Nonlinear Rheological Properties of Dense Colloidal Dispersions Close to a Glass Transition Under Steady Shear

  • Matthias FuchsEmail author
Part of the Advances in Polymer Science book series (POLYMER, volume 236)


The nonlinear rheological properties of dense colloidal suspensions under steady shear are discussed within a first principles approach. It starts from the Smoluchowski equation of interacting Brownian particles in a given shear flow, derives generalized Green–Kubo relations, which contain the transients dynamics formally exactly, and closes the equations using mode coupling approximations. Shear thinning of colloidal fluids and dynamical yielding of colloidal glasses arise from competition between a slowing down of structural relaxation because of particle interactions, and enhanced decorrelation of fluctuations caused by the shear advection of density fluctuations. The integration through transients (ITT) approach takes account of the dynamic competition, translational invariance enters the concept of wavevector advection, and the mode coupling approximation enables one to explore quantitatively the shear-induced suppression of particle caging and the resulting speed-up of the structural relaxation. Extended comparisons with shear stress data in the linear response and in the nonlinear regime measured in model thermo-sensitive core-shell latices are discussed. Additionally, the single particle motion under shear observed by confocal microscopy and in computer simulations is reviewed and analysed theoretically.


Colloidal dispersion Flow curve Glass transition Integration through transients approach Linear viscoelasticity Mode coupling theory Nonlinear rheology Non-equilibrium stationary state Shear modulus Steady shear 



It is a great pleasure to thank all my colleagues for the enjoyable and fruitful collaboration on this topic. I especially thank Mike Cates for introducing me to rheology, and Matthias Ballauff for his inspiring studies. Kind hospitality in the group of John Brady, where part of this review was written, is gratefully acknowledged. Financial support is acknowledged by the Deutsche Forschungsgemeinschaft in SFB-TR6, SFB 513, IRTG 667, and via grant Fu 309/3.


  1. 1.
    Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, New YorkGoogle Scholar
  2. 2.
    Götze W, Sjögren L (1992) Rep Prog Phys 55:241CrossRefGoogle Scholar
  3. 3.
    Russel WB, Saville DA, Schowalter WR (1989) Colloidal dispersions. Cambridge University Press, New YorkCrossRefGoogle Scholar
  4. 4.
    Laun HM, Bung R, Hess S, Loose W, Hess O, Hahn K, Hädicke E, Hingmann R, Schmidt F, Lindner P (1992) J Rheol 36:743CrossRefGoogle Scholar
  5. 5.
    Brady JF (1993) J Chem Phys 99:567CrossRefGoogle Scholar
  6. 6.
    Sollich P, Lequeux F, Hébraud P, Cates ME (1997) Phys Rev Lett 78:2020CrossRefGoogle Scholar
  7. 7.
    Sollich P (1998) Phys Rev E 58:738CrossRefGoogle Scholar
  8. 8.
    Fielding S, Sollich P, Cates ME (2000) J Rheol 44:323CrossRefGoogle Scholar
  9. 9.
    Berthier L, Barrat J-L, Kurchan J (2000) Phys Rev E 61:5464CrossRefGoogle Scholar
  10. 10.
    Berthier L, Barrat J-L (2002) J Chem Phys 116:6228CrossRefGoogle Scholar
  11. 11.
    Pusey PN, van Megen W (1987) Phys Rev Lett 59:2083CrossRefGoogle Scholar
  12. 12.
    Megen W, Pusey PN (1991) Phys Rev A 43:5429CrossRefGoogle Scholar
  13. 13.
    van Megen W, Underwood SM (1993) Phys Rev Lett 70:2766CrossRefGoogle Scholar
  14. 14.
    van Megen W, Underwood SM (1994) Phys Rev E 49:4206CrossRefGoogle Scholar
  15. 15.
    van Megen W, Mortensen TC, Müller J, Williams SR (1998) Phys Rev E 58:6073CrossRefGoogle Scholar
  16. 16.
    Hébraud P, Lequeux F, Munch J, Pine D (1997) Phys Rev Lett 78:4657CrossRefGoogle Scholar
  17. 17.
    Beck C, Härtl W, Hempelmann R (1999) J Chem Phys 111:8209CrossRefGoogle Scholar
  18. 18.
    Bartsch E, Eckert T, Pies C, Sillescu H (2002) J Non-Cryst Solids 802:307Google Scholar
  19. 19.
    Eckert T, Bartsch E (2003) Faraday Discuss 123:51CrossRefGoogle Scholar
  20. 20.
    Weeks ER, Crocker JC, Levitt AC, Schofield A, Weitz DA (2000) Science 287:627CrossRefGoogle Scholar
  21. 21.
    Mason TG, Weitz DA (1995) Phys Rev Lett 75:2770CrossRefGoogle Scholar
  22. 22.
    Zackrisson M, Stradner A, Schurtenberger P, Bergenholtz J (2006) Phys Rev E 73:011408CrossRefGoogle Scholar
  23. 23.
    Senff H, Richtering W (1999) J Chem Phys 111:1705CrossRefGoogle Scholar
  24. 24.
    Senff H, Richtering W, Norhausen Ch, Weiss A, Ballauff M (1999) Langmuir 15:102CrossRefGoogle Scholar
  25. 25.
    Petekidis G, Vlassopoulos D, Pusey P (1999) Faraday Discuss 123:287CrossRefGoogle Scholar
  26. 26.
    Petekidis G, Vlassopoulos D, Pusey PN (2004) J Phys Condens Matter 16:S3955CrossRefGoogle Scholar
  27. 27.
    Petekidis G, Moussaid A, Pusey PN (2002) Phys Rev E 66:051402;CrossRefGoogle Scholar
  28. 28.
    Petekidis G, Vlassopoulos D, Pusey PN (2003) Faraday Discuss 123:287CrossRefGoogle Scholar
  29. 29.
    Pham KN, Petekidis G, Vlassopoulos D, Egelhaaf SU, Pusey PN, Poon WCK (2006) Europhys Lett 75:624CrossRefGoogle Scholar
  30. 30.
    Besseling R, Weeks ER, Schofield AB, Poon WC (2007) Phys Rev Lett 99:028301CrossRefGoogle Scholar
  31. 31.
    Crassous JJ, Siebenbürger M, Ballauf M, Drechsler M, Henrich O, Fuchs M (2006) J Chem Phys 125:204906CrossRefGoogle Scholar
  32. 32.
    Crassous JJ, Siebenbürger M, Ballauf M, Drechsler M, Hajnal D, Henrich O, Fuchs M (2008) J Chem Phys 128:204902CrossRefGoogle Scholar
  33. 33.
    Siebenbürger M, Fuchs M, Winter H, Ballauff M (2009) Viscoelasticity and shear flow of concentrated, non-crystallizing colloidal suspensions: Comparison with Mode-Coupling Theory. J Rheol 53:707–726CrossRefGoogle Scholar
  34. 34.
    Phung T, Brady J, Bossis G (1996) J Fluid Mech 313:181CrossRefGoogle Scholar
  35. 35.
    Strating P (1999) Phys Rev E 59:2175CrossRefGoogle Scholar
  36. 36.
    Doliwa B, Heuer A (2000) Phys Rev E 61:6898CrossRefGoogle Scholar
  37. 37.
    Purnomo EH, van den Ende D, Mellema J, Mugele F (2006) Europhys Lett 76:74CrossRefGoogle Scholar
  38. 38.
    Götze W (1991) In: Hansen JP, Levesque D, Zinn-Justin J (eds) Liquids, freezing and glass transition. Session LI of Les Houches summer schools of theoretical physics, North-Holland, Amsterdam, 287, 1989Google Scholar
  39. 39.
    Götze W (1999) J Phys Condens Matter 11:A1CrossRefGoogle Scholar
  40. 40.
    Miyazaki K, Reichman DR (2002) Phys Rev E 66:050501CrossRefGoogle Scholar
  41. 41.
    Miyazaki K, Reichman DR, Yamamoto R (2004) Phys Rev E 70:011501CrossRefGoogle Scholar
  42. 42.
    Kobelev V, Schweizer KS (2005) Phys Rev E 71:021401CrossRefGoogle Scholar
  43. 43.
    Fuchs M, Cates ME (2002) Phys Rev Lett 89:248304CrossRefGoogle Scholar
  44. 44.
    Fuchs M, Cates ME (2005) J Phys Condens Matter 17:S1681CrossRefGoogle Scholar
  45. 45.
    Fuchs M, Cates ME (2009) A mode coupling theory for Brownian particles in homogeneous steady shear flow. J Rheol 53:957–1000CrossRefGoogle Scholar
  46. 46.
    Dhont JKG (1996) An introduction to dynamics of colloids. Elsevier, AmsterdamGoogle Scholar
  47. 47.
    Risken H (1989) The Fokker–Planck equation. Springer, BerlinCrossRefGoogle Scholar
  48. 48.
    Dhont JKG, Briels W (2008) J Rheol Acta 47:257–281CrossRefGoogle Scholar
  49. 49.
    Bender J, Wagner NJ (1996) J Rheol 40:899CrossRefGoogle Scholar
  50. 50.
    Varnik F, Bocquet L, Barrat JL (2004) J Chem Phys 120:2788CrossRefGoogle Scholar
  51. 51.
    Ganapathy R, Sood AK (2006) Phys Rev Lett 96:108301CrossRefGoogle Scholar
  52. 52.
    Ballesta P, Besseling R, Isa L, Petekidis G, Poon WCK (2008) Phys Rev Lett 101:258301CrossRefGoogle Scholar
  53. 53.
    Van Kampen NG (2007) Stochastic processes in physics and chemistry. North Holland, AmsterdamGoogle Scholar
  54. 54.
    Forster D (1975) Hydrodynamic fluctuations, broken symmetry, and correlation functions. WA Benjamin, Reading, MAGoogle Scholar
  55. 55.
    Götze W, Sjögren L (1987) Z Phys B 65:415CrossRefGoogle Scholar
  56. 56.
    Schofield J, Oppenheim I (1992) Physica A 187:210CrossRefGoogle Scholar
  57. 57.
    Nägele G, Bergenholtz J (1998) J Chem Phys 108:9893CrossRefGoogle Scholar
  58. 58.
    Miyazaki K, Wyss HM, Reichman DR, Weitz DA (2006) Europhys Lett 75:915CrossRefGoogle Scholar
  59. 59.
    Brader JM, Voigtmann Th, Cates ME, Fuchs M (2007) Phys Rev Lett 98:058301CrossRefGoogle Scholar
  60. 60.
    Lionberger RA, Russel WB (1994) J Rheol 38:1885CrossRefGoogle Scholar
  61. 61.
    Onuki A, Kawasaki K (1979) Ann Phys (NY) 121:456CrossRefGoogle Scholar
  62. 62.
    Kawasaki K, Gunton JD (1973) Phys Rev A 8:2048CrossRefGoogle Scholar
  63. 63.
    Indrani AV, Ramaswamy S (1995) Phys Rev E 52:6492CrossRefGoogle Scholar
  64. 64.
    Bergenholtz J, Fuchs M (1999) Phys Rev E 59:5706CrossRefGoogle Scholar
  65. 65.
    Dawson K, Foffi G, Fuchs M, Gotze W, Sciortino F, Sperl M, Tartaglia P, Voigtmann T, Zaccarelli E (2001) Phys Rev E 63:011401CrossRefGoogle Scholar
  66. 66.
    Fabbian L, Götze W, Sciortino F, Tartaglia P, Thiery F (1999) Phys Rev E 59:R1347–R1350CrossRefGoogle Scholar
  67. 67.
    Sciortino F (2009) Nonlinear rheological properties of dense colloidal dispersions close to a glass transition under steady shear. Adv Polymer Sci. doi:10.1007/12_2009_30Google Scholar
  68. 68.
    Sciortino F (2003) Nat Mater 1:145–146CrossRefGoogle Scholar
  69. 69.
    Pham KN, Puertas AM, Bergenholtz J, Egelhaaf SU, Moussaid A, Pusey PN, Schofield AB, Cates ME, Fuchs M, Poon WCK (2002) Science 296:104–106CrossRefGoogle Scholar
  70. 70.
    Poon WCK, Pham KN, Egelhaaf SU, Pusey PN (2003) J Phys Cond Matt 15:S269–S275CrossRefGoogle Scholar
  71. 71.
    Fuchs M, Ballauff M (2005) Colloids Surf A 270/271:232Google Scholar
  72. 72.
    Franosch T, Fuchs M, Götze W, Mayr MR, Singh AP (1997) Phys Rev E 55:7153CrossRefGoogle Scholar
  73. 73.
    Fuchs M, Mayr MR (1999) Phys Rev E 60:5742CrossRefGoogle Scholar
  74. 74.
    Franosch T, Götze W, Mayr MR, Singh AP (1998) J Non-Cryst Solids 235/237:71Google Scholar
  75. 75.
    Verberg R, de Schepper IM, Feigenbaum MJ, Cohen EGD (1997) J Stat Phys 87:1037CrossRefGoogle Scholar
  76. 76.
    Henrich O, Pfeifroth O, Fuchs M (2007) J Phys Condens Matter 19:205132CrossRefGoogle Scholar
  77. 77.
    Szamel G (2001) J Chem Phys 114:8708CrossRefGoogle Scholar
  78. 78.
    Johnson SJ, de Kruif CG, May RP (1988) J Chem Phys 89:5909CrossRefGoogle Scholar
  79. 79.
    Lionberger RA, Russel WB (2000) Adv Chem Phys 111:399CrossRefGoogle Scholar
  80. 80.
    Fuchs M, Cates ME (2003) Faraday Discuss 123:267CrossRefGoogle Scholar
  81. 81.
    Varnik F, Henrich O (2006) Phys Rev B 73:174209CrossRefGoogle Scholar
  82. 82.
    Fuchs M, Götze W, Hofacker I, Latz A (1991) J Phys Condens Matter 3:5047–5071CrossRefGoogle Scholar
  83. 83.
    Zausch J, Horbach J, Laurati M, Egelhaaf SU, Brader JM, Voigtmann Th, Fuchs M (2008) J Phys Condens Matt 20:404210CrossRefGoogle Scholar
  84. 84.
    Angelani L, Leonardo RD, Ruocco G, Scala A, Sciortino F (2000) Phys Rev Lett 85:5356CrossRefGoogle Scholar
  85. 85.
    Broderix K, Bhattachrya KK, Cavagna A, Zippelius Z (2000) Phys Rev Lett 85:5360CrossRefGoogle Scholar
  86. 86.
    Hajnal D, Fuchs M (2009) Eur Phys J E, in print. doi:10.1140/epje/i2008-10361-0; also at arXiv:0807.1288Google Scholar
  87. 87.
    Götze W (1984) Z Phys B 56:139CrossRefGoogle Scholar
  88. 88.
    Fuchs M, Götze W, Hildebrand S, Latz A (1992) J Phys Condens Matter 4:7709CrossRefGoogle Scholar
  89. 89.
    Saltzman EJ, Yatsenko G, Schweizer KS (2008) J Phys Condens Matter 20:244129CrossRefGoogle Scholar
  90. 90.
    Pusey PN (1978) J Phys A 11:119CrossRefGoogle Scholar
  91. 91.
    Varnik F (2006) J Chem Phys 125:164514CrossRefGoogle Scholar
  92. 92.
    Henrich O, Varnik F, Fuchs M (2005) J Phys Condens Matter 17:S3625CrossRefGoogle Scholar
  93. 93.
    Götze W, Voigtmann Th (2003) Phys Rev E 67:021502CrossRefGoogle Scholar
  94. 94.
    Foffi G, Götze W, Sciortino F, Tartaglia P, Voigtmann Th (2003) Phys Rev Lett 91:085701CrossRefGoogle Scholar
  95. 95.
    Kob W, Andersen HC (1995) Phys Rev E 51:4626; 52:4134Google Scholar
  96. 96.
    Gleim T, Kob W, Binder K (1998) Phys Rev Lett 81:4404CrossRefGoogle Scholar
  97. 97.
    Flenner E, Szamel G (2005) Phys Rev E 72:011205CrossRefGoogle Scholar
  98. 98.
    Brader JM, Cates ME, Fuchs M (2008) Phys Rev Lett 101:138301CrossRefGoogle Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Fachbereich PhysikUniversität KonstanzKonstanzGermany

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