} + R_{1}^{3} k_{1} k_{3} k_{4} \gamma _{2} +R_{1}^{3} k_{3}^{2} k_{4} \gamma _{2} +R_{1}^{2} R_{2} k_{1} k_{3}^{2} \gamma _{1} +R_{1}^{2} R_{2} k_{1} k_{3} k_{4} \gamma _{1} +R_{1} R_{2}^{2} k_{1} k_{3} k_{4} \gamma _{2} \nonumber \\{} & {} - R_{1} R_{2}^{2} k_{3}^{2} k_{4} \gamma _{2} -R_{2}^{3} k_{1} k_{3}^{2} \gamma _{1} +R_{2}^{3} k_{1} k_{3} k_{4} \gamma _{1} {+ }R_{1}^{3} R_{2} k_{1} k_{3} +R_{1}^{3} R_{2} k_{1} k_{4} +R_{1}^{3} R_{2} k_{3}^{2} \nonumber \\{} & {} + R_{1}^{3} R_{2} k_{3} k_{4} +R_{1} R_{2}^{3} k_{1} k_{3} -R_{1} R_{2}^{3} k_{1} k_{4} -R_{1} R_{2}^{3} k_{3}^{2} +R_{1} R_{2}^{3} k_{3} k_{4} ) \end{aligned}$$
(95)
$$\begin{aligned} D_{1}= & {} 4A_{1} R_{1}^{3} R_{2} k_{1} k_{3} /(R_{1}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} -R_{2}^{2} k_{1} k_{3}^{2} k_{4} \gamma _{1} \gamma _{2} \nonumber \\{} & {} + R_{1}^{3} k_{1} k_{3} k_{4} \gamma _{2} +R_{1}^{3} k_{3}^{2} k_{4} \gamma _{2} +R_{1}^{2} R_{2} k_{1} k_{3}^{2} \gamma _{1} +R_{1}^{2} R_{2} k_{1} k_{3} k_{4} \gamma _{1} +R_{1} R_{2}^{2} k_{1} k_{3} k_{4} \gamma _{2} \nonumber \\{} & {} - R_{1} R_{2}^{2} k_{3}^{2} k_{4} \gamma _{2} -R_{2}^{3} k_{1} k_{3}^{2} \gamma _{1} +R_{2}^{3} k_{1} k_{3} k_{4} \gamma _{1} {+ }R_{1}^{3} R_{2} k_{1} k_{3} +R_{1}^{3} R_{2} k_{1} k_{4} +R_{1}^{3} R_{2} k_{3}^{2} \nonumber \\{} & {} + R_{1}^{3} R_{2} k_{3} k_{4} +R_{1} R_{2}^{3} k_{1} k_{3} -R_{1} R_{2}^{3} k_{1} k_{4} -R_{1} R_{2}^{3} k_{3}^{2} +R_{1} R_{2}^{3} k_{3} k_{4} ) \end{aligned}$$
(96)
$$\begin{aligned} E_{2}= & {} G_{4} (2G_{1} G_{3} K_{3} R_{1}^{4} T_{1} P_{4} +2G_{1} G_{3} K_{3} R_{1}^{2} R_{2}^{2} T_{2} P_{4} +2G_{1} G_{3} R_{1}^{2} R_{2}^{2} T_{2} P_{4} \nonumber \\{} & {} + 2G_{1} G_{3} R_{2}^{4} T_{1} P_{4} -G_{1} K_{3} R_{1}^{4} P_{2} P_{4} +G_{1} K_{3} R_{2}^{4} P_{2} P_{4} -2G_{3}^{2} R_{1}^{4} T_{1} +2G_{3}^{2} R_{2}^{4} T_{1} \nonumber \\{} & {} + G_{3} K_{3} R_{1}^{2} R_{2}^{2} P_{3} +G_{3} K_{3} R_{2}^{4} P_{2} +G_{3} R_{1}^{4} P_{2} +G_{3} R_{1}^{2} R_{2}^{2} P_{3} )/(R_{2}^{2} (G_{1} G_{3} K_{3} K_{4} R_{1}^{4} P_{4} \nonumber \\{} & {} + G_{1} G_{4} K_{3} R_{1}^{4} P_{1} P_{4} -G_{1} G_{4} K_{3} R_{2}^{4} P_{1} P_{4} +G_{1} G_{3} K_{4} R_{2}^{4} P_{4} \nonumber \\{} & {} - G_{3} G_{4} K_{3} R_{2}^{4} P_{1} -G_{3}^{2} K_{4} R_{1}^{4} +G_{3}^{2} K_{4} R_{2}^{4} -G_{3} G_{4} R_{1}^{4} P_{1} )) \end{aligned}$$
(97)
$$\begin{aligned} F_{-2}= & {} -R_{1}^{2} G_{3} R_{2}^{2} (-2G_{1} G_{4} K_{3} R_{1}^{2} T_{1} P_{1} P_{4} -2G_{1} G_{4} K_{3} R_{2}^{2} T_{2} P_{1} P_{4} +2G_{1} G_{3} K_{4} R_{2}^{2} T_{2} P_{4} \nonumber \\{} & {} - G_{1} K_{3} K_{4} R_{1}^{2} P_{2} P_{4} +2G_{3} G_{4} R_{1}^{2} T_{1} P_{1} -G_{4} K_{3} R_{2}^{2} P_{1} P_{3} +G_{3} K_{4} R_{1}^{2} P_{2} +G_{3} K_{4} R_{2}^{2} P_{3} )/ \nonumber \\{} & {} ( G_{1} G_{3} K_{3} K_{4} R_{1}^{4} P_{4} +G_{1} G_{4} K_{3} R_{1}^{4} P_{1} P_{4} -G_{1} G_{4} K_{3} R_{2}^{4} P_{1} P_{4} \nonumber \\{} & {} + G_{1} G_{3} K_{4} R_{2}^{4} P_{4} -G_{3} G_{4} K_{3} R_{2}^{4} P_{1} -G_{3}^{2} K_{4} R_{1}^{4} +G_{3}^{2} K_{4} R_{2}^{4} -G_{3} G_{4} R_{1}^{4} P_{1} ) \end{aligned}$$
(98)
$$\begin{aligned} H_{2}= & {} G_{3} (2G_{1} G_{3} K_{4} R_{1}^{2} T_{2} P_{4} +2G_{1} G_{4} R_{1}^{2} T_{2} P_{1} P_{4} +2G_{1} G_{4} R_{2}^{2} T_{1} P_{1} P_{4} \nonumber \\{} & {} + G_{1} K_{4} R_{2}^{2} P_{2} P_{4} +2G_{3} G_{4} R_{2}^{2} T_{1} P_{1} +G_{3} K_{4} R_{1}^{2} P_{3} +G_{3} K_{4} R_{2}^{2} P_{2} \nonumber \\{} & {} + G_{4} R_{1}^{2} P_{1} P_{3} )/(G_{1} G_{3} K_{3} K_{4} R_{1}^{4} P_{4} +G_{1} G_{4} K_{3} R_{1}^{4} P_{1} P_{4} -G_{1} G_{4} K_{3} R_{2}^{4} P_{1} P_{4} \nonumber \\{} & {} + G_{1} G_{3} K_{4} R_{2}^{4} P_{4} -G_{3} G_{4} K_{3} R_{2}^{4} P_{1} -G_{3}^{2} K_{4} R_{1}^{4} +G_{3}^{2} K_{4} R_{2}^{4} -G_{3} G_{4} R_{1}^{4} P_{1} ) \end{aligned}$$
(99)
$$\begin{aligned} I_{-2}= & {} -R_{1}^{2} G_{1} (2G_{3} G_{4} K_{3} R_{1}^{2} R_{2}^{2} T_{1} P_{1} +2G_{3} G_{4} K_{3} R_{2}^{4} T_{2} P_{1} +2G_{3}^{2} K_{4} R_{1}^{4} T_{2} \nonumber \\{} & {} - 2G_{3}^{2} K_{4} R_{2}^{4} T_{2} +2G_{3} G_{4} R_{1}^{4} T_{2} P_{1} +2G_{3} G_{4} R_{1}^{2} R_{2}^{2} T_{1} P_{1} +G_{3} K_{3} K_{4} R_{1}^{4} P_{3} \nonumber \\{} & {} + G_{3} K_{3} K_{4} R_{1}^{2} R_{2}^{2} P_{2} +G_{4} K_{3} R_{1}^{4} P_{1} P_{3} -G_{4} K_{3} R_{2}^{4} P_{1} P_{3} +G_{3} K_{4} R_{1}^{2} R_{2}^{2} P_{2} \nonumber \\{} & {} + G_{3} K_{4} R_{2}^{4} P_{3} )/(G_{1} G_{3} K_{3} K_{4} R_{1}^{4} P_{4} +G_{1} G_{4} K_{3} R_{1}^{4} P_{1} P_{4} -G_{1} G_{4} K_{3} R_{2}^{4} P_{1} P_{4} \nonumber \\{} & {} + G_{1} G_{3} K_{4} R_{2}^{4} P_{4} -G_{3} G_{4} K_{3} R_{2}^{4} P_{1} -G_{3}^{2} K_{4} R_{1}^{4} +G_{3}^{2} K_{4} R_{2}^{4} -G_{3} G_{4} R_{1}^{4} P_{1} ) \end{aligned}$$
(100)

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Huang, H., Song, K. Modulation of heat flux and thermal stress at the double interface by nano-coating thickness. Z. Angew. Math. Phys. 75, 20 (2024). https://doi.org/10.1007/s00033-023-02166-8

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