Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions.

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

Drying Drops of Paint Suspension: From "Fried Eggs" to Quasi-Homogeneous Patterns.

Drying of multicomponent sessile drops is a complex phenomenon involving intricate mechanisms. Here, we study the evaporation of drops made of paint suspension and investigate the influence of the substrate temperature and suspension concentration on the resulting deposit patterns. At low concentrations and temperatures, the pigments appear highly concentrated in a narrow area at the center of the drop, a morphology we call "fried eggs". Increasing the temperature or concentration leads to more homogeneous patterns. From a top-view camera used for monitoring the whole evaporative process, we identify three mechanisms responsible for the final pattern: inward/outward flows that convect the pigments, gelation of the paint suspension where pigments accumulate, and final drying of the drop that freezes the location of the pigments onto the substrate. The relative kinetics of these three mechanisms upon concentration and temperature govern the deposit growth and the morphology of the final pattern. These observations are quantitatively supported by rheological measurements highlighting a strong increase of the viscosity with concentration, consistent with the gelation mechanism. Finally, we show that the kinetics of drop drying is controlled by the substrate temperature.

Stress Overshoots in Simple Yield Stress Fluids.

Soft glassy materials such as mayonnaise, wet clays, or dense microgels display a solid-to-liquid transition under external shear. Such a shear-induced transition is often associated with a nonmonotonic stress response in the form of a stress maximum referred to as "stress overshoot." This ubiquitous phenomenon is characterized by the coordinates of the maximum in terms of stress σ_{M} and strain γ_{M} that both increase as weak power laws of the applied shear rate. Here we rationalize such power-law scalings using a continuum model that predicts two different regimes in the limit of low and high applied shear rates. The corresponding exponents are directly linked to the steady-state rheology and are both associated with the nucleation and growth dynamics of a fluidized region. Our work offers a consistent framework for predicting the transient response of soft glassy materials upon startup of shear from the local flow behavior to the global rheological observables.

Continuum modeling of shear startup in soft glassy materials.

Yield stress fluids (YSFs) display a dual nature highlighted by the existence of a critical stress σ_{y} such that YSFs are solid for stresses σ imposed below σ_{y}, whereas they flow like liquids for σ>σ_{y}. Under an applied shear rate γ[over ̇], the solid-to-liquid transition is associated with a complex spatiotemporal scenario that depends on the microscopic details of the system, on the boundary conditions, and on the system size. Still, the general phenomenology reported in the literature boils down to a simple sequence that can be divided into a short-time response characterized by the so-called "stress overshoot," followed by stress relaxation towards a steady state. Such relaxation can be either (1) long-lasting, which usually involves the growth of a shear band that can be only transient or that may persist at steady state or (2) abrupt, in which case the solid-to-liquid transition resembles the failure of a brittle material, involving avalanches. In the present paper, we use a continuum model based on a spatially resolved fluidity approach to rationalize the complete scenario associated with the shear-induced yielding of YSFs. A key feature of our model is to provide a scaling for the coordinates of the stress overshoot, i.e., stress σ_{M} and strain γ_{M} as a function of γ[over ̇], which shows good agreement with experimental and numerical data extracted from the literature. Moreover, our approach shows that the power-law scaling σ_{M}(γ[over ̇]) is intimately linked to the growth dynamics of a fluidized boundary layer in the vicinity of the moving boundary. Yet such scaling is independent of the fate of that layer, and of the long-term behavior of the YSF, i.e., whether the steady-state flow profile is homogeneous or shear-banded. Finally, when including the presence of "long-range" correlations, we show that our model displays a ductile to brittle transition, i.e., the stress overshoot reduces into a sharp stress drop associated with avalanches, which impacts the scaling σ_{M}(γ[over ̇]). This generalized model nicely captures subtle avalanche-like features of the transient shear banding dynamics reported in experiments. Our work offers a unified picture of shear-induced yielding in YSFs, whose complex spatiotemporal dynamics are deeply connected to nonlocal effects.

Mechanical Control of Cell Proliferation Increases Resistance to Chemotherapeutic Agents.

While many cellular mechanisms leading to chemotherapeutic resistance have been identified, there is an increasing realization that tumor-stroma interactions also play an important role. In particular, mechanical alterations are inherent to solid cancer progression and profoundly impact cell physiology. Here, we explore the influence of compressive stress on the efficacy of chemotherapeutics in pancreatic cancer spheroids. We find that increased compressive stress leads to decreased drug efficacy. Theoretical modeling and experiments suggest that mechanical stress decreases cell proliferation which in turn reduces the efficacy of chemotherapeutics that target proliferating cells. Our work highlights a mechanical form of drug resistance and suggests new strategies for therapy.

Interparticle attraction controls flow heterogeneity in calcite gels.

We couple rheometry and ultrasonic velocimetry to study experimentally the flow behavior of gels of colloidal calcite particles dispersed in water, while tuning the strength of the interparticle attraction through physico-chemistry. We unveil, for the first time in a colloidal gel, a direct connection between attractive interactions and the occurrence of shear bands, as well as stress fluctuations.

Mechanisms of Phase Transformation and Creating Mechanical Strength in a Sustainable Calcium Carbonate Cement.

Calcium carbonate cements have been synthesized by mixing amorphous calcium carbonate and vaterite powders with water to form a cement paste and study how mechanical strength is created during the setting reaction. In-situ X-ray diffraction (XRD) was used to monitor the transformation of amorphous calcium carbonate (ACC) and vaterite phases into calcite and a rotational rheometer was used to monitor the strength evolution. There are two characteristic timescales of the strengthening of the cement paste. The short timescale of the order 1 h is controlled by smoothening of the vaterite grains, allowing closer and therefore adhesive contacts between the grains. The long timescale of the order 10-50 h is controlled by the phase transformation of vaterite into calcite. This transformation is, unlike in previous studies using stirred reactors, found to be mainly controlled by diffusion in the liquid phase. The evolution of shear strength with solid volume fraction is best explained by a fractal model of the paste structure.

Apparent Non-Newtonian Behavior of Ionic Liquids.

A significant viscosity variation with the shear rate has been observed for several ionic liquids in rheometry experiments above a critical shear rate. Depending on the liquid and the rheological conditions, both viscosity increase and decrease have been reported. So far, these variations have been interpreted as a signature of a non-Newtonian behavior. However, the measured critical shear rates are orders of magnitude below the ones predicted by numerical simulations. In this work, we perform new rheometry experiments with both ionic liquids and Newtonian liquids to elucidate this discrepancy. For these two types of liquids, both a viscosity decrease and increase have been measured depending on the geometry of the rheometer and the zero-shear viscosity of the liquid. We interpret the viscosity decrease as resulting from viscous heating, since the viscosity of the investigated liquids is also highly temperature-dependent, and the viscosity increase as resulting from the development of instabilities at high shear rates.

Simple ions control the elasticity of calcite gels via interparticle forces.

Suspensions of calcite in water are employed in many industrial fields such as paper filling, pharmaceutics or heritage conservation. Whereas organics are generally used to tune the rheological properties of the paste, we also expect simple ions to be able to control the suspension rheology via the interparticle forces. We have thus investigated the impact of calcium, sodium and hydroxide ions on the elasticity of a colloidal gel of nanocalcite. We confront our macroscopic measurements to DLVO interaction potentials, based on chemical speciation and measurements of the zeta potential. Upon addition of calcium hydroxide, we observe a minimum in shear modulus, correlated to a maximum in the DLVO energy barrier, due to two competing effects: Calcium adsorption onto calcite surface rises the zeta potential, while increasing salt concentration induces stronger electrostatic screening. We also demonstrate that the addition of sodium hydroxide completely screens the surface charge and leads to a more rigid paste. A second important result is that carbonation of the calcite suspensions by the atmospheric CO2 leads to a convergent high elasticity of the colloidal gels, whatever their initial value, also well rationalized by DLVO theory and resulting from a decrease in zeta potential.

Unified Theoretical and Experimental View on Transient Shear Banding.

Dense emulsions, colloidal gels, microgels, and foams all display a solidlike behavior at rest characterized by a yield stress, above which the material flows like a liquid. Such a fluidization transition often consists of long-lasting transient flows that involve shear-banded velocity profiles. The characteristic time for full fluidization τ_{f} has been reported to decay as a power law of the shear rate γ[over ˙] and of the shear stress σ with respective exponents α and ß. Strikingly, the ratio of these exponents was empirically observed to coincide with the exponent of the Herschel-Bulkley law that describes the steady-state flow behavior of these complex fluids. Here we introduce a continuum model, based on the minimization of a "free energy," that captures quantitatively all the salient features associated with such transient shear banding. More generally, our results provide a unified theoretical framework for describing the yielding transition and the steady-state flow properties of yield stress fluids.

Elasticity and yielding of a calcite paste: scaling laws in a dense colloidal suspension.

We address the mechanical characterization of a calcite paste as a model system to investigate the relation between the microstructure and macroscopic behavior of colloidal suspensions. The ultimate goal is to achieve control of the elastic and yielding properties of calcite which will prove valuable in several domains, from paper coating to paint manufacture and eventually in the comprehension and control of the mechanical properties of carbonate rocks. Rheological measurements have been performed on calcite suspensions over a wide range of particle concentrations. The calcite paste exhibits a typical colloidal gel behavior, with an elastic regime and a clear yield strain above which it enters a plastic regime. The yield strain shows a minimum when increasing the solid concentration, connected to a change in the power law scaling of the storage modulus. In the framework of the classical fractal elasticity model for colloidal suspensions proposed by Shih et al. [Phys. Rev. A, 1990, 42, 4772], we interpret this behavior as a switch with the concentration from the strong-link regime to the weak-link regime, which had never been observed so far in one well-defined system without external or chemical forcing.

Structural and cooperative length scales in polymer gels.

Understanding the relationship between the material structural details, the geometrical confining constraints, the local dynamical events and the global rheological response is at the core of present investigations on complex fluid properties. In the present article, this problem is addressed on a model yield stress fluid made of highly entangled polymer gels of Carbopol which follows at the macroscopic scale the well-known Herschel-Bulkley rheological law. First, performing local rheology measurements up to high shear rates ([Formula: see text] s-1)and under confinement, we evidence unambiguously the breakdown of bulk rheology associated with cooperative processes under flow. Moreover, we show that these behaviors are fully captured with a unique cooperativity length [Formula: see text] over the whole range of experimental conditions. Second, we introduce an original optical microscopy method to access structural properties of the entangled polymer gel in the direct space. Performing image correlation spectroscopy of fluorophore-loaded gels, the characteristic size D of carbopol gels microstructure is determined as a function of preparation protocol. Combining both dynamical and structural information shows that the measured cooperative length [Formula: see text] corresponds to 2-5 times the underlying structural size D, thus providing a strong grounding to the "Shear Transformation Zones" modeling approach.

Yield stress and elasticity influence on surface tension measurements.

We have performed surface tension measurements on carbopol gels of different concentrations and yield stresses. Our setup, based on the force exerted by a capillary bridge on two parallel plates, allows us to measure an apparent surface tension of the complex fluid and to investigate the influence of flow history. More precisely the apparent surface tension measured after stretching the bridge is always higher than after compressing it. The difference between the two values is due to the existence of a yield stress in the fluid. The experimental observations are successfully reproduced with a simple elasto-plastic model. The shape of successive stretching-compression cycles can be described by taking into account the yield stress and the elasticity of the gel. We show that the surface tension γLV of yield stress fluids is the mean of the apparent surface tension values only if the elastic modulus is high compared to the yield stress. This work highlights that measurements of thermodynamic quantities are challenged by the fluid out-of-equilibrium state implied by jamming, even at small scales where the shape of the bridge is driven by surface energy. Therefore setups allowing for deformation in opposite directions are relevant for surface tension measurements on yield stress fluids.

Multicellular aggregates: a model system for tissue rheology.

Morphogenetic processes involve cell flows. The mechanical response of a tissue to active forces is linked to its effective viscosity. In order to decouple this mechanical response from the complex genetic changes occurring in a developing organism, we perform rheometry experiments on multicellular aggregates, which are good models for tissues. We observe a cell softening behavior when submitting to stresses. As our technique is very sensitive, we were able to get access to the measurement of a yield point above which a creep regime is observed obtained for strains above 12%. To explain our rheological curves we propose a model for the cytoskeleton that we represent as a dynamic network of parallel springs, which will break under stress and reattach at null strain. Such a simple model is able to reproduce most of the important behavior of cells under strain. We highlight here the importance of considering cells as complex fluids whose properties will vary with time according to the history of applied stress.

Confined flows of a polymer microgel.

In this paper, we probe the influence of confinement on the flows of a polymer microgel, namely Carbopol. We compare its bulk rheological behavior, measured with a rheometer and well described by a Hershel-Bulkley law, to velocity profiles measured in rough microchannels, obtained with a particle tracking velocimetry technique. We show a strong disagreement between the bulk prediction for the velocity profiles and the measured ones in the microchannels. Velocity profiles in confined conditions are successfully analyzed within the framework of a non-local fluidity model introduced recently (J. Goyon et al. Nature, 454, 84 (2008)). This allows to determine a cooperativity length ξ, whose order of magnitude compares with the structure size of the microgel. Moreover, we measure flow curves using a rheometer for different gap conditions and also show that this set of data exhibit a strong effect of the confinement on the measured rheological properties. This is again characterized by a typical length of the same order as the cooperativity length scale ξ. We thus evidence confinement effects with two complementary experiments which both give the same typical length for the rearrangements in the flows.

Probing the fluctuation-dissipation theorem in a Perrin-like experiment.

In this Letter, we present a new experimental approach to investigate the effective temperature concept as a generalization of the fluctuation-dissipation theorem (FDT) for nonequilibrium systems. Simultaneous measurements of diffusion coefficient and sedimentation velocity of heavy colloids, embedded in a Laponite clay suspension, are performed with a fluorescence-recovery-based setup. This nonperturbative dual measurement, performed at a single time in a single sample, allows for a direct application of the FDT to the tracer velocity observable. It thus provides a well-defined derivation of the effective temperature in this ageing colloidal gel. For a wide range of concentrations and ageing times, we report no violation of the FDT, with effective temperature agreeing with bath temperature. This result is consistent with recent theoretical predictions on the coupling between the velocity observable and nonequilibrium gels dynamics.

Transient shear banding in a simple yield stress fluid.

We report a large set of experimental data which demonstrates that a simple yield stress fluid, i.e., which does not present aging or thixotropy, exhibits transient shear banding before reaching a steady state characterized by a homogeneous, linear velocity profile. The duration of the transient regime decreases as a power law with the applied shear rate γ. This power-law behavior, observed here in carbopol dispersions, does not depend on the gap width and on the boundary conditions for a given sample preparation. For γ≲0.1 s(-1), heterogeneous flows could be observed for as long as 10(5) s. These local dynamics account for the ultraslow stress relaxation observed at low shear rates.

Size dependence of tracer diffusion in a laponite colloidal gel.

Using a fluorescence recovery after photobleaching (FRAP) technique, we present measurements of probe diffusion in a colloidal glass-a Laponite suspension. By varying the probe size over 2 orders of magnitude, as well as the concentration of the colloidal glass, we evidence and quantify the deviations of the probe diffusivity from the bulk Stokes-Einstein expectations. These experiments suggest that the probe diffusion in the dynamically arrested Laponite structure is mainly controlled by the ratio between the probe size and the typical clay platelets interdistance. Comparing with a simple hindered diffusion mechanism, the reduction of tracer diffusion is discussed in terms of the hydrodynamic interaction of the probe with the Laponite structure. Finally, these results can be interpreted in terms of a scale dependent viscosity of the colloidal glass.

Influence of boundary conditions on yielding in a soft glassy material.

The yielding behavior of a sheared Laponite suspension is investigated within a 1 mm gap under two different boundary conditions. No-slip conditions, ensured by using rough walls, lead to shear localization as already reported in various soft glassy materials. When apparent wall slip is allowed using a smooth geometry, the sample breaks up into macroscopic solid pieces that get slowly eroded by the surrounding fluidized material up to the point where the whole sample is fluid. Such a drastic effect of boundary conditions on yielding suggests the existence of some macroscopic characteristic length that could be connected to cooperativity effects in jammed materials under shear.

Fluorescent or not? Size-dependent fluorescence switching for polymer-stabilized gold clusters in the 1.1-1.7 nm size range.

The synthesis of fluorescent water-soluble gold nanoparticles by the reduction of a gold salt in the presence of a designed polymer ligand is described, the size and fluorescence of the particles being controlled by the polymer to gold ratio; the most fluorescent nanomaterial has a 3% quantum yield, a 1.1 nm gold core and a 6.9 nm hydrodynamic radius.