Centrifugal compression of soft particle packings: theory and experiment

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.

Under pressure: Hydrogel swelling in a granular medium

Hydrogels hold promise in agriculture as reservoirs of water in dry soil, potentially alleviating the burden of irrigation. However, confinement in soil can markedly reduce the ability of hydrogels to absorb water and swell, limiting their widespread adoption. Unfortunately, the underlying reason remains unknown. By directly visualizing the swelling of hydrogels confined in three-dimensional granular media, we demonstrate that the extent of hydrogel swelling is determined by the competition between the force exerted by the hydrogel due to osmotic swelling and the confining force transmitted by the surrounding grains. Furthermore, the medium can itself be restructured by hydrogel swelling, as set by the balance between the osmotic swelling force, the confining force, and intergrain friction. Together, our results provide quantitative principles to predict how hydrogels behave in confinement, potentially improving their use in agriculture as well as informing other applications such as oil recovery, construction, mechanobiology, and filtration.

Scaling Law for Cracking in Shrinkable Granular Packings

Hydrated granular packings often crack into discrete clusters of grains when dried. Despite its ubiquity, an accurate prediction of cracking remains elusive. Here, we elucidate the previously overlooked role of individual grain shrinkage-a feature common to many materials-in determining crack patterning using both experiments and simulations. By extending classical Griffith crack theory, we obtain a scaling law that quantifies how cluster size depends on the interplay between grain shrinkage, stiffness, and size-applicable to a diverse array of shrinkable granular packings.

Model-Free Rheo-AFM Probes the Viscoelasticity of Tunable DNA Soft Colloids

Atomic force microscopy rheological measurements (Rheo-AFM) of the linear viscoelastic properties of single, charged colloids having a star-like architecture with a hard core and an extended, deformable double-stranded DNA (dsDNA) corona dispersed in aqueous saline solutions are reported. This is achieved by analyzing indentation and relaxation experiments performed on individual colloidal particles by means of a novel model-free Fourier transform method that allows a direct evaluation of the frequency-dependent linear viscoelastic moduli of the system under investigation. The method provides results that are consistent with those obtained via a conventional fitting procedure of the force-relaxation curves based on a modified Maxwell model. The outcomes show a pronounced softening of the dsDNA colloids, which is described by an exponential decay of both the Young's and the storage modulus as a function of the salt concentration within the dispersing medium. The strong softening is related to a critical reduction of the size of the dsDNA corona, down to ≈70% of its size in a salt-free solution. This can be correlated to significant topological changes of the dense star-like polyelectrolyte forming the corona, which are induced by variations in the density profile of the counterions. Similarly, a significant reduction of the stiffness is obtained by increasing the length of the dsDNA chains, which we attribute to a reduction of the DNA density in the outer region of the corona.

Crack formation and self-closing in shrinkable, granular packings

Many clays, soils, biological tissues, foods, and coatings are shrinkable, granular materials: they are composed of packed, hydrated grains that shrink when dried. In many cases, these packings crack during drying, critically hindering applications. However, while cracking has been widely studied for bulk gels and packings of non-shrinkable grains, little is known about how packings of shrinkable grains crack. Here, we elucidate how grain shrinkage alters cracking during drying. Using experiments with model shrinkable hydrogel beads, we show that differential shrinkage can dramatically alter crack evolution during drying-in some cases, even causing cracks to spontaneously "self-close". In other cases, packings shrink without cracking or crack irreversibly. We developed both granular and continuum models to quantify the interplay between grain shrinkage, poromechanics, packing size, drying rate, capillarity, and substrate friction on cracking. Guided by the theory, we also found that cracking can be completely altered by varying the spatial profile of drying. Our work elucidates the rich physics underlying cracking in shrinkable, granular packings, and yields new strategies for controlling crack evolution.

Organization of Microgels at the Air-Water Interface under Compression: Role of Electrostatics and Cross-Linking Density

Poly(N-isopropylacrylamide) (pNIPAM) microgels are soft and deformable particles, which can adsorb at liquid interfaces. In the present paper, we study the two-dimensional organization of charged and quasi-neutral microgels with different cross-linking densities, under compression at the air-water interface and the transfer of the microgel monolayer onto a solid substrate at different surface pressures. At low cross-linking densities, the microgels form highly ordered hexagonal lattices on the solid substrate over large areas, with a unique lattice parameter that decreases continuously as the surface pressure increases. We thus prove that the microgel conformation evolves at the air-water interface. The microgels undergo a continuous transition from a highly flattened state at low surface coverage, where the maximal polymer segments are adsorbed at the interface, to entangled flattened microgels, and finally the thickening of the layer up to a dense hydrogel layer of compacted microgels. Moreover, two batches of microgels, with and without charges, are compared. The contribution of electrostatic interactions is assessed via changing the charge density of the microgels or modulating the Debye length. In both cases, electrostatics does not change the lattice parameter, meaning that, despite the microgel different swelling ratio, charges do not affect neither interactions between particles at the interface nor microgels adsorption. Conversely, the cross-linking density has a strong impact on microgel packing at the interface: increasing the cross-linking density strongly decreases the extent of microgel flattening and promotes the occurrence of coexisting hexagonally ordered domains with different lattice parameters.

Drainage dynamics of thin liquid foam films containing soft PNiPAM microgels: influence of the cross-linking density and concentration

We investigate the drainage dynamics of thin liquid foam films containing PNiPAM microgel suspensions with two cross-linking densities (1.5 and 5 mol% BIS) and at two microgel concentrations (0.1 and 1% wt). For this purpose, we use a thin-film pressure balance apparatus that can apply a controlled and sudden hydrostatic pressure on a film, and record the subsequent film thinning as a function of time. Once the film thickness has reached a stationary value, we test the adhesion between the interfaces of the film by reducing the pressure and measuring the angle between the film and the meniscus. This angle increases on reduction of pressure for adhesive films, which resists the separation of their interfaces. Non-adhesive films separate easily, and the meniscus angle stays constant. At a low microgel concentration, the more densely cross-linked microgels (5 mol% BIS) tend to drain into more adhesive films than the more loosely cross-linked particles (1.5 mol% BIS). The adhesion results from particles that bridge the two air-water interfaces of the film and are shared between them. In these cases, the film, which is initially stabilized by a bilayer of microgel particles, rearrange to a state where the microgels bridge the interfaces. These results are discussed and compared with previous studies at a low concentration of microgels, which have shown that emulsions stabilized with densely cross-linked microgels are more adhesive and less resistant to mechanical stresses than those obtained with lower cross-linking densities. In addition, micron-scale depleted zones with no microgels are observed in the films stabilized with the 5 mol% BIS particles, which eventually lead to the rupture of the films. At 1% wt, the films drain slowly, are not adhesive and have the thickness of a bilayer of microgel; while at 0.1% wt, the films have the thickness of a monolayer of microgel, are adhesive and show bridging. From the thin liquid foam film thicknesses we extract a rough estimation of the radii of adsorbed particles in the thick films before applying the pressure. Our results are consistent with particles being adsorbed in a spread conformation for the 0.1% wt sample and in a compressed conformation for the 1% wt sample. In line with previous studies on emulsions, we conclude that a larger surface coverage may reduce rearrangements, thus preventing bridging.

Compressing a spinodal surface at fixed area: bijels in a centrifuge

Bicontinuous interfacially jammed emulsion gels (bijels) are solid-stabilised emulsions with two inter-penetrating continuous phases. Employing the method of centrifugal compression we find that macroscopically the bijel yields at relatively low angular acceleration. Both continuous phases escape from the top of the structure, making any compression immediately irreversible. Microscopically, the bijel becomes anisotropic with the domains aligned perpendicular to the compression direction which inhibits further liquid expulsion; this contrasts strongly with the sedimentation behaviour of colloidal gels. The original structure can, however, be preserved close to the top of the sample and thus the change to an anisotropic structure suggests internal yielding. Any air bubbles trapped in the bijel are found to aid compression by forming channels aligned parallel to the compression direction which provide a route for liquid to escape.

Flow-induced compaction of a deformable porous medium

Fluid flowing through a deformable porous medium imparts viscous drag on the solid matrix, causing it to deform. This effect is investigated theoretically and experimentally in a one-dimensional configuration. The experiments consist of the downwards flow of water through a saturated pack of small, soft, hydrogel spheres, driven by a pressure head that can be increased or decreased. As the pressure head is increased, the effective permeability of the medium decreases and, in contrast to flow through a rigid medium, the flux of water is found to increase towards a finite upper bound such that it becomes insensitive to changes in the pressure head. Measurements of the internal deformation, extracted by particle tracking, show that the medium compacts differentially, with the porosity being lower at the base than at the upper free surface. A general theoretical model is derived, and the predictions of the model give good agreement with experimental measurements from a series of experiments in which the applied pressure head is sequentially increased. However, contrary to theory, all the experimental results display a distinct and repeatable hysteresis: the flux through the material for a particular applied pressure drop is appreciably lower when the pressure has been decreased to that value compared to when it has been increased to the same value.

Granular dynamics during impact

We study the impact of a projectile onto a bed of 3 mm grains immersed in an index-matched fluid. We vary the amount of prestrain on the sample, strengthening the force chains within the system. We find this affects only the prefactor of the linear depth-dependent term in the stopping force. We propose a simple model to account for the strain dependence of this term, owing to increased pressure in the pile. Interestingly, we find that the presence of the fluid does not affect the impact dynamics, suggesting that dynamic friction is not a factor. Using a laser sheet scanning technique to visualize internal grain motion, we measure the trajectory of each grain throughout an impact. Microscopically, our results indicate that weaker initial force chains result in more irreversible, plastic rearrangements, suggesting static friction between grains does play a substantial role in the energy dissipation.

Rheology of soft colloids across the onset of rigidity: scaling behavior, thermal, and non-thermal responses

We study the rheological behavior of colloidal suspensions composed of soft sub-micron-size hydrogel particles across the liquid-solid transition. The measured stress and strain-rate data, when normalized by thermal stress and time scales, suggest our systems reside in a regime wherein thermal effects are important. In a different vein, critical point scaling predictions for the jamming transition, typical in athermal systems, are tested. Near dynamic arrest, the suspensions exhibit scaling exponents similar to those reported in Nordstrom et al., Phys. Rev. Lett., 2010, 105, 175701. The observation suggests that our system exhibits a glass transition near the onset of rigidity, but it also exhibits a jamming-like scaling further from the transition point. These observations are thought-provoking in light of recent theoretical and simulation findings, which show that suspension rheology across the full range of microgel particle experiments can exhibit both thermal and athermal mechanisms.

Physics in ordered and disordered colloidal matter composed of poly(N-isopropylacrylamide) microgel particles

This review collects and describes experiments that employ colloidal suspensions to probe physics in ordered and disordered solids and related complex fluids. The unifying feature of this body of work is its clever usage of poly(N-isopropylacrylamide) (PNIPAM) microgel particles. These temperature-sensitive colloidal particles provide experimenters with a 'knob' for in situ control of particle size, particle interaction and particle packing fraction that, in turn, influence the structural and dynamical behavior of the complex fluids and solids. A brief summary of PNIPAM particle synthesis and properties is given, followed by a synopsis of current activity in the field. The latter discussion describes a variety of soft matter investigations including those that explore formation and melting of crystals and clusters, and those that probe structure, rearrangement and rheology of disordered (jammed/glassy) and partially ordered matter. The review, therefore, provides a snapshot of a broad range of physics phenomenology which benefits from the unique properties of responsive microgel particles.

Phonons in two-dimensional soft colloidal crystals

The vibrational modes of pristine and polycrystalline monolayer colloidal crystals composed of thermosensitive microgel particles are measured using video microscopy and covariance matrix analysis. At low frequencies, the Debye relation for two-dimensional harmonic crystals is observed in both crystal types; at higher frequencies, evidence for van Hove singularities in the phonon density of states is significantly smeared out by experimental noise and measurement statistics. The effects of these errors are analyzed using numerical simulations. We introduce methods to correct for these limitations, which can be applied to disordered systems as well as crystalline ones, and we show that application of the error correction procedure to the experimental data leads to more pronounced van Hove singularities in the pristine crystal. Finally, quasilocalized low-frequency modes in polycrystalline two-dimensional colloidal crystals are identified and demonstrated to correlate with structural defects such as dislocations, suggesting that quasilocalized low-frequency phonon modes may be used to identify local regions vulnerable to rearrangements in crystalline as well as amorphous solids.

Observation and characterization of the vestige of the jamming transition in a thermal three-dimensional system

We study the dependence on the packing fraction of the pair-correlation function g(r) and particle mobility in a dense three-dimensional packing of soft colloids made of poly N-isopropyl acrylamide (pNIPAM), a thermosensitive gel. We find that g(r) for our samples is qualitatively like that of a liquid at all packing fractions. There is a peak in g(1), the height of the first peak of g(r), as a function of the packing fraction. This peak is identified as a vestige, which remains at finite temperature, of the divergence found at the jamming transition in simulations of soft frictionless spheres at zero temperature. As the density is increased, the particle dynamics slow down and near the packing fraction where there is a peak in g(1) the particles become arrested on the time scale of the experiment.

Dynamical heterogeneity in soft-particle suspensions under shear

We present experimental measurements of dynamical heterogeneities in a dense system of microgel spheres, sheared at different rates and at different packing fractions in a microfluidic channel, and visualized with high-speed digital video microscopy. A four-point dynamic susceptibility is deduced from video correlations, and is found to exhibit a peak that grows in height and shifts to longer times as the jamming transition is approached from two different directions. In particular, the time for particle-size root-mean square relative displacements is found to scale as τ*∼(γΔφ4)(-1), where γ is the strain rate and Δφ = |φ - φ(c)| is the distance from the random close-packing volume fraction. The typical number of particles in a dynamical heterogeneity is deduced from the susceptibility peak height and found to scale as n*∼(γΔφ4)(-0.3). Exponent uncertainties are less than ten percent. We emphasize that the same power-law behavior is found at packing fractions above and below φ(c). Thus our results considerably extend a previous observation of n*∼γ(-0.3) for granular heap flow at fixed packing below φ(c). Furthermore, the implied result n*∼(τ*)(0.3) compares well with the expectation from mode-coupling theory and with prior observations for driven granular systems.

Packings of deformable spheres

We present an experimental study of disordered packings of deformable spheres. Fluorescent hydrogel spheres immersed in water together with a tomography technique enabled the imaging of the three-dimensional arrangement. The mechanical behavior of single spheres subjected to compression is first examined. Then the properties of packings of a randomized collection of deformable spheres in a box with a moving lid are tested. The transition to a state where the packing withstands finite stresses before yielding is observed. Starting from random packed states, the power law dependence of the normal force versus packing fraction or strain at different velocities is quantified. Furthermore, a compression-decompression sequence at low velocities resulted in rearrangements of the spheres. At larger packing fractions, a saturation of the mean coordination number took place, indicating the deformation and faceting of the spheres.

Microfluidic rheology of soft colloids above and below jamming

The rheology near jamming of a suspension of soft colloidal spheres is studied using a custom microfluidic rheometer that provides the stress versus strain rate over many decades. We find non-Newtonian behavior below the jamming concentration and yield-stress behavior above it. The data may be collapsed onto two branches with critical scaling exponents that agree with expectations based on Hertzian contacts and viscous drag. These results support the conclusion that jamming is similar to a critical phase transition, but with interaction-dependent exponents.