A ternary mixture model with dynamic boundary conditions

Shuang Liu; Yue Wu; Xueping Zhao; Shuang Liu; Yue Wu; Xueping Zhao

doi:10.3934/mbe.2024091

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2050-2083. doi: 10.3934/mbe.2024091

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A ternary mixture model with dynamic boundary conditions

1.
Department of Mathematics, University of North Texas, 1155 Union Circle, Denton, Texas 76203-5017, USA
2.
Department of Mathematical Sciences, University of Nottingham Ningbo China, Taikang East Road 199, Ningbo 315100, China

Academic Editor: Jia Zhao

Received: 31 October 2023 Revised: 14 December 2023 Accepted: 21 December 2023 Published: 08 January 2024

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.
- phase-field model,
- ternary mixture,
- dynamic boundary condition,
- energy stability,
- wetting
Citation: Shuang Liu, Yue Wu, Xueping Zhao. A ternary mixture model with dynamic boundary conditions[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2050-2083. doi: 10.3934/mbe.2024091

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Abstract

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

References

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