Global Instability of Rod Eutectic Growth in Directional Solidification
In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutect...
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MDPI AG
2023-03-01
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| author | Yanlin Gan Xiangming Li |
| author_facet | Yanlin Gan Xiangming Li |
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| description | In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo><</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is stable, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is unstable and when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient. |
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| spelling | doaj.art-eea9e68d1b894e3db2c56e1451f6bcb62023-11-17T10:30:19ZengMDPI AGCrystals2073-43522023-03-0113354810.3390/cryst13030548Global Instability of Rod Eutectic Growth in Directional SolidificationYanlin Gan0Xiangming Li1Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, ChinaFaculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, ChinaIn our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo><</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is stable, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is unstable and when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></semantics></math></inline-formula>, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient.https://www.mdpi.com/2073-4352/13/3/548directional solidificationrod eutectic growthperturbation analysisglobal steady (ST) mode |
| spellingShingle | Yanlin Gan Xiangming Li Global Instability of Rod Eutectic Growth in Directional Solidification Crystals directional solidification rod eutectic growth perturbation analysis global steady (ST) mode |
| title | Global Instability of Rod Eutectic Growth in Directional Solidification |
| title_full | Global Instability of Rod Eutectic Growth in Directional Solidification |
| title_fullStr | Global Instability of Rod Eutectic Growth in Directional Solidification |
| title_full_unstemmed | Global Instability of Rod Eutectic Growth in Directional Solidification |
| title_short | Global Instability of Rod Eutectic Growth in Directional Solidification |
| title_sort | global instability of rod eutectic growth in directional solidification |
| topic | directional solidification rod eutectic growth perturbation analysis global steady (ST) mode |
| url | https://www.mdpi.com/2073-4352/13/3/548 |
| work_keys_str_mv | AT yanlingan globalinstabilityofrodeutecticgrowthindirectionalsolidification AT xiangmingli globalinstabilityofrodeutecticgrowthindirectionalsolidification |