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Keywords = convective–diffusive Cahn–Hilliard equation

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26 pages, 2334 KB  
Article
Nonlinear Optical Characteristics of Copper Oxide Thin Films Interpreted Through Soliton Solutions of the Convective–Diffusive Cahn–Hilliard Equation
by Nan Xing, Umair Asghar, Khaleel Ahmad and Luminita-Ioana Cotirla
Mathematics 2025, 13(23), 3799; https://doi.org/10.3390/math13233799 - 26 Nov 2025
Viewed by 615
Abstract
This study investigates the convective–diffusive Cahn–Hilliard equation, a nonlinear model which is used in real-world applications to phase separation and material pattern formation. Using the modified Sardar sub-problem technique, which is an extension of the Sardar sub-equation approach, we derive multiple classes of [...] Read more.
This study investigates the convective–diffusive Cahn–Hilliard equation, a nonlinear model which is used in real-world applications to phase separation and material pattern formation. Using the modified Sardar sub-problem technique, which is an extension of the Sardar sub-equation approach, we derive multiple classes of exact soliton solutions, including bright, dark, kink, and periodic forms. The parametric behaviors of these solutions are examined and visualized through analytical plots generated in Mathematica and Maple. Furthermore, UV–Vis spectrophotometry is employed to examine the optical response of copper oxide (CuO) thin films. The films exhibited a sharp absorption edge around 380–410 nm and an optical band gap of approximately 2.3 eV, confirming their semiconducting nature. The experimentally observed periodic transmission characteristics are linked with the theoretical soliton profiles predicted by the model. Overall, the proposed analytical and experimental framework establishes a clear connection between nonlinear wave theory and thin-film optical characterization, providing new insights into soliton transformation phenomena in complex material systems. Full article
(This article belongs to the Special Issue Advances in Computational Mathematics and Applied Mathematics, 2nd Edition)
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12 pages, 20691 KB  
Article
A Comparative Study of the Diffuse-Interface Model and Sharp-Interface Model in the Soldering Related Wetting Spreading Systems
by Guanpeng Liu, Jianyang Zhang, Min Lei, Yulong Li and Xuewen Li
Metals 2019, 9(9), 944; https://doi.org/10.3390/met9090944 - 28 Aug 2019
Cited by 2 | Viewed by 3576
Abstract
A typical dissolution wetting system, Bi-Sn eutectic filler metal over a Bi substrate in a high-purity argon atmosphere was investigated first using real-time in situ hot stage microscopy for the extensive use of the sharp-interface model and the diffuse-interface model in the modeling [...] Read more.
A typical dissolution wetting system, Bi-Sn eutectic filler metal over a Bi substrate in a high-purity argon atmosphere was investigated first using real-time in situ hot stage microscopy for the extensive use of the sharp-interface model and the diffuse-interface model in the modeling of brazing/soldering related wetting systems. Subsequently, the similarities and differences between the aforementioned models in describing the issues of the wetting and spreading interfaces were discussed in terms of soldering definition and theoretical formula derivation. It is noted that (i) the mutual dissolution diffusion between the liquid Bi-Sn solder and Bi substrate were obvious. As a result, the composition and volume of the liquid solder is constantly changing during the wetting and spreading process; (ii) the sharp-interface model is a special case of the diffuse-interface model of the Cahn-Hilliard nonlinear diffuse-equation under the convective dominant condition; (iii) although there are differences between the sharp-interface model and the diffuse-interface model, both of them could be used in brazing/soldering related processes; and, (iv) the agreement between the experimental and simulation results of the sharp-interface model is not as good as that of the diffuse-interface model, which can be attributed to the effects of the elements’ diffusion and the phase transformation. Full article
(This article belongs to the Special Issue Multiscale Modeling of Materials and Processes)
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