MCR – 508: Diffusion & Sintering
Lecture 2
Solid State Sintering
Solid State Sintering: Introduction
The sintering phenomena in polycrystalline materials are considerably more
complex than those in viscous sintering of amorphous materials.
Because of the availability of several matter transport paths and the presence of
grain boundaries in polycrystalline solids.
Matter transport in solid state sintering can occur by at least six different paths
which define the mechanisms of sintering.
In practice, more than one mechanism may operate during any given regime of
sintering, and the occurrence of multiple mechanisms makes analysis of
sintering rates and the determination of the sintering mechanisms difficult.
The coarsening process provides an alternative route by which the free energy
of the powder system can be reduced, therefore, it reduces the driving force for
densification.
Several theoretical approaches have been used to analyze the densification
process during sintering, including the use of analytical models, scaling laws,
and numerical simulations.
Mechanisms of Sintering
o Sintering of crystalline materials can occur by at least six mechanisms or paths:
vapor transport (evaporation/condensation), surface diffusion, lattice (volume)
diffusion, grain boundary diffusion, and plastic flow.
o They all lead to bonding and growth of necks between the particles, so the
strength of the powder compact increases during sintering.
o Only certain mechanisms lead to shrinkage or densification, and a distinction is
commonly made between densifying and non-densifying mechanisms.
Non-densifying mechanisms: 1, 2,
and 3 produce microstructural
change without causing shrinkage
Densifying mechanisms: 4, 5, and 6
remove materials from the grain
boundary region, leading to
shrinkage
Schematic representation of the sintering
mechanisms for a system of two particles
Mechanisms of Sintering
When the non-densifying mechanisms dominate, coarsening leads to the production
of a porous article, whereas a dense article is favored under conditions when the
densifying mechanisms dominate.
Grain boundary diffusion and lattice diffusion are important densification mechanisms
in metals and ceramics.
Plastic flow, by dislocation motion in response to the sintering stress, plays essentially
no role in the sintering of ceramics because of the low dislocation density.
For amorphous materials (glasses), which
cannot have grain boundaries, neck
growth and densification occur by
viscous flow involving deformation of the
particles.
Figure shows as an example the sintering
of two glass spheres by viscous flow.
Two-particle model made of glass spheres
(3 mm diameter) sintered at 1000 C.
Effects of Grain Boundaries
Because of the presence of grain boundaries in polycrystalline materials, the
energy decrease due to elimination of free surface area does not go totally into
driving the densification process.
Part of the energy decrease goes into driving the grain growth process, leading to
a reduction in the driving force for densification.
The presence of the grain boundaries also dictates the equilibrium shapes of the
pores (and the grains), which can influence matter transport during sintering.
Taking a hypothetical pore surrounded by grains, the forces must balance at the
junction where the surfaces of the pores meet the grain boundary.
(a) The pore with the concave surfaces will shrink while (b) the pore
with the convex surfaces will grow (or become metastable).
Effects of Grain Boundaries
o At the junction, the tension γsv in the solid–vapor interface is tangential to that
interface, while that in the grain boundary, γgb, is in the plane of the boundary.
o The balance of forces leads to
where ψ is the dihedral angle
o A high dihedral angle indicates a low grain boundary energy, making continued
densification favorable.
o On the other hand, a low dihedral angle can lead to inhibition of densification
because the replacement of free surfaces by grain boundaries during sintering
becomes unfavorable.
(a) The pore with the concave surfaces will shrink while (b) the pore
with the convex surfaces will grow (or become metastable).
Effects of Grain Boundaries
If Asv and Agb are the changes in the free surface area and grain boundary area,
respectively, then the change in energy of the system is
where sv is the specific surface energy and gb is the specific grain boundary energy.
The system will evolve towards a provisional equilibrium governed by the
condition:
gb Agb = sv Asv
(a) The pore with the concave surfaces will shrink while (b) the pore
with the convex surfaces will grow (or become metastable).
Analytical Model: Stages of Sintering
Sintering is normally thought to occur in three sequential stages referred to as
1) the initial stage,
2) the intermediate stage, and
3) the final stage.
o A stage represents an interval of time or density over which the microstructure is
considered to be reasonably well defined.
o The below Figure schematically depicts the typical densification curve of a compact
through these stages over sintering time.
o The initial stage is characterized by the
formation of necks between particles and
its contribution to compact shrinkage is
limited to 2–3% at most.
o During the intermediate stage,
considerable densification, up to 93% of
the relative density, occurs before
isolation of the pores.
o The final stage involves densification
from the isolated pore state to the final
densification, ~ 7% shrinkage.
Analytical Model: Stages of Sintering
Some of the main parameters associated with the three idealized stages of sintering are
summarized in the below Table, and examples of the microstructures of real powder
compacts in the initial, intermediate, and final stages are shown in the below Figure.
Initial Stage Intermediate Stage Final Stage
Initial Stage: Mechanisms
The initial stage consists of fairly rapid interparticle neck growth by diffusion,
vapor transport, plastic flow or viscous flow (amorphous materials).
The large initial differences in surface curvature are removed in this stage, and
shrinkage (or densification) accompanies neck growth by the densifying
mechanisms.
Figure shows two geometrical models
for two spherical particles: one
without shrinkage (a) and the other
with shrinkage (b).
In Figure (a), the distance between
the particles does not change but the
neck size increases as the sintering
time increases.
In the model with shrinkage (Figure
(b)), the neck size increases with an
increased sintering time by material
transport between the particles and
hence shrinkage results. Two-particle model for initial stage sintering
(a) without shrinkage and (b) with shrinkage.
Initial Stage: Mechanisms
If the dihedral angle between the particles is 180° and the grain size does not change
during sintering, the radius of neck curvature r, neck area A and neck volume V are
respectively for the geometry without shrinkage.
For the geometry with shrinkage,
Initial Stage: Mechanisms
The driving force of sintering appears as differences in bulk pressure, vacancy
concentration and vapor pressure – parallel phenomena – due to differences in
surface curvature of the particles.
For the geometries in the Figure below, the pressure difference ΔP is
The vacancy concentration difference ΔCv is
and the vapor pressure difference Δp is
Here, γs is the specific surface energy of the solid (solid surface
energy), V’m the molar volume of vacancies and Vm the molar
volume of the solid. In general, V’m is not the same as Vm because
of the relaxation of atoms around the vacancy.
Initial Stage: Mechanisms
The differences in bulk pressure, vacancy concentration, and vapor pressure
induce material transport.
The material transport due to the difference in interface curvature occurs
under the parallel actions of various mechanisms.
The dominant mechanism, however, can vary depending on, for example,
particle size, neck radius, temperature and time for a given system.
Initial Stage: Kinetics Equations
The equations for neck growth and for the densifying mechanisms, shrinkage can be
expressed in the general form
where m and n are numerical exponents that depend on the mechanism of
sintering and H is a function that contains the geometrical and material parameters
of the powder system.
Analytical Model: Intermediate Stage
Intermediate Stage
Intermediate Stage
The geometrical model commonly used for the intermediate stage was proposed
by Coble.
The powder system is idealized by considering it to consist of a space-filling array
of equal-sized tetrakaidecahedra, each of which represents one particle.
The porosity is cylindrical, with the axis of the cylinder coinciding with the edge
of the tetrakaidecahedra (see the below Figure).
This intermediate stage model assumes equal shrinkage of pores in a radial
direction.
Although the model is limited in terms
of describing real sintering, it
reasonably simplifies sintering
complexity and allows the evaluation
of the effect of sintering variables on
sintering kinetics.
Intermediate Stage
o A tetrakaidecahedron is constructed from an octahedron by trisecting each edge and
joining the points to remove the six edges (see below Figure).
o The resulting structure has 36 edges, 24 corners, and 14 faces (8 hexagonal and 6
square).
o The volume of the tetrakaidecahedron is
o Where lp is the edge length of the tetrakaidecahedron. Each pore is shared by three
tetrakaidecahedra, and if r is the radius of the pore, then the total volume of the
porosity per unit cell is
Porosity of the unit cell (Pc)
Intermediate Stage: Kinetic Equations
If the edge effect shown in the geometrical model is neglected, the atom flux towards
the cylindrical pores may be similar to the thermal flux towards the pores provoked by
an electrically heated wire with diameter equal to the grain boundary diameter.
The atom flux per unit length (Jatom) is given by Jatom = -(D/RT)(d/dx) and its solution
Jatom = 4(D/RT).
For the atom flux, two mechanisms are dominant: lattice diffusion and grain
boundary diffusion.
Lattice Diffusion
Since pore shrinkage occurs at all of the 14
surfaces of the grain, the rate of pore volume
change dVp/dt is expressed as:
Intermediate Stage: Kinetic Equations
Therefore, the rate of porosity change dPv/dt is given by
Where l is the length of a grain edge and is the relative density.
where G is the grain diameter which satisfies
( /6)G3 = 82l3.
If grains do not grow during sintering (that is,
when G = const.), the integration of the above
equation is simple.
However, in reality, grain growth usually occurs.
The grain size must then be expressed as a
growth equation in order to integrate the
above equation.
Intermediate Stage: Kinetic Equations
Grain Boundary Diffusion
A similar procedure to that for lattice diffusion gives
where b is the diffusion thickness of grain boundary diffusion.
Therefore,
Analytical Model: Final Stage
Final Stage
Final Stage Sintering
For the final stage sintering of polycrystalline materials, the powder system is
idealized in terms of an array of equal-sized tetrakaidecahedra with spherical
pores of the same size at the corners.
As a geometrical model of final stage sintering, Coble took tetrakaidecahedral
grains with spherical pores with a radius of r at their corners.
The tetrakaidecahedron has 24 pores (one at each corner), and each pore is
shared by four tetrakaidecahedra, so the pore volume associated with a single
tetrakaidecahedron is Vp = (24/4)(4/3)r3, where r is the radius of a pore.
The volume of the tetrakaidecahedron is
Making use of the above equations, the
porosity in a single tetrakaidecahedron is
Final Stage Sintering
A more convenient unit cell of the idealized structure can be chosen as a thick-
walled spherical shell of solid material centered on a single pore of radius r.
The outer radius b of the spherical shell is defined such that the average density of
the unit cell is equal to the density of the powder system, that is,
o The volume of the solid phase in the unit
cell is (4/3)(b3 - r3)
o Since a unit cell contains a single pore, the
number of pores per unit volume of the
solid phase is
Final Stage Sintering: Kinetic Equations
Lattice Diffusion
Coble used a procedure similar to that outlined earlier for the intermediate stage,
but the atomic flux equation was approximated to that corresponding to
diffusion between concentric spherical shells.
The final result is:
This equation indicates that the densification rate is inversely proportional to
the cube of grain size.
This result is the same as that found for the dependence of neck growth and
shrinkage on particle size in the initial stage model.
In Coble’s model, however, a fundamental aspect is not taken into account,
namely, the grain boundary as the atom source for densification.
Final Stage Sintering: Kinetic Equations
Kang and Jung derived the densification rate for both volume diffusion and grain
boundary diffusion.
Since a stress gradient is thought to exist from the pore surface to the centre of the
grain boundary, the gradient may be assumed to be present over a distance of l/2.
For lattice diffusion, the size dependence is the same as that of Coble’s model,
however, it contains a relative density term.
Grain Boundary Diffusion
Thus, the densification rate for grain boundary diffusion is given by
In the case of grain boundary diffusion, the dependence of densification rate on
grain size is the same as that in the initial stage model.
Effect of Sintering Variables on Densification
The sintering rate (densification rate) increases with decreased particle
size and with increased sintering temperature and time, as shown
schematically in Figure.
Effect of Particle Size: Scaling Laws
The effect of particle size on sintering is described by Herring's Scaling Laws.
The scaling laws attempt to answer the following important question:
How does the change in particle size influence the rate of sintering?
When powders with similar shapes but different sizes are sintered under the
same experimental conditions and by the same sintering mechanism, the scaling
law predicts the relative periods of sintering time required to get the same
degree of sintering.
The scaling laws do not assume a specific geometrical model. Instead, the main
assumptions in the model are that during sintering
a) the particle size of any given powder system remains the same, and
b) the geometrical changes remain similar.
Effect of Particle Size: Scaling Laws
Two systems are defined as being geometrically similar if the linear dimension of all
of the features (e.g., grains, pores) of one system (system 1) is equal to a numerical
factor times the linear dimension of the corresponding features in the other system
(system 2):
where is a numerical factor. Geometrically similar systems therefore involve
simply a magnification of one system relative to the other (see below Figure).
An example of two geometrically similar systems consisting of a random arrangement
of circles. The systems differ only in scale and involve a simple magnification of one
relative to the other.
Effect of Particle Size: Scaling Laws
o To illustrate the derivation of the scaling laws, let us now consider a simple system
consisting of two spheres in contact.
o Suppose it takes a time t1 to produce a certain microstructural change (e.g., the
growth of a neck to a certain radius X1) in system 1.
o Now the question is: How long (t2) does it take to produce a geometrically
similar change in system 2?
o For geometrically similar changes, the initial radius of the particle and the neck
radius of the two systems are related by
Geometrically similar models consisting of
two spheres in contact. The linear
dimensions of system 2 are a factor times
those of system 1.
System 1
System 2
Effect of Particle Size: Scaling Laws
The time taken to produce a certain change by diffusional flow of matter can be
expressed as
where V is the volume of matter transported, J is the flux, A is the cross-sectional
area over which matter is transported, and is the atomic volume. We can
therefore write
Geometrically similar models consisting of
two spheres in contact. The linear
dimensions of system 2 are a factor times
those of system 1.
System 1
System 2
Effect of Particle Size: Scaling Laws
Lattice Diffusion
The volume of matter transported is proportional to R3, where R is the radius of
the sphere. Therefore, V2 is proportional to (R)3, or V2 = 3V1.
The area over which matter diffuses is proportional to R2. Therefore, A2 is
proportional to (R)2, or A2 = 2A1.
The flux J is proportional to , the gradient in the chemical potential. For a
curved surface with a radius of curvature r, varies as 1/r. Therefore, J varies as
(1/r) or as 1/r2. Now J2 is proportional to 1/(r)2 so that J2 = J1/ 2.
To summarize, the parameters for lattice diffusion are
If we substitute the above values in the given equation, we obtain
According to this equation, the time taken to produce geometrically similar changes
by a lattice diffusion mechanism increases as the cube of the particle size.
Effect of Particle Size: Scaling Laws
The scaling laws for the other mass transport mechanisms can be derived using a
procedure similar to that outlined above for lattice diffusion.
The laws can be written in the general form:
where m is an exponent that depends on the mechanism of sintering. The
following Table gives the values of m for the different sintering mechanisms.
Effect of Temperature
Since sintering is a thermally activated process, the variables sensitive to temperature
are diffusivity, viscosity, etc. which are expressed as exponential functions of
temperature.
Therefore, lnt (natural log t) is at first proportional to 1/T, where t is the sintering time
to get a given degree of sintering. However, the exact relationship is different for
different mechanisms.
Effect of Pressure
The driving force of sintering is the capillary pressure difference due to curvature
difference.
However, when an external pressure Pappl. is applied, the total sintering pressure Pt is
the sum of the capillary and external pressures:
where f(, geo) is a function of relative density and particle geometry.
Therefore, sintering equations for systems with external pressure are different to
those without it.
The densification rate, however, always increases with increased sintering pressure.
Various techniques are available in pressure-assisted sintering, for example, gas
pressure sintering, hot pressing (HP) and hot isostatic pressing (HIP).
