Sintering Trajectories: Description on How Density, Surface Area, and Grain Size Change
Sintering Trajectories: Description on How Density, Surface Area, and Grain Size Change
RANDALL M. GERMAN 0
0 1.-College of Engineering, San Diego State University , San Diego, CA, USA. 2.-
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
Several mass transport mechanisms act during
sintering, broadly characterized as
eithersurface transport (surface diffusion and
bulk transport (grain boundary diffusion, plastic
flow, dislocation climb, viscous flow, and volume
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
100lm 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-lm 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
Fe2Ni-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.
Energy reduction helps understand sintering.
Contrast the images in Fig. 1; both were taken by
quenching injection molded 17-4 PH stainless steel
compacts during heating.3 As sintering progresses,
surface area declines while density increases, in this
case from 73% to 91%. Grain boundaries form in the
contacts, and over time grain size increases with
densification while surface area declines. Density,
surface area, and grain size are useful sintering
metrics, since there is a natural trajectory evident
over a range of materials, particle sizes, and
processing techniques.4–8 Crystalline materials first
give up surface area to form grain boundaries at the
interparticle bonds. Late in sintering, grain growth
removes grain boundary area as densification
continues. Accordingly, grain boundary area increases,
peaks, and then declines during sintering. Strength
depends on both density and grain size, so
oversintering with a loss of strength occurs with longer
hold times or higher sintering temperatures.
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
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, SM.
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.
On the other hand, quantitative microscopy
measures the surface perimeter on two-dimensional
cross-sections, giving the volume-based surface
area, SV. Convenient units are m2/m3 or cm2/cm3
(inverse length). Volume-based surface area
includes both open and closed pores. Prior to pore
closure the conversion from one measure to the
other is straightforward based on the sintered
SM ¼ qS
Sintered density is related to fractional density
qS = qT f, with f being the fractional density and qT
being the theoretical density for the material.
Several studies have confirmed that specific
surface area depends on sintered density.4,5,9,15,18–22
Figure 2 illustrates such behavior using data for
urania sintering at 1500 C for up to
2000 min(U.19O2T)he specific surface area is given
relative to the starting surface area versus
fractional density with a straight line fit to the data.
The surface area approaches zero at about 10%
porosity, indicating that only closed pores remain.
As another example, Fig. 3 plots the surface area for
170-nm alumina (Al2O3) during sintering at
1325 C.5 Again a linear decline in surface area
During sintering the specific surface area SM falls
from its initial value SO as the fractional sintered
density f increases:1
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.
A favorite metric for sintering is shrinkage, Y,
defined as the change in component size divided by the
initial size, or DL/LO. By convention, a decrease in
component size is positive shrinkage (effectively, a
negative dimensional change is the shrinkage).
Nearly isotropic shrinkage occurs in PIM components.
In those cases, shrinkage links the sintered fractional
density f to the green fractional density fO as,
Accordingly, dilatometer-measured shrinkage
provides a means to assess density during heating.
Related models link shrinkage to other sintering
metrics.18 Using volume conservation calculations,
independent of the atomic transport mechanism,
the normalized surface area SM/SO links to
fractional density.1 Figure 4 plots the results from this
approach for starting densities of 0.50, 0.55, 0.60,
and 0.65. No sintering mechanism is invoked,
simply geometric parameters are employed to link
surface area to densification.
Fig. 4. Geometric volume conservation calculations for surface area
versus sintered density for spherical particles with starting fractional
densities (fO) of 0.50, 0.55, 0.60, or 0.65.1
Other studies verify this behavior. Figure 5
compares the surface area–density trajectory for 0.55
starting density using several studies. The plot from
Fig. 4 is labeled as the ‘‘geometric’’ line. For
comparison, Hare23 simulated three-dimensional
spherical particle sintering, providing results
independent of the diffusion process. The
‘‘computer’’ specific surface area change with density is
included for a starting green density of 0.55. Also
shown are the ‘‘experimental’’ results from 1050 C
copper sintering reported by DeHoff et al.,9
‘‘shrinkage’’ calculations by Kumar,24 and ‘‘energy’’
reduction calculations.1 Similar relationships emerge
from these different approaches.
Since surface transport controlled sintering
reduces surface area without densification, the
surface area trajectory helps identify surface
diffusion versus grain boundary diffusion. Figure 6
shows surface area versus density from constant
heating rate experiments on 140-nm alumina
starting at 0.32 fractional green density.14 Compacts
were extracted at 50 C intervals between 900 C and
1300 C. The trajectory sits between that expected
for sintering by grain boundary diffusion and
surface diffusion. Surface diffusion is the dominant
process, accounting for about 80% of the surface
area loss. As surface area is annihilated and grain
boundary area is created, the dominant process
shifts to grain boundary diffusion.
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
halidedoped atmospheres or in systems sensitive to
oxygen or water partial pressures.
Grain growth in sintering results in the median
grain volume (G3) increasing linearly with heating
G3 ¼ G3O þ Kt
Here, G is the grain size and the starting grain
volume is G3O (often ignored), hold time is t, and K is
the temperature-dependent rate parameter. As a
demonstration, grain size data are plotted in Fig. 7
for copper at 900 C,16 nickel at 900 C,25 and iron at
850 C15 on a log–log basis. Plotting this way ignores
the initial grain volume, but at longer times the GO
term is insignificant. The lines correspond to a slope
of one-third while the individual size measures are
shown as symbols. The fit to Eq. 4 is evident.
For a single phase solid undergoing sintering, the
grain growth rate parameter reflects two factors:
the mass transport rate across the grain boundaries
and the mass transport rate in the vapor phase. The
relative solid–solid interface area is measured by
the contiguity CSS giving:8
K ¼ CSSKSS þ ð1
where KSS is the grain growth rate parameter
associated with grain boundaries and KSV
corresponds to the solid–vapor interface. Contiguity is
the fraction of the grain perimeter consisting of
solid–solid contacts. It is initially zero so early grain
growth depends only on the solid–vapor
contribution. As grain boundary area increases during
sintering, the rate parameter converges to KSS, the
solid–solid behavior. As noted above, the solid–
vapor surface area is a linear function of the
fractional density. Accordingly, Table I captures
relationships between fractional density, grain
coordination number, contiguity, and pore size.26 Note
that for these conditions the contiguity is related to
the square-root of the fractional porosity. Pores
generally retard grain growth, but as pores are
annihilated during sintering, grain size rapidly
GRAIN SIZE TRAJECTORY
Porosity and grain size are related during
sintering, although grain growth continues even after
pore elimination. While pores remain, the mean
grain size tracks with fractional porosity, as
illustrated in Fig. 8. This plot compares copper data16
with the inverse square-root relationship first
proposed by Bruch;27
G ¼ h GpOffiffi
where GO is the initial grain size, e is fractional
porosity (e = 1 f), and h reflects the starting grain
size and porosity condition and is often is near 0.6.
Figure 9 offers examples taken during sintering
4and 0.1-lm iron,28 8-nm zirconia,29 25-lm stainless
steel,30 and 0.1-lm alumina.31 Results from other
studies, even hot pressing and field-assisted or
spark sintering experiments, follow Eq. 6.8 Hence,
grain size and fractional density are related during
sintering, with grain size increasing rapidly as
pores are eliminated.
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.
Many materials exhibit a power law relationship
between sintering density and sintering time:18,32,33
f ¼ fO þ atN
where fO is the starting or green density, t is the
time, and N is often near 1/6 to 1/3. The coefficient a
includes material properties such as diffusivity and
surface energy. Copper sintering data illustrate the
trajectory. The sintered density term (log (f f0))
versus log time agrees with Eq. 7 (correlation of
0.992) as illustrated in Fig. 10.16 In turn, from the
initial conditions, it is possible to predict
parameters such as surface area and grain size versus
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
sus time plotted on a log–log basis using data for copper.16 fO)
verFig. 10. Sintering density change from the initial value (f
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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