A theoretical study of grain growth in porous solids during sintering

The processes of grain boundary migration, pore drag and pore/boundary separation are described on the basis of the phenomenological equations for boundary migration and surface diffusion. Cylindrical pores on triple grain junctions are assumed to represent the open porosity during intermediate-stage sintering. It is found that cylindrical pores can hardly detach from migrating boundaries. Three-dimensional closed pores, however, which predominate during final stage sintering, can separate from migrating grain junctions. The separation process is modelled numerically and the conditions for separation are formulated. Analytical approximations for the pore mobility are shown to describe the numerical results well. They serve to establish effective mobilities of grain boundaries bearing pores in various configurations. Classical theories of grain coarsening are modified by using these effective mobilities. Mechanical constitutive models of sintering contain the grain size as an internal variable. The present analysis leads to an evolution equation for the average grain size, which depends on the volume fraction of the pores and on their configuration.     

Quantitative determination of topological and metric properties during sintering of copper

Quantitative paths of microstructural change, represented as a variation of topological and metric properties with pore volume fraction, were experimentally determined by applying quantitative microscopy to sequences of samples sintered without compaction from two size fractions of spherical copper powder, and one size fraction of dendritic copper powder. The two spherical size fractions followed paths of microstructural change during sintering that were identical except for a scale factor. The connectivity of the pore network first increased slightly, then decreased, reaching zero at a pore volume fraction (V _V ) of about 0.08. Isolated pores begin to appear atV _V = 0.20, and increased in number. The area of the pore-solid interface at first decreased slowly, then more rapidly and ultimately linearly with pore volume fraction, as has been reported in other systems. Total curvature of pore-solid interface decreased from the positive value characteristic of the loose powder stack to a negative value, passed through a minimum, and increased toward zero as full density is approached. The area of grain boundary initially increased slowly, as interparticle contacts grew; at aboutV _V = 0.15, grain growth set in, and the grain boundary area decreased, as the mean grain intercept rapidly increased with densification. The dendritic powder had a highly irregular surface shape, and consequently a loose stack structure containing more than 90 pct porosity. The path that it followed was qualitatively, but not quantitatively, similar to that observed for the spherical powders. These observations are discussed in terms of the unit geometric processes that dominate each of the three stages of loose stack sintering.     

Grain growth inhibition by porosity

A model of pore or other second phase inhibited grain growth has been developed based on shape independent stereologically measurable parameters such as pore surface area and the degree of pore/grain boundary contact. The model prediction of a linear relationship between pore surface area and the inverse of the grain size was observed experimentally with sintered copper, tungsten, Al₂O₃ and UO₂. The model also quantifies the conditions for pore controlled grain growth and pore separation in terms of the pore and grain boundary mobilities and grain boundary curvature and provides a means for quantifying pore mobility from grain growth studies on porous and dense materials.     

Quantitative determination of topological and metric properties during sintering of copper

Quantitative paths of microstructural change, represented as a variation of topological and metric properties with pore volume fraction, were experimentally determined by applying quantitative microscopy to sequences of samples sintered without compaction from two size fractions of spherical copper powder, and one size fraction of dendritic copper powder. The two spherical size fractions followed paths of microstructural change during sintering that were identical except for a scale factor. The connectivity of the pore network first increased slightly, then decreased, reaching zero at a pore volume fraction (V _V ) of about 0.08. Isolated pores begin to appear atV _V = 0.20, and increased in number. The area of the pore-solid interface at first decreased slowly, then more rapidly and ultimately linearly with pore volume fraction, as has been reported in other systems. Total curvature of pore-solid interface decreased from the positive value characteristic of the loose powder stack to a negative value, passed through a minimum, and increased toward zero as full density is approached. The area of grain boundary initially increased slowly, as interparticle contacts grew; at aboutV _V = 0.15, grain growth set in, and the grain boundary area decreased, as the mean grain intercept rapidly increased with densification. The dendritic powder had a highly irregular surface shape, and consequently a loose stack structure containing more than 90 pct porosity. The path that it followed was qualitatively, but not quantitatively, similar to that observed for the spherical powders. These observations are discussed in terms of the unit geometric processes that dominate each of the three stages of loose stack sintering.     

Theory of grain boundary motion in the presence of mobile particles

A theory of grain boundary motion in the presence of mobile particles is put forward. It is shown that the boundary-particle-interaction leads to a hysteresis in the velocity-driving relationship. The extent of the hysteresis depends on particle mobility, which is very sensitive to particle size. The effect of particles is discussed for planar and curved boundaries as well as volume particle distributions. The theory accounts for a smaller limiting grain size during grain growth than predicted by Zener drag. The concept can be generalized to include all kinds of mobile obstacles for boundary migration. In such cases not the distribution of obstacle spacing rather the distribution of obstacle mobilities will control microstructure evolution.     

Modeling of supersolidus liquid phase sintering: II. Densification

Analogous to the viscous-flow principle in rheology, a densification model is developed for super-solidus liquid phase sintering (SLPS). The presence of a threshold value of the liquid film along the grain boundary and its influence on the onset of densification is analyzed. The liquid film thickness is affected by the grain size and liquid volume fraction. Both change with temperature during constant heating, thereby influencing the densification process due to particle rearrangementvia grain frag-mentation. Two cases of densification, one using a constant heating rate and the other under iso-thermal conditions, are discussed. The present model is in good conceptual agreement with the viscous flow-induced densification mechanism and provides a sound basis for a quantitative study of shrinkage behavior in SLPS.     

Grain boundary mobility during recrystallization of copper

Average grain boundary migration rates during recrystallization of cold-deformed copper were estimated from stereological measurements. In the same material, the instantaneous driving forces for boundary migration during recrystallization were calculated from calorimetric measurements of the release of the stored energy of cold work. The migration rate dependence on driving force was analyzed in the context of grain boundary migration rate theory, and within experimental error, a linear dependence was observed. The average mobility of grain boundaries migrating during recrystallization of cold-worked copper at 121°C was calculated to be 6.31×10⁻¹⁰ (m⁴ s⁻¹ MJ⁻¹). This result was found to be consistent with single boundary, curvature-driven grain boundary mobilities measured in copper at higher temperatures. It was also demonstrated that the average grain boundary mobility was reasonably within the expectation (order of magnitude uncertainty) of the Turnbull single process model of boundary migration with a process akin to grain boundary self-diffusivity as the rate-controlling atomic mechanism.     

Grain boundary mobility during recrystallization of copper

Average grain boundary migration rates during recrystallization of cold-deformed copper were estimated from stereological measurements. In the same material, the instantaneous driving forces for boundary migration during recrystallization were calculated from calorimetric measurements of the release of the stored energy of cold work. The migration rate dependence on driving force was analyzed in the context of grain boundary migration rate theory, and within experimental error, a linear dependence was observed. The average mobility of grain boundaries migrating during recrystallization of cold-worked copper at 121 °C was calculated to be 6.31×10⁻¹⁰ (m⁴ s⁻¹ MJ⁻¹). This result was found to be consistent with single boundary, curvature-driven grain boundary mobilities measured in copper at higher temperatures. It was also demonstrated that the average grain boundary mobility was reasonably within the expectation (order of magnitude uncertainty) of the Turnbull single process model of boundary migration with a process akin to grain boundary self-diffusivity as the rate-controlling atomic mechanism.     

Theoretical modeling of densification during activated solid-state sintering

Activated solid-state sintering relies on the addition of low concentrations of grain boundary segre-gating species to increase diffusion rates. In this article, enhanced diffusion through an activated layer at the grain boundaries has been modeled for the case of tungsten sintered with transition element additions. Both constant heating rates and isothermal sintering are considered. As in classical treatments, sintering is divided into three stages, but modifications are proposed based on recent observations and theories regarding packing coordination, pore morphology, pore location, grain growth, and pore-grain boundary separation. The intermediate and final stages of sintering are al-lowed to overlap based on the amount of closed porosity to account for both pore closure early in the process and the gradual increase in packing coordination with densification. Mean curvature theory is used to estimate pore curvature during the intermediate stage of sintering. In the final stage, pores are modeled on both the corners of a tetrakaidecahedron and on its square facets. The pore location has only a small effect on densification, while the grain boundary mobility is more of a factor. The model allows pore-grain boundary separation to match experimentally measured grain sizes. The model predictions are compared to dilatometer curves of pure tungsten and tungsten sintered with additions of Co, Fe, Ni, and Pd. For the Co- and Fe-activated samples, the model is modified to account for an increase in diffusional activation energy due to dissolution of the activator in tungsten. Formerly Director of Materials Development, P/M Lab, Department of Engineering Science and Mechanics, The Pennsylvania State University.     

Spark Plasma Sintering of Cryomilled Nanocrystalline Al Alloy - Part II: Influence of Processing Conditions on Densification and Properties

In this study, nanostructured Al 5083 powders, which were prepared via cryomilling, were consolidated using spark plasma sintering (SPS). The influence of processing conditions, e.g., the loading mode, starting microstructure (i.e., atomized vs cryomilled powders), sintering pressure, sintering temperature, and powder particle size on the consolidation response and associated mechanical properties were studied. Additionally, the mechanisms that govern densification during SPS were discussed also. The results reported herein suggest that the morphology and microstructure of the cryomilled powder resulted in an enhanced densification rate compared with that of atomized powder. The pressure-loading mode had a significant effect on the mechanical properties of the samples consolidated by SPS. The consolidated compact revealed differences in mechanical response when tested along the SPS loading axis and radial directions. Higher sintering pressures improved both the strength and ductility of the samples. The influence of grain size on diffusion was considered on the basis of available diffusion equations, and the results show that densification was attributed primarily to a plastic flow mechanism during the loading pressure period. Once the final pressure was applied, power law creep became the dominant densification mechanism. Higher sintering temperature improved the ductility of the consolidated compact at the expense of strength, whereas samples sintered at lower temperature exhibited brittle behavior. Finally, densification rate was found to be inversely proportional to the particle size.     

Mechanism of steady-state grain growth in aluminum

Grain growth, defined as the increase in volume of the average grain, is found, in its steadystate, to be directly proportional to the time of isothermal annealing. During steady-state grain growth the grain corners are found all to be quadruple points, the grain edges all triple lines and the ratio of corners to faces to edges to be 6:7:12. The rate constant for steady-state grain growth is shown to be calculable from first principles and from properties that can be measured independently of the growth observation. It is the product of four individual constants, namely: 1) a dimensionless topological constant ⊝ that is characteristic of steady-state grain growth in any material, 2) the mobility ώ of the average grain boundary in the specific material, 3) the surface tension y of the average grain boundary in the specific material and 4) a dimensionless structural constant σ which expresses the curvature of surface of the grain boundary in the array of grain forms obtaining in the specific specimen of the material and which can be determined metallographically. The topological changes that constitute steady-state growth are shown to exist as a logical sequence of simple events. Rewritten for publication from theAndrew Carnegie Lecture of the Pittsburgh Section of the American Society for Metals; presented by Frederick N. Rhines on November 9, 1972.     

An effect of chemisorbing surface reaction poisons on the transition from internal to external oxidation

We consider the effect of a chemisorbing surface reaction poison on the transition from internal to external oxidation for a model binary alloy subject to selective oxidation. We solve the diffusion equations for internal oxidation, using the rate of a surface reaction as a boundary condition. The gas-phase concentration of a strongly chemisorbing poison appears in the analysis through its retarding effect on that surface reaction. It is shown that the slow surface reaction can promote the formation of a protective oxide scale by slowing down the initial uptake of oxygen and allowing metal atoms with the highest oxygen affinity sufficient time to diffuse to the surface. As a result, a larger volume fraction of oxide forms near to the surface (with that volume fraction initially inversely proportional to the rate of the surface reaction), and the critical volume fraction required for the transition to an external scale is more easily exceeded. Formerly with Exxon Research and Engineering Company     

Pore-boundary interactions and evolution equations for the porosity and the grain size during sintering

The retardation of grain boundary migration by a pore is analysed numerically and by an analytical approximation, which becomes exact in the limiting case of small velocities. At high velocities, the numerical solution describes separation of the pore from the boundary. Based on the analysis of a single migrating boundary facet dragging a pore, evolution equations for the grain size and the porosity are suggested for the final stage of sintering. It turns out that three-dimensional pores on two-grain junctions have an only moderate effect on the rate of grain coarsening: pores with a high mobility follow the migrating grain boundary easily, while those with a low mobility detach from the boundary. It is also shown that a two-dimensional pore does not detach and can therefore reduce the coarsening rate to very low values if the pore mobility is low. This strong dependence on the pore configuration suggests that other possible configurations need to be analysed, before final evolution equations can be formulated.     

Discussion of grain growth in aluminum

The data of Rhines and Craig (RC) (Met Trans., 1974, vol. 5, pp. 413-25) are partially reinterpreted. For the first regime of grain growth, average grain volume is proportional to time, in accordance with RC, but for the second, it is proportional to the three-halves power of time. The structural factor σ is found to be constant only in the first regime, and its value is redetermined from the data to be about 2.65. Theoretical derivations of this value are also given and are found to be related to grain form (e.g. a tetrakaidecahedron). Grain boundary motion in the second regime is controlled by drag due to lattice diffusion of solute atoms from the boundary, and an activation energy of about 27 kcal/mole (113 kJ/mole) for this process is calculated from the data.     

Stress corrosion crack velocity and grain boundary precipitates in an Al-Zn-Mg alloy

The fracture kinetics of Al-5.5 Zn-2.5 Mg alloys submersed in 3 pct NaCl-H₂O solutions were varied by heat treatment. The steady state velocity, on a plot of velocity vs stress intensity, was compared with microstructure and it was found to be inversely proportional to the volume of MgZn₂ in the grain boundary. This behavior suggests that grain boundary precipitates can act as sacrificial anodes to retard intergranular stress corrosion cracking. Formerly Graduate Student at the University of Connecticut     

Discussion of grain growth in aluminum

The data of Rhines and Craig (RC) (Met Trans., 1974, vol. 5, pp. 413-25) are partially reinterpreted. For the first regime of grain growth, average grain volume is proportional to time, in accordance with RC, but for the second, it is proportional to the three-halves power of time. The structural factor σ is found to be constant only in the first regime, and its value is redetermined from the data to be about 2.65. Theoretical derivations of this value are also given and are found to be related to grain form (e.g. a tetrakaidecahedron). Grain boundary motion in the second regime is controlled by drag due to lattice diffusion of solute atoms from the boundary, and an activation energy of about 27 kcal/mole (113 kJ/mole) for this process is calculated from the data. This work was performed while the author was on sabbatical leave at the School of Applied Sciences, University of Sussex, England.     

Solid-state contributions to densification during liquid-phase sintering

Densification via liquid-phase sintering generally requires transport of substantial amounts of dissolved solid through the liquid. However, in composite systems, such as W-Cu, solid solubility in the liquid is almost negligible, and densification is hindered by the low amount of total mass transport. In this case, solid-state sintering of the skeletal solid structure in the presence of the liquid is a significant densification mechanism. In this article, the relative contributions to densification of both liquid and solid mass transport mechanisms are considered. A computer simulation is constructed to predict the densification behavior and concurrent microstructural development of liquidphase sintered composites for realistic heating cycles. Governing differential equations for densification are derived from idealized models of the microstructure, considering grain size, diffusion distance from vacancy source to sink, pore size, and pore morphology. Temperature-dependent terms, including the diffusivity, solubility, and surface energy, govern densification and microstructural parameters, such as the grain size, dihedral angle, and contiguity. Predictions for the sintered density, grain size, and contiguity are compared to experimental results for the W-Cu and W-Cu-Ni systems with approximately 20 vol pct liquid. For W-Cu, which has almost no intersolubility, solid-state sintering of W in the presence of liquid Cu is the dominant densification mechanism. Nickel additions increase solid solubility in the liquid and improve typical liquid-phase sintering contributions to densification. Alternatively, high sintered densities can be achieved in the absence of solubility with a sufficiently small particle size due to the solid-state contribution.     

Microstructural coarsening during supersolidus liquid phase sintering of alpha brass

Coarsening of the grains and pores during sintering has a pronouncedly negative effect on the densification of prealloyed brass powder compacts. This investigation examines the role of sintering variables in realising the complicated effects on densification and microstructure. Experiments were designed to model and evaluate the effect of sintering parameters such as temperature, time and furnace atmosphere on densification, grain and pore intercept as well as pore number. The study of microstructures suggests that there is a good correlation between grain and pore intercepts. It is concluded that pore coarsening is a result of supersolidus liquid phase sintering of Cu28Zn powder, and it can retard densification, which is in acceptable agreement with the experimental data.     

Effect of shape of sulfide inclusions on anisotropy of inclusion spacings and on directionality of ductility in hot-rolled C-Mn steels

The effects of sulfur content (0.004 or 0.013 pct) and sulfide morphology (stringered or globular) on anisotropy of tensile ductility and Charpy shelf energy were investigated in a series of 0.1 and 0.2 pct carbon, 1.0 pct manganese steels. The effect of sulfide inclusions on fracture strain or Charpy shelf energy correlated with the projected area of inclusions per unit volume,A _v , on a plane perpendicular to the tensile direction and the mean free distance between inclusions, λ, in a direction parallel to the tensile direction regardless of the amount or the shape of the inclusions or the test direction: longitudinal, transverse, or through-thickness. The magnitude ofA _v is directly proportional to the volume fraction of inclusions and inversely proportional to the inclusion dimension parallel to the tensile direction. The mean free distance, λ, is inversely proportional toA _v . Approximate relations were obtained for the nearest-neighbor distances between inclusions on the longitudinal, transverse, and through-thickness planes. These distances were incorporated into a model for ductile fracture based on an adaptation of a previously proposed criterion for the linkage of voids nucleated at second-phase particles. The agreement between the observed and predicted fracture strains for longitudinal, transverse, and through-thickness specimens of the steels studied is encouraging. Formerly with United States Steel Corporation, Research Laboratory, Monroeville, PA 15146