Granular Suspensions Research Articles

Success and failure of the spreading law for large drops of dense granular suspensions

The spreading of large viscous drops of density-matched suspensions of non-Brownian spheres on a smooth solid surface is investigated experimentally at the global drop scale. The focus is on dense suspensions with a solid volume fraction equal to or greater than $40\,\%$ , and for drops larger than the capillary length, i.e. for which the spreading is governed by the balance of gravitational and viscous forces. Our findings indicate that all liquids exhibit a power-law behaviour typical of gravity-driven dynamics, albeit with an effective suspension viscosity that is smaller than the bulk value. When the height of the drop is of the order of the particle size, the power law breaks down as the particles freeze while the contact line continues to advance.

Nonmonotonic rheology and stress heterogeneity in confined granular suspensions

We systematically investigated the impact of boundary confinement on the shear-thickening rheology of dense granular suspensions. Under highly confined conditions, dense suspensions were found to exhibit size-dependent or even rarely reported nonmonotonic (S-shaped) flow curves in steady states. By performing in situ boundary stress microscopy measurements, we observed enhanced flow heterogeneities in confined suspensions, where concentrated high-stress domains propagated stably either along or against the shear direction. By comparing the boundary stress microscopy results with macroscopic flow responses, we revealed the connection between nonmonotonic rheology and stress heterogeneity in confined suspensions. These findings suggest the possibility of controlling suspension rheology by imposing different boundary confinements.

Engineered Shape-Morphing Transitions in Hydrogels Through Suspension Bath Printing of Temperature-Responsive Granular Hydrogel Inks.

4D printing of hydrogels is an emerging technology used to fabricate shape-morphing soft materials that are responsive to external stimuli for use in soft robotics and biomedical applications. Soft materials are technically challenging to process with current 4D printing methods, which limits the design and actuation potential of printed structures. Here, a simple multi-material 4D printing technique is developed that combines dynamic temperature-responsive granular hydrogel inks based on hyaluronic acid, whose actuation is modulated via poly(N-isopropylacrylamide) crosslinker design, with granular suspension bath printing that provides structural support during and after the printing process. Granular hydrogels are easily extruded upon jamming due to their shear-thinning properties and their porous structure enables rapid actuation kinetics (i.e., seconds). Granular suspension baths support responsive ink deposition into complex patterns due to shear-yielding to fabricate multi-material objects that can be post-crosslinked to obtain anisotropic shape transformations. Dynamic actuation is explored by varying printing patterns and bath shapes, achieving complex shape transformations such as 'S'-shaped and hemisphere structures. Furthermore, stepwise actuation is programmed into multi-material structures by using microgels with varied transition temperatures. Overall, this approach offers a simple method to fabricate programmable soft actuators with rapid kinetics and precise control over shape morphing.

Rheology of dense granular suspensions across flow regimes

Rheology of dense granular suspensions across flow regimes

Origins of complexity in the rheology of Soft Earth suspensions

When wet soil becomes fully saturated by intense rainfall, or is shaken by an earthquake, it may fluidize catastrophically. Sand-rich slurries are treated as granular suspensions, where the failure is related to an unjamming transition, and friction is controlled by particle concentration and pore pressure. Mud flows are modeled as gels, where yielding and shear-thinning behaviors arise from inter-particle attraction and clustering. Here we show that the full range of complex flow behaviors previously reported for natural debris flows can be reproduced with three ingredients: water, silica sand, and kaolin clay. Going from sand-rich to clay-rich suspensions, we observe continuous transition from brittle (Coulomb-like) to ductile (plastic) yielding. We propose a general constitutive relation for soil suspensions, with a particle rearrangement time that is controlled by yield stress and jamming distance. Our experimental results are supported by models for amorphous solids, suggesting that the paradigm of non-equilibrium phase transitions can help us understand and predict the complex behaviors of Soft Earth suspensions.

Edible structuring agent shaped via interfacial precipitation on solid template: Crosslinked starch colloidosome

Water-permeable hollow starch particles alter the rheological behavior of their granular suspensions. However, their thin shells can rupture limiting applications. In this study, we used amaranth starch as building blocks (1 μm) to craft a crosslinked superstructure. Pickering emulsions were used as the templates where starch coated the droplets. Emulsions were heated at 75 °C to induce interpenetration of the polymers followed by precipitation in ethanol to trigger colloidal fusion. Particles were then crosslinked by sodium tri-metaphosphate; hollow particles formed after the interior template was removed by hexane. When canola oil was used, the particles ruptured at pH 11.5 due to the repulsion between the strands. In contrast, palm oil, emulsified at 50 °C, formed a rigid core after cooling, locked the starch at the surface and retained the structure. The crosslinked colloidosomes were larger (89 μm) and exhibited higher viscosity, and stronger stability. Larger particles (>100 μm) were produced using higher templating volume. Gentle centrifugation to harvest the particles kept the shells intact. The hollow structure exhibited jamming transition above 10 w/w%, which could serve as a super-thickener. This work demonstrates that microarchitecture plays a critical role in shaping material functionality.

Undular bores in granular suspensions: a dynamical systems study

Undular bores in granular suspensions: a dynamical systems study

Percolation to jamming in polymethylvinylsiloxane/silica nanocomposites

Researchers have conducted extensive experimental and theoretical studies on the jamming transition of colloidal and granular suspensions, but the existence and characteristics of jamming transition in nanoparticle-filled polymers remain elusive. In this work, two kinds of polymethylvinylsiloxane/silica nanocomposites were studied. A liquid-solid transition was found at the silica volume fraction of about 5 vol%, corresponding to the percolation of particle agglomerates. At higher silica content (about 10 vol%), a solid-solid transition was found in yield stress, equilibrium modulus, and the contact stress between agglomerates. The characteristic length of the particle network and inter-molecular distance also showed a transition at this transition. This transition corresponds to the simultaneous jamming of particles inside agglomerates and between agglomerates. This bi-level jamming behavior is distinct from the well-known jamming behavior in soft or hard particulate suspensions. Moreover, the dependence of the yield stress and equilibrium modulus on the distance to the jamming concentration differs from the simulations using simple interaction models like harmonic and Hertzian models. It implies that the interfacial viscoelastic layer may offer additional attractive and adhesive interactions between nanoparticles, which is critical in the jamming of polymer nanocomposites.

Crafting Tunable Hollow Particles Using Antisolvent‐Driven Interlocking of Micron‐Sized Building Blocks

AbstractThe contribution of a hollow structure on the rheological behavior of granular suspensions remains un‐investigated due to the challenge of water impermeability. Here starch is used to fabricate water‐permeable hollow particles, as a model for granular suspensions and investigated the resulting microstructures and rheological behavior. The hollow structure is fabricated based on a bottom‐up method by assembling micron‐sized building blocks into a superstructure. A Pickering emulsion is heated to fuse the starch interface, then, upon antisolvent precipitation, the polymer strands interlock to form a rigid shell around the oil template. When the template is removed a hollow particle remained. These particles exhibited a specific volume >5‐times higher than unmodified starch and consequently a higher viscosity. Larger particles showed higher viscosity but are also more fragile. The template structure can be manipulated to fine‐tune their functionality. These micro‐sized building blocks made from edible materials can be used as the next generation of texturizers. Additionally, these water‐permeable colloidosomes present an innovative approach to understanding how micro‐architectures impact the rheological behavior of granular suspensions.

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model.

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

Maximal submergence in dense granular suspensions

How far can a person sink downwards in quicksand? Experience would seem to suggest that there is low risk of submerging completely, but it is not easy to demonstrate this because of the complex rheology of granular suspensions. We study several mathematical models for the sinking of a vertical cylinder downwards into quicksand, finding that an approach with a buoyancy equation modified by drag force gives an unphysical answer. We instead argue that our proposed conclusion is supported by considering the dynamics of vibration-induced compactification of liquid-saturated granular suspensions. We compare quicksand with other non-Newtonian fluids, emphasising that in this case the same model does not apply and that the risk of drowning could be much more significant. We finish by suggesting some relevant experiments that can be performed in a classroom setting.

Immiscible invasion of granular suspension by high-viscosity liquid

We investigate the morphology and dynamics of the pattern of immiscible invasion by injecting a high-viscosity liquid into a granular suspension consisting of movable solid grains in a low-viscosity liquid. Laboratory experiments conducted in a Hele–Shaw cell shed light on how the frictional forces of the grains and the viscous forces of the liquids affect the instability of the liquid–liquid interface and the formation of viscous fingers. The frictional force increases with an increase in either the volume fraction or the size of the grains, leading to higher resistance to the invading pattern. Upon changing the grain shape from spherical to irregular, both the frictional force and the rotational energy of the grains increase, resulting in more numerous but narrower fingers. Increasing either the injection rate or the viscosity of the injected liquid increases the viscous pressure within the fingers, promoting the splitting of the pattern. Although the defending liquid always has a lower viscosity than the invading liquid in this study, the former's viscous force becomes non-negligible as the viscosity ratio of the invading liquid to the defending liquid decreases to near unity, which destabilizes the fluid–fluid interface and causes a transition to an asymmetric pattern.

Mobility and diffusion of intruders in granular suspensions: Einstein relation

The Enskog kinetic equation is considered to determine the diffusion D and mobility λ transport coefficients of intruders immersed in a granular gas of inelastic hard spheres (grains). Intruders and grains are in contact with a thermal bath, which plays the role of a background gas. As usual, the influence of the latter on the dynamics of intruders and grains is accounted for via a viscous drag force plus a stochastic Langevin-like term proportional to the background temperature T b. In this case, the starting kinetic equations are the Enskog and Enskog–Lorentz equations for grains and intruders, respectively, with the addition of Fokker–Planck terms to each one of the above equations. The transport coefficients λ and D are determined by solving the Enskog–Lorentz kinetic equation by means of the Chapman–Enskog method adapted to inelastic collisions. As for elastic collisions, both transport coefficients are given in terms of the solutions of two integral equations, which are approximately solved up to the second order in a Sonine polynomial expansion. Theoretical results are compared against numerical solutions of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. In general, good agreement between theory and simulations is found, especially in the case of the second Sonine approximation. Knowledge of the coefficients λ and D allows us to assess the departure of the (conventional) Einstein relation ϵ=D/(Tbλ) from 1. As expected from previous results for driven granular gases, it is shown that when the bath temperature T b is replaced by the intruder temperature T 0 in the Einstein relation, the origin of the deviation of ε from 1 is only due to the non-Maxwellian behavior of the reference state of intruders (measured by the cumulant c 0). Since the magnitude of c 0 is in general very small, deviations of the (modified) Einstein relation ϵ0=D/(T0λ) from 1 cannot be detected in computer simulations of dilute granular gases. This conclusion agrees well with previous computer simulation results.

Granular aqueous suspensions with controlled interparticular friction and adhesion.

We present a simple route to obtain large quantities of suspensions of non-Brownian particles with stimuli-responsive surface properties to study the relation between their flow and interparticle interactions. We perform an alkaline hydrolysis reaction on poly(methyl methacrylate) (PMMA) particles to obtain poly(sodium methacrylate) (PMAA-Na) particles. We characterize the quasi-static macroscopic frictional response of their aqueous suspensions using a rotating drum. The suspensions are frictionless when the particles are dispersed in pure water. We relate this state to the presence of electrosteric repulsion between the charged surfaces of the ionized PMAA-Na particles in water. Then we add monovalent and multivalent ions (Na+, Ca2+, La3+) and we observe that the suspensions become frictional whatever the valency. For divalent and trivalent ions, the quasi-static avalanche angle θc at large ionic strength is greater than that of frictional PMMA particles in water, suggesting the presence of adhesion. Finally, a decrease in the pH of the suspending solution leads to a transition between a frictionless plateau and a frictional one. We perform atomic force microscopy (AFM) to relate our macroscopic observations to the surface features of the particles. In particular, we show that the increase in friction in the presence of multivalent ions or under acidic conditions is driven by a nanoscopic phase separation and the bundling of polyelectrolyte chains at the surface of the particle. Our results highlight the importance of surface interactions in the rheology of granular suspensions. Our particles provide a simple, yet flexible platform to study frictional suspension flows.

Multiphase granular suspension flow of fresh concrete during pumping in complex geometries

AbstractDuring the pumping process of multiphase granular suspensions, such as concrete, through pipe systems, many parameters can influence the material's ability to flow. To understand the hydrodynamics and particle mechanics in granular, non‐Newtonian suspensions under nonlinear flow conditions, the authors have been working on a joint project within the Priority Program SPP 2005 “Opus Fluidum Futurum ‐ Rheology of Reactive, Multiscale, Multiphase Construction Materials” funded by the German Research Foundation (DFG). Based on the results obtained, it is possible to describe the relationships between concrete composition and the rheological properties of the fresh mixture for pumping applications. The effects of particle geometry and flow conditions on flow‐induced particle migration (FIPM), subsequently the formation of a lubricating layer (LL), and eventually on pumping pressure were also shown. While the behavior of concrete and other complex, multiscale suspensions in generic flows (Couette, Poiseuille) is still part of current research and only partially understood, the behavior in complex geometries is even less explored so far. Here, additional effects based on the curvature of streamlines and mass inertia of the particle phase come into play. In the framework of the project, these effects have been investigated experimentally and numerically. Various experimental approaches have been developed to observe the behavior of particles within their carrier fluid in an opaque, dense, and enclosed moving system, yielding insightful results. Currently, models are being developed and experimentally validated to describe particle migration quantitatively, allowing conclusions about the local rheological behavior to be drawn. This information is essential to predict the flow behavior necessary for modern processes such as 3D printing and energy‐optimized pumping.

Multi-Sized Granular Suspension Transport Modeling for the Control of Lost Circulation and Formation Damage in Fractured Oil and Gas Reservoirs

Transport and retention of multi-sized suspended granules are common phenomena in fracture media of oil, gas and geothermal reservoirs. It can lead to severe permeability damage and productivity decline, which has a significant impact on the efficient development of underground resources. However, the granule transport and retention behaviors remain not well understood and quantified. The novel stochastic model is proposed for the multi-sized suspended granule transport in naturally fractured reservoirs accounting for granule retention and fracture clogging kinetics. A percolation fracture network is proposed considering fracture connectivity evolution during suspended granule transport. Granule retention and fracture clogging dynamics equations are proposed to account for incomplete fracture clogging by retained granules. The microscale stochastic model is allowed for upscaling to predict the multi-sized granule transport behavior in naturally fractured reservoirs. The model solution exhibits preferential plugging of fractures with sizes equal to or below the granule size. Multi-sized suspended granule shows great advantages over mono-sized suspended granule in the control of permeability damage induced by granule retention and fracture clogging. The retained granule concentration and permeability damage rate decrease with fracture network connectivity improvement. The experimental investigation on size-exclusion suspended granule flow has been performed. The model-based prediction of the retained granule concentration and permeability variation history shows good agreement with the experimental data, which verifies the developed model.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation.

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equationby means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equationfor the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equationby means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Tracer diffusion coefficients in a moderately dense granular suspension: Stability analysis and thermal diffusion segregation

The diffusion transport coefficients of a binary granular suspension where one of the components is present in tracer concentration are determined from the (inelastic) Enskog kinetic equation. The effect of the interstitial gas on the solid particles is accounted for in the kinetic equation through two different terms: (i) a viscous drag force proportional to the particle velocity and (ii) stochastic Langevin-like term defined in terms of the background temperature. The transport coefficients are obtained as the solutions of a set of coupled linear integral equations recently derived for binary granular suspensions with arbitrary concentration [Gómez González et al., “Enskog kinetic theory for multicomponent granular suspensions,” Phys. Rev. E 101, 012904 (2020)]. To achieve analytical expressions for the diffusion coefficients, which can be sufficiently accurate for highly inelastic collisions and/or disparate values of the mass and diameter rations, the above integral equations are approximately solved by considering the so-called second Sonine approximation (two terms in the Sonine polynomial expansion of the distribution function). The theoretical results for the tracer diffusion coefficient D0 (coefficient connecting the mass flux with the gradient of density of tracer particles) are compared with those obtained by numerically solving the Enskog equation by means of the direct simulation Monte Carlo method. Although the first-Sonine approximation to D0 yields, in general, a good agreement with simulation results, we show that the second-Sonine approximation leads to an improvement over the first-Sonine correction, especially when the tracer particles are much lighter than the granular gas. The expressions derived here for the diffusion coefficients are also used for two different applications. First, the stability of the homogeneous steady state is discussed. Second, segregation induced by a thermal gradient is studied. As expected, the results show that the corresponding phase diagrams for segregation clearly differ from those found in previous works when the effect of gas phase on grains is neglected.

The characterization of the particle normal stresses of concentrated granular suspensions by local rheometry

The normal and shear viscosities of non-Brownian suspensions are measured by optical suspension imaging for particle volume fractions $\phi$ between $0.3\phi _m$ and $0.98\phi _m$ , where $\phi _m$ is the jamming fraction. Two distinct refractive-index-matched suspensions, made with the same polymethyl methacrylate spherical particles dispersed in a mixture of water and Triton X-100, are studied. One is density-matched while the other one is negatively buoyant. They are both sheared in a Couette rheometer where the velocity and particle volume fraction fields are measured. The shear viscosity and the second particle normal stress $\varSigma _{22}^p$ are determined through the study of these profiles in the neutrally buoyant suspension, while the third particle normal stress $\varSigma _{33}^p$ is deduced from the analysis of the vertical $\phi$ profiles measured in the negatively buoyant suspension. Our results indicate that the shear viscosity decreases with shear stress $\varSigma _{12}$ , and that this shear-thinning behaviour can be captured by the variation of $\phi _m$ with $\varSigma _{12}$ . We show that $\varSigma _{33}^p$ is proportional to $\varSigma _{12}$ , and that $\varSigma _{33}^p/\eta _0\dot {\gamma }$ is a function of only $\phi /\phi _m(\varSigma _{12})$ . The values of $\varSigma _{22}^p$ deduced from the radial $\phi$ profiles are consistent with the results of Zarraga et al. (J. Rheol., vol. 44, 2000, pp. 185–220). We conclude by discussing our results in the framework of the $\mu (J)$ rheology for viscous numbers $J$ ranging from $2\times 10^{-4}$ to $3\times 10^1$ . We obtain very good agreement with the results obtained by Boyer et al. for $J\lesssim 10^{-1}$ (Phys. Rev. Lett., vol. 107, 2011, 188301) and by Zarraga et al. for $J\gtrsim 10^{-1}$ .

Experiment and simulation study on coarse aggregates settlement in vibrated concrete based on transparent granular suspensions

It is difficult to directly obtain the movement of coarse aggregates (CAs) since concrete is opacity. This paper mixes a transparent granular suspensions similar to fresh concrete for the vibration experiment, using white oil, silica powder, fused quartz sand and fused quartz stone. The vibration destroys the initial network structure of the granular suspensions causing the support force acting on CAs to decrease and CAs to rapidly accelerate sinking. Then, the granular suspensions gradually form a steady network structure making the support force increase as a logarithmic function and the sinking speed of CAs decrease. The variation law is combined with numerical simulation and concrete vibration experiments to obtain CAs settlement process in concrete with different vibration parameters. Under the same vibration frequency and different amplitudes, unit mass of concrete absorbs similar energy to reach the same distribution state of CAs and vibrated state, and this energy increases with the increase of vibration frequency. Therefore, the low-frequency and high-amplitude has higher construction efficiency than high-frequency and low-amplitude vibration. However, the high-frequency and low-amplitude vibration has larger range of optimal vibration time. The proposed method is helpful to better understand the settlement process of CAs and the energy can be used to evaluate the compactness and segregation state of vibrated concrete.