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Systematic extension of the Cahn-Hilliard model for motility-induced phase separation

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Abstract.

We consider a continuum model for motility-induced phase separation (MIPS) of active Brownian particles (ABP) (J. Chem. Phys. 142, 224149 (2015)). Using a recently introduced perturbative analysis (Phys. Rev. E 98, 020604(R) (2018)), we show that this continuum model reduces to the classic Cahn-Hilliard (CH) model near the onset of MIPS. This makes MIPS another example of the so-called active phase separation. We further introduce a generalization of the perturbative analysis to the next higher order. This results in a generic higher-order extension of the CH model for active phase separation. Our analysis establishes the mathematical link between the basic mean-field ABP model on the one hand, and the leading order and extended CH models on the other hand. Comparing numerical simulations of the three models, we find that the leading-order CH model agrees nearly perfectly with the full continuum model near the onset of MIPS. We also give estimates of the control parameter beyond which the higher-order corrections become relevant and compare the extended CH model to recent phenomenological models.

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References

  1. T. Vicsek, A. Zafeiris, Phys. Rep. 517, 71 (2012)

    Article  ADS  Google Scholar 

  2. A. Cavagna, I. Giardina, Annu. Rev. Condens. Matter Phys. 5, 183 (2014)

    Article  ADS  Google Scholar 

  3. A.A. Hyman, C.A. Weber, F. Jülicher, Annu. Rev. Cell Dev. Biol. 30, 39 (2014)

    Article  Google Scholar 

  4. M.E. Cates, J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)

    Article  ADS  Google Scholar 

  5. S. Zhou, A. Sokolov, O.D. Lavrentovich, I.S. Aranson, Proc. Natl. Acad. Sci. U.S.A. 111, 1265 (2014)

    Article  ADS  Google Scholar 

  6. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichardt, G. Volpe, G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)

    Article  ADS  Google Scholar 

  7. Y. Shin, C.P. Brangwynne, Science 357, 4382 (2017)

    Article  Google Scholar 

  8. F. Jülicher, S.W. Grill, G. Salbreux, Rep. Prog. Phys. 81, 076601 (2018)

    Article  ADS  Google Scholar 

  9. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell (Garland Science, New York, 2002)

  10. A. Jilkine, A.F. Marée, L. Edelstein-Keshet, Bull. Math. Biol. 69, 1943 (2007)

    Article  MathSciNet  Google Scholar 

  11. M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki, S. Kuroda, PLoS Comput. Biol. 3, e108 (2007)

    Article  ADS  Google Scholar 

  12. N.W. Goehring, S.W. Grill, Trends Cell Biol. 23, 72 (2013)

    Article  Google Scholar 

  13. P.K. Trong, E.M. Nicola, N.W. Goehring, K.V. Kumar, S.W. Grill, New J. Phys. 16, 065009 (2014)

    Article  Google Scholar 

  14. S. Alonso, M. Bär, Phys. Biol. 7, 046012 (2010)

    Article  ADS  Google Scholar 

  15. N.W. Goehring, P.K. Trong, J.S. Bois, D. Chowdhury, E.M. Nicola, A.A. Hyman, S.W. Grill, Science 334, 1137 (2011)

    Article  ADS  Google Scholar 

  16. F. Bergmann, L. Rapp, W. Zimmermann, Phys. Rev. E 98, 020603(R) (2018)

    Article  ADS  Google Scholar 

  17. M.J. Tindall, P.K. Maini, S.L. Porter, J.P. Armitage, Bull. Math. Biol. 70, 1570 (2008)

    Article  MathSciNet  Google Scholar 

  18. T. Hillen, K.J. Painter, J. Math. Biol. 58, 183 (2009)

    Article  MathSciNet  Google Scholar 

  19. M. Meyer, L. Schimansky-Geier, P. Romanczuk, Phys. Rev. E 89, 022711 (2014)

    Article  ADS  Google Scholar 

  20. B. Liebchen, D. Marenduzzo, I. Pagonabarraga, M.E. Cates, Phys. Rev. Lett. 115, 258301 (2015)

    Article  ADS  Google Scholar 

  21. I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, L. Bocquet, Phys. Rev. Lett. 108, 268303 (2012)

    Article  ADS  Google Scholar 

  22. J. Palacci, S. Sacanna, A.P. Steinberg, D.J. Pine, P.M. Chaikin, Science 339, 936 (2013)

    Article  ADS  Google Scholar 

  23. I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, T. Speck, Phys. Rev. Lett. 110, 238301 (2013)

    Article  ADS  Google Scholar 

  24. Y. Fily, M.C. Marchetti, Phys. Rev. Lett. 108, 235702 (2012)

    Article  ADS  Google Scholar 

  25. J. Stenhammer, A. Tiribocchi, R.J. Allen, D. Marenduzzo, M.E. Cates, Phys. Rev. Lett. 111, 145702 (2013)

    Article  ADS  Google Scholar 

  26. G.S. Redner, M.F. Hagan, A. Baskaran, Phys. Rev. Lett. 110, 055701 (2013)

    Article  ADS  Google Scholar 

  27. T. Speck, J. Bialké, A.M. Menzel, H. Löwen, Phys. Rev. Lett. 112, 218304 (2014)

    Article  ADS  Google Scholar 

  28. Q.-X. Liu, A. Doelman, V. Rottschäfer, M. de Jager, P.M. Herman, M. Rietkerk, J. van de Koppel, Proc. Natl. Acad. Sci. U.S.A. 110, 11905 (2013)

    Article  ADS  Google Scholar 

  29. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

    Article  ADS  Google Scholar 

  30. T. Speck, A.M. Menzel, J. Bialke, H. Löwen, J. Chem. Phys. 142, 224109 (2015)

    Article  ADS  Google Scholar 

  31. R. Wittkowski, A. Tiribocchi, J. Stenhammar, R.J. Allen, D. Marenduzzo, M.E. Cates, Nat. Commun. 5, 4351 (2014)

    Article  ADS  Google Scholar 

  32. E. Tjhung, C. Nardini, M.E. Cates, Phys. Rev. X 8, 031080 (2018)

    Google Scholar 

  33. J. Tailleur, M.E. Cates, Phys. Rev. Lett. 100, 218103 (2008)

    Article  ADS  Google Scholar 

  34. A.J. Bray, Adv. Phys. 43, 357 (1994)

    Article  ADS  Google Scholar 

  35. R.C. Desai, R. Kapral, Dynamics of Self-Organized and Self-Assembled Structures (Cambridge University Press, Cambridge, 2009)

  36. A.P. Solon, J. Stenhammar, M.E. Cates, Y. Kafri, J. Tailleur, Phys. Rev. E 97, 020602(R) (2018)

    Article  ADS  Google Scholar 

  37. M.C. Cross, H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009)

  38. J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  39. J.W. Cahn, Acta Metall. 9, 795 (1961)

    Article  Google Scholar 

  40. L.M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer, Berlin, 2006)

  41. E. Meron, Nonlinear Physics of Ecosystems (CRC Press, Boca Raton, Florida, 2015)

  42. C. Misbah, Complex Dynamics and Morphogenesis (Springer, Berlin, Germany, 2016)

  43. L. Edelstein-Keshet, W.R. Holmes, M. Zajac, M. Dutot, Philos. Trans. R. Soc. B 368, 20130003 (2013)

    Article  Google Scholar 

  44. L. Rapp, F. Bergmann, W. Zimmermann, EPL 113, 28006 (2016)

    Article  ADS  Google Scholar 

  45. F. Bergmann, L. Rapp, W. Zimmermann, New J. Phys. 20, 072001 (2018)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Theoretische Physik I, Universität Bayreuth, 95440, Bayreuth, Germany

    Lisa Rapp, Fabian Bergmann & Walter Zimmermann

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  1. Lisa Rapp
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  2. Fabian Bergmann
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  3. Walter Zimmermann
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Correspondence to Walter Zimmermann.

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Rapp, L., Bergmann, F. & Zimmermann, W. Systematic extension of the Cahn-Hilliard model for motility-induced phase separation. Eur. Phys. J. E 42, 57 (2019). https://doi.org/10.1140/epje/i2019-11825-8

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