Rheologica Acta

, Volume 48, Issue 7, pp 747–753 | Cite as

An empirical constitutive law for concentrated colloidal suspensions in the approach of the glass transition

  • H. Henning Winter
  • Miriam Siebenbürger
  • David Hajnal
  • Oliver Henrich
  • Matthias Fuchs
  • Matthias Ballauff
Original Contribution


Concentrated, non-crystallizing colloidal suspensions in their approach of the glass state exhibit distinct dynamics patterns. These patterns suggest a powerlaw rheological constitutive model for near-glass viscoelasticity, as presented here. The rheological parameters used for this model originate in the mode-coupling theory. The proposed constitutive model provides explicit expressions for the steady shear viscosity, the steady normal stress coefficient, the modulus-compliance relation, and the α peak of G″. The relaxation pattern distinctly differs from gelation.


Glass transition Mode-coupling theory Colloidal glass BSW spectrum Linear viscoelasticity Near-glass dynamics 



HHW acknowledges NSF support (CBET-0651888) and the 2007 Summer School at the Aspen Center for Physics. We also thank the Deutsche Forschungsgemeinschaft, Forschergruppe 608 “Nichtlineare Dynamik,” Bayreuth, and IRTG 667 “Soft Condensed Matter Physics” at Konstanz, for financial support.


  1. Abdel-Goad M, Pyckhout-Hintzen W, Kahle S, Allgaier J, Richter D, Fetters LJ (2004) Rheological properties of 1,4-polyisoprene over a large molecular weight range. Macromolecules 37:8135–8144CrossRefADSGoogle Scholar
  2. Baumgärtel M, Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28:511–519CrossRefGoogle Scholar
  3. Baumgärtel M, Schausberger A, Winter HH (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 29:400–408CrossRefGoogle Scholar
  4. Baumgärtel M, DeRosa ME, Machado J, Masse M, Winter HH (1992) The relaxation time spectrum of nearly monodisperse polybutadiene melts. Rheol Acta 31:75–82CrossRefGoogle Scholar
  5. Bengtzelius U, Götze W, Sjölander A (1984) Dynamics of supercooled liquids and the glass transition. J Phys C 17:5915CrossRefADSGoogle Scholar
  6. Boltzmann L (1874) Zur Theorie der elastischen Nachwirkungen. Sitzungsber kaiserlich Akad Wissen Math Naturwissen 70:275–306Google Scholar
  7. Brambilla G, El Masri D, Pierno M, Berthier L, Cipelletti L, Petekidis G, Schofield AB (2009) Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition. Phys Rev Let 102:085703CrossRefADSGoogle Scholar
  8. Carri G, Winter HH (1997) Mapping of the relaxation patterns of polymer melts with linear flexible molecules of uniform length. Rheol Acta 36:330–344Google Scholar
  9. Chambon F, Winter HH (1985) Stopping of crosslinking reaction in a PDMS polymer at the gel point. Polym Bull 13:499–503CrossRefGoogle Scholar
  10. Chambon F, Winter HH (1987) Linear viscoelasticity at the gel point of a crosslinking PDMS with imbalanced stoichiometry. J Rheol 31:683–697CrossRefADSGoogle Scholar
  11. Crassous JJ, Siebenbürger M, Ballauff M, Drechsler M, Hajnal D, Henrich O, Fuchs M (2006a) Thermosensitive core-shell particles as model systems for studying the flow behavior of concentrated colloidal dispersions. J Chem Phys 125:204906PubMedCrossRefADSGoogle Scholar
  12. Crassous JJ, Ballauff M, Drechsler M, Schmidt J, Talmon Y (2006b) Imaging the volume transition in thermosensitive core-shell particles by cryo-transmission electron microscopy. Langmuir 22:2403PubMedCrossRefGoogle Scholar
  13. Crassous JJ, Siebenbürger M, Ballauff M, Drechsler M, Hajnal D, Henrich O, Fuchs M (2008a) Shear stresses of colloidal dispersions at the glass transition in equilibrium and in flow. J Chem Phys 128:204902PubMedCrossRefADSGoogle Scholar
  14. Crassous JJ, Wittemann A, Siebenbürger M, Schrinner M, Drechsler M, Ballauff M (2008b) Direct imaging of temperature-sensitive core-shell latexes by cryogenic transmission electron microscopy. Colloid Polym Sci 286:805CrossRefGoogle Scholar
  15. Doi M (1981) Explanation for the 3.4 power law of viscosity of polymeric liquids on the basis of the tube model. J Polym Sci Polym Lett Ed 19:265CrossRefGoogle Scholar
  16. Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, New YorkGoogle Scholar
  17. Franosch T, Götze W (1999) Relaxation rate distributions for supercooled liquids. J Phys Chem B 103:4011CrossRefGoogle Scholar
  18. Friedrich C, Waizenegger F, Winter HH (2007) Relaxation patterns of long, linear, flexible, monodisperse polymers: BSW spectrum revisited. Rheol Acta 47(8):909–916CrossRefGoogle Scholar
  19. Fuchs M, Cates ME (2002) Theory of nonlinear rheology and yielding of dense colloidal suspensions. Phys Rev Lett 89:248304PubMedCrossRefADSGoogle Scholar
  20. Fuchs M, Hofacker I, Latz A (1992) Primary relaxation in a hard-sphere system. Phys Rev A 45:898PubMedCrossRefADSGoogle Scholar
  21. Götze W (1991) Liquids, freezing and glass transition. In: Hansen JP, Levesque D, Zinn-Justin J (eds) Session LI of Les Houches summer schools of theoretical physics. North-Holland, Amsterdam, pp 287Google Scholar
  22. Götze W, Sjögren L (1992) Relaxation processes in supercooled liquids. Rep Prog Phys 55:241CrossRefGoogle Scholar
  23. Koumakis N, Schofield AB, Petekidis G (2008) Effects of shear induced crystallization on the rheology and ageing of hard sphere glasses. Soft Matter 4:2008–2018CrossRefGoogle Scholar
  24. Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, OxfordGoogle Scholar
  25. Likhtman A, McLeish TCB (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332CrossRefADSGoogle Scholar
  26. Mason TG, Weitz DA (1995) Linear viscoelasticity of hard sphere suspensions near the glass transition. Phys Rev Lett 75:2770–2773PubMedCrossRefADSGoogle Scholar
  27. Mayer P, Miyazaki K, Reichman DR (2006) Cooperativity beyond caging: generalized mode coupling rheory. Phys Rev Lett 97:095702CrossRefADSGoogle Scholar
  28. Milner ST, McLeish TCB (1998) Reptation and contour-length fluctuations in melts of linear polymers. Phys Rev Lett 81:725CrossRefADSGoogle Scholar
  29. Schweizer KS (2007) Dynamical fluctuation effects in glassy colloidal suspensions. Curr Opin Colloid Interface Sci 12:297CrossRefGoogle Scholar
  30. Schweizer KS, Saltzman EJ (2003) Entropic barriers, activated hopping, and the glass transition in colloidal suspensions. J Chem Phys 119:1181–1196CrossRefADSGoogle Scholar
  31. Siebenbürger M, Fuchs M, Winter HH, Ballauff M (2009) Viscoelasticity and shear flow of concentrated, non-crystallizing colloidal suspensions: comparison with mode-coupling theory. J Rheol 53:707–726CrossRefADSGoogle Scholar
  32. van Megen W, Underwood SM (1994) Glass-transition in colloidal hard spheres—measurement and modecoupling-theory analysis of the coherent intermediate scattering function. Phys Rev E 49:4206–4220CrossRefADSGoogle Scholar
  33. van Megen W, Underwood SM, Pusey PN (1991) Non-ergodicity parameters of colloidal glasses. Phys Rev Lett 67:1586–1589PubMedCrossRefADSGoogle Scholar
  34. von Schweidler E (1907) Studien über die Anomalien im Verhalten der Dielektrika. Ann Phys (Leipzig) 24:711–770ADSGoogle Scholar
  35. Winter HH (1987) Evolution of rheology during chemical gelation. Prog Colloid Polym Sci 75:104–110CrossRefGoogle Scholar
  36. Winter HH, Chambon F (1986a) Analysis of linear viscoelasticity of a crosslinking polymer at the gel point. J Rheology 30:367–382CrossRefADSGoogle Scholar
  37. Winter HH, Chambon F (1986b) Rheology of crosslinking polymers at the gel point. Proc Bi-annual Meeting Polymer Networks Group, ElsinoreGoogle Scholar
  38. Winter HH, Mours M (1997) Rheology of polymers near their liquid–solid transitions. Adv Polym Sci 134:165–234CrossRefGoogle Scholar
  39. Winter HH, Mours M (2006) The cyber infrastructure initiative for rheology. Rheol Acta 45:331–338CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • H. Henning Winter
    • 1
  • Miriam Siebenbürger
    • 2
  • David Hajnal
    • 3
  • Oliver Henrich
    • 3
  • Matthias Fuchs
    • 3
  • Matthias Ballauff
    • 2
  1. 1.Chemical EngineeringUniversity of MassachusettsAmherstUSA
  2. 2.Physikalische Chemie IUniversität BayreuthBayreuthGermany
  3. 3.Fachbereich PhysikUniversität KonstanzKonstanzGermany

Personalised recommendations