Structural and cooperative length scales in polymer gels

Low yield stress measurements with a microfluidic rheometer

Yield stress, τy, is a key rheological property of complex materials such as gels, dense suspensions, and dense emulsions. While there is a range of established techniques to measure τy in the order of tens to thousands of pascals, the measurement of low τy, specifically below 1 Pa, remains underexplored. In this article, we present the measurement of low apparent τy using a Hele-Shaw microfluidic extensional flow device (MEFD). Using the MEFD, we observe a gradient in shear stress, τ, such that τ is lower near the center or stagnation point, and higher away from the stagnation point. For a yield stress fluid, we observe that, below a certain flow rate, τ exceeds τy only in the outer region, leading to stagnation or unyielding of the fluid in the inner region. We use scaling analysis based on a Hele-Shaw linear extensional flow to deduce τy by measuring the size of the unyielded region, S. We validate this scaling relationship using Carbopol solutions with concentrations ranging between 0.015 to 0.3%, measuring τy as low as ∼10 mPa to ∼1 Pa, and comparing it with τy measured using a standard rheometer. While the experimental lower limit of our technique is 5 mPa, modifying the geometry or improving the image analysis can reduce this limit to the order of 10-4 Pa. The MEFD facilitates rapid measurement of τy, allowing for its real-time assessment. We further report τy of human blood samples between 30 to 80 mPa with their hematocrit ranging between 14 to 63%. Additionally, we determine τy for a mucus simulant (∼0.7 Pa), and lactic drink (∼7 mPa) to demonstrate the versatility of the MEFD technique.

Complexity in Wetting Dynamics

The spreading dynamics of a droplet of pure liquid deposited on a rigid, nonsoluble substrate has been extensively investigated. In a purely hydrodynamic description, the dynamics of the contact line is determined by a balance between the energy associated with the capillary driving force and the energy dissipated by the viscous shear in the liquid. This balance is expressed by the Cox-Voinov law, which relates the spreading velocity to the contact angle. More recently, complex situations have been examined in which dissipation and/or the driving force may be strongly modified, leading to sometimes spectacular changes in wetting dynamics. We review recent examples of effects at the origin of deviations from the hydrodynamic model, which may involve physical or chemical modifications of the substrate or of the wetting liquid, occurring at scales ranging from the molecular to the mesoscopic.

Capillary Flow-MRI: Quantifying Micron-Scale Cooperativity in Complex Dispersions

Strongly confined flow of particulate fluids is encountered in applications ranging from three-dimensional (3D) printing to the spreading of foods and cosmetics into thin layers. When flowing in constrictions with gap sizes, w, within 102 times the mean size of particles or aggregates, d, structured fluids experience enhanced bulk velocities and inhomogeneous viscosities, as a result of so-called cooperative, or nonlocal, particle interactions. Correctly predicting cooperative flow for a wide range of complex fluids requires high-resolution flow imaging modalities applicable in situ to even optically opaque fluids. To this goal, we here developed a pressure-driven high-field magnetic resonance imaging (MRI) velocimetry platform, comprising a pressure controller connected to a capillary. Wall properties and diameter could be modified respectively as hydrophobic/hydrophilic, or within w ∼ 100-540 μm. By achieving a high spatial resolution of 9 μm, flow cooperativity length scales, ξ, down to 15 μm in Carbopol with d ∼ 2 μm could be quantified by means of established physical models with an accuracy of 13%. The same approach was adopted for a heterogeneous fat crystal dispersion (FCD) with d and ξ values up to an order of magnitude higher than those for Carbopol. We found that for strongly confined flow of Carbopol in the 100 μm capillary, ξ is independent of flow conditions. For the FCD, ξ increases with gap size and applied pressures over 0.25-1 bar. In both samples, nonlocal interactions span domains up to about 5-8 particles but, at the highest confinement degree explored, ∼8% for FCD, domains of only ∼2 particles contribute to cooperative flow. The developed flow-MRI platform is easily scalable to ultrahigh field MRI conditions for chemically resolved velocimetric measurements of, e.g., complex fluids with anisotropic particles undergoing alignment. Future potential applications of the platform encompass imaging extrusion under confinement during the 3D printing of complex dispersions or in in vitro vascular and perfusion studies.

Quantifying cooperative flow of fat crystal dispersions

We quantify the cooperative flow behaviour of fat crystal dispersions (FCDs) upon varying crystallization conditions. The latter enabled altering the multiscale microstructure of the FCDs, from the nanometer-sized platelets, and the dispersed fractal aggregates, up to the strength of the mesoscopic weak-link network. To the goal of characterizing strongly-confined flow in these optically-opaque materials, we acquire high-resolution rheo-magnetic-resonance-imaging (rheo-MRI) velocimetry measurements using an in-house developed 500 μm gap Couette cell (CC). We introduce a numerical fitting method based on the fluidity model, which yields the cooperativity length, ξ, in the narrow-gap CC. FCDs with aggregates sizes smaller than the confinement size by an order of magnitude were found to exhibit cooperativity effects. The respective ξ values diverged at the yield stress, in agreement with the Kinetic Elasto-Plastic (KEP) theory. In contrast, the FCD with aggregates sizes in the order of the gap size did not exhibit any cooperativity effect: we attribute this result to the correspondingly decreased mobility of the aggregates. We foresee that our optimized rheo-MRI measurement and fitting analysis approach will propel further similar studies of flow of other multi-scale and optically-opaque materials.

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.

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.

Carbomer microgels as model yield-stress fluids

The review presents current research results for Carbopol-based microgels as yield-stress materials, covering three aspects: chemical, physical and rheological. Such a joint three-aspect study has no analog in the literature. The chemical aspects of Carbopol polymers are presented in terms of a cross-linking polymerization of acrylic acid, their molecular structure, microgel formulation, polyacid dissociation and neutralization, osmotic pressure and associated immense microgel swelling. The physical characterization is focused on models of the shear-induced solid-to-liquid transition of microgels, which are formed of mesoscopic particles typical for soft matter materials. Models that describe interparticle effects are presented to explain the energy states of microgel particles at the mesoscale of scrutiny. Typical representatives of the models utilize attributes of jamming dispersions, micromechanical and polyelectrolyte reactions. Selected relationships that result from the models, such as scaling rules and nondimensional flow characteristics are also presented. The rheological part presents the discussion of problems of yield stress in 2D and 3D deformations, appearance and magnitude of the wall slip. The theory and characteristics of Carbopol microgel deformation in rotational rheometers are presented with graphs for the steady-state measurements, stress-controlled oscillation and two types of transient shear deformation. The review is concluded with suggestions for future research.

Investigation of the swollen state of Carbopol molecules in non-aqueous solvents through rheological characterization

We explore how different types of solvent influence the rheological properties of non-aqueous Carbopol dispersions from the dilute to the jammed state. In novel non-aqueous formulations, polar solvents are used more and more frequently, because they can form Carbopol microgels without the need of any neutralizing agents. However, the swelling behaviour of Carbopol molecules in the absence of water, when ionic forces are weak, is still poorly understood. To this end, we study the swelling behaviour of Carbopol 974P NF in different polar solvents, i.e. glycerol, PEG400 and mixtures of the two solvents, by mapping the rheological behaviour of Carbopol suspensions from very dilute to highly concentrated conditions. The rheological study reveals that the onset of the jamming transition occurs at different critical polymer concentrations depending on the solvents used. Nevertheless, once the jammed state is reached, both elastic and yielding behaviours are scalable with the particle volume fraction. These results suggest that the type of solvent influences the final volume of the single Carbopol particles but does not alter the interactions between the particles. The final radius of the swollen particles is estimated from shear rheology measurements in dilute conditions, showing a decrease of the final swelling ratio of Carbopol molecules of almost 50% for PEG400 solutions, a result that confirms the shift to higher values of the critical jamming concentration obtained from linear viscoelasticity for the same solutions.

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 β. 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.

Confinement and activity regulate bacterial motion in porous media

Understanding how bacteria move in porous media is critical to applications in healthcare, agriculture, environmental remediation, and chemical sensing. Recent work has demonstrated that E. coli, which moves by run-and-tumble dynamics in a homogeneous medium, exhibits a new form of motility when confined in a disordered porous medium: hopping-and-trapping motility, in which cells perform rapid, directed hops punctuated by intervals of slow, undirected trapping. Here, we use direct visualization to shed light on how these processes depend on pore-scale confinement and cellular activity. We find that hopping is determined by pore-scale confinement, and is independent of cellular activity; by contrast, trapping is determined by the competition between pore-scale confinement and cellular activity, as predicted by an entropic trapping model. These results thus help to elucidate the factors that regulate bacterial motion in porous media, and could help aid the development of new models of motility in heterogeneous environments.

A mesoscale study of creep in a microgel using the acoustic radiation force

We study the motion of a sphere of diameter 330 μm embedded in a Carbopol microgel under the effect of the acoustic radiation pressure exerted by a focused ultrasonic field. The sphere motion within the microgel is tracked using videomicroscopy and compared to conventional creep and recovery measurements performed with a rheometer. We find that under moderate ultrasonic intensities, the sphere creeps as a power law of time with an exponent α ≃ 0.2 that is significantly smaller than the one inferred from global creep measurements below the yield stress of the microgel (α ≃ 0.4). Moreover, the sphere relaxation motion after creep and the global recovery are respectively consistent with these two different exponents. By allowing a rheological characterization at the scale of the sphere with forces of the order of micronewtons, the present experiments pave the way for acoustic "mesorheology" which probes volumes and forces an intermediate between standard macroscopic rheology and classical microrheology. They also open new questions about the effects of the geometry of the deformation field and of the sphere size and surface properties on the creep behaviour of soft materials.

Polyelectrolyte scaling laws for microgel yielding near jamming

Micro-scale hydrogel particles, known as microgels, are used in industry to control the rheology of numerous different products, and are also used in experimental research to study the origins of jamming and glassy behavior in soft-sphere model systems. At the macro-scale, the rheological behaviour of densely packed microgels has been thoroughly characterized; at the particle-scale, careful investigations of jamming, yielding, and glassy-dynamics have been performed through experiment, theory, and simulation. However, at low packing fractions near jamming, the connection between microgel yielding phenomena and the physics of their constituent polymer chains has not been made. Here we investigate whether basic polymer physics scaling laws predict macroscopic yielding behaviours in packed microgels. We measure the yield stress and cross-over shear-rate in several different anionic microgel systems prepared at packing fractions just above the jamming transition, and show that our data can be predicted from classic polyelectrolyte physics scaling laws. We find that diffusive relaxations of microgel deformation during particle re-arrangements can predict the shear-rate at which microgels yield, and the elastic stress associated with these particle deformations predict the yield stress.