Thermodynamically consistent viscoelastic phase separation: numerical analysis and simulation
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Abstract
Soft matter is a significant topic of modern research, since especially polymers and liquid crystals are important in a wide range of technological applications. In this context, it is essential to understand the thermodynamics of polymer solutions. In this work, we present and validate new, efficient, thermodynamically consistent numerical schemes for the simulation of phase separation of polymer-solvent mixtures. The proposed mathematical models are based on a viscoelastic (non-Newtonian) phase-field model by Zhou, Zhang and E (Physical Review E 73, 2006). It consists of the Cahn-Hilliard equation, describing the dynamics of a diffusive interface separating polymer and solvent phase, and extended Oldroyd-B equations for the complex hydrodynamics of a polymer solution. This macroscopic model is isothermal and dissipates energy over time. Therefore, it is consistent with the second law of thermodynamics. Further, it is the first thermodynamically consistent model which reproduces all essential features of experimentally observed viscoelastic phase separation. The main goal of this dissertation is to derive energy-stable numerical schemes for such a complex phase-field model, which are both accurate and computationally efficient. Thus, the proposed schemes shall satisfy the conservation of mass and preserve the thermodynamic consistency of the model equations while suitably linearizing all nonlinear terms. To this end, several problem-specific time and space discretizations will be proposed, and their properties will be discussed. Furthermore, various numerical experiments will be conducted, including experimental convergence tests, to verify the reliability of the proposed numerical schemes. Additionally, to investigate the quality of our numerical solutions describing the physics of viscoelastic phase separation, we perform a comparison to computationally vastly more expensive simulation results of a thoroughly validated mesoscopic model describing the same physical problem. The latter is realized through our collaboration with the Max Planck Institute for Polymer Research in Mainz.