Random Suspensions Research Articles

Fully optimized second-order estimates for the macroscopic behavior and field statistics of particle-reinforced viscoplastic composites

This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions c, the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as c tends to zero, the effective yield stress is shown to depend linearly on c for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of O(c/lnc) for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As c increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

Comprehensive quasi-steady force correlations for compressible flow through random particle suspensions

Models for quasi-steady drag, quasi-steady drag variation, and transverse forces for compressible flows through random, fixed, particle suspensions are presented. Correlations are formulated using drag force measurements obtained from particle-resolved simulation data spanning subsonic to supersonic flow conditions. The newly proposed drag models are extensions of existing models for incompressible dense gas-solid flows to finite Mach numbers, while the transverse particle force model is new and incorporates the covariation between streamwise and transverse forces.

Quantitative homogenization theory for random suspensions in steady Stokes flow

This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

Particle Scale Investigation of Influencing Factors on Heat Transfer in Nonspherical Particle–Fluid System

Abstract Heat transfer characteristics of random suspensions of 0.25 aspect ratio (AR) cylinders are investigated for Reynolds numbers (Re) between 10 and 300 and solid fraction (φ) ranging from 0.1 to 0.3 using particle resolved simulations. The effect of particle inclination with respect to flow and particle clustering on heat transfer is investigated. The Nusselt number decreases with an increase in inclination angle and the dependence becomes stronger as φ and Re increase. On the other hand, while prolate ellipsoid suspensions of AR 2.5 follow the same trend, the Nusselt number increases with inclination angle as AR increases to 5 and 10 and as φ increases. Local particle clustering nominally decreases the Nusselt number because of the dominance of thermal wakes. At low φ, this effect is felt only at low Re, but as φ increases, the effect spreads to higher Re. Similar but weaker trends are also found in suspensions of prolate ellipsoids of AR 2.5, 5, and 10. High AR, low Re prolate ellipsoids exhibit the greatest dependence of Nusselt number on local solid fraction. Implementation of two independent definitions of reference length, i.e., volume equivalent sphere diameter deq for ellipsoids and diameter dp of the cylindrical particle in the correlation of Tavassoli et al. (2015, “Direct Numerical Simulation of Fluid-Particle Heat Transfer in Fixed Random Arrays of Non-Spherical Particles,” Chem. Eng. Sci., 129, pp. 42–48) provides good estimates of the respective suspension mean Nusselt numbers.

A novel method for investigating the isotropy of response for randomly generated unit cells of particulate composites

This work investigates the isotropy of mechanical response of randomly generated unit cells of particulate composites. We propose a novel algorithm for assessing the isotropic behavior of such microstructures by applying different loading in random planes in space. The Random Plane Evaluation of Isotropy (RPEI) is based on calculating the averaged stress and strain under periodic boundary conditions and finding the maximum angle difference between the principal directions of them as a measure for deviation from coaxiality/isotropy. We show that an average of such data at a statistical level can be used for assessing closeness to isotropy. To put the RPEI in use, we have also proposed a new method for generating unit cells containing a random distribution of ellipsoidal particles. Then the isotropy of mechanical behavior for various unit cells enclosing random suspensions of spheres and ellipsoids was investigated. Our results show that (i) Merely random distribution of particles does not guarantee the isotropy of the response (ii) Checking the isotropy in limited directions can be misleading and data at a statistical level is needed (iii) RPEI proves a reliable tool for checking the isotropy of response of unit cells of heterogeneous materials for a variety of behaviors.

Effective viscosity of random suspensions without uniform separation

This work is devoted to the definition and the analysis of the effective viscosity associated with a random suspension of small rigid particles in a steady Stokes fluid. While previous works on the topic have conveniently assumed that particles are uniformly separated, we relax this restrictive assumption in the form of mild moment bounds on interparticle distances.

Sedimentation of random suspensions and the effect of hyperuniformity

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension $$d=3$$ should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature.

The nonlinear viscoelastic response of suspensions of rigid inclusions in rubber: I—Gaussian rubber with constant viscosity

A numerical and analytical study is made of the macroscopic or homogenized viscoelastic response of suspensions of rigid inclusions in rubber under finite quasistatic deformations. The focus is on the prototypical case of random isotropic suspensions of equiaxed inclusions firmly embedded in an isotropic incompressible Gaussian rubber with constant viscosity. From a numerical point of view, a robust scheme is introduced to solve the governing initial–boundary-value problem based on a conforming Crouzeix–Raviart finite-element discretization of space and a high-order accurate explicit Runge–Kutta discretization of time, which are particularly well suited to deal with the challenges posed by finite deformations and the incompressibility constraint of the rubber. The scheme is deployed to generate sample solutions for the basic case of suspensions of spherical inclusions of the same (monodisperse) size under a variety of loading conditions. From a complementary point of view, analytical solutions are worked out in the limits: (i) of small deformations, (ii) of finite deformations that are applied either infinitesimally slowly or infinitely fast, and (iii) when the rubber loses its ability to store elastic energy and reduces to a Newtonian fluid. Strikingly, in spite of the fact that the underlying rubber matrix has constant viscosity, the solutions reveal that the viscoelastic response of the suspensions exhibits an effective nonlinear viscosity of shear-thinning type. The solutions further indicate that the viscoelastic response of the suspensions features the same type of short-range-memory behavior — as opposed to the generally expected long-range-memory behavior — as that of the underlying rubber. Guided by the asymptotic analytical results and the numerical solutions, a simple yet accurate approximate analytical solution for the macroscopic viscoelastic response of the suspensions is proposed.

Derivation of the Batchelor-Green formula for random suspensions

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction ϕ. The goal is to derive rigorously a o(ϕ2) formula for the effective viscosity. In [17,19], such formula was given for rigid spheres satisfying the strong separation assumption dmin≥cϕ−13r, where dmin is the minimal distance between the spheres and r their radius. It was then applied to both periodic and random configurations with separation, to yield explicit values for the O(ϕ2) coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying softer assumptions of separation, and long range decorrelation. We justify in this setting the famous Batchelor-Green formula [3]. Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption dmin>(2+ε)r, ε>0.

Development of drag correlation for suspensions of ellipsoidal particles

To model drag the current state-of-the-art is to use isolated non-spherical particle drag correlations modified by a solid fraction correlation that is based on experimental or simulation results of spherical particle suspensions. It is shown that this practice can lead to substantial inaccuracies when the particle geometry deviates significantly from a spherical geometry. In this paper particle resolved simulations (PRS) are conducted for ellipsoids of aspect ratio 5 (AR5) and 10 (AR10) in random suspensions with no preferential orientation. Simulations are performed for a Reynolds number Re = 10 to 200, and solid fraction φ = 0.1 to 0.3 and 0.1 to 0.2 for AR5 and AR10 suspensions, respectively. Combined with PRS data from past studies for spherical particle suspensions and ellipsoids with AR2.5, a drag correlation is developed for the mean drag force in suspension as a function of Re, φ , aspect ratio, and inclination angle θ.

A hemicyanine-based optical probe for biomembranes and intracellular pH sensing

The hemicyanine-based fluorescent probe L3 was demonstrated as a suitable membrane physiological pH probe. L3 easily interacted with the lipid components and anchored in inner membrane and gave a red fluorescence signal (~ 650 nm) in neutral or acidic pH, while the emission switching off when intramembrane pH varied to basic. Moreover, the absorption color of L3 changed from purple red to colorless with increasing pH. Thus L3 served as the pH-dependent modulator, a dual optical biomembranes pH sensing probe, showing good linear relationship in physiological pH range 7.0–8.5 or 7.5–9.0. The liposomes prepared under random pH and suspensions of gastric cancer cells (HGC-27) samples were used for verification of proposed intracellular pH sensing method, and satisfied results were obtained.

Is one-size-fits-all strategy adequate for platelet storage?

BackgroundPlatelet suspensions (PSs) are stored at room temperature. However, recent reports show that PSs stored at 4 °C possess superior hemostatic properties. We compared the viabilities and thrombin generation capacities of PSs stored either at 4 °C or 22 °C hours. Materials and MethodsTwenty units of apheresis derived platelets (ADPs) from 20 male donors and 20 units of random platelet suspensions (RPSs) from another 20 male donors were obtained. Half of the ADPs and half of the RPSs (10 units/per group) were stored at 4 °C, the other halves of ADPs and RPSs (10 units/per group) were stored in agitators at 22 °C for 48 hours. The flow cytometric viability tests and thrombin generation tests of the PSs were assessed. ResultsThe viabilities of both ADPs and RPSs group platelets, stored either at 4 °C or 22 °C for 48 hours, were not statistically significantly different. The ADPs and RPSs stored at 4 °C generated significantly higher peak thrombin levels than the platelets stored at 22 °C. Moreover, the ADPs group stored at 4 °C showed significantly shorter time to thrombin generation and reach peak levels. ConclusionThe PSs stored at 4 °C showed higher and faster thrombin generation capacities than the room temperature PSs. Given the superior hemostatic properties of refrigerated platelets, creating different storage temperature capabilities for specific transfusion purposes may be a prudent approach, especially for improving the outcome of bleeding trauma casualties.

Random sets of stadiums in square and collective behavior of bacteria.

Collective motion of swimmers can be detected by hydrodynamic interactions through the eective (macroscopic) viscosity. It follows form the general hydrodynamics that the eective viscosity of non-dilute random suspensions depends on the shape of particles and of their spacial probabilistic distribution. Therefore, a comparative analysis of disordered and collectively interacting particles of the bacteria shape can be done in terms of the probabilistic geometric parameters which determine the eective viscosity. In this paper, we develop a quantitative criterion to detect the collective behavior of bacteria. This criterion is based on the basic statistic moments (e-sums or generalized Eisenstein-Rayleigh sums) which characterize the high-order correlation functions. The locations and the shape of bacteria are modeled by stadiums randomly embedded in medium without overlapping. This shape models can be considered as improvement of the previous segment model.We calculate the e-sums of the simulated disordered sets and of the observed experimental locations of bacteria subtilis. The obtained results show a dierence between these two sets that demonstrates the collective motion of bacteria.

Large-scale calculation of hydrodynamic transport properties for random suspensions of hard-sphere particles

A numerical method based on fast multipole summation scheme is used to calculate hydrodynamic interactions in random suspensions of non-colloidal hard-sphere particles. The calculation is carried out for suspensions of 1,024 particles randomly placed in periodic unit cell to determine hydrodynamic transport properties such as permeability of a viscous flow through porous medium, effective viscosity of suspension, and sedimentation velocity of the suspended particles. The particle volume fraction o ranges from 0.01 to 0.25. Effect of particle number N on the transport properties was examined through the numerical calculations with N=64-1,024. It is shown that sedimentation velocity increases with N approaching an estimate for infinite N, and the finite N effect is negligible in effective viscosity and permeability problems. The present scheme is quite useful for obtaining a statistically-averaged quantity for random suspensions. As an example, ensemble-averaged velocity when position of one particle is fixed is numerically obtained in sedimentation problem. The numerical results are shown to be in excellent agreement with theoretical prediction.

Colloidal crystallization utilizing interfaces of unidirectionally growing ice crystals

We show the possibility of unidirectional freezing of colloidal random suspensions as a novel method for growing colloidal crystals. First we confirmed the rejection of polystyrene particles from unidirectionally growing ice–water interfaces by observing color gradations of colloidal crystals formed in front of the interfaces. The rejection of particles from the growing interfaces increased particle concentration in the colloidal crystals. Then we succeeded in achieving colloidal crystallization in front of ice–water interfaces during the unidirectional freezing of water suspensions of polystyrene particles. The colloidal crystals thus obtained exhibited a columnar shape that is typically observed in the unidirectional colloidal crystallization using centrifugation.

Roles of particle-wall and particle-particle interactions in highly confined suspensions of spherical particles being sheared at low Reynolds numbers

The roles of particle-wall and particle-particle interactions are examined for suspensions of spherical particles in a viscous fluid being confined and sheared at low Reynolds numbers by two parallel walls moving with equal but opposite velocities. Both particle-wall and particle-particle interactions are shown to decrease the rotational velocity of the spheres, so that in the limit of vanishingly small gaps between the spheres and the walls, the spheres acquire a rotational slip relative to the walls. The presence of the walls also increases the particle stresslet and, therefore, the total viscous dissipation. In the limit of vanishingly small gaps, the increased viscous dissipation in the gaps between pairs of spheres aligned in the flow direction is largely compensated by the reduction in the dissipation in the gaps between the spheres and the walls due to a reduction in the rotational velocity of the spheres. As a result, the effect of short-range particle interactions on the stresslet is generally insignificant. On the other hand, the channel-scale particle interactions in the shear flow induced by the moving walls decrease the particle stresslet, primarily because the fraction of pairs of spheres that are aligned parallel to the flow (the presence of which in a shear flow reduces the stresslet) is relatively higher than in unbounded suspensions. Expressions are also derived for the total stress in dilute random suspensions that account for both the particle-wall and the channel-scale particle-particle interactions in determining the rotational velocities and stresses. The latter are shown to be consistent with recent numerical [Y. Davit and P. Peyla, Europhys. Lett. 83, 64001 (2008)] and experimental [P. Peyla and C. Verdier, Europhys. Lett. 94, 44001 (2011)] findings according to which, for a range of sphere radius to gap width ratios, the effect of particle-particle interactions is to decrease the total dissipation.

Local transient rheological behavior of concentrated suspensions

This paper reports experiments on the shear transient response of concentrated non-Brownian suspensions. The shear viscosity of the suspensions is measured using a wide-gap Couette rheometer equipped with a Particle Image Velocimetry (PIV) device that allows measuring the velocity field. The suspensions made of PMMA particles (31$\mu$m in diameter) suspended in a Newtonian index- and density-matched liquid are transparent enough to allow an accurate measurement of the local velocity for particle concentrations as high as 50%. In the wide-gap Couette cell, the shear induced particle migration is evidenced by the measurement of the time evolution of the flow profile. A peculiar radial zone in the gap is identified where the viscosity remains constant. At this special location, the local particle volume fraction is taken to be the mean particle concentration. The local shear transient response of the suspensions when the shear flow is reversed is measured at this point where the particle volume fraction is well defined. The local rheological measurements presented here confirm the macroscopic measurements of Gadala-Maria and Acrivos (1980). After shear reversal, the viscosity undergoes a step-like reduction, decreases slower and passes through a minimum before increasing again to reach a plateau. Upon varying the particle concentration, we have been able to show that the minimum and the plateau viscosities do not obey the same scaling law with respect to the particle volume fraction. These experimental results are consistent with the scaling predicted by Mills and Snabre (2009) and with the results of numerical simulation performed on random suspensions [Sierou and Brady (2001)]. The minimum seems to be associated with the viscosity of an isotropic suspension, or at least of a suspension whose particles do not interact through non-hydrodynamic forces, while the plateau value would correspond to the viscosity of a suspension structured by the shear where the non-hydrodynamic forces play a crucial role.

Memory effects for the heat conductivity of random suspensions of spheres

The presence of a particulate phase defines the effective response of a suspension to changes of the average heat flux. Using the random-point approximation we show that within the first order in the concentration, one needs to solve the problem for the temperature field created by a single inclusion in a matrix subject to an unsteady temperature gradient at infinity. We solve this problem by means of Laplace transform and use the solution as the first-order kernel in the functional expansion. From this kernel, we find the statistical average for the heat flux, which turns out to be a memory integral of the spatially averaged time-dependent temperature gradient. Thus, we discover that the constructive relationship between the average flux and averaged temperature gradient is not local in time, but rather involves a convolution integral that represents the memory due to the heterogeneity of the system. This is a novel result, which inter alia gives a rigorous justification to the usage of generalizations of the heat conduction law involving fractional time derivatives. The decay of the kernel is very close to t −1/2 for dimensionless times lesser than one and abruptly changes to t −3/2 for larger times.

Numerical Modeling of Dielectric Breakdown in Solid Propellant Microstructures

This study addresses the industrial issue of dielectric breakdown in aluminized solid propellants. Direct simulations at the microscale level are performed by considering monodisperse hard sphere systems obtained by a packing algorithm. Steady Maxwell equations are solved using a finite differences technique, and breakdown dynamics is modeled by a quasi-steady succession of local breakdowns between particles. Validations of the numerical procedure are successfully carried out on the prediction of the effective electrical conductivity of random suspensions of spheres. Computations of the breakdown field for monodisperse random dispersions accurately match some experimental data as well as an existing mixture law based on the mean nearest-neighbor distance. This mixture law is then applied to four industrial propellants, for which realistic microstructures have been generated and breakdown experimental are data available. It appears that this law is reasonably accurate to predict the breakdown strength of real solid propellants.

Effects of aggregation on the permittivity of random media containing monodisperse spheres

Numerical simulations were used to calculate the effective permittivities of three-dimensional random particle suspensions containing up to 2440 particles and exhibiting two types of particle aggregation. The particles were modeled as 200 μm spheres that were aggregated into either large spherical clusters or into foam-type microstructures with large spherical voids. Multiple scattering of 0.01–10.0 GHz electromagnetic fields was simulated using a first-principles iterative multipole approach with matrix and particle permittivities of 1.0 and 8.5, respectively. The computational results showed both significant and highly significant trends. Aggregation into spherical clusters decreased the effective permittivity by up to 3.2±0.2%, whereas aggregation into foam-type microstructures increased the effective permittivity by up to 3.0±1.6%. The effective permittivity trends exhibited little change with frequency. These results were compared to effective medium approximations that predicted higher permittivities than those from the simulations and showed opposite trends for cluster aggregation. Three theories are proposed to explain the simulation results. The first theory invokes a waveguidelike mechanism. The simulations indicate that the wave fields propagate more through the continuous paths of greater or lesser particle density created by aggregation, rather than through the isolated particle clusters or large voids. This quasicontinuous phase, or quasimatrix, therefore behaves like a random waveguide structure in the material. A second theory is proposed where the quasicontinuous phase governs the behavior of the system by a percolationlike process. In this theory, the multipole interactions are modeled as the percolation of virtual charges tunneling from one particle to another. A third mechanism for the permittivity changes is also proposed involving collective polarization effects associated with the particle clusters or large voids. The simulation results challenge the general applicability of the quasistatic limit for heterogeneous media by showing how microstructural changes much smaller than the electromagnetic wavelength can alter the effective permittivity by a statistically significant degree. The results also provide a quantitative indication of the effects of aggregation and hierarchical microstructures on the electromagnetic properties of random media and have application to the remote and in situ sensing of soils, the rational design and nondestructive evaluation of composites, and the study of biological tissues and other random materials.