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  • Open Access

Loop expansion in polymer field theory: Application to phase separation

Kiyoharu Kawana1,* and Kyosuke Adachi2,3,4,†

  • *Contact author: kkiyoharu@kias.re.kr
  • †Contact author: kyosuke.adachi@riken.jp
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Phys. Rev. Research 8, 033343 – Published 21 September, 2026

DOI: https://doi.org/10.1103/pz3r-hjkc

Abstract

Liquid-liquid phase separation underlies phenomena ranging from protein condensate formation to the phase coexistence of synthetic polymers. Although the random phase approximation (RPA) is widely used to predict such phase behavior, its quantitative accuracy for binodals of polymer solutions, particularly outside the high-density regime, remains incompletely characterized. Here, we develop a field-theoretic loop expansion in homopolymer systems by identifying the inverse polymer density ρ−1 as the Planck constant ℏ in quantum field theory. We calculate the leading-order and next-to-leading-order corrections to the RPA free energy, denoted as RPA+ and RPA++, respectively. Testing the binodal predicted by the RPA+ against molecular dynamics simulations of bead-spring chains with Gaussian pair interactions, we find that the RPA+ qualitatively improves the dilute-phase coexistence density over the RPA, while the critical point error remains comparable to that of the RPA. Our results establish the loop expansion as a systematic route for refining the RPA-based binodal predictions for polymer phase separation.

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References (46)

  1. C. P. Brangwynne, P. Tompa, and R. V. Pappu, Polymer physics of intracellular phase transitions, Nat. Phys. 11, 899 (2015).
  2. S. F. Banani, H. O. Lee, A. A. Hyman, and M. K. Rosen, Biomolecular condensates: Organizers of cellular biochemistry, Nat. Rev. Mol. Cell Biol. 18, 285 (2017).
  3. J. Berry, C. P. Brangwynne, and M. Haataja, Physical principles of intracellular organization via active and passive phase transitions, Rep. Prog. Phys. 81, 046601 (2018).
  4. J.-M. Choi, A. S. Holehouse, and R. V. Pappu, Physical principles underlying the complex biology of intracellular phase transitions, Annu. Rev. Biophys. 49, 107 (2020).
  5. E. W. Martin, A. S. Holehouse, I. Peran, M. Farag, J. J. Incicco, A. Bremer, C. R. Grace, A. Soranno, R. V. Pappu, and T. Mittag, Valence and patterning of aromatic residues determine the phase behavior of prion-like domains, Science 367, 694 (2020).
  6. A. Bremer, M. Farag, W. M. Borcherds, I. Peran, E. W. Martin, R. V. Pappu, and T. Mittag, Deciphering how naturally occurring sequence features impact the phase behaviours of disordered prion-like domains, Nat. Chem. 14, 196 (2022).
  7. A. Feito, I. Sanchez-Burgos, I. Tejero, E. Sanz, A. Rey, R. Collepardo-Guevara, A. R. Tejedor, and J. R. Espinosa, Benchmarking residue-resolution protein coarse-grained models for simulations of biomolecular condensates, PLoS Comput. Biol. 21, e1012737 (2025).
  8. A. Rizuan, S. Rekhi, Y. C. Kim, S. Najafi, J.-E. Shea, and J. Mittal, Computational modeling of biomolecular phase separation: Current progress and open challenges, Annu. Rev. Phys. Chem. 77, 557 (2026).
  9. G. H. Fredrickson, V. Ganesan, and F. Drolet, Field-theoretic computer simulation methods for polymers and complex fluids, Macromolecules 35, 16 (2002).
  10. G. H. Fredrickson and K. T. Delaney, Field-Theoretic Simulations in Soft Matter and Quantum Fluids, International Series of Monographs on Physics (Oxford University Press, Oxford, 2023).
  11. J. McCarty, K. T. Delaney, S. P. O. Danielsen, G. H. Fredrickson, and J.-E. Shea, Complete phase diagram for liquid-liquid phase separation of intrinsically disordered proteins, J. Phys. Chem. Lett. 10, 1644 (2019).
  12. G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, International Series of Monographs on Physics, Vol. 134 (Oxford University Press, Oxford, 2006).
  13. Y.-H. Lin, J. D. Forman-Kay, and H. S. Chan, Sequence-specific polyampholyte phase separation in membraneless organelles, Phys. Rev. Lett. 117, 178101 (2016).
  14. C. Ellis, X. Fang, C. Balzer, T. Quah, M. S. Shell, G. H. Fredrickson, and M. Gu, Fast phase prediction of charged polymer blends by white-box machine learning surrogates, Macromolecules 59, 50 (2026).
  15. S. F. Edwards, The theory of polymer solutions at intermediate concentration, Proc. Phys. Soc. 88, 265 (1966).
  16. V. Y. Borue and I. Y. Erukhimovich, A statistical theory of weakly charged polyelectrolytes: Fluctuations, equation of state and microphase separation, Macromolecules 21, 3240 (1988).
  17. K. A. Mahdi and M. Olvera de la Cruz, Phase diagrams of salt-free polyelectrolyte semidilute solutions, Macromolecules 33, 7649 (2000).
  18. Y. O. Popov, J. Lee, and G. H. Fredrickson, Field-theoretic simulations of polyelectrolyte complexation, J. Polym. Sci. B Polym. Phys. 45, 3223 (2007).
  19. K. T. Delaney and G. H. Fredrickson, Theory of polyelectrolyte complexation—Complex coacervates are self-coacervates, J. Chem. Phys. 146, 224902 (2017).
  20. Y.-H. Lin, J. P. Brady, J. D. Forman-Kay, and H. S. Chan, Charge pattern matching as a ‘fuzzy’ mode of molecular recognition for the functional phase separations of intrinsically disordered proteins, New J. Phys. 19, 115003 (2017).
  21. Y.-H. Lin, J. P. Brady, H. S. Chan, and K. Ghosh, A unified analytical theory of heteropolymers for sequence-specific phase behaviors of polyelectrolytes and polyampholytes, J. Chem. Phys. 152, 045102 (2020).
  22. M. Muthukumar and S. F. Edwards, Extrapolation formulas for polymer solution properties, J. Chem. Phys. 76, 2720 (1982).
  23. T. Ohta and A. Nakanishi, Theory of semi-dilute polymer solutions. I. Static property in a good solvent, J. Phys. A: Math. Gen. 16, 4155 (1983).
  24. P. Grzywacz, J. Qin, and D. C. Morse, Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts, Phys. Rev. E 76, 061802 (2007).
  25. J. Wessén, S. Das, T. Pal, and H. S. Chan, Analytical formulation and field-theoretic simulation of sequence-specific phase separation of protein-like heteropolymers with short- and long-spatial-range interactions, J. Phys. Chem. B 126, 9222 (2022).
  26. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986).
  27. D. C. Morse, Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory, Ann. Phys. 321, 2318 (2006).
  28. H. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
  29. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, Massachusetts, 1995).
  30. J. P. Hansen and I. R. McDonald, Theory of Simple Liquids: With Applications to Soft Matter, 4th ed. (Academic Press, Amsterdam, Netherlands, 2013).
  31. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed., Course of Theoretical Physics, Vol. 5 (Butterworth-Heinemann, Oxford, UK, 1980).
  32. Y.-H. Lin, J. Wessén, T. Pal, S. Das, and H. S. Chan, Numerical techniques for applications of analytical theories to sequence-dependent phase separations of intrinsically disordered proteins, Methods Mol. Biol. 2563, 51 (2023).
  33. J. Gaite, Effective sextic field theory for tricritical-critical crossover, Nucl. Phys. B 1030, 117604 (2026).
  34. A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO—The Open Visualization Tool, Modell. Simul. Mater. Sci. Eng. 18, 015012 (2010).
  35. D. Ruelle, Statistical Mechanics: Rigorous Results (World Scientific, Singapore, 1999).
  36. J. A. Anderson, J. Glaser, and S. C. Glotzer, HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations, Comput. Mater. Sci. 173, 109363 (2020).
  37. F. J. Blas, L. G. MacDowell, E. de Miguel, and G. Jackson, Vapor-liquid interfacial properties of fully flexible Lennard-Jones chains, J. Chem. Phys. 129, 144703 (2008).
  38. G. L. Dignon, W. Zheng, Y. C. Kim, R. B. Best, and J. Mittal, Sequence determinants of protein phase behavior from a coarse-grained model, PLoS Comput. Biol. 14, e1005941 (2018).
  39. G. Tesei, T. K. Schulze, R. Crehuet, and K. Lindorff-Larsen, Accurate model of liquid-liquid phase behavior of intrinsically disordered proteins from optimization of single-chain properties, Proc. Natl. Acad. Sci. USA 118, e2111696118 (2021).
  40. J. A. Joseph, A. Reinhardt, A. Aguirre, P. Y. Chew, K. O. Russell, J. R. Espinosa, A. Garaizar, and R. Collepardo-Guevara, Physics-driven coarse-grained model for biomolecular phase separation with near-quantitative accuracy, Nat. Comput. Sci. 1, 732 (2021).
  41. Y.-H. Lin, T. H. Kim, S. Das, T. Pal, J. Wessén, A. K. Rangadurai, L. E. Kay, J. D. Forman-Kay, and H. S. Chan, Electrostatics of salt-dependent reentrant phase behaviors highlights diverse roles of ATP in biomolecular condensates, eLife 13, RP100284 (2025).
  42. A. L. Kholodenko and K. F. Freed, Renormalization group treatment of excluded volume effects in a polyelectrolyte chain in the weak electrostatic coupling limit, J. Chem. Phys. 78, 7412 (1983).
  43. J. Song, J. Li, and H. S. Chan, Small-angle X-ray scattering signatures of conformational heterogeneity and homogeneity of disordered protein ensembles, J. Phys. Chem. B 125, 6451 (2021).
  44. A. M. Nemirovsky, M. G. Bawendi, and K. F. Freed, Lattice models of polymer solutions: Monomers occupying several lattice sites, J. Chem. Phys. 87, 7272 (1987).
  45. J. Dudowicz and K. F. Freed, Effect of monomer structure and compressibility on the properties of multicomponent polymer blends and solutions: 1. Lattice cluster theory of compressible systems, Macromolecules 24, 5076 (1991).
  46. K. W. Foreman and K. F. Freed, Lattice cluster theory of multicomponent polymer systems: Chain semiflexibility and specific interactions, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice, Advances in Chemical Physics, Vol. 103 (John Wiley & Sons, Inc., 1998), pp. 335–390.

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Phys. Rev. Research 8, 033343– Published 21 September, 2026
  • Received 10 May 2026
  • Accepted 1 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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