Finite Volume Fraction Research Articles

Finite volume fraction effect on self-induced velocity in two-way coupled Euler-Lagrange simulations

Finite volume fraction effect on self-induced velocity in two-way coupled Euler-Lagrange simulations

Non‐Hierarchical Architected Materials with Extreme Stiffness and Strength

AbstractStretch‐dominated truss and plate microstructures are contenders in the quest for realizing architected materials with extreme stiffness and strength. In the low volume fraction limit, closed‐cell isotropic plate microstructures meet theoretical upper bounds on stiffness but have low buckling strength, whereas open‐cell truss microstructures have high buckling strength at the cost of significantly reduced stiffness. At finite volume fractions, the picture becomes less clear but both are outperformed by hollow truss lattice and hierarchical microstructures in terms of buckling strength. Despite significant advances in manufacturing methods, hollow and multi‐scale hierarchical microstructures are still challenging to build. The question is if there exist realizable microstructures providing stiffness and strength matching or even beating hard‐to‐realize hollow or hierarchical microstructures?Herein, single‐scale non‐hierarchical (first order) microstructures that beat the buckling strength of hollow truss lattice structures by a factor of 2.4 and first‐ and second‐order plate microstructures by factors of 5 and 1.4, respectively, are systematically designed, built, and tested. Stiffness of the microstructures is within 40% of theoretical bounds and beats both truss and second order plate microstructures. The microstructures are realized with 3D printing. Experiments validate theoretical predictions and additional insight is provided through numerical modeling of a CT‐scanned sample.

Ostwald ripening of aqueous microbubble solutions.

Bubble solutions are of growing interest because of their various technological applications in surface cleaning, water treatment, and agriculture. However, their physicochemical properties, such as the stability and interfacial charge of bubbles, are not fully understood yet. In this study, the kinetics of radii in aqueous microbubble solutions are experimentally investigated, and the results are discussed in the context of Ostwald ripening. The obtained distributions of bubble radii scaled by mean radius and total number were found to be time-independent during the observation period. Image analysis of radii kinetics revealed that the average growth and shrinkage speed of each bubble is governed by diffusion-limited Ostwald ripening, and the kinetic coefficient calculated using the available physicochemical constants in the literature quantitatively agrees with the experimental data. Furthermore, the cube of mean radius and mean volume exhibit a linear time evolution in agreement with the Lifshitz-Slezov-Wagner (LSW) theory. The coefficients are slightly larger than those predicted using the LSW theory, which can be qualitatively explained by the effect of finite volume fraction. Finally, the slowdown and pinning of radius in the shrinkage dynamics of small microbubbles are discussed in detail.

Revisiting transient coarsening kinetics: a new framework in the Lifshitz-Slyozov-Wagner space

Phase coarsening in multiphase materials is a fundamental and crucial process in the study of microstructure evolution, for which a thorough understanding of kinetics can make a significant contribution to material processing and performance design. In this work, a new framework in the dimensionless Lifshitz-Slyozov-Wagner space is developed to study the kinetics of transient coarsening co-controlled by interface and matrix diffusion. The dynamic equations for individual particles are derived from the thermodynamic extremal principle, in consideration of energy dissipations by matrix diffusion, interface migration/reaction, and trans-interface diffusion. The effects of initial particle size distribution and finite volume fractions are clarified. Different from the conventional viewpoints, the time for transient coarsening is found to change non-monotonically with the width and tail length of the initial distribution. The ultralong transient coarsening with ‘quasi-steady’ distributions can occur for systems initiated from both the ‘wide & long tail’ and ‘narrow & short tail’ distributions. Furthermore, this work proves the existence of a unique attractor state for the steady stage and reveals the numerical origin of ‘quasi-steady’ distributions, which answers Brown's unsolved puzzle for years [LC Brown. Acta Metall 1989;37:71]. Finally, an increase in volume fraction is shown to directly shorten the transient stage dominated by single mechanism (matrix diffusion or interface), but indirectly delay the transition from the interface-dominated state to the diffusion-controlled stage. These findings not only offer new insights into the previous observations in microgravity experiments [VA Snyder, J Alkemper, PW Voorhees, Acta Mater 2001;23:699], but also theoretically describe the limiting coarsening behaviors at ultrahigh volume fractions [H Yan, KG Wang, ME Glicksman. Acta Mater 2022;117964].

Large and small field inflation from hyperbolic sigma models

Long standing themes in inflation include the issue of large field vs. small field inflation as well as the question what fraction of phase space leads to sufficient inflation and furthermore is compatible with the experimental data. In the present paper, these issues are discussed in the context of modular inflation, a specialization of the framework of automorphic nonlinear $\ensuremath{\sigma}$ models associated to homogeneous spaces $G/K$ in which the continuous shift symmetry group $G$ is weakly broken to discrete subgroups $\mathrm{\ensuremath{\Gamma}}$. The target spaces of these theories inherit a curved structure from the group $G$, which in the case of modular invariant inflation leads to a hyperbolic field space geometry. It is shown that in this class of models the symmetry structure leads to both large and small field inflationary trajectories within a single modular inflation model. The present paper analyzes the concrete model of $j$-inflation, a hyperbolic model with nontrivial inflaton interactions. It describes in some detail the structure of the initial conditions, including a systematic analysis of several phenomenological functions on the target space, leading to constraints on the curvature scalar of the field space by upcoming experiments, as well as a discussion of the scaling behavior of the spectral index, the finite volume fraction of the field space leading to sufficient inflation, the attractor behavior of $j$-inflation, and a comparison of inflaton trajectories vs. target space geodesics. The tensor-ratio analysis shows that $j$-inflation is an interesting target for upcoming ground and satellite experiments.

Shear-induced lift force on spheres in a viscous linear shear flow at finite volume fractions

Several studies have shown a significant increase in drag on a distribution of solid spherical particles within a fluid with increasing particle volume fraction. As a result, many empirical drag laws accounting for the dependence on the Reynolds number and volume fraction can be found in the literature. This study investigates the possibility of a similar effect of the particle volume fraction on the mean hydrodynamic lift force on randomly distributed spherical particles in a linear shear flow. Particle-resolved direct numerical simulations are performed to evaluate the mean lift force, and the results are compared with the case of an isolated particle in a linear shear flow for the same Reynolds number and shear rate. The mean lift force acting on the particles appears to remain nearly the same as that on an isolated particle. However, due to the influence of neighboring particles, there is a substantial force variation in transverse directions on each individual particle, whose magnitude is comparable to the mean drag force. The distribution of drag force in a linear shear flow is shown to be nearly the same as in a uniform flow at the same volume fraction and Reynolds number. A simple stochastic model based on a Gaussian distribution is presented for the lift force variation, and its performance is compared to the prediction of the deterministic pairwise interaction extended point-particle model.

Nonlinear compressive stability of hyperelastic 2D lattices at finite volume fractions

A framework is introduced for benchmarking periodic microstructures in terms of their ability to maintain their stiffness under large deformations, accounting in a unified manner both for buckling and softening due to geometric and material nonlinearities. The proposed framework is applied to three classical 2D lattice microstructures at different volume fractions as well as to an optimized hierarchical microstructure from the literature. The high slenderness of the structure members, often assumed in analyses, is demonstrated not to be valid at volume fractions of 10% and above, with the infinitesimal volume fraction solutions underestimating the actual buckling resistance considerably. The performed analyses provide useful and quantitative insight regarding the compressive load carrying capacity of materials with a moderately dense periodic microstructure, in a rather universal and practical form.

High-Temperature Static Coarsening of Gamma-Prime Precipitates in NiAlCr-X Single Crystals

The high-temperature coarsening behavior of γ′ precipitates in a series of NiAlCr, NiAlCrTi, NiAlCrW, and NiAlCrTa single-crystal alloys was determined at temperatures between 1158 K and 1473 K (885 °C and 1200 °C). For this purpose, samples were supersolvus solution treated, water quenched, and then subsolvus aged for times between 0.067 and 96 hours. All of the measurements revealed an r3 dependence of the average precipitate radius on time, thus suggesting bulk-diffusion control of the coarsening process. Coarsening kinetics were fastest for NiCrAl and slowest for NiCrAlTa. The observations were interpreted in terms of the classical Lifshitz–Slyosov–Wagner (LSW) theory modified to account for the finite volume fraction of particles, the composition of the precipitates, and the multicomponent nature of the alloys. By this means, an effective diffusivity for the coarsening process was determined and found to lie between 0.6 and 1.5 times that for the impurity diffusivity of chromium in nickel. Furthermore, the modified LSW theory in conjunction with experimental measurements suggested that the effective diffusivity controlling γ′ coarsening at high temperatures in multi-component nickel-base superalloys lay in the lower portion of this range.

Outsized Amplitude-Modulated Structure of Very-Long-Range Charge Inversions in Model Colloidal Dispersions.

Most theoretical and simulation studies on charged suspensions are at infinite dilution and are focused on the electrolyte structure around one or two isolated particles. Some classic experimental studies with latex particle solutions exhibit interesting phenomenology which imply very-long-range correlations. Here, we apply an integral equation theory to a model charged macroion suspension, at finite volume fraction, and find an amplitude-modulated charge inversion structure, with outsized amplitudes and of very-long-range extension. These inversions are different from the standard charge inversions in that they occur at finite macroions' volume fraction, far away from the central macroion, are outsized, and increase, not decrease, with increasing particle charge and distance to the central particle, which is indicative of long-range correlations. We find our results to be in agreement with our Monte Carlo simulations and qualitatively consistent with existing experimental results.

A general micromechanical framework of effective moduli for the design of nonspherical nano- and micro-particle reinforced composites with interface properties

The morphological and physical properties of interfaces can significantly affect the overall mechanical properties of nano- and micro-particle reinforced composites. In this work, we devise a general micromechanical framework to predict the effective elastic moduli of particle-reinforced composites containing ellipsoidal nano- or micro-particles with varying interface properties (e.g., both hard and soft). Specifically, the interface is treated as an interphase perfectly bonding the particle and matrix with a finite thickness and volume fraction. The morphological characteristics of the interface (e.g., volume fraction) are quantified using a statistical geometry approach and subsequently incorporated into the Mori-Tanaka average scheme with Eshelby's equivalent inclusion theory to derive the general micromechanical framework for the three-phase composites with non-spherical inclusions. We show that our new framework leads to predictions of the elastic moduli of a wide spectrum of nano- and micro-particle reinforced composites with both soft and hard interfaces to a reasonable accuracy by comparing with available experimental data and predictions from other theoretical frameworks. The general framework provides a robust and convenient predictive toolkit for composite design and evaluation.

Ripening kinetics of bubbles: A molecular dynamics study.

The ripening kinetics of bubbles is studied by performing molecular dynamics simulations. From the time evolution of a system, the growth rates of individual bubbles are determined. At low temperatures, the system exhibits a t1/2 law and the growth rate is well described by classical Lifshitz-Slyozov-Wagner (LSW) theory for the reaction-limited case. This is direct evidence that the bubble coarsening at low temperatures is reaction-limited. At high temperatures, although the system exhibits a t1/3 law, which suggests that it is diffusion-limited, the accuracy of the growth rate is insufficient to determine whether the form is consistent with the prediction of LSW theory for the diffusion-limited case. The gas volume fraction dependence of the coarsening behavior is also studied. Although the behavior of the system at low temperatures has little sensitivity to the gas volume fraction up to 10%, at high temperatures it deviates from the prediction of LSW theory for the diffusion-limited case as the gas volume fraction increases. These results show that the mean-field-like treatment is valid for a reaction-limited system even with a finite volume fraction, while it becomes inappropriate for a diffusion-limited system since classical LSW theory for the diffusion-limited case is valid at the dilute limit.

Finite-element modeling of soft solids with liquid inclusions

Surface tension is an important factor in the behavior of fluids but typically has a minimal or negligible effect in solids. However, when a solid is soft and its characteristic dimension is small, forces due to surface tension can become important and significantly affect elastic deformation, leading to interesting elasto-capillary phenomena. We have developed a finite-element formulation accounting for surface tension and large deformations in three-dimensional settings and demonstrate the simulation capability by examining a class of problems involving fluid-filled droplet inclusions in a soft solid matrix. Specifically, we (1) consider the response of isolated droplets under far-field loading and (2) micromechanically model composite materials made up of a finite volume fraction of fluid-filled inclusions in a soft solid matrix. In the latter case, recent experimental work in the literature has shown that when the matrix material is sufficiently compliant, the presence of droplets leads to stiffening–counter to the intuitive notion of the presence of fluid-filled inclusions leading to a more compliant composite material. We show that our numerical simulation capability predicts all experimentally observed phenomena related to fluid-filled inclusions in soft solids. Furthermore, we consider the large-deformation response of composite materials with fluid-filled inclusions–a situation difficult to address using analytical methods.

Nonmodal instability of a stratified plane-channel suspension flow with fine particles.

We consider the nonmodal instability and transient growth of small disturbances in a plane-channel suspension flow with a nonuniform concentration profile of fine noncolloidal particles accumulated in two localized layers, symmetric about the channel axis. A single-velocity model of an effective Newtonian fluid with a finite particle volume fraction is employed. It is established that fine particles distributed nonuniformly in the main flow significantly modify the growth rate of the first mode in a wide range of governing parameters. The most pronounced destabilizing effect is produced by the particles localized in the vicinity of the walls. A parametric study of the so-called optimal disturbances showed that they are streaks elongated in the flow direction, similar to the optimal disturbances in the flow devoid of particles. The transverse wave number of the optimal disturbances depends strongly on the location of the particle layers. Even when the particle mass concentration (averaged over the channel cross section) is small (of the order of a percent) and the particles are localized in the middle between the walls and the channel axis, the energy of the optimal disturbances is by several orders of magnitude larger than in dusty-gas and pure-fluid flows. When the particle layers are located in the vicinity of the walls or the channel axis, the nonmodal instability mechanism is less pronounced, as compared to the flow devoid of particles.

Topology of pair-sphere trajectories in finite inertia suspension shear flow and its effects on microstructure and rheology

The relative motion of pairs of spheres in finite inertia shear flow is examined. The pair relative trajectory is studied, using lattice-Boltzmann simulation, both for pairs which are isolated and for pairs in suspension of solid volume fraction of ϕ ≤ 0.3. For the suspension, the average trajectory and aspects of its dispersion are considered. For neutrally buoyant particles in simple-shear flow, particle-scale inertia is represented by a Reynolds number Re=ργ̇a2μ, where γ̇ is the shear rate, a is the particle radius, and ρ and μ are the density and viscosity of the Newtonian suspending fluid. The pair trajectories in a dilute inertial suspension have the same basic features as the streamlines around an isolated particle at similar Re: reversing, in-plane and off-plane spiraling, and open but fore-aft asymmetric trajectories are observed. The origin of the off-plane spirals is examined in detail in this work. The average pair trajectory space in a suspension of finite volume fraction is found to be qualitatively similar to the dilute suspension pair trajectories, as the spiraling and reversing zones are retained; the influence of ϕ and Re on the extension of different zones is described. The role of the average pair trajectory in setting the microstructure of the suspension is analyzed through a convection equation description of the pair distribution function, showing extreme accumulation at contact consistent with sampling from simulation. The influence of the microstructure on the bulk rheology is considered, with a focus on the combined effects of ϕ and Re on the near contact structural effects on rheology.

THE LONGITUDINAL PERTURBATED FLUID VELOCITY OF THE DUSTY FLUID IN THE INCOMPRESSIBLE FLOW IN CYLINDRICAL POLAR COORDINATE

The effect of finite volume fraction of suspended particulate matter on axially symmetrical jet mixing of incompressible dusty fluid has been considered. However, this assumption is not justified when the fluid density is high or particle mass fraction is large Here we are assuming the velocity and temperature in the jet to differ only slightly from that of surrounding stream, a perturbation method has been employed to linearized the equation those have been solved by using Laplace Transformation technique. Numerical computations have been made to discuss the longitudinal perturbated fluid velocity.

Effective moduli of nanoparticle reinforced composites considering interphase effect by extended double-inclusion model – Theory and explicit expressions

In this work, an extended double-inclusion model incorporating finite matrix volume fraction is derived by the method of Eshelby’s equivalent inclusion and Mori–Tanaka’s average theory, which can be used to predict overall mechanical properties of nanocomposites considering particle/matrix interphase effects. For convenience of engineering applications, the explicit expressions of the effective elastic moduli for composites with distributions of unidirectionally aligned, three- and two-dimension randomly oriented spheroidal particles, respectively, are obtained. The results can be used to predict effective properties of nanocomposites including interphase effects.

Model for growth of bubbles during proofing of viscoelastic dough

A model for growth of bubbles during proofing of bread dough is presented for infinitely dilute as well as finite volume fraction gas dispersions. The model accounts for diffusion of CO2, produced by fermentation, along with the resistance to bubble expansion by the viscoelasticity of the dough as described by Oldroyd-B constitutive equation. The model predicts evolution of bubble size, bubble growth rate and the profiles of CO2 concentration and normal stresses in the vicinity of the gas bubble. The growth of bubbles exhibits a lag time followed by an exponential growth phase. The lag time is found to be smaller for larger dough relaxation time, larger surface tension, smaller dough viscosity, higher temperature, higher gas volume fraction and higher CO2 production rate. The bubble growth rate is larger for smaller bubbles resulting in crossover of bubble size at longer times.

Granular polymer composites

To advance our understanding of molecular packing in glassy nanocomposites, we adopt a macroscopic granular model—comprising spherical inclusions dispersed in ball-chain polymers—and ascertain from measurements of the bulk density the partial molar sphere volume at small but finite sphere volume fractions. By systematically varying the chain length, sphere size, and mixture composition, the ratio of the sphere radius to the chain loop size was found to be the primary dimensionless parameter that bridges solvent (small inclusion size) and continuum (large inclusion) scaling regimes. In the intermediate regime, where the inclusion and chain loop radius are comparable, the primary sphere–chain interaction increases free volume—up to twice the intrinsic inclusion volume per particle—and the primary sphere–sphere interaction decreases free volume. The results gleaned from these ‘granular nanocomposites’ may help to understand how geometrical constraints influence packing in dense granular systems and glassy molecular nanocomposites.

Recent developments in hydrodynamic stability of two-phase flows

In the framework of two-continuum model, the stability of plane-parallel dispersed flows is analyzed. Several flow configurations are considered and several new factors are analyzed. The factors include: particle velocity slip and particle concentration non-uniformity in the main flow, non-Stokesian components of the interphase force and finite volume fraction of the dispersed phase. It is found that the new factors modify significantly the parameters of the fastest growing mode and change the critical Reynolds number of two-phase flows. A method for studying algebraic (non-modal) instability and optimal disturbances to dispersed flows is proposed. While studying the non-modal instability of the dusty-gas boundary-layer flow with a non-uniform particle concentration, we found that the disturbances with the maximum energy gain at a limited time interval are streamwise-elongated structures (streaks). As compared to the flow of a particle-free fluid, optimal disturbances to the dusty-gas flow gain much larger kinetic energy even at the boundary layer width-averaged mass concentration of ten percent, which leads to significant amplification of non-modal instability mechanism due to the presence of suspended particles.

The influence of monomeric resin and filler characteristics on the performance of experimental resin-based composites (RBCs) derived from a commercial formulation

ObjectiveTo explore experimental RBCs derived from a successful commercially available RBC (Grandio) to investigate resin monomer blend and filler parameters (volume fraction, density and diameter) on RBC performance. MethodSix experimental RBCs modified from a commercial RBC were tested. The three-point flexure strength (σ3) and modulus (E) data was determined for groups of 20bar-shaped specimens, prepared in a custom-made knife-edge split aluminum mold and irradiated using a modification of the ISO 4049 protocol. The biaxial flexure strength (BFS) and top and bottom Vickers hardness number (VHN) determination was performed on disc-shaped specimens (n=20). Normal distribution of the σ3, E, BFS and top and bottom VHN data was verified using the Shapiro–Wilk's test. Paired groups were compared using independent samples t-test for the individual tests investigated (σ3, E, BFS and top and bottom VHN) at a significance level of P<0.05. ResultsA significant decrease in the mean σ3 (P<0.011) and mean E (P<0.001) were identified on increasing the filler fraction (from 71.4 to 74.5vol%) or increasing the mean filler diameter (from 1.5 to 2.5μm). Increasing the filler density resulted in a significant increase in the mean σ3 (P<0.001), mean E (P<0.001) and mean top VHN (P≤0.001). Replacing the monomeric blend with an ormocer blend significantly reduced the mean σ3, E, BFS and top and bottom VHNs (all P<0.001). SignificanceFor each RBC monomeric resin and filler (type, density and diameter) combination employed, a finite filler volume fraction exists. Operating at the finite filler volume fraction increases the difficulty in improving the mechanical properties of the RBCs tailored from a commercial product.