Termodinâmica da Sinterização

Prévia do material em texto

© Woodhead Publishing Limited, 2010
3
1
Thermodynamics of sintering
R. M. GERMAN, San Diego State University, USA
Abstract: Particles bond together when heated by a sintering process that is 
a combination of several atomic level events that include diffusion, creep, 
viscous flow, plastic flow and evaporation. Significant strengthening occurs in 
powder compacts due to sintering. Sintering consumes surface energy to build 
bonds between those particles. Small particles have more surface energy and 
sinter faster than large particles. Since atomic motion increases with 
temperature, sintering is accelerated by high temperatures. The thermodynamic 
driving force for sintering then is found in the surface area, interfacial energies 
and curvature gradients in the particle system. Actual atomic motion is by 
several transport mechanisms with concomitant microstructure changes.
Key words: surface energy, surface area, diffusion, creep, viscous flow, plastic 
flow, particle size, interfacial energy, dihedral angle, contact angle, wetting, 
curvature.
1.1 Introduction
Sintering acts to bond particles together into strong, useful shapes. It is used to fire 
ceramic pots and in the fabrication of complex, high-performance shapes, such as 
medical implants. Sintering is irreversible since the particles give up surface 
energy associated with small particles to build bonds between those particles. 
Prior to sintering the particles flow easily while after sintering the particles are 
bonded into a solid body. From a thermodynamic standpoint sinter bonding is 
driven by the surface energy reduction. Small particles have more surface energy 
and sinter faster than large particles. Since atomic motion increases with 
temperature, sintering is accelerated by high temperatures.
The driving force for sintering comes from the high surface energy and curved 
surface inherent to a powder. The initial stage of sintering corresponds to neck 
growth between contacting particles where curvature gradients normally dictate the 
sintering behavior. The intermediate stage corresponds to pore rounding and the 
onset of grain growth. During the intermediate stage the pores remain interconnected, 
so the component is not hermetic. Final stage sintering occurs when the pores 
collapse into closed spheres, giving a reduced impediment to grain growth. Usually 
the final stage of sintering starts when the component is more than 92% dense. 
During all three stages, atoms move by several transport mechanisms to create the 
microstructure changes, including surface diffusion and grain boundary diffusion.
Sintering models include parameters such as particle size and surface area, 
temperature, time, green density, pressure and atmosphere. Further, the addition 
of a wetting liquid induces faster sintering. Accordingly, most sintering is 
4 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
performed with a liquid phase present during the heating cycle. These basic 
thermodynamic attributes are treated in this first chapter.
1.2 The sintering process
Sintering is fundamentally a one-way event. Once sintering starts, surface energy 
is consumed through particle bonding, resulting in increased compact strength and 
often a dimensional change. Accordingly, the definition of sintering is as follows:1
Sintering is a thermal treatment for bonding particles into a coherent, predominantly 
solid structure via mass transport events that often occur on the atomic scale. The 
bonding leads to improved strength and lower system energy.
The bonding between particles is evident in the scanning electron microscope in 
terms of the newly formed solid necks between contacting particles. Figure 1.1 
illustrates spherical bronze particles after sintering at 800 °C. Necks grow between 
the contacting spheres, providing strength and rigidity. Longer sintering gives a 
larger neck and usually more strength. The emergence of the necks between is 
driven by the system thermodynamics, while the rate of sintering depends mostly 
on the temperature. At room temperature, the atoms in a material such as bronze 
are not noticeably mobile, so the particles do not sinter. However, when heated to 
a temperature near the melting range, the atoms are very mobile. Atomic motion 
1.1 Scanning electron micrograph of the sintering neck formed 
between 26 µm bronze particles after sintering at 800 °C.
 Thermodynamics of sintering 5
© Woodhead Publishing Limited, 2010
increases with temperature and eventually this motion induces bonding that 
reduces the overall system energy.
The energy changes in sintering are usually small, so the rate of change during 
sintering is slow. In the case of the 26 µm bronze powder shown in Fig. 1.1, which 
has a solid–vapor surface energy of 1.7 J/m2, the energy per unit mass stored as 
excess surface area is about 50 J/kg. But not all of this energy can be consumed 
during sintering, since the structure usually fails to sinter to full density and other 
interfaces emerge, such as grain boundaries, which add energy into the system. 
The total surface energy increases as the particle size decreases, so with nanoscale 
powders smaller than 0.1 µm there is a large driving force for sintering, meaning 
faster sintering or a lower sintering temperature.
Early models for sintering realized that a sphere affixed to a flat plate presented 
a large energy difference, since the sphere has much more surface area and by 
implication more surface energy. Accordingly, early sintering studies measured 
the neck size between spheres and plates, and subsequently between contacting 
spheres. The two-sphere model considers two equal-sized spheres in point contact 
that subsequently fuse to form a single larger sphere with a diameter 1.26 times 
the starting sphere diameter, as sketched in Fig. 1.2.
The rate of particle bonding during sintering depends on temperature, materials, 
particle size and several processing factors.2 Small particles are more energetic, 
so they sinter faster. Thus, the thermodynamics of sintering show the importance 
of smaller powders, while the kinetics of sintering emphasizes the importance of 
temperature.
Sintering occurs in stages, as illustrated in Fig. 1.3. Without compaction a 
model powder system starts at a packing density of 64%, the dense random 
packing. In the initial sintering stage, the interparticle neck grows to the point 
where the neck size is less than one-third of the particle size. Often there is little 
dimensional change so at the most 3% linear shrinkage is seen in the initial stage. 
For loose spheres this generally corresponds to a density below 70% of theoretical. 
Intermediate stage sintering implies the necks are larger than one-third the particle 
size, but less than half the particle size. For a system that densifies, this corresponds 
to a density range from 70% to 92% for spheres. During the intermediate stage the 
pores are tubular in character and connected (open) to the external surface. The 
sintering body is not hermetic so gas can pass in or out during firing. Final stage 
sintering corresponds to the elimination of the last 8% porosity, where the pores 
are no longer open to the external surface. Isolated pores, associated with final 
stage sintering, are filled with the process atmosphere.
1.3 Surface energy
Surface energy is the thermodynamic cause of sintering. A model of a surface is 
generated by starting with an ideal crystal, such as shown in Fig. 1.4, where each 
atomic species occupies specific, repeating sites. Between atoms are bonds, 
6 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
represented by lines. If scissors were used to snip these atomic bonds, then the 
resulting surface would consist of broken bonds. These bonds provide the atomic 
interaction responsible for the surface energy. Figure 1.5 illustrates this concept, 
where the free surface is covered with broken bonds. Surface energy relates to thedensity of broken bonds (bonds per unit area), so it varies with crystal orientation. 
Also, since stronger bonding is associated with a higher melting temperature, 
surface energy is higher for high melting temperature materials.
D D
Neck
Grain
boundary
Initial
point
contact
Spherical particle
D = diameter
Early stage
neck growth
(short time)
Late stage
neck growth
(long time)
Terminal
condition
fully coalesced
(infinite time)1.26 D
1.2 Two-sphere sintering model, where the two spheres grow a neck 
during sintering that grows to the point where the spheres fuse into a 
single sphere that is 1.26 times the diameter of the starting spheres.
 Thermodynamics of sintering 7
© Woodhead Publishing Limited, 2010
Loose powder Initial stage
Intermediate stage Final stage
1.3 Illustration of the sintering stages with a focus on the changes in 
pore structure during sintering.
1.4 An illustration of a perfect crystal where each atom is in a repeating 
position and atomic bonds are linking the atoms.
8 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
An atomic model for the grain boundary would be similar, where broken atomic 
bonds from the two crystal lattices only partly match. As illustrated in Fig. 1.6, 
some misorientations lead to more disrupted bonding and high grain boundary 
energies, while other misorientations lead to less disruption and lower grain 
boundary energy. Thus, depending on the misorientation between the two crystals, 
the grain boundary energy might be high (much disrupted bonding) or low (good 
bond matching).
Some grain orientations are higher energy than others, so touching grains rotate 
or rearrange during sintering to reduce their grain boundary energy. In a sintering 
structure consisting of solid particles and pores, a variety of grain boundary 
configurations are possible between the randomly assembled particles. Further, a 
range of solid–vapor surface energies come from the range of crystal surface 
orientations. With a liquid, the inventory of surface energies increases to include 
grain boundary, solid–vapor, liquid–vapor, and solid–liquid combinations. All of 
these are distributed properties and not single-valued. Rather than dealing with 
this level of detail, sintering models rely on average values that reflect the millions 
of different combinations.
For many engineering materials, the average solid–vapor surface energy is 
in the 1 to 2 J/m2 range, while grain boundary energies are even lower. For a 
single-phase solid, sintering is slow since the energy release on sintering is low. 
Similar to other chemical reactions, as the surface energy is consumed, then the 
driving force for continued sintering is diminished and the process continually 
slows.
1.5 An illustration of how a free surface for a crystalline material 
results in disrupted atomic bonding; it is the dangling atomic bonds 
that give surface energy.
 Thermodynamics of sintering 9
© Woodhead Publishing Limited, 2010
1.4 Sintering stress
The capillary stress arising from the surface energy acts to move surfaces during 
sintering. The neck between contacting particles is associated with a large change 
in curvature over distance. For example in Fig. 1.1, the base of the neck is concave. 
A concave surface acts to pull itself into a flat surface. On the sphere surface away 
from the neck, the curvature is convex with an opposite curvature. The Laplace 
equation gives the stress s associated with a curved surface as,
σ = γ ( 1 + 1 ) R1 R2 [1.1]
where γ is the energy associated with the curved surface (for example solid–liquid, 
solid–vapor, or liquid–vapor surface energy), and R1 and R2 are the radii of 
curvature for the surface. For a sphere, both radii are the same and equal to the 
radius of the sphere so the stress is uniform, but during sintering the two radii vary 
with position in the microstructure and often are opposite in sign. This is evident 
with the saddle surface seen in the sintering neck. The sintering microstructure 
consists of a mixture of convex and concave surfaces, and the shift from tension 
to compression occurs over distances smaller than the particle size. The natural 
tendency is to remove the gradients during sintering.
Because the stress in the neck region is different from the neighboring region, 
the curvature gradient gives a thermodynamic gradient that drives mass flow 
during sintering. Atomic motion takes place to remove the gradient. When heated 
Surface
Neck Atoms
Grain boundary
energy
Misorientation
angle
Misorientation
angle
Grain
boundary
1.6 Grain boundary misorientation and the relative energy from the 
misorientation, such that during sintering the random particle to 
particle contacts result in a wide array of microstructure relations.
10 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
to where atomic motion occurs, the atoms naturally flow from the convex to 
concave surfaces. Atom motion is faster at higher temperatures and small particles 
have large gradients. Accordingly, particles sinter faster when they are small and 
heated to high temperatures.
As sketched in Fig. 1.7, a concave solid surface tends to fill and a convex 
surface tends to flatten. In a powder compact consisting of a mixture of pores and 
particles, sintering acts to remove the curvature gradients – namely to smooth the 
pores. The convex particles represent mass sources and pores represent mass 
sinks that fill using mass from the convex regions. From the instantaneous 
pore-grain geometry it is possible to quantify these parameters and assess the 
sintering stress and its dissipation over time. As with many reactions, sintering 
goes more slowly as it progresses, simply because the action of sintering is to 
remove the gradients.
In the initial stage of sintering, the saddle surface formed between particles has 
a sharp curvature at the root. Assuming isotropic surface energy and spherical 
particles, then for a small neck size a substitution into Equation 1.1 gives the 
sintering stress σ as follows:
σ = γSV
 [- 2 + 4(D – X)] X X2 [1.2]
where X is the neck diameter, D is the particle (sphere) diameter, and γSV is the 
solid–vapor surface energy. This relation is valid in the initial stage of sintering 
when X/D < 0.3. This stress induces particle bonding as a natural part of sintering. 
Although surface energy is consumed as the neck grows, not all surface energy is 
available for sintering. For a crystalline solid, nearly every particle contact forms 
a grain boundary. The grain boundaries are defective regions with high atomic 
mobility. For most inorganic powders, diffusion along the grain boundary proves 
to be a dominant sintering mechanism. As the neck grows to remove surface 
Concave surface
Convex surface
Surface mass
under compression
Mass flowSurface mass
under tension
1.7 The action of sintering is to remove concave and convex surfaces 
and to move toward flat surfaces, as schematically illustrated here. The 
mass from the convex transports to fill in the convex surface.
 Thermodynamics of sintering 11
© Woodhead Publishing Limited, 2010
energy the grain boundary grows and adds interfacial energy, so sintering only 
continues as long as the rate of surface energy annihilation exceeds the rate of 
grain boundary annihilation.
Heat stimulates the atomic motion that allows sintering to proceed. Most 
sintering processes are thermally activated, meaning that input energy is necessary 
for mass flow. For example, sintering by diffusion depends on the energy to 
create a vacancy and the energy to move an atom into that vacancy. The population 
of vacant atomic sites and the number of atoms with sufficient energy to move 
into those sites both vary with an Arrhenius temperature relation. The Arrhenius 
relation determines the probability that an atom has enough energy to move, 
as determined by the activation energy Q. For example,the volume diffusion 
coefficient DV is determined from the atomic vibrational frequency D0, absolute 
temperature T, universal gas constant R, and the activation energy Q, 
which corresponds to the energy required to induce atomic diffusion via vacancy 
exchange,
DV = D0 exp (- Q ) RT [1.3]
Sintering is faster at higher temperatures, because of the increased number of 
active atoms and available sites. Thus, temperature is a dominant parameter in 
defining a sintering cycle. Other important factors include the particle size, applied 
pressure, formation of a liquid phase, sintering time, heating rate, and process 
atmosphere.
Another important source of sintering stress comes from wetting liquids. About 
80% of all sintering occurs with a liquid or glassy phase. The liquid causes the 
powder to agglomerate since significant capillary stress is generated by a wetting 
liquid. Wetting refers to a liquid that spreads over a surface. We rely mostly on the 
contact angle to measure wetting. Also known as the wetting angle, the contact 
angle is formed at the intersection of liquid, solid, and vapor phases. When gravity 
is ignored, the contact angle θ is defined by the horizontal equilibrium of surface 
energies, as illustrated in Fig. 1.8. The general consensus is to measure the contact 
angle on a surface perpendicular to the gravity vector. Then ignoring gravity the 
horizontal solution is known as Young’s equation,
γSV = γSL + γLV cos(θ) [1.4]
where γSV is the solid–vapor surface energy, γSL is the solid–liquid energy, and γLV 
is the liquid–vapor surface energy. Wetting liquids are associated with contact 
angles near zero and nonwetting liquids are associated with contact angles over 
90°. During spreading or retraction of a liquid over a solid surface, the contact is 
not in equilibrium. Further, various corrections exist for the effect of surface 
roughness, since finely textured solid surfaces will induce wetting even though 
the contact angle predicts nonwetting.
12 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
Finally, the dihedral angle describes the grain boundary structure. The angle 
formed by a grain boundary where it intersects with another solid, pore, or liquid 
during sintering is described by a thermodynamic balance termed the dihedral 
angle. As illustrated in Fig. 1.9, it is determined by a vertical surface energy 
balance. For the case of a grain boundary in contact with a liquid during liquid 
phase sintering the vector balance gives,
φ
φ = Dihedral angle
γSL γSL
γSS
φ
Liquid
Solid Solid
Grain boundary
γSS = 2γSL cos (φ_2)
1.9 The dihedral angle is defined based on a solid–solid grain 
boundary energy intersecting a liquid or vapor phase.
V-vapor
L-liquid
S-solid
θ = Contact angle
θ
θ
γSV = γSL + γLV cos (θ)
γLV
γSVγSL
1.8 Contact or wetting angle definition based on a droplet sitting flat on 
a surface so the vertical forces are balanced.
 Thermodynamics of sintering 13
© Woodhead Publishing Limited, 2010
γSS = 2γSL cos (ϕ) 2 [1.5]
where γSS is the solid–solid interfacial energy (grain-boundary energy) and γSL is 
the solid–liquid interfacial energy. Alternatively,
ϕ = 2 arccos( γSS ) 2γSL [1.6]
In the case of a grain boundary in contact with the free surface, a thermal groove 
forms and the dihedral angle is determined by the solid–vapor surface energy γSV. 
In materials held at high temperature for a prolonged time the dihedral angle is 
evident at all surfaces and exposed grain boundaries. Grain boundary grooving on 
a free surface is a reflection of the dihedral angle. Since segregation changes grain 
boundary and surface energies, the dihedral angle exhibits a time dependence 
related to the diffusion of species to or from grain boundaries and free surfaces.
1.5 Atomistic changes in sintering
The surface stress associated with a curved surface gives a nonequilibrium 
vacancy concentration. A flat surface free of stress is at equilibrium. In sintering, 
microstructure curvature drives mass flow by taking both the concave and convex 
surfaces toward a flat state. Mass from the convex surface moves to fill in the 
concavity. The vacancy concentration C under a curved surface depends on the 
local curvature,
C = C0 [1 – γ Ω ( 1 + 1 )] kT R1 R2 [1.7]
where C0 is the equilibrium vacancy concentration associated with a flat surface 
at the same temperature, γ is the surface energy (either solid–liquid or solid–
vapor), Ω is the atomic volume, k is Boltzmann’s constant, and T is the absolute 
temperature. The equilibrium concentration increases on heating. As shown in 
Fig. 1.10, two perpendicular arcs pass through at any point on the surface. These 
arcs have radii of curvature designated as R1 and R2. The more highly curved the 
surface, the smaller R1 and R2 and the departure from equilibrium. For a concave 
surface, the vacancy concentration is higher than equilibrium; for a convex surface 
it is lower; thus, atomic flow is from regions of vacancy deficiency – convex – to 
regions of vacancy excess – concave. When a radius of curvature is located inside 
the solid it is deemed negative while a radius located outside the solid is positive. 
A concave surface is a source of vacancies that works with a counter flow of 
atoms to fill the concavity.
Atomic motion (volume diffusion) depends on atomic exchange with 
neighboring vacancies. For diffusion to occur, an atom must have sufficient 
energy QB to break existing bonds with neighboring atoms and then additional 
energy to exchange its position with a neighboring vacant site. The probability of 
14 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
a neighboring atomic site being vacant depends on the vacancy formation energy 
QN. In other words, volume diffusion requires both the formation of a vacancy 
and the provision of sufficient energy to break an atom free so that it can jump 
into the vacant site. As an approximation to the rate of atomic diffusion, the 
Arrhenius equation gives the relative number of active atoms NA compared with 
the total number of atoms N0 as follows:
NA = N0 exp (- QB + QN) RT [1.8]
where R is the gas constant and T is the absolute temperature. Most typically, the rate 
of atomic diffusion is termed the diffusivity, which depends on several parameters 
including the frequency of atomic vibration, crystal class, lattice parameter and 
similar factors. The resulting form for the diffusion coefficient is an Arrhenius 
equation given earlier as Equation 1.3. The activation energy Q is the sum QN + QB. 
In turn, for a given crystal structure both activation energies can be rationalized to 
Curved
surface
R1
R1, R2 = Principal radii
 of curvature
R2
1.10 The definition of surface curvature in terms of the radii of the two 
perpendicular arcs passing through a point on a curved surface.
 Thermodynamics of sintering 15
© Woodhead Publishing Limited, 2010
the number of atomic bonds that must be broken to form a vacancy and the number 
of atomic bonds that must be broken to move an atom. Many handbooks compile 
data diffusion data as D0 and Q, which allows calculation of D at any temperature.
Similar to vacancy creation and annihilation at free surfaces, the grain boundaries 
are important to sintering. Diffusion on a grain boundary undergoes a rapid increase 
with a modest temperature increase. Further, impurities preferentially segregate to 
grain boundaries, so often the fast diffusion observed along the grain boundaries is 
a reflection of the segregated impurities. At very high temperatures the impurities 
are more soluble in the materials being sintered, so there is less effect. But at 
intermediate temperatures, segregation is more severe and leads to significant 
changes in sintering rates. This is true in systems such as tungsten doped with 
nickel, where low concentrationsof nickel doped into the tungsten greatly lower 
the sintering temperature.3
1.6 Sintering changes prior to interfacial energy 
equilibrium
During sintering, shrinkage causes grains to come into contact with each other 
and form new sinter bonds, at times much delayed from the initial bonding. Grain 
rearrangement is observed due to the grain boundary torque.4 The motion of 
grains or particles into new and higher density packing positions is frequent in 
liquid phase sintering. During heating, the liquid spreads to wet the solid grains as 
soon as it forms, dissolving existing solid–solid necks. The resulting loose grain 
structure with a wetting liquid produces a capillary force that acts to pull the 
separated grains together. The individual rearrangement events happen very 
quickly when the liquid forms, so the grains literally jump into new positions. 
However, the formation of liquid requires heat transport through the porous 
compact, which tends to be a slow step.5 For this reason most powder compacts 
show a slow rearrangement step that is controlled by heat transport. Each 
individual bond undergoes rearrangement in a split second, but the thermal wave 
needed to form the liquid propagates through the compact over a few minutes.
An especially important nonequilibrium transient occurs in liquid phase sintering. 
Newly formed liquid spreads, and if it has solubility for the solid, then it penetrates 
the solid–solid interfaces on liquid formation. This results in a dimensional change, 
usually swelling, where the amount of swelling varies with the liquid flow into the 
surrounding pores. The liquid flow is estimated as a function of hold time as follows:
X2 =
 dP γLVt cos θ
 4η 
[1.9]
where x is the depth of liquid penetration, dP is the pore size, γLV is the liquid–
vapor surface energy, θ is the contact angle, t is the hold time, and η is the liquid 
viscosity. Several aspects of sintering are explained by this transient liquid 
penetration of grain boundaries. The solid skeleton formed during heating 
16 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
dissolves, reducing compact rigidity, and in turn this allows for distortion. A 
second is that liquid can be stranded on grain boundaries, leading to what is 
termed a necklace microstructure. Finally, the dihedral angle and other equilibrium 
thermodynamic properties vary during sintering.
1.7 Microstructure gradients
A natural affinity exists between the pores and grain boundaries. Because of the 
solid–vapor surface energy, a pore contributes surface energy. At the same time a 
grain boundary has grain boundary energy. If the pore sits on the grain boundary, 
then the configurational energy is lower; effectively the pore-boundary combination 
is pinned. Thus, there is a high probability for a pore to attach to a grain boundary, 
even during grain growth. However, sintering works to minimize energy and this 
usually means a reduction in grain boundary energy through an increase in grain 
size. Thus, a dynamic exists where pores are attached to grain boundaries while at 
the same time grains are growing to reduce grain boundary area. Mobile pores 
remain with the moving grain boundaries and sintering progresses to full density. 
On the other hand, if the pores and grain boundaries separate, then a porous sintered 
body results. It is effectively impossible to shrink a pore that is removed from a 
grain boundary.
At high sintered densities the pores are mostly associated with the largest 
grains. In final stage sintering the relation between grain size G, pore diameter dP, 
and fractional porosity ε is given as:
 G 
=
 K
dP Rε 
[1.10]
where R expresses the ratio of attached pores to randomly placed pores, and K is 
a geometric constant. Values of R range from 1.7 to 5.7 for various sintering 
materials.6 The degree of boundary-pore contact remains essentially constant 
during most of the sintering cycle.
Consequently, grain size tracks with porosity during sintering; grain size 
increases, porosity decreases, and pore size initially decreases, but late in sintering 
might increase. In the initial stage of sintering the pores pin the grain boundaries 
to retard grain growth. If a grain boundary were to move then it must drag the pore 
and that is a slow event. In the intermediate stage of sintering the pores are smaller 
yet located on the grain edges. The pore surface area declines while the grain size 
enlarges, effectively making the pore diameter smaller and the pore length longer. 
Accordingly, a fundamental relation is observed where the grain size and solid–
vapor surface area per unit volume SV tracks with the square of the porosity ε, and 
the grain size G tracks with the inverse of the remaining surface area [7, 8]:
G ≈
 1 
≈
 1
 SV ε
2 
[1.11]
 Thermodynamics of sintering 17
© Woodhead Publishing Limited, 2010
Of course this predicts an infinite grain size at full density. The terminal condition 
in sintering is a single crystal, or one grain, so this is not overly incorrect. As 
porosity declines the pore surface area that retards grain growth decreases, so 
grain growth occurs with decreasing impediment. Thus, grain size increases 
rapidly as full density is approached [1, 9]. As plotted in Fig. 1.11, the declining 
surface energy associated with pores diminishes grain boundary pinning, so there 
is little resistance to a rapid rise in grain size as full-density is approached.
Typically the assumed grain geometry during late stage sintering is the 
tetrakaidecahedron, a 14-sided polyhedron consisting of squares and hexagons. 
Figure 1.12 shows a sketch of this grain shape. In intermediate stage sintering the 
pores exist as tubes on the grain edges and in the final stage of sintering the pores 
are spheres located at the grain corners.
During intermediate stage sintering the pores form a tubular network that is 
attached to the grain boundaries. As densification occurs the pores shrink while 
simultaneous grain growth stretches the pores. As this continues, eventually the 
elongated and thinning pores pinch off into closed spherical pores, a process 
termed pore closure. Based on energy reduction, a calculation of the instability of 
a cylindrical pore of length l and diameter dP gives the critical condition for 
closure into separate pores as follows:
l ≥ dPπ [1.12]
For a cylindrical pore occupying the edges of tetrakaidecahedron grains this 
instability occurs at a porosity of approximately 8%. In reality, due to distributions 
in initial particle sizes, the instability that induces pore closure occurs over a 
0.6
0
4
8
12
0.7 0.8
Fractional sintered density
Copper
8 µm powder
Progressive
sintering
Grain size,
µm
0.9
Final
stage
Intermediate
stage
Initial
stage
1.0
1.11 Grain size plotted as a function of porosity during the sintering of 
8 µm copper powder to illustrate how rapid grain growth occurs as 
pores are removed with a reduction in the grain boundary pinning effect 
that retards grain growth at higher porosities.
18 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
broad range of densities from 85 to 95% and final stage sintering occurs with 
rapid grain growth and slow densification from that point.
1.8 Chemical and strain gradients
Sintering is a process where energy is consumed and the material is relaxed. If 
the powder is milled and has stored strain energy, then the release of that strain 
during sintering increases the sintering rate. Indeed, thermal stresses from rapid 
heating will improve the sintering rate, but often damage the component. Phase 
transformations are another means of including strains to alter the sintering rate. 
Adding energy to the material increases the sintering rate, so radiation and 
electromagnetic fields have beneficial effects.
Chemical gradients are important to mixed powder sintering. For example, 
mixed powders are compacted and heated to form an alloysuch as bronze or 
stainless steel. Most common is the formation of steel using mixed iron and 
graphite powder. During the sintering cycle the carbon goes into solution. Other 
examples include mixed copper and tin powders used to form bronze, and mixed 
silica and alumina used to form mullite. The number of combinations is large.
1.12 The tetrakaidecahedron is a 14-sided polygon with 35 edges and 
24 corners that packs to full density. It consists of six squares and eight 
hexagons, and pores then occupy the edges during intermediate stage 
sintering or the corners during final stage sintering.
 Thermodynamics of sintering 19
© Woodhead Publishing Limited, 2010
Mixed powder sintering is biased by the phase diagram thermodynamics. If an 
intermetallic is produced, typically it involves an exothermic reaction, such as 
NiTi, MoSi2, or Ni3Al. Some reactions are so strong that the system self-heats 
once initiated. Phase diagrams are equilibrium depictions of the phases that form 
versus temperature and composition, but mixed powders are not necessarily at 
equilibrium during heating. The tendency to react, swell, densify, or otherwise 
change traces to the phase diagram. For example, the iron-aluminum system 
shows solubility of aluminum in iron, but not the reverse solubility. Thus, during 
heating the aluminum melts and diffuses into the iron, leaving a pore behind at the 
prior aluminum particle sizes. Figure 1.13 is the Fe-Al phase diagram and the 
resulting microstructures below the phase diagram show images taken during 
heating as the core aluminum particle first melts, reacts, and then diffuses out to 
create a pore, giving compact swelling.
0
600
700
660
800
900
1000Te
m
pe
ra
tu
re
, º
C
at. %Al
(A
l)
Fe
1100
1200
1300
1400
1500
1600
10
L
Fe
4A
l 13
Fe
Al
2.
8
Fe
6.
5A
l 11
.5
rt
?F
e 2
Al
3h
t
Fe
Al (Fe) rt
1160
1170 1160
1155
1100
1230
1310
1540
655
20 30 40 50 60 70 80 90 100
1.13 The pore evolution in sintering mixed iron and aluminum, showing 
the reaction around the aluminum particle to form an intermediate 
compound. The phase diagram shows that the intermetallic phases are 
very stable, thus the chemical reaction dominated sintering.
20 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
Other features of great importance to sintering can be identified from phase 
diagrams. These include solubility, dissolution, and liquid phase formation. Several 
of the important reactions are categorized elsewhere [1, 3, 10, 11]. One of the more 
difficult sintering tasks is to manage reactive systems, especially where a transient 
phase forms. Copper-tin is the most famous of these, where during heating tin 
melts, forms intermetallic compounds, and the compounds subsequently melt or 
dissolve. Although complicated, this system is fundamental to sintered oil-less 
bronze bearings and control of the events is crucial to successful functioning.
1.9 Thermodynamics, stages and mechanisms of 
mass flow
Transport mechanisms tell how mass flows to lower the system energy during 
sintering. There are two classes of sintering mechanisms: surface transport and 
bulk transport. Each is composed of several atomistic events that contribute to 
bonding. The pores are large accumulations of vacancies, so the sintering 
mechanisms describe vacancy motion and annihilation during heating. Vacancies 
and atoms move along particle surfaces (surface diffusion), across pores 
(evaporation-condensation), along grain boundaries (grain boundary diffusion), 
and through the lattice (viscous flow or volume diffusion). Also, vacancies couple 
with dislocations via plastic flow and dislocation climb.
Surface transport processes give neck growth without a change in particle 
spacing (no shrinkage or densification) since the mass flow originates and 
terminates at the particle surface. The atoms are rearranged, but no annihilation of 
vacancies takes place. Surface diffusion and evaporation-condensation are two 
contributors to surface transport controlled sintering. Surface diffusion dominates 
the low-temperature sintering of many metals and ceramics, while evaporation-
condensation is active when the vapor pressure is high.
Bulk transport processes promote neck growth and shrinkage during sintering. 
For densification to occur, the mass must originate from the particle interior with 
deposition at the neck. The vacancy annihilation takes place on the grain boundary 
by particle rotation and rearrangement. Bulk transport mechanisms include 
volume diffusion, grain boundary diffusion, dislocation climb, plastic flow and 
viscous flow. Plastic flow is important during the heating period, especially for 
compacted powders where the initial dislocation density is high. Without rapid 
heating, surface tension stresses are generally insufficient to generate new 
dislocations and the dislocations are annihilated once they intersect a grain 
boundary or free surface. Thus, the role of plastic flow decreases as the dislocations 
are annealed out at high temperatures. In contrast, amorphous materials, such as 
glasses and polymers, sinter by viscous flow, where the particles coalesce at a rate 
that depends on the particle size and material viscosity. A form of viscous flow is 
also possible for metals with liquid phases on the grain boundaries. Grain 
boundary diffusion is fairly important to densification for most crystalline 
 Thermodynamics of sintering 21
© Woodhead Publishing Limited, 2010
materials, and appears to dominate the densification of many common systems. 
Volume diffusion is most active in cooperation with dislocation climb. Relative to 
the melting temperature, bulk transport processes are dominant at higher tem-
peratures and surface transport processes are dominant at lower temperatures.
The sinter bond between the contacting particles is the critical region. It is the 
point where atoms are deposited to reduce the surface energy. Generally all of the 
key sintering measures relate to the mass transport rates and how they influence 
neck growth and change the pores and grains. Models for solid–state sintering 
have subdivided the treatments into specific combinations of the sintering stage 
and mass transport mechanism, such as surface diffusion during initial stage 
sintering or grain boundary diffusion during intermediate stage sintering.
Amorphous materials exhibit a decreasing viscosity (increased flow) as 
temperature increases. Under the action of an applied stress a viscous material 
flows. Both glasses and polymers densify by viscous flow. The lower the viscosity 
the more rapid the sintering process, so temperature is a key control parameter. If 
an external stress is applied, then the rate of sintering increases in proportion to 
the applied stress [1].
The two-particle sintering situation is different for amorphous materials when 
compared to crystalline solids, since amorphous materials lack grain boundaries. 
As neck growth occurs, amorphous materials reach a neck size ratio of 
approximately X/D = 2/3 where sintering often stops. In most cases the powder is 
fully densified by this point. Many early models of sintering associated viscous 
flow with creep and volume diffusion processes. The Stokes-Einstein relation 
effectively relates the volume diffusion coefficient to an effective viscosity, so 
such an idea is widely accepted. If a sintering body is measured for effective 
viscosity during periods of rapid sintering, the viscosity of 10 GPa-s is about the 
same as that of a viscous child’s toy known as Silly Putty®.
Vapor transport during sintering leads to the repositioning of atoms located on 
the particle surface, without densification. Evaporation preferentially occurs from 
a convex surface and transport takes place across the pores to deposit mass on a 
nearby convex surface. The result is a reduction in the surface area as bonds grow 
between touching particles, but there is no change in the distance betweenparticle 
centers. The fraction of atoms on surface sites decreases over time as the concave 
surfaces are filled using mass from the convex surfaces. Vapor pressure increases 
with temperature, following the Arrhenius behavior. Higher temperatures give a 
higher vapor pressure and more vapor phase transport during sintering. Because 
the vapor pressure changes with surface curvature, deposition occurs at the 
concave necks between particles where the vapor pressure is slightly below 
equilibrium. Materials with a large sintering contribution from evaporation-
condensation include NaCl, PbO, TiO2, H2O, Si3N4, BN and ZrO2. All of these 
systems exhibit weight loss during sintering.
In several situations the sintering atmosphere induces vapor transport, even 
when the vapor pressure of the material being sintered is low. Chemical species in 
22 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
the sintering atmosphere (hydrogen, water, oxygen, carbon monoxide, chlorine 
and fluorine as examples) initiate considerable vapor phase transport, for both 
metals and ceramics. Sintering in vacuum stops vapor transport.
No matter what the transport mechanism, once the neck size reaches a 
thermodynamic equilibrium dictated by the solid–vapor dihedral angle, further 
neck growth only occurs if there is grain growth. Neck growth occurs until the 
surface energy, dihedral angle and grain boundary energy attain a balance. From 
this point on, neck growth follows grain growth and generally both increase with 
the cube-root of time.
In final stage sintering, closed pores become distorted by pore migration since 
pores try to stay on moving grain boundaries. As illustrated in Fig. 1.14, a 
migrating pore-boundary combination leads to a differential curvature between 
the leading and lagging faces. The corresponding vapor pressure gradient allows 
the pore to move with the grain boundary. Mass evaporates from the lower 
curvature surface and deposits on the higher curvature surface. Final stage 
Pore
Grain
boundary
Dihedral
angle
1.14 Pore pinning of grain boundaries is possible if the pore has 
differing front and rear surface curvature gradients that enable 
transport in the pore to allow motion with the grain boundary.
 Thermodynamics of sintering 23
© Woodhead Publishing Limited, 2010
densification critically depends on minimized grain growth and attachment of 
the pores to the grain boundaries. Vapor transport provides one of the means for 
this process.
Surfaces of crystalline solids are usually not smooth, but consist of defects that 
include ledges, kinks, vacancies and adatoms. Surface diffusion involves the 
motion of atoms between the surface defects. The population of sites and the 
motion between sites are both thermally activated, meaning temperature has a 
significant influence on surface diffusion. Secondary consideration is given to the 
crystal orientation, since some orientations favor diffusion. A typical surface 
diffusion event involves three steps that might be rate controlling. The first is 
breaking an atom away from existing bonds, typically at surface defect. The 
population of kinks depends on both the surface orientation and temperature. 
Once dislodged, the atom moves with a random motion across the surface, usually 
as a fast step. Finally, the atom must reattach at an available surface site, possibly 
again at a kink. The populations of sites and the ease of motion determine the 
surface diffusion rate. There is an activation energy associated with the slowest 
step that is known as the surface diffusion activation energy, which often changes 
with temperature. Highly curved surfaces and high temperatures increase the 
defective site population, leading to more surface diffusion.
Surface diffusion is active during heating to the sintering temperature. The 
activation energy for surface diffusion is less than that for other mass transport 
processes. Consequently, it initiates at a low temperature. Surface diffusion slows 
as the surface defect structure is consumed or as the available surface area is lost 
to sintering bonds. It does not produce shrinkage. For this reason surface diffusion 
works against densification, and rapid heating is one means to circumvent the 
problem. Surface diffusion is an initial contributor to the sintering of almost all 
materials. Boron and several covalent ceramics such as SiC exhibit surface 
diffusion dominance. Other examples include very small oxide powders at low 
temperatures and some metals when the particle size is small.
Volume diffusion, or lattice diffusion, involves the motion of vacancies through 
a crystalline structure. The rate of volume diffusion depends on temperature, 
composition and particle size. In compounds, temperature and stoichiometry are 
the controlling parameters. There are three vacancy diffusion paths in sintering. 
One path is from the neck surface, through the particle interior, with subsequent 
emergence at the particle surface. A net result is deposition of mass at the neck 
surface. This is effectively transport from a surface source to a surface site so 
there is no densification or shrinkage. It is termed volume diffusion adhesion to 
distinguish it from the densification process. Although treated theoretically, there 
is little evidence for this occurring at significant levels in most sintering cycles.
The second path is termed volume diffusion densification and involves vacancy 
flow to the interparticle grain boundary from the neck surface. This produces 
shrinkage and densification since effectively a layer of atoms moves in the 
opposite direction to the contact between the particles, allowing the centers to 
24 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
approach as the sinter bond grows. A cooperative grain boundary accommodation 
step of rotation or slip is implied with this transport path. Dispersoids and phase 
boundaries are other interfacial vacancy sources that are important to the sintering 
of multiphase materials.
Finally, the vacancies can be emitted or annihilated by dislocations, via a process 
termed dislocation climb. It involves cooperative action by both dislocations and 
vacancies. This process occurs during heating, and is especially active in compacted 
powders. The vacancy path is in the opposite direction to the atomic flux in each case.
For compounds there is an additional factor beyond temperature that controls 
the vacancy population, that being stoichiometry. Off-stoichiometric ionic 
compounds contain excess vacancies to neutralize charge. The flux by volume 
diffusion is then the combined action of the thermally induced vacancies 
and those induced by the loss of stoichiometry. An excess of ionic vacancies 
associated with the slow-moving species accelerates sintering. The stoichiometric 
effect is accessible through the original compound formulation or through the 
process atmosphere or by chemical additions. For example, in sintering UO2, a 
hyperstoichiometric oxygen level (2.02 oxygen atoms for each uranium atom), 
gives the highest sintered density. Sintering in a reducing atmosphere lowers the 
oxygen excess, resulting in retarded sintering. Alternatively, sintering in nitrogen 
preserves the oxygen excess and lowers the sintering temperature. Similar results 
are evident in other ionic materials, where small compositional changes result in 
large densification changes.
Late in the sintering process, the remaining pores exist as nearly smooth, 
spherical collections of vacancies. A difference in size between neighboring pores 
leads to a vacancy concentration gradient. Consequently, large pores are vacancy 
sinks and small pores are vacancy sources, leading to progressive coarsening of 
the large pore and the eventual elimination of the small pore. It is important to 
sustain vacancy annihilation sites, such as grain boundaries, to avoid this form of 
pore coarsening during late stage sintering. Thus,attention is directed toward 
grain growth control and the coupling of pores to grain boundaries to achieve full 
densification.
Although volume diffusion is active in most materials at high temperatures, it 
is often not the dominant mass transport process during sintering, especially for 
small powders. The activation energy for surface diffusion is typically lower, and 
in many cases grain boundary diffusion has an activation energy intermediate 
between surface and volume diffusion. Consequently, interfacial diffusion 
processes (surface and boundary diffusion) are generally more active. If the 
material has a small grain size or small particle size, then the effective transport 
via interfacial paths dominates sintering. Volume diffusion is a controlling process 
in the sintering of narrow stoichiometry compounds, such as BeO, CaO, Cr2O3, 
CuO, TiO2 UO2 and Y2O3.
Grain boundary diffusion is relatively important to the sintering densification 
of most metals and many compounds. Grain boundaries form in the sinter bond 
 Thermodynamics of sintering 25
© Woodhead Publishing Limited, 2010
between individual particles due to misaligned crystals as a collection of repeated 
misorientation steps. The defective character of the grain boundary allows mass 
flow along the boundary with an activation energy that is usually intermediate 
between surface diffusion and volume diffusion. The net impact depends on the 
grain size. As surface area is consumed and surface diffusion declines in 
importance, the simultaneous emergence of new grain boundaries increases the 
role of grain boundary diffusion. But grain growth reduces the importance of 
grain boundary diffusion.
During sintering, transport also takes place between pores via the grain 
boundary, leading to pore coarsening. This is most active late in sintering when 
the grain boundary is an inefficient vacancy sink: Vacancy accumulation on a 
grain boundary requires motion of the boundary, and this is resisted by contacting 
neighbors.
Grain boundary diffusion controlled sintering is most prevalent. It is well 
documented for metals, including Ni, W, Mo, Fe, Cu and various alloys. For 
compounds, a grain boundary segregant often acts to accelerate sintering; 
examples of this are in ZrO2 with Er2O3 additives, Ni3Al with small quantities of 
B, and Al2O3 with TiO2 additives.
Dislocations play two roles in sintering: vacancy absorption (dislocation climb) 
and dislocation glide (slip). Dislocations participate in sintering during heating, 
especially if the powders were subjected to plastic deformation during compaction. 
Dislocations interact with vacancies during sintering to improve mass transport. 
The dislocations climb by the absorption of vacancies emitted from the pores, 
leading to annihilation of the vacancies and dislocation motion to a new slip 
plane. In this case, densification by volume diffusion does not require an efficient 
vacancy sink at the grain boundary. Unfortunately, one consequence of dislocation 
climb is that the dislocation population declines, thereby halting the process. 
Dislocation flow is restricted to the early stage of sintering near the neck surface 
for small powders. As the sintering neck enlarges, the shear stress declines below 
the flow stress and the process becomes inactive.
Plastic flow contributions to sintering are transients that are favored by rapid 
heating (over 10°C/min), smaller particles (less than 100 µm), or pressure-assisted 
sintering. Dislocation motion can also be induced by phase transformations during 
heating, but this is restricted to polymorphic materials. Plastic flow has been 
observed during sintering for a variety of materials – Al2O3, Ag, CaF2, CoO, Cu, 
Fe, MgO, NaCl, Ni, Pb, ThO2, Ti, W and Zn. But in each case the contribution 
occurred during the application of a stress or during heating and was not sustained 
under isothermal conditions.
There are several possible mass transport paths in sintering. The two main 
categories are surface transport and bulk transport. It is the latter which is 
responsible for densification during sintering. Both contribute to bonding. 
Evaporation-condensation and surface diffusion are the common surface transport 
processes. Materials with high vapor pressures or those that form a volatile 
26 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
species by reacting with the sintering atmosphere are candidates for evaporation-
condensation controlled sintering. A weight loss (beyond that normally 
encountered by evaporation of surface contaminants) is an indication of 
evaporation-condensation. For most materials the vapor phase transport con-
tributions are small and can be ignored, but for low sublimation enthalpy 
compounds this is not true. In reactive atmospheres (including hydrogen, oxygen, 
halides and water) a high vapor pressure can be generated to sustain surface area 
loss during sintering, without densification. Surface diffusion also produces a loss 
of surface area during neck growth, but fails to induce shrinkage or densification. 
It is an initial contributor to the sintering and microstructure coarsening of many 
materials, especially those with a low activation energy for surface diffusion. 
Covalent ceramics exhibit surface diffusion controlled sintering so it is common 
to add grain boundary dopants to induce liquid phase sintering to attain sintering 
densification. Surface transport processes are involved in pore smoothing and 
migration during the latter stages of sintering densification.
1.10 Microstructure links to sintering thermodynamics
Atomic motion during sintering is not directly visible, so various monitors are 
used, often based on the microstructure. However, studies have been able to image 
particles and necks during sintering [12–14]. Neck size and its change with time 
or temperature is the most important aspect of sintering. The neck-size ratio X/D, 
defined as the neck diameter X divided by the particle diameter D, is the 
fundamental monitor, as evident in Fig. 1.1. If the powder is irregular, compacted, 
or far from this ideal, still the conceptualization is valid. From the neck size ratio 
come many other measures, some of which are easier to measure.
The surface area declines rapidly during sintering and is tracked with a 
dimensionless parameter ∆S/So (change in surface area normalized to starting 
surface area). Surface area is measured using microscopic analysis, gas adsorption, 
or gas permeability techniques. Also it is tracked based on quantitative microscopy. 
Related to the surface area are parameters such as thermal conductivity, electrical 
conductivity, corrosion behavior, and even catalytic activity.
Many powder compacts change dimensions during sintering, as well as density, 
strength, hardness, and elastic modulus. A good example is illustrated in Fig. 1.15, 
showing the size of a teacup prior to and after sintering. Simple experiments can 
be performed using interrupted cycles where the component is cooled and 
measured for size or density. More preferred is dilatometry, where the sample size 
is measured in situ during a firing cycle. However, not all neck growth in sintering 
gives dimensional change and not all dimensional change is associated with neck 
growth. Thus, although convenient, shrinkage and dimensional change need to be 
used with caution in trying to identify the sintering mechanism. In a related 
manner, bulk properties are used to follow the sintering process, and show similar 
changes with temperature and time. Sintering shrinkage is coupled to density 
 Thermodynamics of sintering 27
© Woodhead Publishing Limited, 2010
changes and the elimination of pores. Shrinkage, ∆L/Lo, is the change in compact 
length divided by the initial dimension. Because of shrinkage, the compact 
densifies from the fractional green density ρG to the fractional sintered density ρS 
according to the relation,
ρS =
 ρG
 (1 – ∆L)Lo 
3
 
[1.13]
This is a mass-conservation equation that assumes no mass loss. In reality, powder 
compacts usually have contaminants and polymers that burn out during sintering, 
so the green density needs to be corrected for such mass loss. High green densities 
result in high final densities, even with small sintering shrinkages.
Not all forms of sintering lead to densification and some lead to swelling. The 
latter is especially true for reactive systems where two powders undergo a 
dissolution or solvation event during heating.
Porosity is the remaining void space. For filters the final porosity might be 25% 
and in some distended materials, such as sintered aluminum foams, the porosity 
might be 95%. Curiously, this material is formed by adding titanium hydride to 
the aluminum powder, and during heating the hydrogen evolved from the hydride 
blossoms many a gas pocket in the compact. This is an instance where the green 
density might be 80% of theoretical and the final density is 5% of theoretical, so 
obvious swelling has occurred.
1.15 A picture of two teacups, before and after sintering, to illustrate the 
shrinkage common to sintering.
28 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
Another parameter is densification Ψ, defined as the change in fractional 
density due to sintering divided by the fractional density change that is needed to 
attain a pore-free solid:
Ψ = 
 ρS – ρG
 1 – ρG 
[1.14]
Densification, final density, neck size, surface area, and shrinkage are related 
measures of the particle bonding and pore elimination during sintering.
Although densification is associated with many sintering cycles, it is not a 
guarantee that the pores will shrink. Porosity might decline as the pore size 
increases, with a concomitant decrease in the number of pores. As a rough guide, a 
pore size less than half the grain size is needed to sustain densification in most 
materials. Consequently, broad pore size distributions, due to agglomeration or 
poor consolidation, lead to sintering difficulties. The narrow distribution associated 
with a high packing density inhibits grain growth and allows rapid densification. 
Thus, smaller pores, higher green densities, and narrow pore size distributions are 
precursors to rapid sintering densification and high final densities; consequently, 
narrow particle size distributions (which usually give a narrow range of pore sizes) 
prove easier to sinter to full density [15].
As sintering progresses the individual particles are blurred and the grain 
structure becomes evident. Not all grains are the same size or shape. Most sintered 
materials are assessed for grain structure using two-dimensional sections. Larger 
grains will have more faces, but the average will be between four and six in 
two dimensions and 13 to 15 faces in three dimensions. As noted earlier, the 
tetrakaidecahedron is commonly assumed as the best model for the grain shape 
during sintering, where pores occupy the corners of this polyhedron with 14 faces, 
36 edges, and 24 corners. Figure 1.16 shows an example of such a grain.
The sintered grain structure is not random, since smaller grains tend to cluster. 
Pores tend to collect on grain faces when the grain is growing and on corners when 
the grain is shrinking. The grain size distribution for sintered materials follows 
an exponential distribution function. In the cumulative form this is a Weibull 
distribution given by F(G) = 1 – exp[− ln(2) (G/Gm)
m], where F(G) is the cumulative 
fraction of grains up to size G, where Gm is the median size corresponding to half 
of the grains being smaller, and m is an exponent that is 2 for two-dimensional 
grain size measures and m = 3 for three-dimensional grain size measures. Early 
work suggested an exponential probability density function given by a related 
function where P(G) is the probability of finding grains of size G [1]:
P(G) = Pm exp[-α ( G )2]  Gm – 1 [1.15]
where Pm is the peak in the frequency distribution (the amount at the mode size), 
G is the grain size, Gm is the mode grain size, and α is typically between 2 and 6. 
 Thermodynamics of sintering 29
© Woodhead Publishing Limited, 2010
Since sintering produces a self-similar distribution (shape of the distribution is 
the same, only shifting by the location of the size scale) the mode is usually 17% 
larger than the median size. This model works for both solid and liquid phase 
sintered materials. Both broad and narrow initial particle size distributions result 
in similar grain size distributions after sintering to full density [16]. Hence, 
sintering is a process that moves the microstructure toward a normalized condition, 
independent of the starting attributes. Figure 1.17 plots several grain size 
distributions measured in two dimensions for liquid phase sintered materials with 
a normalization to show how similar these distributions become. There are big 
differences in the ease of sintering various powders, but the morphological 
attributes of the sintered product tend to converge. This convergence is often 
termed ‘self-similar’ in that no matter where we start in microstructure, the 
thermodynamics of the sintering system seem to be attracted toward the same 
final state. Of course the time to reach this point is long, so most sintered materials 
represent only partial marches along the natural sintering trajectory.
1.11 Conclusion
Sintering concepts are best developed for the case of loose, monosized spherical 
powders sintering by solid-state diffusion. In this case, the thermodynamic driving 
force is well understood and the stages are easily identified. Unfortunately, only a 
200 μm 
1.16 Scanning electron micrograph of the polyhedral grains associated 
with sintering. Compare these actual grains with the idealized 
tetrakaidecahedron shown earlier.
30 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
small portion of sintering practice relies on solid-state sintering of loose, 
monosized spheres. More common is to start with multiple phases, nonspherical 
particles, and broad particle size distributions, where one of the ingredients forms 
a liquid during the heating cycle. Further, an external pressure might be added 
to enhance densification. Although much of the effort here relates to engineered 
products, the reader must realize that sintering is pervasive and occurs for 
example in nature during the transformation of snow into glaciers and the 
transformation of certain mineral phases in the presence of magma melts. Indeed, 
the microstructures seen in geological and glacial samples are identical to those 
seen in products formed in the materials laboratory.
Single phase, solid-state sintering is applicable to pure substances such as 
nickel, ice, alumina, or copper. Usually, faster sintering is induced by adding 
phases that form liquids between the solid particles, usually by wetting the grain 
boundaries. If there is solid solubility in the liquid, then significant increases in 
mass transport rates are possible with a further benefit from capillary forces 
pulling the particles together in a manner that is similar to the role of an external 
pressure. Over 70% of the sintered products are formed using a liquid phase and 
they constitute 90% of the commercial sintered product value. The most 
important application is in the fabrication of hard materials also known as 
cemented carbides, such as WC-Co, TiC-Fe, and mixtures such as WC-TaC-
TiC-Co. Other examples are encountered in almost all areas of engineering, and 
include stainless steels, superalloys, Si3N4– based compositions, steel and 
1 – exp(–0.7 L2)
W-Ni-Fe
TiC-Mo-Ni
BaTiO3-TiO2
VC-Ni
Fe-Cu
Co-Cu
Sn-Pb
0.1
0
20
40
60
80
100
0.2 0.4
Relative intercept size, mm (L = G/G50)
C
um
ul
at
iv
e 
pe
rc
en
t
0.6 1 2 3
1.17 Cumulative grain size distributions for several liquid phase 
sintered materials toshow how the normalized distributions become 
self-similar when the size is normalized to the median grain size, each 
follows a Weibull distribution with M = 2.
 Thermodynamics of sintering 31
© Woodhead Publishing Limited, 2010
bronze, intermetallics such as silicides and aluminides, tool steels, many 
electronic compositions, most carbides, oxides, borides, nitrides, and a wide 
variety of composites such as AlN-Y2O3, TiC-Fe, ZnO-Bi2O3, WC-Co, Fe-P, 
Mo-Cu, W-Ag, Al-SiC, and W-Ni-Fe.
Sintering is critical to many industries and contributes significantly to the 
advanced materials area. As the sintering process is mastered, we find the products 
being tailored for a wide range of engineering property combinations – literally 
from high-temperature rocket nozzles forming hafnium carbide to low-temperature 
copper-based solders for electronic circuits. In turn the applications range from 
mundane bathroom fixtures to magnetic recording devices, cutting tools, home 
appliances, wristwatches, musical instruments, sporting equipment, bearings, 
filters, heat sinks, hard disk drives, hand tools, rechargeable batteries, and 
electrical capacitors. Some of these devices require high surface areas, so it is 
desirable to obtain sintered strength without densification. In other cases, sintering 
is performed under conditions where near full density is obtained. In the latter 
cases, sintering requires a high temperature, small particles, a liquid phase, or 
external pressure to ensure densification. Such processing flexibility is unparalleled 
in materials science.
1.12 Sources of further information and advice
J. R. Blackford, ‘Sintering and Microstructure of Ice: A Review,’ Journal of Physics D: 
Applied Physics, 2007, vol. 40, pp. R355–R385.
E. A. Olevsky, V. Tikare, and T. Garino, ‘Multi-Scale Study of Sintering: A Review,’ 
Journal of the American Ceramic Society, 2006, vol. 89, pp. 1914–22.
R. M. German, Sintering Theory and Practice, John Wiley and Sons, 1996, New York, NY.
R. M. German, P. Suri, and S. J. Park, ‘Review: Liquid Phase Sintering,’ Journal of 
materials Science, 2009, vol. 44, pp. 1–39.
S. J. L. Kang, Sintering Densification, Grain Growth, and microstructure, Elsevier, 
Oxford, United Kingdom, 2005.
Z. A. Munir, U. Anselmi-Tamburini, and M. Ohyanagi, ‘The Effect of Electric Field and 
Pressure on the Synthesis and Consolidation of Materials: A Review of the Spark 
Plasma Sintering Method,’ Journal of materials Science, 2006, vol. 41, pp. 763–77.
A. P. Savitskii, Liquid Phase Sintering of the Systems with Interacting Components, 
Russian Academy of Sciences, Tomsk, Russia, 1993.
N. J. Shaw, ‘Densification and Coarsening During Solid State Sintering of Ceramics: A 
Review of the Models, I. Densification,’ Powder metallurgy International, 1989, vol. 
21, no. 3, pp. 16–21.
B. Uhrenius, J. Agren, and S. Haglund, ‘On the Sintering of Cemented Carbides,’ Sintering 
Technology, R. M. German, G. L. Messing and R. G. Cornwall (eds.), Marcel Dekker, 
New York, NY, 1996, pp. 129–39.
1.13 References
 1. R. M. German, Sintering Theory and Practice, John Wiley and Sons, 1996, New 
York, NY.
32 Sintering of advanced materials
© Woodhead Publishing Limited, 2010
 2. P. W. Lee, Y. Trudel, R. Iacocca, R. M. German, B. L. Ferguson, W. B. Eisen, 
K. Moyer, D. Madan, and H. Sanderow (eds.), Powder metallurgy Technologies and 
Applications, vol. 7 ASM Handbook, ASM International, Materials Park, OH, 1998.
 3. A. P. Savitskii, ‘Relation between Shrinkage and Phase Diagram,’ Science of 
Sintering, 1991, vol. 23, pp. 3–17.
 4. G. Petzow, and H. E. Exner, ‘Particle Rearrangement in Solid State Sintering,’ 
Zeitschrift fur metallkunde, 1976, vol. 67, pp. 611–18.
 5. A. Belhadjhamida, and R. M. German, ‘A Model Calculation of the Shrinkage 
Dependence on Rearrangement During Liquid Phase Sintering,’ Advances in Powder 
metallurgy and Particulate materials – 1993, vol. 3, Metal Powder Industries 
Federation, Princeton, NJ, 1993, pp. 85–98.
 6. Y. Liu and B. R. Patterson, ‘A Stereological Model of the Degree of Grain Boundary-
Pore Contact During Sintering,’ metallurgical Transactions, 1993, vol. 24A, 
pp. 1497–505.
 7. Y. Liu and B. R. Patterson, ‘Grain Growth Inhibition by Porosity,’ Acta metallurgica 
et materialia, 1993, vol. 41, pp. 2651–6.
 8. O. Blaschko, R. Glas, G. Krexner and P. Weinzierl, ‘Stages of Surface and Pore 
Volume Evolution During Sintering,’ Acta metallurgica et materialia, 1994, vol. 42, 
pp. 43–50.
 9. R. L. Coble and T. K. Gupta, ‘Intermediate Stage Sintering,’ Sintering and Related 
Phenomena, G. C. Kuczynski, N. A. Hooton and C. F. Gibbon (eds.), Gordon and 
Breach, New York, NY, 1967, pp. 423–41.
10. R. M. German, ‘The Identification of Enhanced Sintering Systems Through 
Phase Diagrams,’ modern Developments in Powder metallurgy, vol. 15, E. N. Aqua 
and C. I. Whitman (eds.), Metal Powder Industries Federation, Princeton, NJ, 1985, 
pp. 253–73.
11. K. G. Nickel and G. Petzow, ‘Phase Diagrams – Key to Advanced Ceramics 
Development,’ Sintering ’91, A. C. D. Chaklader and J. A. Lund (eds.), Trans Tech 
Publ., Brookfield, VT, 1992, pp. 11–22.
12. A. Vagnon, J. P. Riviere, J. M. Missiaen, D. Bellet, M. Di Michiel, C. Josserond, and 
D. Bouvard, ‘3D Statistical Analysis of a Copper Powder Sintering Observed In Situ 
by Synchrotron Microtomography,’ Acta materialia, 2008, vol. 56, pp. 1084–93.
13. P. Lu. J. L. Lannutti, P. Klobes, and K. Meyer, ‘X-Ray Computed Tomograph and 
Mercury Porosimetry for Evaluation of Density Evolution and Porosity Distribution,’ 
Journal of the American Ceramic Society, 2000, vol. 83, pp. 518–22.
14. M. Nothe, K. Pischang, P. Ponizil, R. Bernhardt, and B. Kieback, ‘3D Analysis of 
Sinter Processes by X-Ray Computer Tomography,’ Advances in Powder metallurgy 
and Particulate materials – 2002, Metal Powder Industries Federation, Princeton, NJ, 
2002, pp. 13.176–13.184.
15. A. Petersson and J. Agren, ‘Sintering Shrinkage of WC-Co Materials with Bimodal 
Grain Size Distributions,’ Acta materialia, 2005, vol. 53, pp. 1665–71.
16. Z. Fang and B. R. Patterson, ‘Influence of Particle Size Distribution on Liquid Phase 
Sintering of W-Ni-Fe Alloy,’ Tungsten and Tungsten Alloys Recent Advances, 
A. Crowson and E. S. Chen (eds.), The Minerals, Metals and Materials Society, 
Warrendale, PA, 1991, pp. 35–41.