Understanding the microscopic origin of the rheological behavior of soft matter is a long-lastingendeavour. While early efforts concentrated mainly on the relationship between rheology and structure,current research focuses on the role of microscopic dynamics. We present in two companion papers athorough discussion of how Fourier space-based methods may be coupled to rheology to shed light onthe relationship between the microscopic dynamics and the mechanical response of soft systems. In thisfirst companion paper, we report a theoretical, numerical and experimental investigation of dynamiclight scattering coupled to rheology. While in ideal solids and simple viscous fluids the displacement fieldunder a shear deformation is purely affine, additional non-affine displacements arise in many situationsof great interest, for example in elastically heterogeneous materials or due to plastic rearrangements.We show how affine and non-affine displacements can be separately resolved by dynamic lightscattering, and discuss in detail the effect of several non-idealities in typical experiments.

We discuss in two companion papers how Fourier-space measurements may be coupled to rheologicaltests in order to elucidate the relationship between mechanical properties and microscopic dynamics insoft matter. In this second companion paper, we focus on Differential Dynamic Microscopy (DDM)under shear. We highlight the analogies and the differences with dynamic light scattering coupled torheology, providing a theoretical approach and practical guidelines to separate the contributions toDDM arising from the affine and the non-affine part of the microscopic displacement field. We showthat in DDM under shear the coherence of the illuminating source plays a key role, determining theeffective sample thickness that is probed. Our theoretical analysis is validated by experiments on 2Dsamples and 3D gels.

By combining rheology, light scattering, small angle X-ray scattering and simulations we explore the glassy dynamics of soft colloids using microgels and charged particles interacting by steric and screened Coulomb interactions, respectively. In the supercooled regime, the structural relaxation time (SRT) of both systems grows steeply with volume fraction, reminiscent of the behavior of colloidal hard sphere systems [1]. This suggests that softness has no impact on the growth of the SRT on approaching the glass transition, as confirmed by computer simulations. We show that the onset of a finite yield-stress, occurring within the supercooled regime, systematically corresponds to a minimum of the stretching exponent characterizing the decay of the two-time intensity autocorrelation function.Softness becomes relevant only at very large packing fractions, when the system falls out of equilibrium. In this non-equilibrium regime, the SRT depends surprisingly weakly on packing fraction and time correlation functions exhibit compressed exponential decays consistent with internal stress-driven relaxation. The transition to this novel regime coincides with the onset of an anomalous decrease of local order with increasing density, reminiscent of the reentrant glass transition predicted theoretically in ultrasoft systems [2,3], and corresponds to a weakening of the volume fraction dependence of the plateau modulus of the suspensions. We propose that these peculiar dynamics result from a competition between the non-equilibrium aging dynamics of the glassy state and the tendency of soft systems to refluidize at high packing fractions.

In our work we report the measurement of non-equilibrium interfacial tension of polymer and hard sphere suspensions in contact with their own solvent. By visualizing fingering instability (VF) in radial Hele-Shaw geometry, appearing when the solvent displaces suspensions of colloids or polymers, we measure interfacial tensions in function of the volume fraction of the suspended objects, showing that the internal degrees of freedom of the particles drive the low volume fraction behavior (Figure 1). Our results support the existence of a positive tension between miscible fluids, confirm the quadratic scaling predicted by Korteweg [4] for long linear and crosslinked polymers and show a positive rapidly growing tension for hard sphere suspensions up to maximum packing, whose description necessitates a theoretical framework going beyond the classic square gradient model. We rationalize our findings assuming the suspension/solvent interface in local thermodynamic equilibrium, computing explicitly the square gradient contribution to the interfacial tension for polymer/solvent and simple molecular liquid mixtures and proposing a phenomenological model capturing the compositional dependence of the interfacial tension for large concentration gradients. Finally we include and analyze data reported in literature and obtained via spinning drop tensiometry that validate the model and we propose the analysis of fluid dynamic instability as a new tool to probe interfacial stresses.