A Unified Approach to Phase Equilibria in Multi-component Polymer Systems with Effective Chemical Potential

Wei-Ling Huang; Yu-Chen Zhang; Yi-Xin Liu

doi:10.1007/s10118-025-3369-8

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A Unified Approach to Phase Equilibria in Multi-component Polymer Systems with Effective Chemical Potential

RESEARCH ARTICLE | Updated：2025-08-25

- A Unified Approach to Phase Equilibria in Multi-component Polymer Systems with Effective Chemical Potential
- Chinese Journal of Polymer Science Vol. 43, Issue 9, Pages: 1681-1689(2025)
- Affiliations：
  
  State Key Laboratory of Molecule Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
- Author bio：
  
  lyx@fudan.edu.cn
- Funds：
- DOI：10.1007/s10118-025-3369-8
  CLC：
- Received：24 February 2025，
  
  Accepted：02 May 2025，
  
  Online First：30 June 2025，
  
  Published：05 September 2025
- Accepted：
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Huang, W. L.; Zhang, Y. C.; Liu, Y. X. A unified approach to phase equilibria in multi-component polymer systems with effective chemical potential. Chinese J. Polym. Sci. 2025, 43, 1681–1689

Wei-Ling Huang, Yu-Chen Zhang, Yi-Xin Liu. A Unified Approach to Phase Equilibria in Multi-component Polymer Systems with Effective Chemical Potential[J]. Chinese Journal of Polymer Science, 2025, 43(9): 1681-1689.
Huang, W. L.; Zhang, Y. C.; Liu, Y. X. A unified approach to phase equilibria in multi-component polymer systems with effective chemical potential. Chinese J. Polym. Sci. 2025, 43, 1681–1689 DOI： 10.1007/s10118-025-3369-8.

Wei-Ling Huang, Yu-Chen Zhang, Yi-Xin Liu. A Unified Approach to Phase Equilibria in Multi-component Polymer Systems with Effective Chemical Potential[J]. Chinese Journal of Polymer Science, 2025, 43(9): 1681-1689. DOI： 10.1007/s10118-025-3369-8.

摘要

This study introduces a unified framework for determining phase equilibria in multi-component polymer systems through effective chemical potentials. The ensemble-agnostic approach decouples free energy calculations from phase equilibrium computations

enabling direct solution of multiphase coexistence within the canonical ensemble while maintaining compatibility with existing SCFT solvers.

Abstract

Multi-component polymer systems exhibit exceptional versatility and structural diversity

making them indispensable in the polymer industry as well as in advanced and high performance applications. However

constructing accurate phase diagrams for these systems remains challenging due to inhomogeneous structures arising from the introduction of block copolymer components. Here

we present a unified and model-agnostic framework for computing phase equilibria in multi-component polymeric systems based on the concept of “effective chemical potential”. This approach directly connects key thermodynamic variables in the canonical ensemble to other ensembles

unifying phase coexistence determination without requiring the reformulation of self-consistent field theory (SCFT) calculations across different ensembles. By decoupling phase equilibrium determination from specific ensemble formulations

our approach enables the reuse of existing SCFT solvers. Moreover

it provides a useful framework to develop highly efficient phase equilibrium solvers for multi-component polymer systems.

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references

Clarke, R. W.; Sandmeier, T.; Franklin, K. A.; Reich, D.; Zhang, X.; Vengallur, N.; Patra, T. K.; Tannenbaum, R. J.; Adhikari, S.; Kumar, S. K.; Rovis, T.; Chen, E. Y. X. Dynamic crosslinking compatibilizes immiscible mixed plastics. Nature 2023 , 616 , 731−739..

[Gdowski, Z. M.; Swartzendruber, S. M.; Mahanthappa, M. K.; Bates, F. S. Diblock copolymer blends do not mimic metal alloys. Macromolecules 2024 , 57 , DOI: 10.1021/acs.macromol.4c01146..

Coughlin, M. L.; Huang, D. E.; Edgar, C. M.; Kotula, A. P.; Migler, K. B. Composition dependence of flow-induced crystallization in high-density polyethylene/isotactic polypropylene blends. Macromolecules 2024 , 57 , 9751−9765..

Wang, Y.; Li, G.; Zhu, H.; Wan, Z.; Zhang, X.; Ma, Y.; Xie, L. Morphological evolution and structural orientation of poly(ϵ-caprolactone)/poly(lactic acid) biodegradable blend films under complex coupling flow. Macromolecules 2024 , 57 , 4345−4356..

Wang, Q.; Yu, X.; Chen, X.; Gao, J.; Shi, D.; Shen, Y.; Tang, J.; He, J.; Li, A.; Yu, L.; Ding, J. A facile composite strategy to prepare a biodegradable polymer based radiopaque raw material for “visualizable” biomedical implants. ACS Appl. Mater. Interfaces 2022 , 14 , 24197−24212..

Li, Q.; Woo, D.; Kim, J. K.; Li, W. Truly “Inverted” cylinders and spheres formed in the A(AB) 3 /AC blends of B/C hydrogen bonding interactions. Macromolecules 2022 , 55 , 6525−6535..

Li, W.; Duan, C.; Shi, A. C. Nonclassical spherical packing phases self-assembled from AB-type block copolymers. ACS Macro Lett. 2017 , 6 , 1257−1262..

He, W.; Wang, F.; Qiang, Y.; Pan, Y.; Li, W.; Liu, M. Asymmetric binary spherical phases self-assembled by mixing AB diblock/abc triblock copolymers. Macromolecules 2023 , 56 , 719−729..

Chen, J.; Rong, C.; Lin, T.; Chen, Y.; Wu, J.; You, J.; Wang, H.; Li, Y. Stable co-continuous PLA/PBAT blends compatibilized by interfacial stereocomplex crystallites: toward full biodegradable polymer blends with simultaneously enhanced mechanical properties and crystallization rates. Macromolecules 2021 , 54 , 2852−2861..

Hassan, M.; Mohanty, A. K.; Misra, M. Additive manufacturing of a super toughened biodegradable polymer blend: structure-property-processing correlation and 3D printed prosthetic part development. ACS Appl. Polym. Mater. 2024 , 6 , 3849−3863..

Maiti, M.; Jasra, R. V.; Kusum, S. K.; Chaki, T. K. microcellular foam from ethylene vinyl acetate/polybutadiene rubber (EVA/BR) based thermoplastic elastomers for footwear applications. Indus. Eng. Chem. Res. 2012 , 51 , 10607−10612..

Peng, B.; Yang, Y.; Ju, T.; Cavicchi, K. A. Fused Filament Fabrication 4D printing of a highly extensible, self-healing, shape memory elastomer based on thermoplastic polymer blends. ACS Appl. Mater. Interfaces 2020 , 13 , 12777−12788..

Liu, Y. X.; Delaney, K. T.; Fredrickson, G. H. Field-Theoretic simulations of fluctuation-stabilized aperiodic “bricks-and-Mortar” mesophase in miktoarm star block copolymer/homopolymer blends. Macromolecules 2017 , 50 , 6263−6272..

Riera-Galindo, S.; Leonardi, F.; Pfattner, R.; Mas-Torrent, M. Organic semiconductor/polymer blend films for organic field-effect transistors. Adv. Mater. Technol. 2019 , 4 , 1900104..

Kang, J.; Shin, N.; Jang, Y. P.; Prabhu, V. M.; Yoon, D. Y. Structure and properties of small molecule-polymer blend semiconductors for organic thin film transistors. J. Am. Chem. Soc. 2008 , 130 , 12273−12275..

Alshammari, A. H.; Alshammari, M.; Ibrahim, M.; Alshammari, K. Taha new hybrid PVC/PVP polymer blend modified with Er 2 O 3 nanoparticles for optoelectronic applications. Polymers 2023 , 15 , 684−684..

Jiang, L.; Yang, L.; Yuan, Y.; Zhu, Q.; Wu, W.; Wang, X.; Xu, W.; Qiu, L. Flexible optoelectronic synapses based on conjugated polymer blends for ultra broadband spectrum light perception. ACS Mater. Lett. 2024 , 6 , 1606−1615..

Pucci, A.; Bizzarri, R.; Ruggeri, G. Polymer composites with smart optical properties. Soft Matter 2011 , 7 , 3689..

Guo, X.; Liu, L.; Zhuang, Z.; Chen, X.; Ni, M.; Li, Y.; Cui, Y.; Zhan, P.; Yuan, C.; Ge, H.; Wang, Z.; Chen, Y. A new strategy of lithography based on phase separation of polymer blends. Scientific Reports 2015 , 5 , DOI: 10.1038/srep15947..

Huang, C. Z.; Moosmann, M.; Jin, J.; Heiler, T.; Walheim, S.; Schimmel, T. Polymer blend lithography: a versatile method to fabricate nanopatterned selfassembled monolayers. Beilstein J. Nanotechnol. 2012 , 3 , 620−628..

Robeson, L. M. Applications of polymer blends: emphasis on recent advances. Polym. Eng. Sci. 1984 , 24 , 587−597..

Imre, B.; Pukánszky, B. Compatibilization in bio-based and biodegradable polymer blends. Eur. Polym. J. 2013 , 49 , 1215−1233..

Okuzono, T.; Ozawa, K.; Doi, M. Simple model of skin formation caused by solvent evaporation in polymer solutions. Phys. Rev. Lett . 2006 , 97 , 136103..

Zhou, J.; Jiang, Y.; Doi, M. Cross interaction drives stratification in drying film of binary colloidal mixtures. Phys. Rev. Lett. 2017 , 118 , 108002..

Buxton, G. A.; Clarke, N. Ordering polymer blend morphologies via solvent evaporation. Europhysics Letters (EPL) 2007 , 78 , 56006..

Wodo, O.; Ganapathysubramanian, B. Modeling morphology evolution during solventbased fabrication of organic solar cells. Comput. Mater. Sci. 2012 , 55 , 113−126..

Zheng, C.; Zhang, B.; Bates, F. S.; Lodge, T. P. Effect of poly[oligo(ethylene glycol) methyl ether methacrylate ] side chain length on the brush swelling behavior in A/B/A-B ternary blends with polystyrene. Soft Matter 2023 , 19 , 4519−4525..

Xie, S.; Zhang, B.; Bates, F. S.; Lodge, T. P. Phase behavior of salt-doped A/B/AB ternary polymer blends: the role of homopolymer distribution. Macromolecules 2021 , 54 , 6990−7002..

Zhao, M.; Li, W. Laves phases formed in the binary blend of AB 4 miktoarm star copolymer and A-homopolymer. Macromolecules 2019 , 52 , 1832−1842..

Zhao, F.; Dong, Q.; Li, Q.; Liu, M.; Li, W. Emergence and stability of exotic “binary” HCP-type Spherical phase in binary AB/AB blends. Macromolecules 2022 , 55 , 10005−10013..

Xie, J.; Shi, A. C. Formation of complex spherical packing phases in diblock copolymer/homopolymer blends. Giant 2021 , 5 , 100043..

Bates, F. S.; Fredrickson, G. H. Block copolymers—designer soft materials. Phys. Today 1999 , 52 , 32−38..

Boles, M. A.; Engel, M.; Talapin, D. V. Self-Assembly of colloidal nanocrystals: from intricate structures to functional materials. Chem. Rev. 2016 , 116 , 11220−11289..

[Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers ; Oxford University Press, 2006 ..

[Rubinstein, M.; Colby, R. H. Polymer Physics ; OxfordUniversity Press, 2003 ..

Mao, S.; Kuldinow, D.; Haataja, M. P.; Košmrlj, A. Phase behavior and morphology of multicomponent liquid mixtures. Soft Matter 2019 , 15 , 1297−1311..

Janert, P. K.; Schick, M. Phase behavior of ternary homopolymer/diblock blends: microphase unbinding in the symmetric system. Macromolecules 1997 , 30 , 3916−3920..

Huang, C.; Cruz, M. O. d. l.; Swift, B. W. Phase separation of ternary mixtures: symmetric polymer blends. Macromolecules 1995 , 28 , 7996−8005..

Yadav, M.; Bates, F. S.; Morse, D. C. Effects of segment length asymmetry in ternary diblock co-polymer–homopolymer mixtures. Macromolecules 2019 , 52 , 4091−4102..

Matsen, M. W. Stabilizing new morphologies by blending homopolymer with block copolymer. Phys. Rev. Lett. 1995 , 74 , 4225−4228..

Matsen, M. W. New Fast SCFT algorithm applied to binary diblock copolymer/homopolymer blends. Macromolecules 2003 , 36 , 9647−9657..

Mester, Z.; Lynd, N. A.; Fredrickson, G. H. Numerical self-consistent field theory of multicomponent polymer blends in the Gibbs ensemble. Soft Matter 2013 , 9 , 11288..

Park, S. J.; Bates, F. S.; Dorfman, K. D. Alternating Gyroid in Block Polymer Blends. ACS Macro Lett. 2022 , 11 , 643−650..

Xie, J.; Shi, A. C. Theory of complex spherical packing phases in Diblock copolymer/homopolymer blends. Macromolecules 2023 , 56 , 10296−10312..

Hong, K. M.; Noolandi, J. Theory of inhomogeneous multicomponent polymer systems. Macromolecules 1981 , 14 , 727−736..

[Legendre, A. M. Mémoire sur l’intégration de quelques équations aux différences partielles ; Imprimerie royale, 1787 ; pp 309–351..

Zhang, Y.; Huang, W.; Liu, Y. X. Automated chain architecture screening for discovery of block copolymer assembly with graph enhanced self-consistent field theory. Commun. Mater . 2024 , 5 , 266..

Guo, Z.; Zhang, G.; Qiu, F.; Zhang, H.; Yang, Y.; Shi, A. C. Discovering ordered phases of block copolymers: new results from a generic fourier-space approach. Phys. Rev. Lett . 2008 , 101 , 028301..

Xu, W.; Jiang, K.; Zhang, P.; Shi, A. C. A strategy to explore stable and metastable ordered phases of block copolymers. J. Phys. Chem. B 2013 , 117 , 5296−5305..

Arora, A.; Morse, D. C.; Bates, F. S.; Dorfman, K. D. Accelerating self-consistent field theory of block polymers in a variable unit cell. J. Chem. Phys. 2017 , 146 , 244902..

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