DOI 10.1007/s00397-008-0306-z ORIGINAL CONTRIBUTION A new interpretation for the dynamic behaviour of complex fluids at the sol–gel transition using the fractional calculus Stéphane Warlus · Alain Ponton Received: 7 April 2008 / Accepted: 11 August 2008 / © Springer-Verlag 2008 Abstract We propose to analyse power law shear stress relaxation modulus observed at the sol–gel transition (SGT) in many gelling systems in terms of fractional calculus. We show that the critical gel (gel at SGT) can be associated to a single fractional element and the gel in the post-SGT state to a fractional Kelvin– Voigt model. In this case, it is possible to give a physical interpretation to the fractional derivative order. It is associated to the power law exponent of the shear modulus related to the fractal dimension of the critical gel. A preliminary experimental application to silica alkoxide-based systems is given. Keywords Stress relaxation · Small amplitude oscillatory shear · Fractional calculus · Gel · Power law Introduction In recent years, much attention has been focused on the relationship between structure and dynamic prop- erties of complex materials. In particular, relaxation processes in disordered systems have attracted much attention because the observed relaxation present some general characteristics which are independent of the studied materials and deviate from an exponential re- laxation behaviour. S. Warlus · A. Ponton (B ) Laboratoire Matière et Systèmes Complexes UMR 7057, Université Paris Diderot-Paris 7, Bâtiment Condorcet CC 7056 10 rue Alice Domon et Léonie Duquet, Paris Cedex 13, 75205, France e-mail: alain.ponton@univ-paris-diderot.fr A shear stress power law relaxation with non-entire exponent over a wide time window has been ob- served by rheological investigations in critical gel near the sol–gel transition (SGT; Winter and Chambon 1986; Chambon and Winter 1987; Mours and Winter 1996), gelling Laponite clay suspensions (Cocard et al. 2000), micro-gels (English et al. 1999), polymers melts (Larson 1985), bio-polymer gel network (Rodd et al. 2001) and cross-linked actin networks (Tempel et al. 1996). Dynamic light scattering (DLS) in gelling solutions of polymers (Martin and Wilcoxon 1988; Martin et al. 1991) and X-ray scattering in colloidal aggregates (Schaefer and Keefer 1984), measurements also, show a power law relaxation of the dynamic struc- ture factor. Diffusing wave spectroscopy was used to measure a power law frequency dependence in con- centrated dispersed suspensions (Mason et al. 1997), in actin filament networks (Palmer et al. 1999) and critical power law behaviour in sol–gel transition of concentrated colloidal suspensions (Romer et al. 2000). Power law behaviours are also observed at var- ious levels of organisation in biological phenomena with different techniques: micro-rheology in living cells by magnetic twisting cytometry (Puig-de-Morales et al. 2001; Fabry et al. 2001), magnetic bead micro- rheometry (Bausch et al. 1998) and atomic force mi- croscopy (Alcaraz et al. 2003). This frequency (or time) dependence of dynamic properties is difficult to de- scribe with classical viscoelasticity. The behaviour of viscoelastic materials is intermediate between classi- cal Hookean solids and Newtonian fluids. They show damping (or energy dissipation) properties for which the actual stress depend both on the actual strain and the entire strain history. Viscoelastic materials are said to possess memory. Classical viscoelastic models based Rheol Acta (2009) 48:51–58 Published online: 1 ber 2008 1 Septem