Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors
Abstract
:1. Introduction
2. Materials and Methods
2.1. HXPFC-s2 Formalism
2.1.1. Free Energy Functional and Advected XPFC Equation
2.1.2. Navier–Stokes Momentum Coupling
2.2. HXPFC-s2 Dimensionless Form
2.3. Significant Dimensionless Numbers
2.4. Numerical Methods
2.5. Simulation Setup
3. Results
3.1. Effective Temperature-Dependent Crystal Growth
3.2. Peclet-Dependent Crystal Growth
3.3. Crystal Plane Growth Dependence
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
HXPFC-s2 | Hydrodynamic Structural Phase Field Crystal semi-two-way Coupled |
PFC | Phase Field Crystal |
XPFC | Structural Phase Field Crystal |
DCF | Direct Correlation Function |
DDFT | Dynamic Density Functional Theory |
HNC | Hypernetted Chain |
MSA | Mean Spherical Approximation |
RPA | Random Phase Approximation |
Appendix A. General HXPFC Formalism Derivation
Appendix A.1. Free Energy Functional
Appendix A.2. Advected PFC Equation
Appendix A.3. Navier–Stokes Coupling
Appendix B. Dimensionless HXPFC-s2 Form Derivation
Appendix B.1. Convective Timescale
Appendix B.2. Diffusive Timescale
Appendix C. Derivation of Theoretical Growth Rate Equation
References
- Yang, H.; Wang, H.; Wang, K.; Liu, D.; Zhao, L.; Chen, D.; Zhu, W.; Zhang, J.; Zhang, C. Recent Progress of Film Fabrication Process for Carbon-Based All-Inorganic Perovskite Solar Cells. Crystals 2023, 13, 679. [Google Scholar] [CrossRef]
- Soto-Montero, T.; Soltanpoor, W.; Morales-Masis, M. Pressing challenges of halide perovskite thin film growth. APL Mater. 2020, 8, 110903. [Google Scholar] [CrossRef]
- Li, Z.; Klein, T.R.; Kim, D.H.; Yang, M.; Berry, J.J.; Van Hest, M.F.A.M.; Zhu, K. Scalable fabrication of perovskite solar cells. Nat. Rev. Mater. 2018, 3, 18017. [Google Scholar] [CrossRef]
- Weerasinghe, H.C.; Macadam, N.; Kim, J.E.; Sutherland, L.J.; Angmo, D.; Ng, L.W.T.; Scully, A.D.; Glenn, F.; Chantler, R.; Chang, N.L.; et al. The first demonstration of entirely roll-to-roll fabricated perovskite solar cell modules under ambient room conditions. Nat. Commun. 2024, 15, 1656. [Google Scholar] [CrossRef]
- Guo, Q.; Gong, X.; Shen, Z.; Hao, X.; Zhang, J. Numerical Simulation on Preparing Uniform and Stable Perovskite Wet Film in Slot-Die Coating Process. ACS Omega 2023, 8, 19547–19555. [Google Scholar] [CrossRef]
- Wang, X.; Zhou, Y. Research on Single Crystal Preparation via Dynamic Liquid Phase Method. Crystals 2023, 13, 1150. [Google Scholar] [CrossRef]
- Małysiak, A.; Orda, S.; Drzazga, M. Influence of Supersaturation, Temperature and Rotational Speed on Induction Time of Calcium Sulfate Crystallization. Crystals 2021, 11, 1236. [Google Scholar] [CrossRef]
- Mura, F.; Zaccone, A. Effects of shear flow on phase nucleation and crystallization. Phys. Rev. E 2016, 93, 042803. [Google Scholar] [CrossRef]
- Elder, K.R.; Katakowski, M.; Haataja, M.; Grant, M. Modeling Elasticity in Crystal Growth. Phys. Rev. Lett. 2002, 88, 245701. [Google Scholar] [CrossRef] [PubMed]
- Elder, K.R.; Grant, M. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 2004, 70, 051605. [Google Scholar] [CrossRef]
- Ramakrishnan, T.V.; Yussouff, M. First-principles order-parameter theory of freezing. Phys. Rev. B 1979, 19, 2775–2794. [Google Scholar] [CrossRef]
- Elder, K.R.; Provatas, N.; Berry, J.; Stefanovic, P.; Grant, M. Phase-field crystal modeling and classical density functional theory of freezing. Phys. Rev. B 2007, 75, 064107. [Google Scholar] [CrossRef]
- Gao, N.; Zhao, Y.; Xia, W.; Liu, Z.; Lu, X. Phase-Field Crystal Studies on Grain Boundary Migration, Dislocation Behaviors, and Topological Transition under Tension of Square Polycrystals. Crystals 2023, 13, 777. [Google Scholar] [CrossRef]
- Hirvonen, P.; Ervasti, M.M.; Fan, Z.; Jalalvand, M.; Seymour, M.; Vaez Allaei, S.M.; Provatas, N.; Harju, A.; Elder, K.R.; Ala-Nissila, T. Multiscale modeling of polycrystalline graphene: A comparison of structure and defect energies of realistic samples from phase field crystal models. Phys. Rev. B 2016, 94, 035414. [Google Scholar] [CrossRef]
- Ankudinov, V.; Galenko, P.K. Noise-Induced Defects in Honeycomb Lattice Structure: A Phase-Field Crystal Study. Crystals 2023, 14, 38. [Google Scholar] [CrossRef]
- Podmaniczky, F.; Gránásy, L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. Crystals 2021, 11, 437. [Google Scholar] [CrossRef]
- Praetorius, S.; Voigt, A. A Phase Field Crystal Approach for Particles in a Flowing Solvent. Macromol. Theory Simul. 2011, 20, 541–547. [Google Scholar] [CrossRef]
- Praetorius, S.; Voigt, A. A Navier-Stokes phase-field crystal model for colloidal suspensions. J. Chem. Phys. 2015, 142, 154904. [Google Scholar] [CrossRef]
- Heinonen, V.; Achim, C.; Kosterlitz, J.; Ying, S.C.; Lowengrub, J.; Ala-Nissila, T. Consistent Hydrodynamics for Phase Field Crystals. Phys. Rev. Lett. 2016, 116, 024303. [Google Scholar] [CrossRef]
- Podmaniczky, F.; Gránásy, L. Molecular scale hydrodynamic theory of crystal nucleation and polycrystalline growth. J. Cryst. Growth 2022, 597, 126854. [Google Scholar] [CrossRef]
- Dean, D.S. Langevin equation for the density of a system of interacting Langevin processes. J. Phys. A Math. Gen. 1996, 29, L613–L617. [Google Scholar] [CrossRef]
- Marconi, U.M.B.; Tarazona, P. Dynamic density functional theory of fluids. J. Chem. Phys. 1999, 110, 8032–8044. [Google Scholar] [CrossRef]
- Van Teeffelen, S.; Backofen, R.; Voigt, A.; Löwen, H. Derivation of the phase-field-crystal model for colloidal solidification. Phys. Rev. E 2009, 79, 051404. [Google Scholar] [CrossRef]
- Emmerich, H.; Löwen, H.; Wittkowski, R.; Gruhn, T.; Tóth, G.I.; Tegze, G.; Gránásy, L. Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: An overview. Adv. Phys. 2012, 61, 665–743. [Google Scholar] [CrossRef]
- Greenwood, M.; Provatas, N.; Rottler, J. Free Energy Functionals for Efficient Phase Field Crystal Modeling of Structural Phase Transformations. Phys. Rev. Lett. 2010, 105, 045702. [Google Scholar] [CrossRef]
- Greenwood, M.; Rottler, J.; Provatas, N. Phase-field-crystal methodology for modeling of structural transformations. Phys. Rev. E 2011, 83, 031601. [Google Scholar] [CrossRef] [PubMed]
- Cavaterra, C.; Grasselli, M.; Mehmood, M.A.; Voso, R. Analysis of a Navier–Stokes phase-field crystal system. Nonlinear Anal. Real World Appl. 2025, 83, 104263. [Google Scholar] [CrossRef]
- An, J.; Zhang, J.; Yang, X. A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model. Comput. Math. Appl. 2022, 113, 70–85. [Google Scholar] [CrossRef]
- Yang, X.; He, X. Numerical approximations of flow coupled binary phase field crystal system: Fully discrete finite element scheme with second-order temporal accuracy and decoupling structure. J. Comput. Phys. 2022, 467, 111448. [Google Scholar] [CrossRef]
- Yang, J.; Kim, J. Consistent energy-stable method for the hydrodynamics coupled PFC model. Int. J. Mech. Sci. 2023, 241, 107952. [Google Scholar] [CrossRef]
- Rauscher, M. DDFT for Brownian particles and hydrodynamics. J. Phys. Condens. Matter 2010, 22, 364109. [Google Scholar] [CrossRef] [PubMed]
- Archer, A.J. Dynamical density functional theory for molecular and colloidal fluids: A microscopic approach to fluid mechanics. J. Chem. Phys. 2009, 130, 014509. [Google Scholar] [CrossRef] [PubMed]
- Willis, L.W.; Rao, R.R.; Liu, L.L. Towards a multiscale computational framework for simulating flow-mediated crystallization based on phase-field crystal formalisms. In Proceedings of the 16th World Congress on Computational Mechanics and 4th Pan American Congress on Computational Mechanics, Vancouver, BC, Canada, 21–26 July 2024. [Google Scholar] [CrossRef]
- Huang, Y.; Wang, J.; Wang, Z.; Li, J.; Guo, C.; Guo, Y.; Yang, Y. Existence and forming mechanism of metastable phase in crystallization. Comput. Mater. Sci. 2016, 122, 167–176. [Google Scholar] [CrossRef]
- Gardiner, C.W. Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, study ed., 2. ed., 6. print ed.; Number 13 in Springer Series in Synergetics; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Hansen, J.P.; McDonald, I.R. Theory of Simple Liquids: With Applications to Soft Matter, 4th ed.; Elsevier/AP: Amstersdam, The Netherlands, 2013. [Google Scholar]
- Liu, Z.; Zhu, Y.; Clausen, J.R.; Lechman, J.B.; Rao, R.R.; Aidun, C.K. Multiscale method based on coupled lattice-Boltzmann and Langevin-dynamics for direct simulation of nanoscale particle/polymer suspensions in complex flows. Int. J. Numer. Methods Fluids 2019, 91, 228–246. [Google Scholar] [CrossRef]
- Rasaiah, J.C.; Card, D.N.; Valleau, J.P. Calculations on the “Restricted Primitive Model” for 1–1 Electrolyte Solutions. J. Chem. Phys. 1972, 56, 248–255. [Google Scholar] [CrossRef]
- Solana, J.R. Perturbation Theories for the Thermodynamic Properties of Fluids and Solids; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2013. [Google Scholar]
- Te Vrugt, M.; Löwen, H.; Wittkowski, R. Classical dynamical density functional theory: From fundamentals to applications. Adv. Phys. 2020, 69, 121–247. [Google Scholar] [CrossRef]
- Lovett, R.; Mou, C.Y.; Buff, F.P. The structure of the liquid–vapor interface. J. Chem. Phys. 1976, 65, 570–572. [Google Scholar] [CrossRef]
- Henderson, D.; Boda, D. Mean spherical approximation for the Yukawa fluid radial distribution function. Mol. Phys. 2011, 109, 1009–1013. [Google Scholar] [CrossRef]
- Kundu, M.; Howard, M.P. Dynamic density functional theory for drying colloidal suspensions: Comparison of hard-sphere free-energy functionals. J. Chem. Phys. 2022, 157, 184904. [Google Scholar] [CrossRef] [PubMed]
- Taira, K.; Colonius, T. The immersed boundary method: A projection approach. J. Comput. Phys. 2007, 225, 2118–2137. [Google Scholar] [CrossRef]
- Tanaka, H.; Araki, T. Simulation Method of Colloidal Suspensions with Hydrodynamic Interactions: Fluid Particle Dynamics. Phys. Rev. Lett. 2000, 85, 1338–1341. [Google Scholar] [CrossRef] [PubMed]
- Sanz, E.; Vega, C.; Espinosa, J.R.; Caballero-Bernal, R.; Abascal, J.L.F.; Valeriani, C. Homogeneous Ice Nucleation at Moderate Supercooling from Molecular Simulation. J. Am. Chem. Soc. 2013, 135, 15008–15017. [Google Scholar] [CrossRef]
- Fletcher, C.A.J. Computational Techniques for Fluid Dynamics 2: Specific Techniques for Different Flow Categories; Scientific Computation, Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar] [CrossRef]
- Trefethen, L.N. Spectral Methods in MATLAB; Software, Environments, Tools; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2000. [Google Scholar]
- Strikwerda, J.C. Finite Difference Schemes and Partial Differential Equations, 2nd ed.; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2004. [Google Scholar]
- Qu, G.; Zhao, X.; Newbloom, G.M.; Zhang, F.; Mohammadi, E.; Strzalka, J.W.; Pozzo, L.D.; Mei, J.; Diao, Y. Understanding Interfacial Alignment in Solution Coated Conjugated Polymer Thin Films. ACS Appl. Mater. Interfaces 2017, 9, 27863–27874. [Google Scholar] [CrossRef] [PubMed]
- Calabrese, V.; Haward, S.J.; Shen, A.Q. Effects of Shearing and Extensional Flows on the Alignment of Colloidal Rods. Macromolecules 2021, 54, 4176–4185. [Google Scholar] [CrossRef]
- Kristupas, T. peaks2-find peaks in 2D data without additional toolbox.
- Kelton, K.; Greer, A. Interpretation of Experimental Measurements of Transient Nucleation. In Rapidly Quenched Metals; Elsevier: Amsterdam, The Netherlands, 1985; pp. 223–226. [Google Scholar] [CrossRef]
- Kelton, K. Crystal Nucleation in Liquids and Glasses. In Solid State Physics; Elsevier: Amsterdam, The Netherlands, 1991; Volume 45, pp. 75–177. [Google Scholar] [CrossRef]
- Kelton, K.; Greer, A. Transient nucleation effects in glass formation. J. Non-Cryst. Solids 1986, 79, 295–309. [Google Scholar] [CrossRef]
- Mermin, N.D. Crystalline Order in Two Dimensions. Phys. Rev. 1968, 176, 250–254. [Google Scholar] [CrossRef]
- Radhakrishnan, R.; Trout, B.L. Order parameter approach to understanding and quantifying the physico-chemical behavior of complex systems. In Handbook of Materials Modeling: Methods; Springer: Dordrecht, The Netherlands, 2005; pp. 1613–1626. [Google Scholar]
- Athreya, B.P.; Goldenfeld, N.; Dantzig, J.A.; Greenwood, M.; Provatas, N. Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model. Phys. Rev. E 2007, 76, 056706. [Google Scholar] [CrossRef]
- Wu, Y.L.; Derks, D.; Van Blaaderen, A.; Imhof, A. Melting and crystallization of colloidal hard-sphere suspensions under shear. Proc. Natl. Acad. Sci. USA 2009, 106, 10564–10569. [Google Scholar] [CrossRef]
- Li, W.; Peng, Y.; Still, T.; Yodh, A.G.; Han, Y. Nucleation kinetics and virtual melting in shear-induced structural transitions. Rep. Prog. Phys. 2025, 88, 010501. [Google Scholar] [CrossRef] [PubMed]
- Stefan-Kharicha, M.; Kharicha, A.; Zaidat, K.; Reiss, G.; Eßl, W.; Goodwin, F.; Wu, M.; Ludwig, A.; Mugrauer, C. Impact of hydrodynamics on growth and morphology of faceted crystals. J. Cryst. Growth 2020, 541, 125667. [Google Scholar] [CrossRef]
- Stefan-Kharicha, M.; Kharicha, A.; Zaidat, K.; Reiss, G.; Eßl, W.; Goodwin, F.; Wu, M.; Ludwig, A.; Mugrauer, C. Hydrodynamically driven facet kinetics in crystal growth. J. Cryst. Growth 2022, 584, 126557. [Google Scholar] [CrossRef]
- Qazi, S.J.S.; Rennie, A.R.; Tucker, I.; Penfold, J.; Grillo, I. Alignment of Dispersions of Plate-Like Colloidal Particles of Ni(OH)2 Induced by Elongational Flow. J. Phys. Chem. B 2011, 115, 3271–3280. [Google Scholar] [CrossRef]
Parameter | Description |
---|---|
Order-parameter, representing density fluctuations | |
Kinematic Viscosity | |
Dimensionless Kinematic Viscosity Field | |
Reference liquid kinematic viscosity | |
Dynamic viscosity | |
Reference liquid number density | |
Average liquid order-parameter 1 | |
Effective temperature 1 | |
m | Atomic mass |
M | Interface mobility |
Optional thermal noise | |
Interplanar spacing 2 | |
Atomic density within plane 2 | |
Interfacial properties 2 | |
Number of planes in planar family 2 | |
Pressure normalized by | |
Dimensionless free energy, normalized by | |
Dimensionless chemical potential, normalized by |
Parameter Definition | Analogy |
---|---|
− 1 | |
− 1 | |
Analogous to Peclet Number | |
Analogous to Euler Number 2 | |
Analogous to Schmidt Number | |
Analogous to Deborah Number 2 | |
Dimensionless Convective Timescale | |
Dimensionless Diffusive Timescale |
Parameter | Value |
---|---|
1 | |
4 | |
4 | |
1 | |
1 | |
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Willis, L.C.; Janicki, T.D.; Rao, R.R.; Liu, Z.L. Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors. Crystals 2025, 15, 157. https://doi.org/10.3390/cryst15020157
Willis LC, Janicki TD, Rao RR, Liu ZL. Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors. Crystals. 2025; 15(2):157. https://doi.org/10.3390/cryst15020157
Chicago/Turabian StyleWillis, L. Connor, Tesia D. Janicki, Rekha R. Rao, and Z. Leonardo Liu. 2025. "Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors" Crystals 15, no. 2: 157. https://doi.org/10.3390/cryst15020157
APA StyleWillis, L. C., Janicki, T. D., Rao, R. R., & Liu, Z. L. (2025). Unraveling the Determinant Mechanisms in Flow-Mediated Crystal Growth and Phase Behaviors. Crystals, 15(2), 157. https://doi.org/10.3390/cryst15020157