Sintering Trajectories: Description on How Density, Surface Area, and Grain Size Change
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Sintering is a mainstay production step in forming metal, ceramic, polymer, and composite components from particles. Since the 1940s, the sintering process is treated using a matrix of mathematical relationships that include at least seven atomic transport mechanisms, several options on powder characteristics, and three pore–grain morphology options. The interplay of these relationships is handled by numerical solutions to predict property development. An alternative approach is to track the sintering trajectory using relatively simple relationships based on bulk measures. Energy minimization dictates that initial stage sintering acts to reduce surface area. In late stage sintering, the energy minimization turns to grain boundary area reduction via grain growth. Accordingly, relationships result between density, surface area, and grain size, which largely ignore mechanistic details. These relationships are applicable to a wide variety of materials and consolidation conditions, including hot pressing, and spark sintering.
Sintering reduces surface area by growing bonds between contacting particles during heating. Due to random orientations for the particles, the bond forms with an embedded grain boundary accommodating the crystal misorientation between particles. Effectively, early sinter bonding replaces surface area with lower energy grain boundary area. As surface area is annihilated the driving force declines, resulting in slower sintering rates.1 Bond size between particles is one monitor of sintering; however, it is a tedious measure, especially for small particles. On the other hand, density, surface area, shrinkage, and properties (hardness and strength) are measures that average over many particle–particle bonds. These attributes are easier to measure and follow trajectories that require only a few experiments to map the sintering process.2
surface transport (surface diffusion and evaporation–condensation), or
bulk transport (grain boundary diffusion, plastic flow, dislocation climb, viscous flow, and volume diffusion).
Bulk transport processes contribute to densification, but surface transport only gives bonding. Early sintering initiates bonding by surface transport, but as surface area is converted into grain boundary area the opportunity for densification increases. Small particles, longer sintering times, and higher sintering temperatures increase sintering densification and improve properties. For example, traditional ferrous powder metallurgy relies on nominally 100-µm particles compacted to 85–90% density, followed by sintering for up to 30 min at 1120°C. This combination minimizes densification to avoid component warpage that would arise from the density gradients induced by uniaxial compaction. Alternatively, powder injection molding (PIM) relies on binder-assisted hydrostatic forming using 5-µm particles sintered at higher temperatures (1250°C) for longer times (120 min). The 60% dense PIM shape densifies to about 98% density, with isotropic shrinkage to avoid distortion. Sintered properties reflect the density difference. For example, after heat treatment, a Fe-2Ni-0.5C steel delivers 650 MPa yield strength by conventional powder metallurgy, but 1230 MPa by injection molding. This strength difference comes from the higher density attained with the smaller particles, higher temperature, and longer time.
Sintering reduces energy by elimination of surface area due to bond growth, partially offset by a concomitant increase in grain boundary area and energy. Both aspects are linked to density. DeHoff et al.9 proposed a linear relationship between surface area and sintered density, assuming densification work was derived from the surface energy release. A similar conceptualization is embedded in treatments of sintering by viscous flow10 and grain boundary diffusion.11
Late in sintering, surface area loss is slow, but grain coalescence continues to reduce grain boundary area. Sensibly, an energy cascade occurs. First, solid–vapor energy is converted into grain boundary energy by bond growth. Subsequently, grain boundary energy is eliminated by grain growth. The details of the sintering trajectory depend on the relative surface transport and bulk transport rates. Some cases lose surface area without densification, such as boron sintering in vacuum12 or zirconia sintering in hydrogen chloride,13 others lose surface area with some densification such as alumina in argon14 or iron in hydrogen,15 while yet others sinter with considerable densification such as copper in hydrogen16 or urania in hydrogen.17 In all cases involving densification, surface area declines in proportion to the gain in density.
Surface Area: Density Trajectory
Surface area is a means to track energy release during sintering. Measures are either area per unit mass or per unit volume. Surface area per unit mass, specific surface area, is measured by gas absorption or fluid permeability. These measures only access open pores, so sealed internal pores are not included in the specific surface area, S M. Common units are m2/g or cm2/g. The absorption or permeability measurements are effective up to pore closure at fractional densities typically from 0.90 to 0.95.
Sintered density is related to fractional density ρ S = ρ T f, with f being the fractional density and ρ T being the theoretical density for the material.
The constants a and b depend on the powder. Spherical PIM powder with an initial fractional density of 0.64 would give a = 3.3 and b = 3.6.
Grain Boundary Area
Surface area is an effective monitor for sintering. However, the loss of surface area (energy) is offset by the growth of grain boundary area (energy); subsequently, grain growth acts to remove grain boundary area. For polycrystalline particles, initial grain growth is rapid until the grain size reaches the particle size, but then slows in the presence of pores.8
Two coarsening options operate while pores exist. The first is when the vapor phase in the pores is inactive, corresponding to most sintering practice. Grain growth then depends on transport across the solid–solid interface at the grain contacts. The second case is when the pores contain an active vapor phase, providing evaporation–condensation transport across pores. This occurs with halide-doped atmospheres or in systems sensitive to oxygen or water partial pressures.
Geometric relationships between sintering microstructure parameters
Grain coordination (N C)
Fractional density (f)
Solid–solid contiguity (C SS)
Pore size (d)
1–1.4 ε 1/2
0.4 G ε 1/2
1–1.5 ε 1/2
0.4 G ε 1/2
1–1.6 ε 1/2
0.4 G ε 1/2
1–1.7 ε 1/2
0.5 G ε 1/3
Grain Size Trajectory
Energy reduction during sintering leads to a competition within a sintering structure.1,9,31 Bond growth is initially dominant while the grain boundary area is small. Grain growth relies on grain boundary formation in the bonds between contacting grains. Late in a sintering grain growth acts to eliminate grain boundary area and becomes a dominant aspect of sintering. From a few experiments, it is possible to link the key sintering parameters. For example, knowing the time–temperature required to reach final density allows calculation of the expected grain size.
Most powders sinter by a combination of densification and nondensification mechanisms, usually surface diffusion and grain boundary diffusion. Both reduce surface area during bond growth. Surface diffusion is important to early sintering when there is little grain boundary area. Subsequently, grain boundary diffusion produces densification. Depending on the material, various trajectories of surface area versus density result. A few time–temperature experiments help isolate the trajectory for density.34 In turn, surface area and grain size variations with sintered density are similar over a wide range of materials.
Early sintering concepts focused on mass transport mechanisms, particle bonding and the associated shrinkage, densification, and pore structure changes. Computer simulations help track the resulting complex interactions and events. In spite of the complexities, a simple view comes from following energy reduction by surface area loss and subsequently grain boundary loss.
Tracking sintered density is sufficient to estimate many sintering parameters. Sinter density changes with a log–log relationships to sintering time. Initially, surface area is eliminated as bonds grow between contacting particles. Grain boundaries form in those bonds to accommodate the crystal orientation difference between grains. Specific surface area decreases linearly as density increases. At the same time, grain boundary area increases, enabling more densification by grain boundary diffusion, but energy reduction drives grain growth and the elimination of grain boundary area. As a consequence grain boundary area peaks near 80–85% density. While pores remain, grain size varies with the inverse square-root of fractional porosity. Over a broad array of materials, the sintering trajectories follow a characteristic trajectory, where specific surface area, grain size, and fractional density are related.
Prof. Viplava Kumar of Mahatma Gandhi Institute of Technology sparked renewed interest in morphological models for sintering. Funding for research on sintering is provided by the National Aeronautics and Space Administration (NNX14AB31G) under the management of Drs. James Patton Downey and Biliyar Bhat at the Marshall Space Flight Center.
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