Biomedical Engineering Reference
In-Depth Information
13.2 Thermodynamic driving force for the sintering of
nanosized particles
In general, the sintering of nanosized or nanocrystalline powders follows the
same path as larger grain powders. However, compared to conventional
micron-sized or submicron-sized particles, the densification behavior of
nano particles during sintering is notably different. From the perspectives of
thermodynamics, the driving force for sintering particles of any size is the
reduction of surface energy. Based on conventional sintering theories, the
driving force of sintering can be given by
7
1
R
1
þ
1
R
2
s
¼
gk
¼
g
½
13
:
1
where
g
is the surface energy of the material,
k
is the curvature of a surface,
which is defined by
k
1
1
R
2
(for a convex surface, it is taken to be
positive; for a concave surface, it is taken to be negative), and where R
1
and
R
2
are the principal radii of the curvature. The driving force for the sintering
of nanosized particles is, therefore, inversely proportional to the size of the
particles. This relationship would lead to a much higher driving force for the
sintering of nanosized particles compared to micron-sized particles. For
example, based on equation 13.1, the driving force for a ten nanometer
particle is two magnitudes higher than that for a one micron particle.
The large driving force of sintering of nano particles can be even higher
than the result of equation 13.1 if the non-linear dependency of vacancy
concentrations on the particle size is considered. During sintering, mass
transport, usually mediated by vacancies, is driven by the difference in
vacancy concentration
¼
R
1
þ
D
C
n
¼
C
n
C
n
0
, where C
n
is the vacancy concentra-
tion for a surface with curvature of
, and C
n
0
is vacancy concentration for a
flat surface. Based on the Gibbs-Thomson equation,
8
κ
gk
O
kT
C
n
¼
C
n
0
exp
½
13
:
2
where
is the atomic volume, k is Boltzmann's constant, and T is the
absolute temperature. For micron-sized particles, the term
gk
O
W
kT
<<
1in
, therefore
gkO
kT
equation 13.2 becomes linear, C
n
&
C
n
0
1
C
n
0
gk
O
kT
D
C
n
&
½
13
:
3
However, when particle size approaches nanoscale, the linear approxima-
tion is no longer valid. The correct expression for the driving force of mass
