The theoretical treatment of depletion gels with central interactions often involves expanding the free energy around a stress-free reference state to derive a constitutive relation between global stress and strain. The premise upon which the previous continuum theories are based, i.e., the stress-free reference state and the affine deformation, both of which do not hold in the context of amorphous gel materials. Gels never reach a true global minimum in the potential energy landscape and contain local regions of significant compressive and tensile stress, interspersed with zero-stress regions. Hence, expansion of free energy around a stressed reference state will produce scalar terms in harmonic expansion, the effects of which are qualitatively different from the terms appearing in the expansion around an unstressed reference state. In this study, we demonstrate the limitations of traditional continuum theories and propose simple constitutive relations that better capture the mechanical response of gel materials. The robustness of the proposed relations is established through large-scale numerical simulations of depletion and frictional gels across a vast parameter space.

In this study, we employ large scale numerical simulations complemented by a micromechanical model to investigate the consolidation process of a strong depletion gel. We clearly demonstrate that the origin of the normal stress difference in such depletion gel systems is a direct consequence of the mechanical anisotropy in the force networks, which cannot be captured by traditional continuum models that start with a stress-free reference state. The history-dependent nonlinear effects of prestress are better captured by tracking the evolution of essential state parameters, such as force anisotropy, mean normal force, and the average bond number, which govern different aspects of the consolidation process. A simple micromechanical constitutive relationship is proposed between the different stress tensor components and the internal state parameters, which is in excellent agreement with the simulation observations. Intriguingly, our findings strongly indicate the dominance of particle length scale phenomena in dictating the mechanical response of the consolidating depletion gel. This observation directly contradicts the existing literature's assumption, namely, that a characteristic cluster length scale larger than the particle length scale determines the mechanical response.

We systematically investigate the impact process on the granular material in the high speed limit and show the breakdown of the conventional drag models. Our results also suggest that spatial dimensionality influences the early-stage peak force.

The Eshelby problem refers to the response of a two-dimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the so-called Eshelby kernel, which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here, we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modifications are needed for modeling correctly the spatial responses to plastic events in amorphous solids.

In this work, we thoroughly investigate the impact process on the granular media in the limit when the ratio of the impact velocity to the acoustic speed becomes of the order of 0.01-1, which is far greater than the existing literature (0.0001-0.001). We show that the energy dissipation is largely due to the energy cost associated with the exploration between different metastable states via large scale reorganization of the force chain network. In this regime, the conventional drag force models break down, and the drag force can not be decomposed into a depth dependent static pressure and a depth independent inertial drag as proposed in the existing literature. The high dynamical stress generates acoustic pulses, which propagate longer distances rather than decaying exponentially, as observed in the previous works. In the latter stage of the impact process, the boundary also plays an essential role in the reorganization of the force chains as the reflected acoustic pulses interact with the original impact pulses. Furthermore, we study the scaling of the early stage peak forces with the impact velocity and find that spatial dimensionality strongly influences the scaling.

When amorphous solids are subjected to simple or pure strain, they exhibit elastic increase in stress, punctuated by plastic events that become denser (in strain) upon increasing the system size. It is customary to assume in theoretical models that the stress released in each plastic event is redistributed according to the linear Eshelby kernel, causing avalanches of additional stress release. Here we demonstrate that contrary to the uniform affine strain resulting from simple or pure strain, each plastic event is associated with a non-uniform strain that gives rise to a displacement field that contains quadrupolar and dipolar charges that typically screen the linear elastic phenomenology and introduce anomalous length-scales and influence the form of the stress redistribution. An important question that opens up is how to take this into account in elasto-plastic models of shear induced phenomena like shear-banding.

Predicting the mechanical response of soft gel materials under external deformation is of paramount importance in many areas, such as foods, pharmaceuticals, solid-liquid separations, cosmetics, aerogels, and drug delivery. Most of the understanding of the elasticity of gel materials is based on the concept of fractal scaling with very few microscopic insights. Previous experimental observations strongly suggest that the gel material loses the fractal correlations upon deformation and the range of packing fraction up to which the fractal scaling can be applied is very limited. In addition, correctly implementing the fractal modeling requires identifying the elastic backbone, which is a formidable task. So far, there is no clear understanding of the gel's elasticity at high packing fractions or the correct length scale that governs its mechanical response. In this work, we undertake extensive numerical simulations to elucidate the different aspects of stress transmission in gel materials. We observe the existence of two percolating networks of compressive and tensile normal forces close to the gel point. We also find that the probability distribution for the compressive and tensile parts normalized by their respective mean shows a universal behavior irrespective of various values of interaction potential and thermal energy and different particle size distributions. Interestingly, there are also a large number of contacts with zero normal force, and, consequently, a peak in the normal force distribution is observed at fn ≈ 0 even at higher pressures. We also identify the critical internal state parameters, such as the mean normal force, force anisotropies, and the average coordination number, and propose simple constitutive relations that relate different components of stress to internal state parameters. The agreement between our model prediction and the simulation observation is excellent. It is shown that the anisotropy in the force networks gives rise to the normal stress difference in soft gel materials. Our results strongly demonstrate that the mechanical response of the gel system is governed mainly by the particle length scale phenomena, with a complex interplay between the compressive and tensile forces at the particle contact.

In this work, we systematically investigate for the first time the nature of stress correlations in soft colloidal gel materials which support tensile and compressive forces as well as finite rolling torque, as a function of system pressure. Similar to previous studies on frictional granular matter with only compressive forces and without any rolling torque, the full stress autocorrelation matrix is dictated by the pressure and torque autocorrelations due to mechanical balance and material isotropy constraints. Surprisingly, it is observed that the gel materials do not behave as a normal elastic solid close to the gel point as assumed loosely in the literature because the real space pressure fluctuations decay slower than the normal. We also demonstrate that at low pressure the fractal like structural correlation determines the pressure fluctuations and this is manifested in the real space in terms of inhomogeneous and anisotropic force networks formed due to large voids. Far away from the gel point, as the voids collapse under compression, the force chain network becomes homogeneous and isotropic and the pressure fluctuations become normal leading to normal elastic decay at long range, behaving similar to frictionless granular matter and glass. We also observe that the torque autocorrelation is not hyperuniform in the presence of rolling resistance close to the gel point. Furthermore, we link the abnormal pressure fluctuations to the non-hyperuniform behaviour of the system with respect to the local packing fraction fluctuations, thus relating the deviations from the normal elastic behaviour across various non-equilibrium systems under a common framework.

The concept of mechanical screening is widely applied in solid-state systems. Examples include nucleation of defects in crystalline materials, scars and pleats in curved crystals, wrinkles in strongly confined thin sheets, and cell-rearrangements in biological tissue. Available theories of such screening usually contain a crucial ingredient, which is the existence of an ordered reference state, with respect to which screening elements nucleate to release stresses. In contradistinction, amorphous materials in which a unique reference state does not exist, nevertheless exhibits plastic events that act as screening geometric charges with significant implications on the mechanical response. In a recent paper [Phys. Rev. E 104, 024904] it was proposed that mechanical strains in amorphous solids can be either weakly or strongly screened by the formation of low or high density of plastic events. At low densities the screening effect is reminiscent of the role of dipoles in dielectrics, in only renormalizing the elastic moduli. The effect of high density screening has no immediate electrostatic analog and is expected to change qualitatively the mechanical response, as seen for example in the displacement field. On the basis of experiments and simulations, we show that in granular matter, strong screening results in significant deviation from elasticity theory. The theoretical analysis, which accounts for an emergent inherent length scale, the experimental measurements and the numerical simulations of frictional granular amorphous assemblies are in agreement with each other, and provide a strong support for the novel continuum theory.

This paper investigates whether in frictional granular packings, like in Hamiltonian amorphous elastic solids, the stress autocorrelation matrix presents long range anisotropic contributions just as elastic Green's functions. We find that in a standard model of frictional granular packing this is not the case. We prove quite generally that mechanical balance and material isotropy constrain the stress auto-correlation matrix to be fully determined by two spatially isotropic functions: the pressure and torque auto-correlations. The pressure and torque fluctuations being respectively normal and hyperuniform force the stress autocorrelation to decay as the elastic Green's function. Since we find the torque fluctuations to be hyper-uniform, the culprit is the pressure whose fluctuations decay slower than normally as a function of the system's size. Investigating the reason for these abnormal pressure fluctuations we discover that anomalous correlations build up already during the compression of the dilute system before jamming. Once jammed these correlations remain frozen. Whether this is true for frictional matter in general or is it the consequence of the model properties is a question that must await experimental scrutiny and possible alternative models.

Theoretical treatments of frictional granular matter often assume that it is legitimate to invoke classical elastic theory to describe its coarse-grained mechanical properties. Here, we show, based on experiments and numerical simulations, that this is generically not the case since stress autocorrelation functions decay more slowly than what is expected from elasticity theory. It was theoretically shown that standard elastic decay demands pressure and torque density fluctuations to be normal, with possibly one of them being hyperuniform. However, generic compressed frictional assemblies exhibit abnormal pressure fluctuations, failing to conform with the central limit theorem. The physics of this failure is linked to correlations built in the material during compression from a dilute configuration prior to jamming. By changing the protocol of compression, one can observe different pressure fluctuations, and stress autocorrelations decay at large scales.

Chandana Mondal, Michael Moshe, Itamar Procaccia∗,3 Saikat Roy, Jin Shang, and Jie Zhang Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem, Israel 9190 Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel, ∗Center for OPTical IMagery Analysis and Learning, Northwestern Polytechnical University, Xi’an, 710072 China Department of Chemical Engineering, Indian Institute of Technology Ropar, Punjab 140001, India School of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China Institute of Natural Sciences and School of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China

Shear yield stress is an important parameter in the processing of colloidal suspensions as it characterizes the solid-to-fluid transition. Although shear rheology of colloidal gel is of widespread academic and industrial interest, first principles theory that connects the microscopic properties to the macroscopic mechanical response in a self-consistent manner is lacking. In this work, we derive a constitutive relation to predict the yield stress for a strongly attractive gel undergoing quasi-static shear deformation as a function of volume fraction, inter-particle potential, contact scale properties and the microstructure of a strongly-aggregated colloidal gel. The model also predicts the strain at which the colloidal gel network will yield under shear load. To test the model, discrete element simulation is performed using a non-central potential with friction while accounting for the rolling resistance, which is important in real colloidal gel systems. The theoretical predictions are not only in good agreement with the simulation results, but also with previous experiments.

This work is aimed at understanding the basic principles of adsorption process in great details as adsorptive separation process has broad applications in the industry. To this end, a simple mathematical model has been used to describe transient fixed bed physical adsorption process. Governing equations are solved numerically to obtain breakthrough curves for single component and multi-component monolayer adsorption. Desorption of a saturated bed by an inert fluid is also considered. A full parametric study is performed to analyze the effects of different parameters such as bed length, velocity, diffusivity, particle radius and isotherm properties on the nature of the breakthrough curve. Analysis of these results led to the development of the generic breakthrough curve for a single component monolayer adsorption which will enable us to tell the nature of breakthrough curve for different process parameters without recourse to the numerical simulation or experiment. Thus this study will be of great interest in the industrial separation process.

When compressed frictional granular media are decompressed, generically a fragile configuration is created at low pressures. Typically this is accompanied by a giant frictional slippage as the fragile state collapses. We show that this instability is understood in terms of a scaling theory with theoretically computable amplitudes and exponents. The amplitude diverges in the thermodynamic limit hinting to the possibility of huge frictional slip events in decompressed granular media. The physics of this slippage is discussed in terms of the probability distribution functions of the tangential and normal forces on the grains which are highly correlated due to the Coulomb condition.

We study agitated frictional disks in two dimensions with the aim of developing a scaling theory for their diffusion over time. As a function of the area fraction ϕ and mean-square velocity fluctuations 〈v^{2}〉 the mean-square displacement of the disks 〈d^{2}〉 spans four to five orders of magnitude. The motion evolves from a subdiffusive form to a complex diffusive behavior at long times. The statistics of 〈d^{n}〉 at all times are multiscaling, since the probability distribution function (PDF) of displacements has very broad wings. Even where a diffusion constant can be identified it is a complex function of ϕ and 〈v^{2}〉. By identifying the relevant length and time scales and their interdependence one can rescale the data for the mean-square displacement and the PDF of displacements into collapsed scaling functions for all ϕ and 〈v^{2}〉. These scaling functions provide a predictive tool, allowing one to infer from one set of measurements (at a given ϕ and 〈v^{2}〉) what are the expected results at any value of ϕ and 〈v^{2}〉 within the scaling range.

We report a joint experimental and theoretical investigation of cyclic training of amorphous frictional granular assemblies, with special attention to memory formation and retention. Measures of dissipation and compactification are introduced, culminating with a proposed scaling law for the reducing dissipation and increasing memory. This scaling law is expected to be universal, insensitive to the details of the elastic and frictional interactions between the granules.

We report a joint experimental and theoretical investigation of the probability distribution functions (PDFs) of the normal and tangential (frictional) forces in amorphous frictional media. We consider both the joint PDF of normal and tangential forces together, and the marginal PDFs of normal forces separately and tangential forces separately. A maximum entropy formalism is utilized for all these cases after identifying the appropriate constraints. Excellent agreements with both experimental and simulation data are reported. The proposed joint PDF predicts giant slip events at low pressures, again in agreement with observations.

Consolidation of colloidal dispersions under external load is a complex process involving inter-particle interactions, thermal forces and hydrodynamics. Despite its importance in diverse industrial applications, past studies involving experiments, scaling approaches and simulations are yet to provide a comprehensive understanding of how the microstructure determines the mechanical response in three dimensional colloidal gels. Here, we develop a model that accounts for the microstructural details and predicts the mechanical response under slow, uniaxial compression of a strongly aggregated three dimensional colloidal gel. The particle network assumes a fractal structure that is independent of the strength of inter-particle interactions. While the yield strain changes negligibly during the entire process, the yield stress increases by several orders of magnitude. The predicted yield stress and strain are in close agreement with those observed in simulations and experiments with diverse colloidal systems, suggesting a universality in the consolidation process.

We derive a constitutive relation to describe the deformation of a two-dimensional strongly aggregated colloidal system by incorporating the interparticle colloidal forces and contact dynamics. The theory accounts for the plastic events that occur in the form of rolling/sliding during the deformation along with elastic deformation. The theory predicts a yield stress that is a function of volume fraction of the colloidal packing, the coordination number, the interparticle potential, coefficient of friction, and the normal and the tangential stiffness coefficients. The predicted yield strain was independent of the particle volume fraction although the compressive yield stress exhibited a power-law relation with the volume fraction. The power-law exponent, however, was lower than that obtained from simulations reported in a paper by Roy and Tirumkudulu [“Yielding in a strongly aggregated colloidal gel. Part I: 2D simulations,” J. Rheol. 60(4), 559–574 (2016)]. The cause for the discrepancy was identified to be the nonaffine deformation of the network. To account for such effects, a constitutive relation based on a simple fractal model was developed that predicts yield stress profile close to those obtained from simulations.