Hard Spherical Particles Research Articles

Numerical prediction of particle mixing behavior in a bubbling fluidized bed

In this article the hard-sphere discrete particle model (DPM) is used to study the mixing behavior of particles in the 2-D fluidized bed. Different flow patterns in the bed for two kinds of inlet configurations, namely free bubbling and jet bubbling mode, are captured by the numerical model, under specific superficial gas velocities. To examine the degree of particle mixing, the Fan index is applied. The numerical results show that the rate of particle mixing is larger in the jet bubbling than that in the free bubbling mode. The gross circulations of particles in the jet bubbling bed give a higher degree of mixing because of the involvement of a greater number of particles.

Novel polarization and detection of NMR and MRI for porous materials

A theoretical model for prediction of permeate flux during crossflow membrane filtration of rigid hard spherical solute particles is developed. The model utilizes the equivalence of the hydrodynamic and thermodynamic principles governing the equilibrium in a concentration polarization layer. A combination of the two approaches yields an analytical expression for the permeate flux. The model predicts the local variation of permeate flux in a filtration channel, as well as provides a simple expression for the channel-averaged flux. A criterion for the formation of a filter cake is presented and is used to predict the downstream position in the filtration channel where cake layer build-up initiates. The predictions of permeate flux using the model compare remarkably well with a detailed numerical solution of the convective diffusion equation coupled with the osmotic pressure model. Based on the model, a novel graphical technique for prediction of the local permeate flux in a crossflow filtration channel has also been presented.

Steric stabilization of spherical colloidal particles: Implicit and explicit solvent

We present the results of Monte Carlo simulations and density functional theory treatment of interactions between spherical colloidal brushes both in implicit (good) solvent and in an explicit polymeric solution. Overall, theory is seen to be in good agreement with simulations. We find that interactions between hard-sphere particles grafted with hard-sphere chains are always repulsive in implicit solvent. The range and steepness of the repulsive interaction is sensitive to the grafting density and the length of the grafted chains. When the brushes are immersed in an explicit solvent of hard-sphere chains, a weak mid-range attraction arises, provided the length of the free chains exceeds that of the grafted chains.

Fracture mechanisms and fracture toughness in semicrystalline polymer nanocomposites

Polymer nanocomposites, especially nanoclay composites, have received much attention in recent years. While it is the functional properties of these composites that are the driving force, good mechanical properties are necessary in most applications. Although claims are made that the mechanical properties of nanocomposites should be excellent, in practice the mechanical properties are often disappointing. Soft or hard spherical particles can toughen semicrystalline polymers, but plate-like nanoclay particles have not produced any significant toughening action and frequently cause significant embrittlement. In this paper the toughening of semicrystalline polymers with nanoparticles, and particularly the difference in behaviour of spherical and plate-like particles is reviewed.

Formation and evolution of sediment layers in an aggregating colloidal suspension

The coupled aggregation and sedimentation of colloidal particles with short-range attractions are investigated. Nonadsorbing polymer is used to induce depletion interactions between the hard-sphere particles. Gravitational forces, caused by a density mismatch between the particles and the suspending fluid, result in the sedimentation of particles and aggregates, as well as the compaction and rearrangement of the final sediment layer. At low polymer concentrations CP, or low initial volume fractions phio, clusters formed during the period of fast sedimentation are small, and the structure of the final sediment is dense. Conversely, at high CP, or high phio, large clusters form during sedimentation, and the resulting sediment structure is significantly less dense, with large void volumes. The size of the presediment aggregates depends on CP and phio with a functional form that resembles other thermally activated barrier hopping processes in colloidal systems, such as the delayed sedimentation of colloidal gels. Finally, when the particles are weakly attractive, gravitational stresses are found to induce compaction of the sediment over long periods of time. However, sediments composed of particles that are strongly attractive resist rearrangements and compaction, even when the sediment layers have a relatively large amount of free volume.

Direct imaging of repulsive and attractive colloidal glasses

Coherent anti-Stokes Raman scattering microscopy is performed on glassy systems of poly(methylmethacrylate) colloidal particles in density- and refractive-index-matched solvents. Samples are prepared with varying amounts of linear polystyrene, which induces a depletion driven attraction between the nearly hard-sphere particles. Images collected over several hours confirm the existence of a reentrant glass transition. The images also reveal that the dynamics of repulsive and attractive glasses are qualitatively different. Colloidal particles in repulsive glasses exhibit cage rattling and escape, while those in attractive glasses are nearly static while caged but exhibit large displacements upon (infrequent) cage escape.

Plane Poiseuille flow of a sedimenting suspension of Brownian hard-sphere particles: Hydrodynamic stability and direct numerical simulations

The pressure driven flow of a suspension of sedimenting Brownian hard-sphere particles in a plane channel is considered. The balance of gravity and Brownian forces leads to a stationary state where a concentration profile of the particles is established in the channel, with a transition from a viscous sediment to clear fluid. The hydrodynamic stability of the flow and the nonlinear evolution of unstable disturbances are studied numerically by spectral element/Fourier expansion techniques. Two modes of instability with different characteristics are identified. The first is of the Tollmien-Schlichting type, similar to the one present in single-fluid parallel shear flows. This instability appears at much lower Reynolds numbers than for a single fluid when the transition in viscosity is gradual and the sediment is receptive to the fluid motion in the bulk, that is, for relatively small colloidal particles well into the submicron range. An interesting feature, observed through three-dimensional numerical simulations, is the formation of longitudinal striation patterns in the sediment, reminiscent of drag reducing surfaces with organized roughness known as riblets. The second type of instability is similar to the interfacial instability in stratified shear flows with a jump in viscosity. This type of instability appears also at low Reynolds numbers, when the transition from sediment to clear fluid is sharper, that is, for relatively larger particles. This instability results in slow waves traveling with velocities characteristic of those in the sediment, and gives rise to a significant resuspension and formation of low concentration regions in the vicinity of the sediment. An interesting feature is the formation of patterns in the sediment reminiscent of sand ripples.

Case of a Boltzmann gas leading to the Smoluchowski coagulation equation

A special model of a rarefied hard-sphere gas is considered. The hard-sphere particles undergo absolutely elastic collisions. It is assumed that particles can collide only if their nonzero velocities are orthogonal to each other. The model makes it possible to proceed from the Boltzmann equation to the Smoluchowski coagulation equation, where coagulation means that the kinetic energies of the colliding particles are added. A Monte Carlo scheme for simulation of the phenomenon is described, and the convergence of the simulation algorithm is proved. The convergence of numerical results to exact solutions of the Smoluchowski equation and to finite-difference solutions is tested.

Dynamical perspective of the freezing transition of a suspension of hard spheres from the velocity autocorrelation function

The velocity autocorrelation function of concentrated colloidal fluids of hard-sphere particles, measured by dynamic light scattering, decays to the experimental noise floor from below. The decay follows a stretched exponential function of delay time for the colloidal fluid in thermodynamic equilibrium, and a power law for the nonequilibrium, undercooled colloidal fluid. Consideration of this difference between the two states points to a possible dynamical mechanism of freezing.

Chemical potential of a hard sphere fluid adsorbed in model disordered polydisperse matrices

We consider a model for adsorption of a simple fluid in disordered polydisperse adsorbents. The fluid consists of hard sphere particles. On the other hand, the adsorbents of this study are modeled as a collection of hard spheres with their diameter obeying a certain distribution function. Our focus is in the evaluation of the chemical potential of the fluid immersed in such a polydisperse material. It permits us to obtain porosity and pore size distribution for the adsorbent, as well as a set of adsorption isotherms. The latter have been calculated theoretically and by grand canonical Monte Carlo simulations. We observe that the width of assumed polydispersity distribution affects all the properties of the system. Nevertheless, the effect of matrix packing is dominant in determining adsorption for this class of models. We are convinced that the matrix structures generated via more sophisticated algorithms would exhibit stronger effects of polydispersity on the entire set of properties of adsorbed simple fluids.

Dynamics of Swelling/Contracting Hard Spheres Surmised by an Irreversible Langevin Equation

The diffusion of molecules through uniform homogeneous materials can readily be described by Brownian motion or generalizations thereof. The further generalization of these models to describe molecular diffusion through heterogeneous and nonstationary solvents is much less understood. Phenomenological nonstationary generalizations of the generalized Langevin equation (GLE) have earlier been developed satisfying the fluctuation-dissipation relationship in quasi-equilibrium limits while exhibiting somewhat complex behavior away from equilibrium. This reduced-dimensional representation should be capable of describing the diffusion of a particle through a colloidal suspension whose average particle size is tuned by an external driving force such as pH. A simple particle model of such a process involves the motion of a hard-sphere particle in an explicit environment of swelling hard spheres. The velocity autocorrelation functions observed in a large number of simulations of the particle model under various swelling rates agree precisely with those of a single form of the nonstationary phenomenological model. Though this procedure is not an explicit projection of the mechanical system onto the nonstationary GLE, it does show that the latter correctly describes the dynamics of the projected coordinate--namely, diffusion of the solute--under nonequilibrium conditions. Both nonequilibrium solvent models lead to behavior reminiscent of beta-relaxation processes at packing fractions substantially below that of the glass transition.

A reference interaction site model approach to depletion forces induced by hard rodlike particles

We study the effective interactions among large hard spherical colloidal particles induced by small hard rodlike particles and compare them with those induced by small hard spherical particles to highlight the specific effects due to the anisotropic shape of the former. This is done by determining the effective pair potentials within the framework of the reference interaction site model approach. The rodlike particles are modeled as N nonoverlapping spherical units arranged in a straight line, so that their total length is N times their transversal diameter. These results are compared against those obtained in the Asakura-Oosawa limit.

Stability of the binary colloidal crystalsAB2andAB13

Suspensions of binary mixtures of hard-sphere poly-methylmethacrylate colloidal particles were studied at six different size ratios alpha. The main aim was to determine the range of size ratios over which the binary colloidal crystals AB2 and AB13 are stable. Combining these results with those of earlier work, we found stability of AB2 for 0.60 approximately > alpha approximately > 0.425, in good agreement with theoretical predictions by computer simulation and cell model methods. AB13 was observed for 0.62 approximately > alpha approximately > 0.485, the lower limit being significantly smaller than the theoretical prediction of about 0.525. Rough measurements of crystallization rates showed that AB2 tended to crystallize fastest at small size ratios, whereas the opposite was true for AB13. These findings should provide a guide to the optimum conditions for materials applications of these binary colloidal crystals.

Extending the simple weighted density approximation for a hard-sphere fluid to a Lennard–Jones fluid: II. Application

A simple weighted density approximation (SWDA) was extended to nonuniform Lennard–Jones fluids by following the spirit of a partitioned density function theory [S. Zhou, Phys. Rev. E 68 (2003) 061201] and mapping the hard-core part onto an effective hard-sphere fluid whose higher order terms beyond the second order of the functional perturbation expansion are treated by the SWDA. The resultant DFT formalism performs well for Lennard–Jones fluids under the influence of diverse external fields. With the present DFT formalism, we investigate in detail the structure and adsorption properties of a low-density LJ gas in a spherical cavity with a wall consisting of hard-sphere or LJ particles. It was found that when the cavity wall exerts an attractive external potential on the LJ particles in the cavity, the excess adsorption decreases as the temperature increases, while when the cavity wall exerts a hard repulsive external potential on the LJ particles in the cavity, the excess adsorption increases as the temperature increases.

Isostructural solid–solid transitions in binary asymmetrical hard sphere system: Based on solvent-mediated potential

Size dependence is imparted onto a modified bridge functional, adopted for a recently proposed semi-analytical hard sphere reference system theory for calculation of solvent-mediated potential (SMP). The SMP for two large hard sphere particles immersed in a small hard sphere solvent bath predicted by the present improved version is in satisfactory agreement with the prediction from a theoretically based fitting formula. Isostructural solid–solid transitions in the binary asymmetrical hard sphere system are investigated based on a single-component macrofluid approximation combined with the improved version. It is found that the isostructural solid–solid transition appears when size asymmetry increases. The limiting asymmetry size ratio is near 1 / 8 . As the size asymmetry increases, critical density for both large and small hard sphere components for the fcc isostructural solid–solid transition increases and decreases, respectively.

Crystallization kinetics of polydisperse colloidal hard spheres. II. Binary mixtures

In this paper we present measurements of the crystallization kinetics of binary mixtures of two different sized hard sphere particles. The growth of the Bragg reflections over time were analyzed to yield the crystallite scattering vector, the total amount of crystal, and the average linear crystal size. It was observed that a particle size distribution skewed to higher sized particles has a less detrimental effect on the crystal structure than a skew to smaller sized particles. In the latter case we observe that initial crystallite growth occurs at only a small number of sites, with further crystallization sites developing at later times. Based on these measurements we elaborate further on the previously proposed growth mechanism whereby crystallization occurs in conjunction with a local fractionation process in the fluid, which significantly affects the kinetic growth of crystallites in polydisperse systems.

Supersonic transverse rarefied gas flow past a strip

The supersonic flow of a monatomic gas consisting of hard spherical particles past a flat strip normal to the flow is investigated using the direct simulation Monte-Carlo (DSMC) method. The calculations are performed over the Knudsen and Mach number ranges 0.015–5 and 1.8–15, respectively. The structure of the compressed layer and the aerodynamic characteristics are systematically studied for the Mach number 5 and various Knudsen numbers. The dependences of the compressed-layer thickness in molecular free paths are found. The nonequilibrium processes in the neighborhood of the strip are described on the basis of the data on the temperature anisotropy with respect to three coordinates.

Erosion and Damage by Hard Spherical Particles on Glass

Solid particle impact of hard spherical particles on glass is of fundamental interest because of the presence of a number of different impact regimes. Understanding the impact of spherical particles is also a step toward modeling the behavior of rounded particles. This paper verifies theoretical models for the transitions between the different impact regimes and process parameters like erosion rate and surface roughness. The work also includes rounded particles. The transitions plotted in a so-called erosion map are validated with single-impact experiments. Data from erosion experiments are used to test the relations for the erosion rate and resulting surface roughness following from the models. Although the theoretical transitions compare reasonably with experiments, the models for erosion rate and surface roughness do not describe the experimentally found behavior. The models not incorporating the interaction between an impact and the damage remaining from earlier impacts might cause this deviation.

Hydrodynamic and Brownian Fluctuations in Sedimenting Suspensions

We use a mesoscopic computer simulation method to study the interplay between hydrodynamic and Brownian fluctuations during steady-state sedimentation of hard sphere particles for Peclet numbers (Pe) ranging from 0.1-15. Even when the hydrodynamic interactions are an order of magnitude weaker than Brownian forces, they still induce backflow effects that dominate the reduction of the average sedimentation velocity with increasing particle packing fraction. Velocity fluctuations, on the other hand, begin to show nonequilibrium hydrodynamic character for Pe>1.

Are thermoresponsive microgels model systems for concentrated colloidal suspensions? A rheology and small-angle neutron scattering study.

The structure of concentrated temperature-sensitive poly(N-isopropylacrylamide) (PNiPAM) microgel suspensions has been investigated employing rheology and small-angle neutron scattering (SANS). A previously described model expression for the particle form factor P(inho)(q) is extended by a model hard sphere structure factor S(q), and the average radial density profiles phi(r) are calculated from the amplitude of the form factor A(q) and the structure factor S(q). By this procedure, a direct real space description of the spatial ordering in the neighborhood of a single particle is obtained. The overall particle size and the correlation length xi of the concentration fluctuations of the internal polymer network decrease with concentration, revealing the increasing compression of the spheres. Thus, the particle form factor P(inho)(q) of the swollen PNiPAM microgels depends on concentration. The particle-particle interaction potential does not change significantly between 25 and 32 degrees C. Even approximately 1 K below the lower critical solution temperature (LCST), the experimental scattering intensity distributions I(q)/c are described very well by the hard sphere structure factor when an equivalent hard sphere particle size R(HS) and volume fraction eta(HS) are used. Microgels with different degrees of cross-linking and particle size resemble true hard sphere behavior up to effective volume fractions of phi(eff) < 0.35. At higher effective volume fractions phi(eff) > 0.35 strong deviations from true hard spheres are observed. Interpenetration of the outer, less cross-linked regions of the soft spheres as well as particle compression occurred at higher concentrations. In agreement with this, the equilibrium colloidal phase behavior and rheology also has some features of soft sphere systems. At temperatures well above the LCST, the interaction potential becomes strongly attractive and the collapsed microgel spheres form aggregates consisting of flocculated particles without significant long-range order. Hence, an attractive interaction potential in concentrated suspensions of PNiPAM microgels leads to distinctively different structures as compared to attractive hard sphere colloids. When the peculiar structural properties of the PNiPAM microgels are considered, they can be used as model systems in colloidal science.