THE MICROSCOPIC SIGNATURE OF THE RHEOLOGY OF AMPORPHOUS SOFT MATERIALS

The context In this thesis, we investigated the relation between the mechanical properties of a material and the state of its microscopic constituents, with a particular focus on the yielding transition in soft materials. Yielding represents one of the key modes of mechanical failure and characterizes the mechanical response of a wide class of materials, from metals to polymers, from biological tissues to emulsions. Despite being a complex phenomenon, encompassing a wide range of length- and time-scales [6], [9], [3], yielding is characterized by an associated phenomenology that displays a number of universal features, that are largely material-independent [8], [2]. It is this universality that makes soft materials very convenient model systems for understanding yielding: on one hand, they usually display a structural organization at length scales that can be probed by means of optical techniques; on the other hand, as they are "soft", it is very easy to obtain large deformations or induce yielding even upon the application of moderate forces. Experimentally, the greatest challenge of a multi-scale (from micro to macro) study of yielding in soft materials lies in imposing a macroscopic deformation on a soft sample while monitoring the microscopic degrees of freedom. Characterization of the microscopic states through far field scattering techniques suffers from lack of spatial resolution, as the output of these experiments typically represent a spatial average performed over a large (and possibly heterogeneous) volume. At the same time, imaging techniques usually suffer from poor statistics, since usually a better spatial resolution comes at the expense of the extension of the field of view. The possibility to combine a good spatial resolution and a statistically robust characterization is crucial in the study of a complex phenomenon, that has an intrinsic stochastic nature, a considerable sample to sample variability and a strong sensitivity to spatial heterogeneity. Our methodological contribution Here we have developed an integrated approach (rheology protocol and image analysis) to study the microscopic dynamics of soft materials, both at rest and under oscillatory shear strain. All the techniques and methods that we have refined provide a quantitative characterization of the statics (e.g. spatial arrangement and positional correlation,...) and dynamics (e.g. relaxation of density modes, determination of particle trajectories, dynamical activity,...) of the soft material under scrutiny. Most of the proposed tools pivot around Differential Dynamic Microscopy that combines the experimental setup of optical microscopy with an image processing aimed at extracting scattering-like information about the sample at different wave vectors. Two basic methodological advances obtained in this work, which were succesfully applied to samples at rest and under shear, include Image Windowing - A pre-processing step for DDM analysis aimed at removing artifacts due to the image finite size. This tool revealed to be essential to extract the correct dynamics over a larger wave-vector range, as detailed in the associated peer-reviewed article [5]. DDM-rheology - Microrheology is a family of techniques aimed at characterizing the linear mechanical properties of a material via the measurement of the MSD of embedded micrometric tracers . The most diffuse route to microrheology is particle tracking, that requires to individually identify the tracers and their trajectories. We have shown how DDM can be used to obtain the same information, under a wide range imaging conditions. More details are available also in the published peer-reviewed article [4]. To study the dynamics under shear, we used a large amplitude oscillatory shear (LAOS) approach, applied to simple yield stress fluids. This choice was motivated by the fact that in LAOS experiments, simple yield stress fluids reach a stationary condition even in the non-linear regime. The so-obtained stationary state allows for a statistically significant characterization of the microscopic state. To achieve this goal we have acted along two main lines: we have developed an echo-DDM approach to monitor and quantify the shear-induced dynamics at different wave-vectors: images of the sample are acquired with a frame rate equal to the deformation rate, in order to be sensible only to irreversible plastic rearrangements. we have monitored intermittency and heterogeneity of the shear-induced dynamics by computing activity maps (in direct space) and q-resolved dynamic susceptibility chi 4(q;dt) (in reciprocal space) to go beyond average quantities like the intermediate scattering function or the particle MSD. Results on the physics of yielding LAOS experiments with the simple yield stress fluid (Carbopol 971 P NF) provided interesting results on the physics behind the yielding transition. Through the study of Lissajous plots we have shown that in the yielding regime the mechanical response of the material is non-linear but yet harmonic, that is, accurately described in terms of phase-independent viscoelastic moduli G'' (loss modulus) and G' (storage modulus), respectively. This property allows to treat the sample in the non-linear regime as an "effective viscoelastic material", with distinct mechanical properties from the sample at rest. In other words, it should be conceptually feasible to study the linear response of the shared material to small harmonic perturbations superimposed on the LAOS, obtaining an effective frequency-dependent dynamic modulus. According to basic concepts of linear rheology, whether this effective dynamic modulus presents or not a relaxation time provides a criterion to discriminate a solid-like to a liquid-like behaviour [7]. In this context, with the additional hypothesis that the storage and loss moduli crosses each other once (if ever), we can thus interpret the crossover point where G'' = G' (occurring for c:o: = 66%, and marked by a red circle in Fig. 1.1) as the point at which the relaxation time becomes equal to the shearing period. As a consequence, under the hypothesis that the relaxation time monotonically decreases with the shear amplitude, we expect the yielding point to be located at gamma crossover. As our experimental setup did not allow a direct measurement of the linear response of the sheared material, we probed the state of the shared material via microscopy observations of the spontaneous dynamics of embedded tracer particles. EchoDDM measurements highlighted the presence of plastic dynamics: in the high-q limit we found a diffusive scaling of the relaxation rate with the wavevector, with an effective diffusion coefficient which is strongly dependent on the shear amplitude (Fig. 1.1 inset top right corner). Experiments with tracers of different sizes show that the shear-induced diffusion coefficient is inversely proportional to the particle radius, in analogy with the Stokes-Einstein relation. The validity of this scaling law is intriguing and opens to the possibility of pushing even further the analogy between a yielding material and viscoelastic material at rest. Because of limits imposed by our setup on the duration of the experiments, we are not always able to unambiguously discriminate between free and constrained diffusion, so that the presence of a diffusive dynamics on small spatial length scales should not be necessarily interpreted as a proxy for the "fluidization" of the sample. We could instead consider "fluidized" a sample where the effective diffusion coefficient D is such to allow a particle with the typical size of a microgel to displace by an amount comparable with its own size within the duration of the experiment. This operative criterion defines a threshold DT for the effective diffusion coefficient (marked by the red line in Fig. 1.1 inset top right corner). By applying this criterion we obtain gamma yielding 45%, which is within the nonlinear regime identified in LAOS experiments, but below the crossover point, consistently with the picture described before. Perspectives and open questions on yielding. The approach proposed in this work enables one overcoming some of the key experimental limitations in the study of soft yield-stress materials. For this reason it should be applied to other systems to understand the generality of the observed behaviour and possibly new aspects of the shear-induced dynamics. One of the open issues remains the exact location of the yielding threshold. Our experiments provides an upper boundary, but further investigation and longer measurements are needed to increase the precision in locating the transition point. Also, the comparison with ad hoc mechanical tests aimed at measuring the linear response to small perturbation in the non-linear regime1 will be key to validate the consistency of the proposed scenario and the identified yielding criterion. We suggest that the proposed scenario for the yielding transition in simple YSF under LAOS forcing could provide an ideal starting point also for understanding more complex non-linear and non-stationary phenomena linked to yielding. For this reason the very next steps for the follow-up to this work is the application of this method to other simple YSF. This work includes also a first body of preliminary results on a different sample: a depletion gel. In particular, we showed that we can impose a controlled homogeneous deformation profile. Unfortunately, EchoDDM experiments on this sample didn’t provide fully satisfactory results. In particular, the genuine dynamic signal originated from shear-induced plastic rearrangement in the sample was almost completely buried by the spurious effects (large-scale drifts or flows) due to the unstable confinement of the sample. Beside optimizing the experimental set-up, a possible strategy to improve the quality of the optical signal could be to study gels formed by larger colloidal particles. This should increase the "effective resolution" of the experiment. In fact, we expect the amplitude of shear-induced displacements to scale with the particle size, which is the only relevant length-scale in the system. Methodological perspectives The self-consistent approach developed for the DDM-microrheology is applicable also outside of the context of microrheology: it is a promising tool for a more accurate characterization of the colloids dynamics. Given the close analogy that we have drawn between sheared samples and viscoelastic materials, valid both at the macroscopic (as shown by Lissajous plots), and at microscopic level (as suggested by the validity of the scaling of the diffusion coefficient with the particle size) we believe that the application of microrheological methods to sheared samples could provide interesting information. In particular, we think that DDM-microrheology could be the ideal tool to extract the MSD of tracers embedded in a sheared sample For example the so-called superposition rheology [10]. and to extract via a suitable GSER an effective dynamic modulus. Another interesting development is to apply the DDM analysis to characterize the in-cycle dynamics. This is expected to be particularly interesting for intermediate shear amplitudes, i.e. in the non-linear regime, below the yielding point. Methods to decouple affine and non affine displacements have been recently proposed [1], but, to the best of our knowledge, they have not been applied yet to yielding systems.