Oscillatory Rheology Measuring the Viscoelastic Behaviour of Soft Materials Soft materials such as emulsions, foams, or dispersions are ubiquitous in industrial products and formulations; they exhibit unique mechanical behaviours that are often key to the way these materials are employed in a particular application. Studying the mechanical behaviour of these materials is complicated by the fact that their response is viscoelastic, intermediate between that of solids and liquids. Oscillatory rheology is a standard experimental tool for studying such behaviour; it provides new insights about the physical mechanisms that govern the unique mechanical properties of soft materials. Photo: 108 (photocase) Soft materials such as colloidal suspensions, Measuring this time dependent stress re- emulsions, foams, or polymer systems are ubiq- sponse at a single frequency immediately reveals uitous in many industries, including foods, phar- key differences between materials, as shown maceuticals, and cosmetics. Their macroscopic schematically in figure 1(b). If the material is an mechanical behaviour is a key property which ideal elastic solid, then the sample stress is pro- often determines the usability of such materials portional to the strain deformation, and the pro- for a given industrial application. Characterising portionality constant is the shear modulus of the the mechanical behaviour of soft materials is material. The stress is always exactly in phase complicated by the fact that many materials are with the applied sinusoidal strain deformation. viscoelastic, so their mechanical properties lie In contrast, if the material is a purely viscous between that of a purely elastic solid and that of fluid, the stress in the sample is proportional to Prof. David Weitz, Dr. Hans Wyss, Ryan Larsen, a viscous liquid. Using oscillatory rheology, it is the rate of strain deformation, where the pro- Harvard University possible to quantify both the viscous-like and the elastic-like properties of a material at differ- ent time scales; it is thus a valuable tool for un- derstanding the structural and dynamic proper- ties of these systems [1, 2]. Oscillatory Rheology The basic principle of an oscillatory rheometer is to induce a sinusoidal shear deformation in the sample and measure the resultant stress re- sponse; the time scale probed is determined by the frequency of oscillation, ω, of the shear de- formation. In a typical experiment, the sample is placed between two plates, as shown in figure 1(a). While the top plate remains stationary, a motor rotates the bottom plate, thereby impos- ing a time dependent strain γ(t)=γ ·sin(ωt) on the sample. Simultaneously, the time dependent Fig. 1: (a) Schematic representation of a typical rheometry setup, with the sample placed between stress σ (t) is quantified by measuring the torque two plates. (b) Schematic stress response to oscillatory strain deformation for an elastic solid, a that the sample imposes on the top plate. viscous fluid and a viscoelastic material. G.I.T. Laboratory Journal 3-4/2007, pp 68-70, GIT VERLAG GmbH & Co. KG, Darmstadt www.gitverlag.com www.PRO-4-PRO.com Fig. 2: Frequency dependence of G’ and G’’ for (a) a suspension of hydrogel particles and (b) for the elastomer blend DC-9040 (Dow Corning), a typical additive in cosmetic and pharmaceutical formulations. Further measurements of DC-9040: (c) Strain dependence of G’ and G’’, (d) Constant-rate measurements at strain rates 0.002 s-1, 0.02 s-1, 0.45 s-1 and 10 s-1, (e) master curve of constant-rate measurements that were scaled both in magnitude and frequency. portionality constant is the viscosity of the fluid. the applied strain deformations are sufficiently The applied strain and the measured stress are small so as not to affect the material properties. out of phase, with a phase angle δ=π/2, as Many soft materials show a strikingly nonlinear shown in the center graph in figure 1(b). stress response as the strain deformation is in- Viscoelastic materials show a response that creased; this behaviour suggests a fundamental contains both in-phase and out-of-phase contri- change of the material properties under shear butions, as shown in the bottom graph of figure from the equilibrium properties at rest. However, 1(b); these contributions reveal the extents of such nonlinear data are often disregarded be- solid-like (red line) and liquid-like (blue dotted cause the physical origins of this behaviour re- line) behavior. As a consequence, the total stress main poorly understood. response (purple line) shows a phase shift δ with respect to the applied strain deformation New Developments in Nonlinear that lies between that of solids and liquids, Rheology 0<δ<π/2. The viscoelastic behaviour of the sys- tem at ω is characterised by the storage modu- Nonlinear viscoelastic measurements have the lus, G’(ω), and the loss modulus, G’’(ω), which potential of revealing new information about respectively characterise the solid-like and fluid- the behavior of soft materials. Recent studies [4, like contributions to the measured stress re- 5] suggest that such nonlinear data provides sponse. For a sinusoidal strain deformation valuable information about the properties of a γ (t)=γ 0 sin(ωt), the stress response of a visco- wide range of soft materials, called soft glassy elastic material is given by σ(t)=G’(ω)γ 0sin(ωt)+ materials [6]. The underlying physical picture G’’(ω)γ0 cos(ωt). suggests an intrinsic link between the slow In a typical rheological experiment, we seek structural relaxation of such glassy systems and to measure G’(ω) and G’’(ω). We make the mea- their linear and nonlinear viscoelastic response. surements as a function of omega because The linear viscoelastic response of soft glassy whether a soft material is solid-like or liquid-like materials is often rather featureless within the depends on the time scale at which it is de- accessible range of frequencies, as shown in the formed. A typical example is shown in figure 2(a), example in figure 2(c); the storage modulus is where we plot G’(ω) and G’’(ω) for a suspension typically larger than the loss modulus and de- of hydrogel particles [3]; at the lowest accessible pends only weakly on frequency. However, the frequencies the response is viscous-like, with a typical nonlinear response shows more pro- loss modulus that is much larger than the stor- nounced features: A robust hallmark of these age modulus while at the highest frequencies ac- systems is that as the strain increases at fixed ω, cessed the storage modulus dominates the re- the loss modulus initially increases to a peak be- sponse, indicating solid-like behaviour. fore ultimately decreasing at the largest strains, as shown in the example in figure 2(c). This un- Linear and Nonlinear Rheology usual behaviour can be explained as a natural consequence of a slow structural relaxation pro- It is important to recognise that the frequency- cess that is expected in these materials at very dependent moduli G’ and G’’ probe the behav- low frequencies [6, 7], as shown schematically iour of a material in an undisturbed state, where in the inset of figure 2(c). Importantly, the time R H EOLO G Y scale of this relaxation is expected to depend on The behaviour observed suggests that valuable the strain rate of deformation; the relaxation information about the structural relaxation of speeds up if the system is subjected to shear [4, soft materials can be accessed from such nonlin- 8–10]. As the strain amplitude of deformation is ear viscoelastic measurements, even if the relax- increased in a strain-sweep measurement, the ation occurs at time scales that are not accessi- strain rate increases and the characteristic fre- ble to linear oscillatory rheology. quency of the slow relaxation process moves to- wards higher frequencies. As a consequence, a Conclusions peak in the loss modulus is observed at the point where this characteristic frequency becomes Oscillatory rheology is a valuable tool for study- comparable to the oscillation frequency of the ing the mechanical behavior of soft materials. strain deformation, ω. Thus, the observed behav- Recent studies suggest that the nonlinear visco- iour at large strains directly reflects the structur- elastic behavior contains valuable information al relaxation of the material. about the dynamics of these systems. Thus mea- This behaviour is exploited in a new ap- surements at large strain deformations should proach to oscillatory rheology, Strain-Rate Fre- lead to a better understanding of the physical quency Superposition (SRFS) [5], where the am- mechanisms that govern their behaviour. plitude of strain rate is kept constant as ω is varied. Examples of such measurements are References shown in figure 2(d) for different strain rate am- [1] W. Macosko: Rheology: Principles, Measurements plitudes of shear deformation. At all strain rates, and Applications, Wiley-VCH, New York, 1994 the frequency dependence of the response [2] Larson R.G.: The structure and rheology of com- shows the behaviour expected for a slow struc- plex fluids, Oxford University Press, New York, tural relaxation process, with a pronounced peak 1999 in the loss modulus. The data show that with in- [3] Hu Z. and Xia X.: Adv. Materials 16, 305 (2004) creasing strain rate amplitude the response [4] Miyazaki H.M. et al.: Europhys. Lett. 75, 915 moves towards higher frequencies; however, the (2006) general shape of the response is surprisingly in- [5] Wyss H.M. et al.: cond–mat/0608151 (2006) sensitive to strain rate, as shown in figure 2(e). [6] Sollich P. et al.: Phys. Rev. Lett. 78, 2020 (1997) By scaling the frequency and the magnitude of [7] Cates M.E. and Sollich P.: J. Rheology 48, 193 each data set, we collapse the data recorded at (2004) different strain rates onto a master curve with [8] Fuchs M.