Simulation of weak polyelectrolytes: a comparison between the constant pH and the reaction ensemble method

An Erratum to this article is available

This article has been updated


The reaction ensemble and the constant pH method are well-known chemical equilibrium approaches to simulate protonation and deprotonation reactions in classical molecular dynamics and Monte Carlo simulations. In this article, we demonstrate the similarity between both methods under certain conditions. We perform molecular dynamics simulations of a weak polyelectrolyte in order to compare the titration curves obtained by both approaches. Our findings reveal a good agreement between the methods when the reaction ensemble is used to sweep the reaction constant. Pronounced differences between the reaction ensemble and the constant pH method can be observed for stronger acids and bases in terms of adaptive pH values. These deviations are due to the presence of explicit protons in the reaction ensemble method which induce a screening of electrostatic interactions between the charged titrable groups of the polyelectrolyte. The outcomes of our simulation hint to a better applicability of the reaction ensemble method for systems in confined geometries and titrable groups in polyelectrolytes with different pK a values.

This is a preview of subscription content, log in to check access.

Change history

  • 21 May 2019

    In the final print version of the article formulas (11) and (12) are wrong. The formulas should read in accordance with reference [1]:

  • 21 May 2019

    In the final print version of the article formulas (11) and (12) are wrong.


  1. 1.

    W. Richtering, Progr. Colloid Polym. Sci. 133, 9 (2006)

    Article  Google Scholar 

  2. 2.

    J.M. Berg, ed., Biochemistry (Freeman, New York, NY, USA, 2015)

  3. 3.

    M. Castelnovo, P. Sens, J.-F. Joanny, Eur. Phys. J. E 1, 115 (2000)

    Article  Google Scholar 

  4. 4.

    C. Shi, J.A. Wallace, J.K. Shen, Biophys. J. 102, 1590 (2012)

    ADS  Article  Google Scholar 

  5. 5.

    M. Lund, B. Jönsson, Biochemistry 44, 5722 (2005)

    Article  Google Scholar 

  6. 6.

    M. Lund, B. Jönsson, Q. Rev. Biophys. 46, 265 (2013)

    Article  Google Scholar 

  7. 7.

    S.E. Harnung, M.S. Johnson, Chemistry and the Environment (Cambridge University Press, Cambridge, UK, 2012)

  8. 8.

    J.N. Butler, Ionic Equilibrium: Solubility and pH Calculations (John Wiley & Sons, New York, NY, USA, 1998)

  9. 9.

    P.W. Atkins, J. de Paula, Physical Chemistry (Oxford University Press, Oxford, UK, 2010)

  10. 10.

    M. Müller, ed. Polyelectrolyte Complexes in the Dispersed and Solid State II: Application Aspects, volume 256 of Adv. Polym. Sci. (Springer, Berlin, Germany, 2013)

  11. 11.

    C.E. Reed, W.F. Reed, J. Chem. Phys. 96, 1609 (1992)

    ADS  Article  Google Scholar 

  12. 12.

    M. Ullner, B. Jönsson, B. Söderberg, C. Peterson, J. Chem. Phys. 104, 3048 (1996)

    ADS  Article  Google Scholar 

  13. 13.

    M. Ullner, B. Jönsson, Macromolecules 29, 6645 (1996)

    ADS  Article  Google Scholar 

  14. 14.

    M. Ullner, C.E. Woodward, Macromolecules 33, 7144 (2000)

    ADS  Article  Google Scholar 

  15. 15.

    B. Jönsson, M. Ullner, C. Peterson, O. Sommelius, B. Söderberg, J. Phys. Chem. 100, 409 (1996)

    Article  Google Scholar 

  16. 16.

    S. Uyaver, C. Seidel, Macromolecules 42, 1352 (2009)

    ADS  Article  Google Scholar 

  17. 17.

    F. Carnal, S. Stoll, J. Chem. Phys. 134, 044909 (2011)

    ADS  Article  Google Scholar 

  18. 18.

    A.K.N. Nair, S. Uyaver, S. Sun, J. Chem. Phys. 141, 134905 (2014)

    ADS  Article  Google Scholar 

  19. 19.

    J. Mongan, D.A. Case, J.A. McCammon, J. Comput. Chem. 25, 2038 (2004)

    Article  Google Scholar 

  20. 20.

    C. Heath Turner, J.K. Brennan, M. Lisal, W.R. Smith, J.K. Johnson, K.E. Gubbins, Mol. Simul. 34, 119 (2008)

    Article  Google Scholar 

  21. 21.

    W. Smith, B. Triska, J. Chem. Phys. 100, 3019 (1994)

    ADS  Article  Google Scholar 

  22. 22.

    J.K. Johnson, A.Z. Panagiotopoulos, K.E. Gubbins, Mol. Phys. 81, 717 (1994)

    ADS  Article  Google Scholar 

  23. 23.

    A. Panagiotopoulos, J. Phys. Cond. Matt. 21, 424113 (2009)

    ADS  Article  Google Scholar 

  24. 24.

    F. Uhlik, P. Kosovan, Z. Limpouchova, K. Prochazka, O.V. Borisov, F.A. Leermakers, Macromolecules 47, 4004 (2014)

    ADS  Article  Google Scholar 

  25. 25.

    F. Uhlík, P. Košovan, E.B. Zhulina, O.V. Borisov, Soft Matter 12, 4846 (2016)

    ADS  Article  Google Scholar 

  26. 26.

    J. Landsgesell, C. Holm, J. Chem. Theory Comput., DOI: 10.1021/acs.jctc.6b00791 (2016)

  27. 27.

    W.K. Hastings, Biometrika 57, 97 (1970)

    MathSciNet  Article  Google Scholar 

  28. 28.

    D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, San Diego, USA, 2002)

  29. 29.

    N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)

    ADS  Article  Google Scholar 

  30. 30.

    J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54, 5237 (1971)

    ADS  Article  Google Scholar 

  31. 31.

    R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles (CRC Press, Boca Raton, USA, 1988)

  32. 32.

    K. Kremer, G.S. Grest, J. Chem. Phys. 92, 5057 (1990)

    ADS  Article  Google Scholar 

  33. 33.

    H.-J. Limbach, A. Arnold, B.A. Mann, C. Holm, Comput. Phys. Commun. 174, 704 (2006)

    ADS  Article  Google Scholar 

  34. 34.

    A. Arnold, O. Lenz, S. Kesselheim, R. Weeber, F. Fahrenberger, D. Roehm, P. Košovan, C. Holm, in M.A. Schweitzer, ed., Meshfree Methods for Partial Differential Equations VI, volume 89 of Lecture Notes in Computational Science and Engineering (Springer, Berlin, Germany, 2013), pp. 1–23

  35. 35.

    R.J. Hunter, Foundations of Colloid Science (Oxford University Press, Oxford, UK, 2001)

  36. 36.

    A. Ramos, S. López, R. López, S. Fiol, F. Arce, J.M. Antelo, Analusis 27, 414 (1999)

    Article  Google Scholar 

Download references

Author information



Corresponding authors

Correspondence to Jonas Landsgesell or Jens Smiatek.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Landsgesell, J., Holm, C. & Smiatek, J. Simulation of weak polyelectrolytes: a comparison between the constant pH and the reaction ensemble method. Eur. Phys. J. Spec. Top. 226, 725–736 (2017).

Download citation