Computer Simulation of Final-Stage Sintering: II, Influence of Initial Pore Size

A two-dimensional Monte Carlo simulation procedure has been used to investigate the effect of the initial pore size on the microstructural evolution and the kinetics of final-stage sintering. The sintering time scales with r ⁴₀/ D_gb and the grain-growth time scales with r ²_O/ D_m . Pores are found to effectively pin the grain boundaries from the beginning of final-stage sintering at a porosity of Φ= 0.09 until Φ= 0.03. For Φ 0.03, the remaining pores do not effectively retard grain-boundary migration and normal grain growth occurs. Small pores were found to be less effective at retarding grain growth than expected on the basis of a simple grain-growth pinning model. The mean pore size was found to be nearly constant throughout the simulations.     

Effect of Pore Distribution on Microstructure Development: III, Model Experiments

Model experiments have been conducted on a series of alumina samples in which the microstructures have been tailored to conform to the classical configuratins depicted in the models of final-stage sintering. Simultaneous measurements of sintered density, grain size, pore number density, and pore size distribution were made as a function of sintering time at constant temperature (1850°C). The data supported a model of grain-boundary-diffusion-controlled densification and surface-diffusion-controlled grain growth. An atom flux equation for grain-boundary diffusion transport was deduced from the data. The kinetics analysis highlights the importance of incorporating the number of pores per grain as an independent variable in mechanistic studies of final-stage sintering. The number of pores per unit volume was identified as a critical factor influencing densification kinetics. The effect of pore distribution on microstructure development was simulated for comparison with the data obtained from the model experiments.     

Influence of Atmosphere on the Final-Stage Sintering Kinetics of Ultra-High-Purity Alumina

The influence of sintering atmosphere on the final-stage sintering of ultra-high-purity alumina has been investigated. Model final-stage microstructures were tailored via a latex sphere impregnation and burnout technique. Critical experiments have been conducted to quantitatively examine the influence of the oxygen partial pressure on the final-stage sintering kinetics. Samples were sintered at 1850°C in either dry hydrogen ( P o₂∼ 3 × 10⁻¹⁷ atm) or wet hydrogen P o₂∼ 5 × 10⁻¹⁰ to 2 × 10⁻¹¹ atm), and their microstructures were characterized as a function of sintering time, Sintering in dry hydrogen decreased the susceptibility of the final-stage microstructure to pore/boundary breakaway. In the kinetic analysis, the variation in the number of pores per grain, N _g, was taken into account. It was found that in both atmospheres, the densification rate was controlled by grain boundary diffusion, and that sintering in dry hydrogen increased the densification rate by a factor of 2.25. In addition, it was determined that the grain growth rate in both atmospheres was controlled by the rate of surface diffusion of matter around the pores and that sintering in dry hydrogen enhanced the grain growth rate by a factor of 5.6. The overall effect of the dry hydrogen atmosphere was that it enhanced the coarsening rate relative to the densification rate by a factor of 2.5, and consequently shifted the grain size-density trajectory to much lower densities for a given grain size.     

Novel model for the sintering of ceramics with bimodal pore size distributions: Application to the sintering of lime

A mathematical model for the sintering of ceramics with bimodal pore size distributions at intermediate and final stages is developed. It considers the simultaneous effects of coarsening by surface diffusion, and densification by grain boundary diffusion and lattice diffusion. This model involves population balances for the pores in different zones determined by each porosimetry peak, and is able to predict the evolution of pore size distribution function, surface area, and porosity over time. The model is experimentally validated for the sintering of lime and it is reliable in predicting the so called “initial induction period” in sintering, which is due to a decrease in intra‐aggregate porosity offset by an increase inter‐aggregate porosity. In addition, a novel methodology for determination of mechanisms based on the analysis of the pore size distribution function is proposed, and with this, it was demonstrated that lattice diffusion is the controlling mechanism in the CaO sintering. © 2016 American Institute of Chemical Engineers AIChE J, 63: 893–902, 2017     

Three-Dimensional Simulation of Grain Growth in the Presence of Mobile Pores

A kinetic, three-dimensional Monte Carlo model for simulating grain growth in the presence of mobile pores is presented. The model was used to study grain growth and pore migration by surface diffusion in an idealized geometry that ensures constant driving force for grain growth. The driving forces, pore size, and pore mobilities were varied to study their effects on grain-boundary mobility and grain growth. The simulations captured a variety of complex behaviors, including reduced grain-boundary velocity due to pore drag that has been predicted by analytical theories. The model is capable of treating far more complex geometries, including polycrystals. We present the capabilities of this model and discuss its limitations.     

Effect of Pore Distribution on Microstructure Development: II, First- and Second-Generation Pores

A sintering model has been developed to predict the consequences of independently varying the grain growth rate in alumina during final-stage sintering of a microstructure containing both small (first-generation) and large (inter-agglomerate second-generation) pores. The model shows that although it may be thermodynamically favorable to increase the grain growth rate, the kinetics of densification are such that it almost always pays to inhibit grain growth. This conclusion was verified by experiments on undoped, MgO-doped, and ZrO₂-doped alumina impregnated with model spherical large pores produced by the burnt-out latex sphere method. A new type of ceramic processing map has also been developed to aid in the selection of the optimum processing conditions for the sintering of ceramics containing large pores.     

Sinter-Forging of Nanocrystalline Zirconia: I, Experimental

Nanocrystalline (15 nm) yttria (3 mol%)-stabilized zirconia (3Y-TZP) was sinter-forged under conditions of varying temperature (1050–1200°C), plastic strain rate (5 × 10⁻⁵ to 2 × 10⁻³s⁻¹), and green density (33–48%), using constant-crosshead-speed tests, constant-load (i.e., load-and-hold) tests, and constant-loading-rate tests. The densification and pore size evolution results indicate that plastic strain is largely responsible for elimination of large pores, while diffusional mechanisms control the elimination of small pores. Grain growth during sinter-forging is observed to be dependent solely on porosity during intermediate-stage sintering. Once the powder compact enters final-stage sintering, however, both static (time- and temperature-dependent) and dynamic (plastic-strain-dependent) grain growth take place, greatly accelerating the overall rate of grain growth. The use of fast strain rates to impose plastic strain before the onset of dynamic grain growth is proposed as a method of preserving small grain sizes during sinter-forging.     

Strain rate dependence of the contribution of surface diffusion to bulk sintering viscosity

Modeling of bulk sintering viscosity usually neglects the contribution of pore surface diffusion with respect to grain-boundary diffusion. This approximation is questionable at the high densification rates used today in advanced fast sintering techniques. A two-dimensional analysis of the problem shows that the influence of surface diffusion on bulk viscosity at high strain rate can be decomposed as the sum of two terms: a term linked to the change in pore surface curvature and a term linked to the change in grain-boundary size. The computational procedure relies on the partition of pore profile evolution into a transient component accounting for non-densifying phenomena and an asymptotic component accounting for strain-rate-controlled phenomena. The largest impact of surface diffusion is found to arise from the change in grain-boundary size. It follows a transition from Newtonian viscosity at low strain rate to non-Newtonian viscosity which, during densification, increases nearly linearly with strain rate. In some conditions, viscosity can then reach more than twice the value estimated when neglecting pore surface diffusion. Reversely, expansion is accompanied by a decrease in grain-boundary size which causes a decrease in viscosity and can lead to grain separation at high strain rate.     

Theory of the Dependence of Densification on Grain Growth During Intermediate-Stage Sintering

A semiempirical model for intermediate-stage sintering is developed based on simultaneously occurring volume and grain-boundary diffusion mechanisms of mass transport and explicitly incorporating the effects of grain growth. The sintering equation derived depends strongly on the reduction of pore number density associated with grain growth and is independent of the mechanism of grain growth. The time and temperature dependencies of densification predicted by the equation, which are tested using data for metal and ceramic powders, agree well with observations. The data indicate that grain-boundary diffusion contributes negligibly to densification.     

Sintering forces acting among particles during sintering by grain-boundary/surface diffusion

The microstructural evolution during sintering involves the formation and pinch-off of pore channels. The sintering of 3 particles in 3 dimensions is a simple model for studying the pinch-off of a pore channel. We simulate the solid-state sintering of 3 particles by using Brakke's Surface Evolver program, which incorporates coupled grain-boundary diffusion and surface diffusion. The pinch-off of pore channel divides the sintering process into 2 stages; the initial stage and the later stage. The contact area has a noncircular shape bounded by both surface triple junction and triple junction in the later stage. A general method is presented to determine the sintering force acting on the noncircular contact. The mechanical approach of densification, where the relative motion of particles is driven by both sintering force and applied force, is applicable not only for the initial stage, but also for the later stage.     

Effect of Magnesia Solute on Surface Diffusion in Sapphire and the Role-of Magnesia in the Sintering of Alumina

Surface diffusion in sapphire and sintering of Al₂O₃ have been studied under identical firing conditions (1593°C, N₂) as a function of MgO solute additions. The effect of MgO on surface diffusion has been investigated directly using the technique of multiple scratch smoothing on the (1123) surface of sapphire. For these conditions MgO additions enhance the decay of periodic profiles by a factor of 4; that is interpreted as reflecting an increase in the surface diffusivity by the same amount. Measurements of the grain sizedensity trajectory during final stages of sintering of Al₂O₃ reveal that MgO enhances the densification rate/coarsening rate ratio by a modest factor of 1.8. Since an increase in surface diffusivity alone would decrease the ratio, it is deduced that the primary role of MgO in the sintering of Al₂O₃ is other than the influence on surface diffusion. These observations in conjunction with independent measurements of the effect of MgO on grain-boundary mobility confirm that the primary role of MgO is the suppression of grainboundary motion.     

Sinter-Forging of Nanocrystalline Zirconia: II, Simulation

A quantitative computer simulation of densification, pore-size evolution, and grain growth during sinter-forging has been developed and applied to the sinter-forging of nanocrystalline yttria (30 mol%)-stabilized zirconia (3Y-TZP) at 1050° and 1100°C. Densification is simulated as a superposition of stress-assisted and plastic-strain-controlled pore-shrinkage mechanisms. Grain growth is simulated as a pore-controlled process during intermediate-stage densification and as a combination of normal (static) grain growth and dynamic grain growth during final-stage densification. The densification portion of the model offers very good agreement with the experiment under a wide variety of imposed forging conditions, despite the absence of adjustable variables. Grain-growth predictions qualitatively illustrate a key feature in the sinter-forging of 3Y-TZP: i.e., the minimization of grain size, as a function of density, under forging conditions that promote high strain rates. This particular effect seems to be due to the quick elimination of large pores by plastic strain while small pores (which shrink by diffusion) are still available to control grain growth.     

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.     

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.     

Behavior of Large Pores During Sintering and Hot Isostatic Pressing

A viscoelastic model that describes the shrinkage rate of large pores within a fine-grained body has been developed. The concept is based on a shrinkage potential derived from the surface and grain-boundary tensions and viscous response determined by the creep characteristics of the polycrystal. Pore removal rates are also derived and used to predict pore removal times during sintering and hot isostatic pressing.     

Particle Rearrangement and Pore Space Coarsening During Solid-State Sintering

Coarsening of porosity during sintering has been observed in powder compacts of metallic, ceramic, and amorphous materials. Monitoring and modelling of the growth of individual (closed) pores in the late sintering stages are well established. Porosity is interconnected up to very high densities. Coarsening of the continuous pore space takes place during the initial and intermediate sintering stages. This coarsening is caused by localized transport of atoms or molecules (diffusion or viscous flow) as well as by bulk particle movement (rearrangement). Its quantitative exploration poses problems both experimentally and theoretically. Ways to characterize the geometry of the interconnected pore space and of closed pores are discussed with emphasis on stereological parameters. Recent and classical approaches, experimental findings with 2D model arrangements (as the formation and opening up of particle contacts, pore coarsening, and particle rearrangement) and some advances of computer simulations are discussed together with open questions.     

Effect of Pore Distribution on Microstructure Development: I, Matrix Pores

A model has been developed to describe the effect of the matrix (first-generation) pore distribution on microstructure development in the final stages of sintering. A model of simultaneous densification and grain growth was used to predict the effects of the number of pores per grain and the pore size distribution on microstructure evolution. Increasing the number of pores per grain was predicted to increase the densification rate, the grain growth rate, and the relative densification rate/grain growth rate ratio. Narrowing the pore size distribution was predicted to inhibit grain growth initially and to increase the densification rate indirectly. Overall, the pore distribution was predicted to have a strong influence on microstructure development and sintering kinetics.     

Effect of Yttrium and Lanthanum on the Final-Stage Sintering Behavior of Ultrahigh-Purity Alumina

Final-stage sintering has been investigated in ultrahigh-purity Al₂O₃ and Al₂O₃that has been doped individually with 1000 ppm of yttrium and 1000 ppm of lanthanum. In the undoped and doped materials, the dominant densification mechanism is consistent with grain-boundary diffusion. Doping with yttrium and lanthanum decreases the densification rate by a factor of ˜11 and 21, respectively. It is postulated that these large rare-earth cations, which segregate strongly to the grain boundaries in Al₂O₃, block the diffusion of ions along grain boundaries, leading to reduced grain-boundary diffusivity and decreased densification rate. In addition, doping with yttrium and lanthanum decreases grain growth during sintering. In the undoped Al₂O₃, surface-diffusion-controlled pore drag governs grain growth; in the doped materials, no grain-growth mechanism could be unambiguously identified. Overall, yttrium and lanthanum decreases the coarsening rate, relative to the densification rate, and, hence, shifted the grain-size-density trajectory to higher density for a given grain size. It is believed that the effect of the additives is linked strongly to their segregation to the Al₂O₃grain boundaries.     

Effect of MgO Solute on Microstructure Development in Al2O3

The effect of MgO as a solid-solution additive in the sintering of Al₂O₃ was studied. The separate effects of the additive on densification and grain growth were assessed. Magnesia was found to increase the densification rate during sintering by a factor of 3 through a raising of the diffusion rate. The grain-size dependence of the densification rate indicated control primarily by grain-boundary diffusion. Magnesia also increased the grain growth rate during sintering by a factor of 2.5. The dependence of the grain growth rate on density and grain size suggested a mechanism of surface-diffusion-controlled pore drag. It was argued, therefore, that MgO enhanced grain growth by raising the surface diffusion coefficient. The effect of MgO on the densification rate/grain growth rate ratio was, therefore, found to be minimal and, consequently, MgO did not have a significant effect on the grain size/density trajectory during sintering. The role of MgO in the sintering of alumina was attributed mainly to its ability to lower the grain-boundary mobility.