Advertisement

Dynamical Density Functional Theory for Brownian Dynamics of Colloidal Particles

  • Hartmut LöwenEmail author
Chapter
Part of the Molecular Modeling and Simulation book series (MMAS)

Abstract

Variational methods play a key role in physics. Density functional theory (DFT) is a special and important example of such a variational formulation: There is a functional of the one-particle density which gives access to the equilibrium thermodynamics when it is minimized with respect to the density. This important theory can be both applied to quantum-mechanical electrons and to classical systems. In this book chapter we shall consider a nonequilibrium situation where an explicit time-dependence comes into play. This addresses the point of nonequilibrium dynamics and is therefore much more complicated than the traditional DFT. In the special case of completely overdamped Brownian dynamics of classical colloidal particles typically described by the Langevin or Smoluchowski equations, a dynamical version of DFT, the so-called dynamical density functional theory (DDFT), is available and makes dynamical predictions which are in good agreement with computer simulations. Here we shall derive DDFT for Brownian dynamics in a tutorial way from the Smoluchowski equation and mention some applications. The theory will then be generalized towards hydrodynamic interactions between the particles and to orientational degrees of freedom describing e.g. rod-like colloids. Finally some recent developments will be discussed including active Brownian particles and the derivation of DDFT from projection operator techniques which can be viewed as a variational problem.

Keywords

Density Functional Theory Hard Sphere Hydrodynamic Interaction Brownian Dynamic Smoluchowski Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

I thank R. Wittkowski, M. Marechal and B. ten Hagen for many helpful suggestions. This work was supported by the DFG within project LO 418/20-1.

References

  1. 1.
    C.V. Achim, M. Schmiedeberg, H. Löwen, Phys. Rev. Lett. 112, 255501 (1–5) (2014)Google Scholar
  2. 2.
    C.V. Achim, R. Wittkowski, H. Löwen, Phys. Rev. E 83, 061712 (1–8) (2011)Google Scholar
  3. 3.
    M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids (Oxford Science Publications, Clarendon Press, Oxford, 1987)zbMATHGoogle Scholar
  4. 4.
    L. Almenar, M. Rauscher, J. Phys.: Condens. Matter 23, 184115 (1–9) (2011)Google Scholar
  5. 5.
    J.G. Anero, P. Español, P. Tarazona, J. Chem. Phys. 139, 034106 (1–14) (2013)Google Scholar
  6. 6.
    A.J. Archer, Phys. Rev. E 72, 051501 (1–7) (2005)Google Scholar
  7. 7.
    A.J. Archer, R. Evans, J. Chem. Phys. 121, 4246–4254 (2004)CrossRefGoogle Scholar
  8. 8.
    A.J. Archer, A.M. Rucklidge, E. Knobloch, Phys. Rev. E 92, 012324 (1–14) (2015)Google Scholar
  9. 9.
    A.J. Archer, M. Rauscher, J. Phys. A 37, 9325–9333 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. Löwen, Phys. Rev. Lett. 102, 238301 (1–4) (2009)Google Scholar
  11. 11.
    J.-L. Barrat, J.-P. Hansen, Basic Concepts for Simple and Complex Liquids (Cambridge University Press, 2003)Google Scholar
  12. 12.
    J. Bleibel, A. Dominguez, M. Oettel, S. Dietrich, Soft Matter 10, 4091–4109 (2014)CrossRefGoogle Scholar
  13. 13.
    P. Bolhuis, D. Frenkel, J. Chem. Phys 106, 666–687 (1997)CrossRefGoogle Scholar
  14. 14.
    J.M. Brader, M. Krüger, Mol. Phys. 109, 1029–1041 (2011)CrossRefGoogle Scholar
  15. 15.
    J.F. Brady, G. Bossis, Ann. Rev. Fluid Mech. 20, 111–157 (1988)CrossRefGoogle Scholar
  16. 16.
    J.K.G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996)Google Scholar
  17. 17.
    M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Oxford Science Publications, Clarendon Press, Oxford, 1986)Google Scholar
  18. 18.
    A. Donev, E. Vanden-Eijnden, J. Chem. Phys. 140, 234115 (1–18) (2014)Google Scholar
  19. 19.
    R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A. Stone, J. Bibette, Nature 437, 862–865 (2005)CrossRefGoogle Scholar
  20. 20.
    J. Dzubiella, H. Löwen, C.N. Likos, Phys. Rev. Lett. 91, 248301 (1–4) (2003)Google Scholar
  21. 21.
    J. Elgeti, R.G. Winkler, G. Gompper, Rep. Prog. Phys. 78, 056601 (1–50) (2015)Google Scholar
  22. 22.
    H. Emmerich, H. Löwen, R. Wittkowski, T. Gruhn, G.I. Tóth, G. Tegze, L. Gránásy, Adv. Phys. 61, 665–743 (2012)CrossRefGoogle Scholar
  23. 23.
    A. Erbe, M. Zientara, L. Baraban, C. Kreidler, P. Leiderer, J. Phys.: Condens. Matter 20, 404215 (1–5) (2008)Google Scholar
  24. 24.
    P. Español, H. Löwen, J. Chem. Phys. 131, 244101 (1–7) (2009)Google Scholar
  25. 25.
    R. Evans, Adv. Phys. 28, 143–200 (1979)CrossRefGoogle Scholar
  26. 26.
    M.X. Fernandes, J.G. de la Torre, Biophys. J. 83, 3039–3045 (2002)CrossRefGoogle Scholar
  27. 27.
    D. Frenkel, in Liquids, Freezing and Glass Transition, ed. by J.P. Hansen, D. Levesque, J. Zinn-Justin (North Holland, Amsterdam, 1991), pp. 689–756Google Scholar
  28. 28.
    D. Frenkel, B.M. Mulder, J.P. McTague, Phys. Rev. Lett. 52, 287–290 (1984)CrossRefGoogle Scholar
  29. 29.
    B.D. Goddard, A. Nold, S. Kalliadasis, J. Chem. Phys 138, 144904 (1–9) (2013)Google Scholar
  30. 30.
    B.D. Goddard, A. Nold, N. Savva, P. Yatsyshin, S. Kalliadasis, J. Phys.: Condens. Matter 25, 035101 (1–14) (2013)Google Scholar
  31. 31.
    B.D. Goddard, A. Nold, N. Savva, G.A. Pavliotis, S. Kalliadasis, Phys. Rev. Lett. 109, 120603 (1–4) (2012)Google Scholar
  32. 32.
    B.D. Goddard, G.A. Pavliotis, S. Kalliadasis, Multiscale Model. Simul. 10, 633–663 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    D. Gottwald, C.N. Likos, G. Kahl, H. Löwen, Phys. Rev. Lett. 92, 068301 (1–4) (2004)Google Scholar
  34. 34.
    H. Graf, H. Löwen, J. Phys.: Condens. Matter 11, 1435–1452 (1999)Google Scholar
  35. 35.
    B. ten Hagen, S. van Teeffelen, H. Löwen, J. Phys.: Condens. Matter 23, 194119 (1–16) (2011)Google Scholar
  36. 36.
    B. ten Hagen, R. Wittkowski, D. Takagi, F. Kümmel, C. Bechinger, H. Löwen, J. Phys.: Condens. Matter 27,194110 (1–10) (2015)Google Scholar
  37. 37.
    A. Härtel, H. Löwen, J. Phys.: Condens. Matter 22, 104112 (1–11) (2010)Google Scholar
  38. 38.
    A. Härtel, R. Blaak, H. Löwen, Phys. Rev. E 81, 051703 (1–5) (2010)Google Scholar
  39. 39.
    J.P. Hansen, I. McDonald, Theory of  Simple Liquids, 3rd edn. (Elsevier, Amsterdam, Academic Press, 2005)Google Scholar
  40. 40.
    H. Hansen-Goos, K. Mecke, Phys. Rev. Lett. 102, 018302 (1–4) (2009)Google Scholar
  41. 41.
    N. Hoffmann, F. Ebert, C.N. Likos, H. Löwen, G. Maret, Phys. Rev. Lett. 97, 078301 (1–4) (2006)Google Scholar
  42. 42.
    J.R. Howse, R.A.L. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian, Phys. Rev. Lett. 99, 048102 (1–4) (2007)Google Scholar
  43. 43.
    J.E. Hug, F. van Swol, C.F. Zukoski, Langmuir 11, 111–118 (1995)CrossRefGoogle Scholar
  44. 44.
    R.J. Hunter, Foundations of Colloid Science, 2nd edn. (Oxford University Press, Oxford, 1989)Google Scholar
  45. 45.
    A.V. Ivlev, H. Löwen, G.E. Morfill, C.P. Royall, Complex Plasmas and Colloidal Dispersions: Particle-Resolved Studies of Classical Liquids and Solids (World Scientific, Singapore, 2012)CrossRefGoogle Scholar
  46. 46.
    G. Kahl, H. Löwen, J. Phys.: Condens. Matter 21, 464101 (1–7) (2009)Google Scholar
  47. 47.
    D.J. Kraft, R. Wittkowski, B. ten Hagen, K.V. Edmond, D.J. Pine, H. Löwen, Phys. Rev. E 88, 050301(R) (1–4) (2013)Google Scholar
  48. 48.
    C. Kreuter, U. Siems, P. Nielaba, P. Leiderer, A. Erbe, Eur. Phys. J. - Special Topics 222, 2923–2939 (2013)CrossRefGoogle Scholar
  49. 49.
    M. Krüger, J.M. Brader, EPL 96 68006 (1–6) (2011)Google Scholar
  50. 50.
    A. Lang, C.N. Likos, M. Watzlawek, H. Löwen, J. Phys.: Condens. Matter 12, 5087–5108 (2000)Google Scholar
  51. 51.
    K. Lichtner, S.H.L. Klapp, Phys. Rev. E 86, 051405 (1–10) (2012)Google Scholar
  52. 52.
    C.N. Likos, Z.T. Németh, H. Löwen, J. Phys.: Condens. Matter 6, 10965–10976 (1994)Google Scholar
  53. 53.
    C.N. Likos, H. Löwen, M. Watzlawek, B. Abbas, O. Jucknischke, J. Allgaier, D. Richter, Phys. Rev. Lett. 80, 4450–4453 (1998)CrossRefGoogle Scholar
  54. 54.
    C.N. Likos, A. Lang, M. Watzlawek, H. Löwen, Phys. Rev. E 63, 031206 (1–9) (2001)Google Scholar
  55. 55.
    H. Löwen, Phys. Rep. 237, 249–324 (1994)CrossRefGoogle Scholar
  56. 56.
    H. Löwen, Phys. Rev. E 50, 1232–1242 (1994)CrossRefGoogle Scholar
  57. 57.
    H. Löwen, J. Phys.: Condens. Matter 13, R415–R432 (2001)Google Scholar
  58. 58.
    H. Löwen, J. Phys.: Condens. Matter 14, 11897–11905 (2002)Google Scholar
  59. 59.
    H. Löwen, J. Phys.: Condens. Matter 22, 364105 (1–6) (2010)Google Scholar
  60. 60.
    H. Löwen, in 3rd Warsaw School of Statistical Physics, (Warsaw University Press, 2010), pp. 87–121Google Scholar
  61. 61.
    H. Löwen, in Understanding Soft Condensed Matter via Modeling and Computation, ed. by W. Hu, A.-C. Shi. Applications of Density Functional Theory in Soft Condensed Matter, Series in Soft Condensed Matter (World Scientific, 2011), vol. 3, pp. 9–45Google Scholar
  62. 62.
    H. Löwen, M. Heinen, Eur. Phys. J. - Special Topics 223, 3113–3127 (2014)CrossRefGoogle Scholar
  63. 63.
    M. Makino, M. Doi, J. Phys. Soc. Jpn. 73, 2739–2745 (2004)CrossRefGoogle Scholar
  64. 64.
    A. Malijevsky, A.J. Archer, J. Chem. Phys. 139 144901 (1–13) (2013)Google Scholar
  65. 65.
    M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R.A. Simha, Rev. Mod. Phys. 85, 1143–1189 (2013)CrossRefGoogle Scholar
  66. 66.
    U.M.B. Marconi, P. Tarazona, J. Chem. Phys. 110, 8032–8044 (1999)CrossRefGoogle Scholar
  67. 67.
    U.M.B. Marconi, P. Tarazona, J. Phys.: Condens. Matter 12, A413–A418 (2000)Google Scholar
  68. 68.
    U.M.B. Marconi, S. Melchionna, J. Chem. Phys. 126 184109 (1–9) (2007)Google Scholar
  69. 69.
    U.M.B. Marconi, P. Tarazona, F. Cecconi, S. Melchionna, J. Phys.: Condens. Matter 20, 494233 (1–6) (2008)Google Scholar
  70. 70.
    U.M.B. Marconi, S. Melchionna, J. Chem. Phys. 131 014105 (1–14) (2009)Google Scholar
  71. 71.
    U.M.B. Marconi, S. Melchionna, J. Phys.: Condens. Matter 22, 364110 (1–8) (2010)Google Scholar
  72. 72.
    U.M.B. Marconi, S. Melchionna, Comm. in Theor. Phys. 62, 596–606 (2014)MathSciNetCrossRefGoogle Scholar
  73. 73.
    M. Marechal, H. Löwen, Phys. Rev. Lett. 110, 137801 (1–5) (2013)Google Scholar
  74. 74.
    M. Marechal, H.-H. Goetzke, A. Härtel, H. Löwen, J. Chem. Phys. 135, 234510 (1–13) (2011)Google Scholar
  75. 75.
    M. Marechal, U. Zimmermann, H. Löwen, J. Chem. Phys. 136, 144506 (1–14) (2012)Google Scholar
  76. 76.
    A.M. Menzel, H. Löwen, Phys. Rev. Lett. 110, 055702 (1–5) (2013)Google Scholar
  77. 77.
    A.M. Menzel, A. Saha, C. Hoell, H. Löwen, J. Chem. Phys. 144, 024115 (1–13) (2016)Google Scholar
  78. 78.
    M. Mukherjee, P. Mishra, H. Löwen, J. Phys.: Condens. Matter 26, 465101 (1–9) (2014)Google Scholar
  79. 79.
    K. Nagao, T. Inuzuka, K. Nishimoto, K. Edagawa, Phys. Rev. Lett. 115, 075501 (1–5) (2015)Google Scholar
  80. 80.
    T. Neuhaus, M. Schmiedeberg, H. Löwen, Phys. Rev. E 88, 062316 (1–6) (2013)Google Scholar
  81. 81.
    T. Neuhaus, M. Schmiedeberg, H. Löwen, New J. Phys. 15, 073013 (1–11) (2013)Google Scholar
  82. 82.
    T. Neuhaus, M. Marechal, M. Schmiedeberg, H. Löwen, Phys. Rev. Lett. 110, 118301 (1–5) (2013)Google Scholar
  83. 83.
    R. Ohnesorge, H. Löwen, H. Wagner, Europhys. Lett. 22, 245–249 (1993)CrossRefGoogle Scholar
  84. 84.
    L. Onsager, Proc. N. Y. Acad. Sci. 51, 627–659 (1949)CrossRefGoogle Scholar
  85. 85.
    D.W. Oxtoby, in Liquids, Freezing and Glass Transition, ed. by J.P. Hansen, D. Levesque, J. Zinn-Justin (North Holland, Amsterdam, 1991), pp. 145–189Google Scholar
  86. 86.
    A. Poniewierski, R. Holyst, Phys. Rev. Lett. 61, 2461 (1988)CrossRefGoogle Scholar
  87. 87.
    S. Praetorius, A. Voigt, R. Wittkowski, H. Löwen, Phys. Rev. E 87, 052406 (1–8) (2013)Google Scholar
  88. 88.
    P.N. Pusey, in Liquids, Freezing and Glass Transition, ed. by J.P. Hansen, D. Levesque, J. Zinn-Justin (North Holland, Amsterdam, 1991), pp. 145–189Google Scholar
  89. 89.
    M. Rauscher, A. Dominguez, M. Krüger, F. Penna, J. Chem. Phys 127, 244906 (1–8) (2007)Google Scholar
  90. 90.
    M. Rauscher, J. Phys.: Condens. Matter 22, 364109 (1–6) (2010)Google Scholar
  91. 91.
    J. Reinhardt, F. Weysser, J. M. Brader, EPL 102, 28011 (1–6) (2013)Google Scholar
  92. 92.
    M. Rex, H. Löwen, Phys. Rev. Lett. 101, 148302 (1–4) (2008)Google Scholar
  93. 93.
    M. Rex, H.H. Wensink, H. Löwen, Phys. Rev. E 76, 021403 (1–10) (2007)Google Scholar
  94. 94.
    M. Rex, H. Löwen, Eur. Phys. J. E 28, 139–146 (2009)CrossRefGoogle Scholar
  95. 95.
    P. Romanczuk, M. Bär, W. Ebeling, B. Linder, L. Schimansky-Geier, Eur. Phys. J. - Special Topics 202, 1–162 (2012)CrossRefGoogle Scholar
  96. 96.
    Y. Rosenfeld, M. Schmidt, H. Löwen, P. Tarazona, Phys. Rev. E 55, 4245–4263 (1997)CrossRefGoogle Scholar
  97. 97.
    R. Roth, J. Phys.: Condens. Matter 22, 063102 (1–18) (2010)Google Scholar
  98. 98.
    R. Roth, K. Mecke, M. Oettel, J. Chem. Phys. 136 081101 (1–4) (2012)Google Scholar
  99. 99.
    E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52(12), 997–1000 (1984)CrossRefGoogle Scholar
  100. 100.
    M. Schmidt, Phys. Rev. E 84, 051203 (1–8) (2011)Google Scholar
  101. 101.
    M. Schmidt, J. Brader, J. Chem. Phys. 138, 214101 (1–8) (2013)Google Scholar
  102. 102.
    D. Stopper, K. Marolt, R. Roth, H. Hansen-Goos, Phys. Rev. E 92, 022151 (1–11) (2015)Google Scholar
  103. 103.
    P. Tarazona, Phys. Rev. Lett. 84, 694–697 (2000)CrossRefGoogle Scholar
  104. 104.
    P. Tarazona, J.A. Cuesta, Y. Martinez-Raton, Density Functional Theories of Hard Particle Systems, Lecture Notes in Physics, vol. 753 (Springer, Berlin, 2008), pp. 247–341Google Scholar
  105. 105.
    S. van Teeffelen, H. Löwen, Phys. Rev. E 78, 020101(R) (1–4) (2008)Google Scholar
  106. 106.
    S. van Teeffelen, N. Hoffmann, C.N. Likos, H. Löwen, Europhys. Lett. 75, 583–589 (2006)CrossRefGoogle Scholar
  107. 107.
    S. van Teeffelen, C.N. Likos, H. Löwen, Phys. Rev. Lett. 100, 108302 (1–4) (2008)Google Scholar
  108. 108.
    S. van Teeffelen, C.V. Achim, H. Löwen, Phys. Rev. E 87, 022306 (1–6) (2013)Google Scholar
  109. 109.
    S. van Teeffelen, R. Backofen, H. Löwen, A. Voigt, Phys. Rev. E 79, 051404 (1–10) (2009)Google Scholar
  110. 110.
    H.H. Wensink, H. Löwen, Phys. Rev. E 78, 031409 (1–4) (2008)Google Scholar
  111. 111.
    R. Wittkowski, H. Löwen, H.R. Brand, Phys. Rev. E 82, 031708 (1–7) (2010)Google Scholar
  112. 112.
    R. Wittkowski, H. Löwen, Mol. Phys. 109, 2935–2943 (2011)CrossRefGoogle Scholar
  113. 113.
    R. Wittkowski, H. Löwen, H.R. Brand, Phys. Rev. E 83, 061706 (1–10) (2011)Google Scholar
  114. 114.
    R. Wittkowski, H. Löwen, H.R. Brand, Phys. Rev. E 84, 041708 (1–9) (2011)Google Scholar
  115. 115.
    R. Wittkowski, H. Löwen, H.R. Brand, J. Chem. Phys. 137, 224904 (1–14) (2012)Google Scholar
  116. 116.
    R. Wittkowski, H. Löwen, H.R. Brand, J. Phys. A: Math. Theor. 46, 355003 (1–17) (2013)Google Scholar
  117. 117.
    U. Zimmermann, F. Smallenburg, H. Löwen, J. Phys.: Condens. Matter 28, 244019 (1–10) (2016)Google Scholar

Copyright information

© Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Institut für Theoretische Physik II: Weiche MaterieHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

Personalised recommendations