Computer Simulation of Final-Stage Sintering: I, Model Kinetics, and Microstructure

A Monte Carlo model for simulating final-stage sintering has been developed. This model incorporates realistic microstructural features (grains and pores), variable surface difusivity, grain-boundary diffusivity, and grain-boundary mobility. A preliminary study of a periodic array of pores has shown that the simulation procedure accurately reproduces theoretically predicted sintering kinetics under the restricted set of assumptions. Studies on more realistic final-stage sintering microstructure show that the evolution observed in the simulation closely resembles microstructures of real sintered materials over a wide range of diffusivity, initial porosity, and initial pore sizes. Pore shrinkage, grain growth, pore breakaway, and reattachment have all been observed. The porosity decreases monotonically with sintering time and scales with the initial porosity and diffusivity along the grain boundary. Deviations from equilibrium pore shapes under slow surface diffusion or fast grain-boundary diffusion conditions yield slower than expected sintering rates.     

A two-dimensional Monte Carlo model for pore densification in a bi-crystal via grain boundary diffusion: Effect of diffusion rate,initial pore distance,temperature, boundary energy and number of pores

A two-dimensional Monte Carlo (MC) model is introduced for simulating the evolution of the pore on a bi-crystal grain boundary via grain boundary diffusion. Simulated pore shrinkage kinetics is found to be consistent with previously reported results over variable grain boundary diffusion rates and initial pore distances while the essential characteristics of the microstructural evolution are simultaneously realized. The influence on the pore densification kinetics of grain boundary motion, boundary energy ratio, simulation temperature and pore interactions in an array is found such that pore shrinkage rate increases as the grain boundary motion, the simulation temperature and the grain boundary energy increase. The interactions of the pores are found to hinder the pore densification. The body of results signify that the more elongated the pore shape and the shaper the pore tip region are favored for the faster pore shrinkage kinetics during the simulated densification process via grain boundary diffusion.     

The sintering kinetics of ultrafine tungsten carbide powders

The sintering kinetics of nano grained tungsten carbide (n-WC) powders has been analyzed by non isothermal and isothermal sintering. Non isothermal sintering experiments reveal a multi staged sintering process in which at least three major sub-stages can be distinguished. The isothermal shrinkage strain also exhibits an asymptotic behavior with time indicating an end point density phenomenon in most of the temperature ranges. Combined microstructural and kinetic data analyses suggest that differences in the sinterability of inter and intra agglomerate pore phases introduce sub-stages in the sintering process which manifest as stagnant density regions in both the isothermal and non isothermal experiments. Kinetic analysis of the data reveals very low activation energies for sintering suggesting that particle rearrangement and agglomeration at low temperatures may be brought about by surface diffusion leading to neck growth and grain rotation. At higher temperatures rapid grain boundary diffusion by overheating along inter particle boundaries induced by sparking may be a dominant sintering mechanism. Although grain growth and densification in conventional WC powders generally obey an inverse relation to each other, in n-WC powders both can act synergistically to increase the net densification rate. In fact, complete densification cannot be achieved in n-WC powders without grain growth as one abets the other.     

Electric Fields Obviate Constrained Sintering

The phenomenon of constrained sintering, where large rigid inclusions of alumina have been shown to significantly reduce the rate of sintering of titania [Bordia and Raj (1988) J. Am. Ceram. Soc. 71, 302–310], is shown to subside nearly completely during flash sintering carried out under modest electrical fields. The result is explained by a different mechanism for volumetric and shear deformation under electric fields. It is proposed that vacancy and interstitials generated within the grains migrate to grain boundaries and pores to produce both volumetric and shear strain at equal rates, since, in this way, the diffusion distance for both modes of deformation becomes the same. In conventional sintering, where transport occurs from one interface to another, the diffusion distance for shear is twice as far as for densification, which retards sintering should it become controlled by shear deformation, as seen in constrained sintering.     

菱镁矿煅烧过程中氧化镁烧结与晶粒生长动力学的研究

用XRD、BET和TEM研究了菱镁矿煅烧过程中氧化镁烧结与晶粒长大动力学。用表面积降低动力学模型和颗粒长大动力学方程对结果进行了分析,得出结果认为菱镁矿煅烧过程中氧化镁微晶的烧结可分为三个阶段:(1)微晶氧化镁结构的调整,伴随着晶粒迅速长大、比表面积急剧下降;(2)团聚体内孔容积保持不变,团聚体也不收缩,这时烧结过程为表面扩散和蒸发-凝聚所控制;(3)烧结过程为体积或晶界扩散所控制,团聚体出现显著收缩。     

Useful Extensions of the statistical theory of sintering

The statistical theory of sintering is modified to account for the new concept of pore coordination number (after Lange) as well as other refinements of the original assumptions (after Kuczynski). When pores are located in the grain boundaries, the theory postulates that for a given dihedral angle, a critical pore coordination number, n_crmc, exists such that when n < n_c, pores shrink, whereas when n > n_c pores grow. A general morphological kinetic equation is developed for uniform microstructures with pores located at the grain boundaries. This equation provides for contributions of grain boundary and volume diffusion to densification and is unrestricted with respect to the type of firing schedule imposed. Using this modified model, one may evaluate the significant parameter x, a measure of ‘the path of microstructural evolution’. For reasonably uniform compacts, the equation also provides a quantitative estimate of the relative width of the pore size distribution, y, during the later stages of sintering.     

Shape Sensitivity of Initial Sintering Equations

Initial sintering equations are shown to depend as much on possible particle shape variations in powder systems as on the mechanism of material transport. Also, competing mechanisms, such as surface diffusion, may alter the generally assumed relations between neck growth and shrinkage and so may affect the sintering kinetics. When shrinkage occurs by volume diffusion, both effects may be overcome by measuring the size of the interparticle boundary as well as the shrinkage during sintering. When grain boundary diffusion is the mechanism of shrinkage, both the cross-sectional area and the radius of curvature of the neck must be measured during sintering. The usefulness of simultaneous measurements of neck growth and shrinkage is demonstrated with literature data for copper.     

Role of Solute Segregation at Grain Boundaries During Final–Stage Sintering of Alumina

The addition of minor amounts of MgO or NiO to Al₂O₃ inhibits grain growth during sintering and allows the sintering process to proceed to theoretical density by maintaining a high diffusion flux of vacancies from the pores to the grain boundaries. The inhibition of grain growth is accomplished by the segregation of solute at the grain boundaries, causing a decrease in the grain–boundary mobility. The segregation of MgO or NiO at the grain boundaries can be inferred from the results of the microhardness studies presented and is substantiated by autoradiographic experiments and also by lattice parameter determinations as a function of grain size.     

Morphological Changes in Process-Related Large Pores of Granular Compacted and Sintered Alumina

Large pore defects clearly develop in Al₂O₃ ceramics during sintering. These large pores originate from voids caused by the incomplete deformation and adhesion of powder particles in collapsed dimples at the centers and boundaries of granules in the green compacts. The coalescence of pores, with limited shrinkage, during densification and grain growth in the late intermediate to final stages of sintering, is considered responsible for the development of the large pores. The mechanism of pore coalescence is explained by thermodynamic arguments, which demonstrate that the largest pores result in a stable system.     

Grain-Boundary Segregation and Final-Stage Sintering of Y2O3

The addition of ThO₂ to Y₂O₃ inhibits grain growth during sintering and allows the sintering process to proceed to theoretical density by maintaining a high diffusion flux of vacancies from the pores to the grain boundaries. The inhibition of grain growth is accomplished by the segregation of ThO₂ solute at the grain boundaries, causing a decrease in the grain-boundary mobility. The segregation of ThO₂ at the grain boundaries can be inferred from the results of the microhardness and grain-growth studies presented. Further evidence for segregation is provided by quenching experiments and surface activity experiments.     

Ionomigration of Pores and Gas Bubbles in Yttria‐Stabilized Cubic Zirconia

Migration of residual pores in partially sintered 8 mol% yttria‐stabilized zirconia under an electric field was investigated. To avoid shrinkage via sintering, Ar‐filled bubbles introduced to dense ceramic were also studied. Pores/bubbles were found to migrate against the field, e.g., under 1.9 V at 875°C, a temperature when cation diffusion is supposed to have frozen according to the prevailing consensus. Pore/bubble movement left contorted grains in some samples, but not at lower temperatures and higher fields when they apparently pass through grain boundaries without causing any visible distortion. These results are explained by a surface diffusion model and a temperature‐pore‐size map that delineates two distinct modes of pore/boundary pinning and breakaway. The implications of these results to solid oxide fuel cells and electrolysis cells are explored.     

Numerical Simulation of Anisotropic Shrinkage in a 2D Compact of Elongated Particles

Microstructural evolution during simple solid-state sintering of two-dimensional compacts of elongated particles packed in different arrangements was simulated using a kinetic, Monte Carlo model. The model used simulates curvature-driven grain growth, pore migration by surface diffusion, vacancy formation, diffusion along grain boundaries, and annihilation. Only the shape of the particles was anisotropic; all other extensive thermodynamic and kinetic properties such as surface energies and diffusivities were isotropic. We verified our model by simulating sintering in the analytically tractable cases of simple-packed and close-packed, elongated particles and comparing the shrinkage rate anisotropies with those predicted analytically. Once our model was verified, we used it to simulate sintering in a powder compact of aligned, elongated particles of arbitrary size and shape to gain an understanding of differential shrinkage. Anisotropic shrinkage occurred in all compacts with aligned, elongated particles. However, the direction of higher shrinkage was in some cases along the direction of elongation and in other cases in the perpendicular direction, depending on the details of the powder compact. In compacts of simple-packed, mono-sized, elongated particles, shrinkage was higher in the direction of elongation. In compacts of close-packed, mono-sized, elongated particles and of elongated particles with a size and shape distribution, the shrinkage was lower in the direction of elongation. The results of these simulations are analyzed, and the implication of these results is discussed.     

Diffusion Sintering: I, Initial Stage Sintering Models and Their Application to Shrinkage of Powder Compacts

Consideration is given to several geometrical models that contribute to shrinkage. Various shapes of particles, vacancy sinks, and diffusion paths are described as they affect sintering shrinkage. These simplified models are extended to compacts of nonuniform particles so that much of the kinetics of sintering of a substance can be determined by measuring shrinkage rates of powder compacts. A nonideal compact may sinter as though it had once been an ideal compact after a specific amount of shrinkage has occurred. This shrinkage is characteristic of the particular compact and its origin and is independent of temperature.     

Final Stage Sintering of ThO2

The addition of CaO to ThO₂ inhibits discontinuous grain growth and allows sintering to proceed to theoretical density by maintaining a high diffusion flux of vacancies from the pores to the grain boundaries. The optimum amount of CaO added is 2.0 mol%, which exceeds the solid solubility limit. A model of second-phase inclusions impeding grain boundary motion was used to explain the results.     

Densification and Shrinkage During Liquid-Phase Sintering

The process of densification and shrinkage during the final stage of liquid-phase sintering is described. The densification occurs by the liquid filling of pores during grain growth. The pore filling results in an instantaneous drop of liquid pressure in the compact and causes gradual accommodation of grain shape. The grain shape accommodation by the growth causes the specimen shrinkage. At the same time, the grains tend to restore their spherical shape, resulting in microstructure homogenization around filled pores. The process of densification and shrinkage appears to be determined by the growth of grains during sintering.     

Viscous Sintering of a Bimodal Pore-Size Distribution

The cylinder model, used previously to analyze the viscous sintering of flame hydrolysis preforms and gels, is shown to be consistent with other models of early- and late-stage sintering. The model is then applied to a bimodal pore-size distribution in which the different shrinkage rates of small and large pores produce local stresses. The effect of these stresses on the sintering rate is determined and shown to be substantial. Initially, the small pores accelerate the densification of the large pores; later, shrinkage of the isolated large pores is resisted by the sintered remains of the small pores. Consequently, the time to reach full density is nearly independent of the initial volume fraction of small pores.     

Effect of Green Density on the Thermomechanical Properties of a Ceramic During Sintering

The thermomechanical properties of a commercial barium titanate were experimentally or theoretically determined for samples with green densities ranging from 45% to 55%. For stresses less than 300 kPa, sample deformation was determined to be linear viscous for all three stages of sintering. The shrinkage rates at a given temperature can differ by up to ∼25% as the green density changes from 45% to 55%, and the maximum shrinkage rate increased with decreasing green density. The increase in shrinkage rate with lower green density samples persisted through the final sintering stage. The viscosity was determined by cyclic loading dilatometry to range from 5 to 6 GPa·s in the initial stage of sintering, to 2 GPa·s in the intermediate stage, and to increase to 10–20 GPa·s for all specimens in the final stage of sintering. Differences in the final-stage viscosity were attributed to grain size differences. Relaxation times for the sintering body were estimated to be less than 1 s, indicating that viscous behavior is dominant throughout the sintering process.     

Effects of Deagglomeration and Zirconia Wear Debris on Sintering Behavior of α-Alumina Powder

This work investigates sintering behaviors of α-alumina powder that were treated by attrition milling with zirconia media. While zirconia wear debris delays initial sintering shrinkage and increases the temperature to reach the maximum shrinkage rate, the agglomerates also influence the final sintering. The final density of the sintered bulk is limited because of the existence of pores among agglomerates. Zirconia wear debris effectively inhibits the grain growth of alumina during the sintering. Varying the degree of deagglomeration and the amounts of zirconia wear debris together can control the microstructure during the final stage of the sintering.     

An Experimental Measurement of Effective Diffusion Distance for the Sintering of Ceramics

The traditional models of sintering predict a pronounced dependence of densification rate on the scale of the microstructure as measured by the grain size. This study evaluates the grain size exponent for densification during isothermal sintering of an aggregated nanocrystalline zirconia powder, and for a submicrometer alumina powder. The results gave grain size exponents that are much higher than those anticipated for the expected sintering mechanisms. Furthermore, microstructural analysis showed that this overestimate of the exponent could be due to the spatial heterogeneity in the microstructure on the scale of the diffusion distance. To assess this issue, pore boundary tessellation was used to determine a new measurement of effective diffusion distance that takes into account the local spatial arrangement of pores. This measurement gives exponents much closer to those expected for the sintering of tetragonal zirconia by volume diffusion, and for the sintering of the alumina by grain-boundary diffusion.     

Thermodynamics of Densification: I, Sintering of Simple Particle Arrays, Equilibrium Configurations, Pore Stability, and Shrinkage

Equilibrium configurations for linear and closed arrays (rings and regular polyhedra containing a single pore) of identical particles (cylinders or spheres) were determined by minimizing the array's surface and grain-boundary energies with the assumption that each particle conserves its mass. The change in free energy between the initial and equilibrium configuration increases with dihedral angle (i.e., the equilibrium angle). More significantly, it is shown that pores will shrink to an equilibrium size if the number ( n ) of coordinating particles is greater than a critical value. The critical pore coordination number ( n _c) increases with the dihedral angle. Only pores with n n _c are thermodynamically unstable during sintering. It is also shown that any mass-transport mechanism can lead to pore shrinkage while a connecting path to the pore surface remains open. The effective sintering "stress" (i.e., driving force) increases with the dihedral angle and decreases to zero as the equilibrium configuration is reached. Sintering stresses increase with decreasing coordination number. It is also shown that the shrinkage strain for closed arrays increases with the pore coordination number. Rearrangement phenomena within a powder compact are discussed with regard to resultant sintering forces on nonsymmetrically coordinated particles.