Biomedical Engineering Reference
In-Depth Information
Kingery's initial concept of pore stability.
62
1
cos
R
c
¼
½
13
:
12
f
e
R
c
is called critical particle size ratio for boundary migration,
f
e
is the
dihedral angle relating surface energy and grain boundary energy. Lange
explained that when the size ratio between two particles is larger than the
critical size ratio R
c
, grain boundary migration will occur, resulting in grain
growth. When actual size ratio is less than R
c
, boundary migration will yield
an increase in the grain boundary area and is energetically unfavorable. In
this situation, inter-particle mass transport will happen first, in order to
increase the size ratio between adjacent particles. This coarsening process
will not stop until the size ratio R = r
1
/r
2
reaches R
c
. Then grain boundary
migration will take over because the condition for grain boundary migration
is now energetically satisfied.
The studies by Lange and Kingery aimed to explain the stability of pores
in the intermediate stage of sintering. Shi further applied the critical size
ratio criteria to the initial sintering of ultrafine particles.
60,61
It was shown
that the driving force for neck growth and inter-particle diffusion are given
respectively as follows:
1
X
1
r
D
m
n
¼
g
s
O
½
13
:
13
1
r
1
1
r
2
D
m
c
¼
2
g
s
O
½
:
13
14
Δ
μ
n
and
Δ
μ
c
are chemical potential for neck formation and mass transport
between two particles;
g
s
is surface energy;
is atomic volume; X is radius of
the neck; r is radius of particles (r
1
and r
2
are radii of two particles with
different sizes).
Equation 13.14 indicates that if a difference in the radius of curvature
exists, mass transport would take place from the area of larger curvature to
the area of smaller curvature. This process is related to the particle
coarsening.
Considering equations 13.13 and 13.14 together, both the neck growth
and coarsening, driven by the surface tension between the particles, can take
place concurrently. However, the magnitude of the driving force for the two
processes is different. Assuming the interface energy is not considered, then
|
W
Δ
μ
n
|
>
|
Δ
μ
c
|, which implies
that neck formation takes place before
coarsening.
On the other hand, if the interface energy between particles is considered
in the analysis of the driving forces as an energy barrier to neck growth,
