Concentrated colloidal suspensions and emulsions are amorphous soft solids, widespread in technological and industrial applications and studied as model systems in physics and materials sciences. They are easily fluidized by applying mechanical stress, undergoing a yielding transition that still lacks a unified description. Here we investigate yielding in three classes of repulsive soft solids and find that at the microscopic level, yielding consists of a transition between two distinct dynamical states. We rationalize this by proposing a lattice model with dynamical coupling between neighbouring sites, leading to a unified state diagram for yielding. Employing the analogy with van der Waals phase diagram for real gases, we show that distance from a critical point plays a major role in the emergence of first-order-like versus second-order-like features in yielding, thereby reconciling previously contrasting observations on the nature of the transition.

Concentrated colloidal suspensions and emulsions are amorphous soft solids, widespread in technological and industrial applications and studied as model systems in physics and materials sciences. They are easily fluidized by applying mechanical stress, undergoing a yielding transition that still lacks a unified description. Here we investigate yielding in three classes of repulsive soft solids and find that at the microscopic level, yielding consists of a transition between two distinct dynamical states. We rationalize this by proposing a lattice model with dynamical coupling between neighbouring sites, leading to a unified state diagram for yielding. Employing the analogy with van der Waals phase diagram for real gases, we show that distance from a critical point plays a major role in the emergence of first-order-like versus second-order-like features in yielding, thereby reconciling previously contrasting observations on the nature of the transition.

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.