Particle Coordination Number Research Articles

Anisotropic shrinkage induced by particle rearrangement in sintering

A microscopic model is presented which describes the anisotropic shrinkage induced by particle rearrangement during sintering. Two types of topological transitions in the rearrangement, that is the formation of a new contact, are illustrated in three-dimensional simulation of sintering of four spheres arranged in a rhombus. The shrinkage rate is shown to become anisotropic as the particle coordination number changes with rearrangement. A micromechanical principle is discussed with the use of sintering force and effective viscosity. The anisotropic shrinkage is caused by the additional sintering force acting on the new grain boundary formed at the rearrangement.

Interaction effects within Langmuir layers and three-dimensional arrays of ε-Co nanoparticles

Magnetostatic interactions are studied in two types of nanoparticle assemblies. Highly ordered monolayers of ε-Co nanoparticles are prepared from Langmuir films and self-assembled three-dimensional (3D) arrays formed by slow precipitation of colloidal dispersions of same nanoparticles. The structure of assemblies is found from electron microscopy and small-angle x-ray scattering. The differences in magnetic behavior are interpreted in terms of the particle moment, interparticle separation, and coordination number. Preferential in-plane magnetization is observed due to interactions within a monolayer. 3D arrays show stronger interactions compared to monolayers due to an increased number of nearest neighbors.

Quantitative computer reconstruction of particulate materials from microtomography images

Traditional particle-characterization techniques require particles to be dispersed from their original form, which destroys important morphologic information in materials such as aggregates or porous materials. X-ray computed microtomography provides a powerful tool for non-destructive analysis. However, robust techniques for comprehensive material characterization of these images have not been developed. In this paper we present a new algorithm for the computer analysis of particulate materials from high-resolution tomography images. The key aspect of the algorithm is the assignment of every solid-phase voxel in the image to its associated particle in a physically representative manner, which is in essence a particle-scale computer reconstruction of the material. Once this digital reconstruction is obtained, a vast amount of morphologic information can be extracted, including parameters obtained by traditional particle-analysis techniques (e.g., particle-size distribution and porosity) as well as parameters not usually available (e.g., spatial correlations in particle size, particle aspect ratios, surface areas, and orientations, particle contacts, particle coordination numbers, and more). Additionally, the computer reconstruction allows for complex manipulations such as the comparison of a specific parameter for two different particle-size classes within the material. The paper includes validation of the algorithm using computer-generated packings, as well as an example using microtomography data from a real material.

Influence of Particle Coordination Number in NanoporousTiO2 Films on the Performance of Dye-Sensitized Solar CellModules

The average particle coordination number, one of the charactersconcerning the pore size distribution in the films, was introduced indye-sensitized solar cell (DSC) modules with the size of 15 cm ×20 cm and the active area of 187.2 cm2 to estimate theperformance of a TiO2 nanoporous film, which is critical to thefuture DSC production. The current–voltage measurement of the DSCmodules indicates that the average particle coordination number in therange of 4–5 typically appears in nanocrystalline TiO2 films usedin the DSC modules and that the average coordination number couldprovide a very valuable way to evaluate the performance of nanoporousTiO2 films.

Modelling of swelling of Fe–Cu compacts sintered at temperatures above the copper melting point

Swelling of Fe–Cu sintered alloy at temperatures above the copper melting point during sintering has been studied for many years. It has been shown that the penetration of liquid copper into iron interparticle boundaries is the dominant mechanism. Based on this mechanism, a model of swelling of Fe–Cu sintered alloys at the copper melting point is established. The particle coordination number is introduced to take into account the porosity. Different heating rates lead to different thicknesses of the diffusion layer between iron and copper particles, which also influences the swelling. In the model, the combined effect of copper content, porosity, particle size and heating rate on the swelling is analysed. The calculated volume and dimensional growths show a qualitative agreement with the published data. The calculated results can be used to predict swelling of Fe–Cu compacts with different copper contents and green densities.

Effect of granule morphology on breakage behaviour during compression

In the area of dry particle breakage, Discrete Element Method (DEM) simulations have been widely used to analyse the sensitivity of various physical parameters to the behaviour of agglomerates during breakage. This paper looks at the effect of agglomerate shape and structure on the mechanisms and extent of breakage of dry agglomerates under compressive load using DEM simulations. In the simulations, a spherical-shaped agglomerate produced within the DEM code is compared with an irregularly shaped agglomerate, whose structure is that of an actual granule that was characterised with X-ray microtomography (μCT). Both agglomerates have identical particle size distribution, coordination number and surface energy values, with only the agglomerate shape and structure differing between the two. The work here details the breakage behaviour with a number of traditional DEM output parameters (i.e., contact/cluster distributions) with showing vastly different behaviour between the two agglomerates.

Flocculation and coagulation kinetics of Al 2O 3 suspensions

The effect of solid content on the structure of Al 2O 3 particle networks is studied by Brownian dynamics simulations using a novel interaction potential with image charges. Significant differences between 1 and 2 vol% suspensions are discussed with pair correlation functions, particle coordination numbers and cluster sizes. It is concluded, that the 2 vol% system is nearer to the percolation limit, thus fast aggregation occur. It is also stated, that the clustering dynamics should be described rather by the flocculation half-life than by the coagulation half-life. In real particle systems the interaction energy is not conservative due to friction with the medium. New expressions for the flocculation and coagulation half-life are derived using the interaction force as determining parameter, without the need for energy conservation. The corresponding coagulation half-life is compared with the classical expression of Fuchs.

The rheology of concentrated suspensions of depletion-flocculated latex particles

The effect of adding non-adsorbing polymer, poly(acrylic acid) (PAA), on the rheological properties of concentrated electrostatically-stabilised polystyrene latex suspensions was investigated. The extrapolated yield stress of the depletion-flocculated suspensions was studied as a function of the concentration and the molecular weight of the free polymer. Specifically, the Bingham yield value was observed to increase with either an increase in the weight fraction of the polymer or through the use of a longer chain polyacid. Unlike the sterically stabilised colloidal suspensions studied in the literature, at high polymer concentrations these charge-stabilised latex particles exhibited plateauing in the yield stress for each polyacid. The energy of separating particles within an aggregate into single units was calculated using the yield stress and average particle co-ordination number. In contrast to what was commonly done in the literature, the latter was not assumed to be constant but estimated from the aggregate mass fractal dimension as obtained from static light scattering techniques. It was found that the separation energy showed similar trends to those in the yield stress with both varying the polymer concentration and molecular weight. These trends also bear remarkable similarities to those in the depth of the secondary potential energy well as measured using atomic force microscopy (AFM).

Densification and strength evolution in solid-state sintering Part II Strength model

A compact gains strength in sintering through low-temperature interparticle bonding, followed by further strength contributions from high-temperature densification. On the other hand, thermal softening substantially reduces a compact's strength at high temperatures. Therefore, the in situ strength during sintering is determined by the competition among interparticle neck growth, densification, and thermal softening. Distortion in sintering occurs when the compact is weak. Most strength models for sintered materials are semi-empirical relations based on the sintered fractional density. These models do not include microstructure or sintering cycle parameters; thus, they do not provide guidelines for thermal cycle design to improve compact dimensional control. A strength evolution model is derived which combines sintering theories and microstructure parameters, including interparticle neck size, solid volume fraction, and particle coordination number. The model predicts sintered strength and when combined with thermal softening gives a good prediction of in situ strength. The validity of the model is verified by comparison to experimental data for sintered and in situ strength of bronze and steel powders.

Relation between Particle Coordination Number and Porosity in Nanoparticle Films: Implications to Dye-Sensitized Solar Cells

The morphology and local three-dimensional structure of mesoporous nanoparticle TiO2 films are simulated by using a simple method based on the random packing of spheres. The pore size distribution ...

Specific low-shear viscosity of a liquid dispersion of solid particles from variational theory for the Fuchs stability ratio

The contribution of a solid phase to the low-shear viscosity of a solid–liquid dispersion, i.e., the specific viscosity, ηSP, is investigated theoretically by applying a variational procedure. The Fuchs stability ratio has been interpreted as a general steady-state equilibrium constant for aggregation and has been extended to a functional form which describes the motion of two Brownian units in a dispersion. Application of the Euler–Lagrange equation under the validity of an adiabatic-like approximation for the Hamiltonian (approximately Brownian kinetic energy and inertial potential field) yields a constraint that involves specific viscosity, solid volume fraction, φ, interparticle energy and correlation functions of the dispersed phase. The Einstein formula is found as the limit of the Saito equation when an infinitely dilute hard-sphere suspension is considered, while a general closed form expression, ηSP=ηSP(φ), is proposed for a concentrated suspension. It depends on the particle coordination number and affinity, returns the low density expansion predicted by effective-medium-type theories for the viscosity, and can be represented as the sum of two dominant contributions, associated respectively with the first peak of the radial distribution function and the second peak of the total correlation function. Application to experimental data, concerning latex particles in cis-decalin and interacting silica–water systems, is presented and discussed.

Early stage consolidation mechanisms during hot isostatic pressing of Ti–6Al–4V powder compacts

Mechanisms contributing to early stage compaction of metal powder compacts were identified in a series of Ti–6Al–4V powder compacts hot isostatically pressed to relative densities ranging from 71% to 100%. The partially dense compacts, consolidated from loose powder in thin-walled containers, were examined using optical microscopy of polished sections and stereo pair scanning electron microscopy of fracture surfaces. Relative particle motion, characterized by small relative movements of particles and clusters of particles, was found to contribute significantly to compaction over a broad range of relative densities (from 63% to 90%). A mechanism was identified by which the preferential deformation of small particles at large–small particle contacts enables rigid body motion of larger particles which, in turn, increases the relative density of the compact. Tensile fractures within the compact occurred between 82% and 90% relative density providing direct evidence of cooperative movement among clusters of particles. In comparisons with mechanistic powder compaction models, measurements of particle deformation, contact areas, and coordination numbers were found to substantiate the importance of the experimentally identified mechanisms.

Ultrasonic velocity and reduction of surface area during solid-state sintering

We studied the relationships between the ultrasonic velocity and the reduction of specific surface area during solid-state sintering and addressed three questions: (1) What must be the correlation between ultrasonic velocity and specific surface area reduction during the solid-state sintering of monosize spheres? (2) How is this correlation affected by the particle-size distribution and/or changes in packing coordination? (3) How do the answers to these questions help us improve the use of ultrasonic measurements to monitor sintering kinetics and microstructure evolution? From both a theoretical and experimental basis, we found a general power law that describes the relationship between ultrasonic velocity and the reduction of specific surface area during solid-state sintering: V P/ V B∝(Δ S/ S o) x . The power-law exponent x depends on three possible scenarios: (1) x<1 for ideal packing of monosize spheres; (2) x=1 for random packing of monosize spheres where the average, particle coordination number changes upon sintering; and (3) x>1 for specimens containing a broad distribution of particle sizes. We also demonstrated that this power law could be used with ultrasonic velocity measurements to obtain reasonable values of the activation energy for solid-state sintering. The power-law relationship is useful to characterize microstructure evolution during solid-state sintering and is useful to classify different materials in a ceramic manufacturing setting. It is also a simple modeling tool that opens the way to optimize control of the sintering process.

Thermodynamics and Densification Kinetics in Solid-state Sintering of Ceramics

Based on the stability analysis of the closed pores in two dimensions which was determined mathematically with their particle coordination number and dihedral angle, the stability of those in three dimensions was determined with a spherical pore model. The model is set up by first excluding the effect of interface tension, so the pore was supposed to be spherical, and then the tensile stress arisen from the interface tension was supposed to act on this hypothesized spherical pore. On the basis of the spherical pore model, microstructure models based on pores, not grains, for the real powder compacts were first established. Densification kinetics were then determined from the models by the densification rate equations, which were derived by relating density to pore size to grain size ratio, for the intermediate and final stages of sintering. The criteria for pore shrinkage were discussed quantitatively. The derived equations can be used to simulate the relation between densification rate and density during heating with a constant rate and for the explanation of the effects of pore size distribution, agglomerates, and green density on sintering.

Effect of compact structure on phase transformation kinetics from anatase phase to rutile phase and microstructure evolution during sintering of ultrafine titania powder compacts

The effect of compact structure both on phase transition kinetics and on the densification during sintering of n-TiO2 compacts has been investigated. The compact structures varied from loose powder pack to very high density using different compaction pressures. The compact structure is verified to significantly affect phase transition kinetics and densification during sintering. The onset temperature of anatase -> rutile phase transition decreased with the increase of the compact density. The highest density compact of 86.9% showed phase transition at the lowest temperature and no detrimental effect on densification. The compact structure, i.e., the coordination number of particles in the compacts, is considered to significantly affect both the phase transition kinetics and the subsequent densification during the sintering of n-TiO2 compacts.

Solid state sintering of ceramics: pore microstructure models, densification equations and applications

The stability of closed pores in two and three dimensions has been discussed and it is found that the stability of pores in two dimension can be determined mathematically from their particle coordination number and dihedral angle; while those in three dimension can be approximately determined by a spherical pore model. This model is set up by first excluding the effect of interface tension, so the pore was supposed to be spherical, and then the tensile stress arising from the interface tension was allowed to act on this hypothesized spherical pore. On the basis of the spherical pore model, pore microstructure models for real powder compacts were set up and the densification equations for the intermediate and final stages of sintering were derived. The criterion for pore shrinkage, and the effect of pore size distribution and green density were discussed according to the derived equations. The densification equations for pressureless solid state sintering can be easily extended to describe the densification behaviour during hot-pressing or hot-isostatic-pressing. Densification characteristics in liquid state sintering were also considered from the result of solid state sintering.

An examination of the interparticle contact area during sintering of W-0.3 Wt Pct Co

As a powder compact sinters, its microstructure evolves. One way to quantify the scale of the microstructure is to consider the interparticle contact area. This study examines two known models for calculating the interparticle contact area: the classic two-sphere model and the Voronoi cell model. Both models have particular assumptions about the microstructure that make them not applicable for treating densification to near full density with concurrent grain growth. The classic two-sphere model assumes a regular packing of particles and a perfectly spherical particle geometry and neglects an increasing particle coordination number with sintering. The Voronoi cell model assumes that the scale of the microstructure remains constant; i.e., as long as the compact is densifying, grain growth does not occur. We propose a modified Voronoi cell that accounts for an increasing grain size, making it applicable to a general case where grain growth occurs during sintering. The three models are compared to the interparticle contact area data, obtained by stereology techniques, for W-0.3 wt pct Co sintered from green state to near full density. The original Voronoi cell model fits the data only at low temperatures, before the onset of grain growth. Below approximately 90 pct relative density, the two-sphere model with an assumed coordination number of six (coordination number in a green compact) and the modified Voronoi cell model provide a good fit to the data. At higher densities, both models overestimate the interparticle contact area.

Computer Studies on free and Confined Amorphous Polymer Films

ABSTRACTWe study surface effects in amorphous polymer systems by means of computer simulation. In the framework of molecular dynamics, we present two different methods to prepare such surfaces. Free surfaces are stabilized solely by van-der-Waals interactions, whereas confined surfaces emerge in the presence of repelling plates. The buildup of density and pressure profiles from zero to their bulk values depends on the surface preparation method. In the case of confined surfaces, we find density and pressure oscillations next to the repelling plates. For free surfaces, we attest chain-end enrichment and present a comparison between density profile and particle coordination number.

Modelling the densification of powder composites by power law creep

A model describing the densification under isotropic pressure of a powder composite, consisting of viscoplastic particles deforming by power law creep mixed with rigid inclusions of the same size, is presented. A description of the behaviour of each type of contacts is first proposed, from which an expression for the densification rate of the mixture is deduced. The evolution of particle coordination numbers and contact areas during densification is then estimated from an extension of Arzt's model [ Acta metall. 30, 1883 (1982)] to powder composites. The model can be used up to a density corresponding to the formation of a supportive inclusion network. Finally the validity of the model is discussed in relation with experimental data.

Simulation of statistical packings of spherical particles

An algorithm has been developed which simulates the process of statistically filling a fixed volume with spherical particles and which allows one to obtain the distributions of the coordination numbers of particles depending on the packing density.