Biomedical Engineering Reference
In-Depth Information
using equation 13.6.
36
Note that the significant changes of the initial
sintering temperature do not occur until the particle size is less than
approximately 20 nm.
Scaling law - dependence of sintering on particle size
In conventional sintering theories, the dependence of densification behavior
on the size of particles is described by the scaling law. In 1950, Herring
39
first introduced the scaling law as follows:
n
D
t
2
¼ l
D
t
1
½
13
:
7
where
= R
2
/R
1
, R
1
and R
2
are particle radius, n depends on specific
diffusion mechanisms of the densification. Specifically, n = 1 for viscous
flow, 2 for evaporation and condensation, 3 for volume diffusion, and 4 for
surface diffusion or grain boundary diffusion. The scaling law states that the
time required to sinter powders with particle radii of R
1
and R
2
is
proportional to the ratio of the particle radius. Although the densification
behavior of nano powders can be qualitatively understood on the basis of
the scaling law, few direct analyses of experimental data exist in the
literature. The few studies that did apply the scaling law used the following
expression to analyze the activation energies of the sintering of nano
powders:
40,41
λ
¼
d
1
d
2
Q
R
1
T
2
1
T
1
n
ln
½
13
:
8
where d
1
, d
2
are particle sizes, T
1
and T
2
are corresponding sintering
temperatures, R is the gas constant, and Q is the activation energy. By using
the above equation, some studies obtained activation energy values that are
closer to grain boundary diffusion, while others obtained values closer to
volume diffusion, which is believed to be unlikely at low temperatures.
These discrepancies can in part be attributed to the assumption inherent in
the scaling law that the particle size of two different powders does not
change during sintering and microstructural changes remain geometrically
similar for the two systems.
42
However, if the values of n and Q/R can be
evaluated by other methods, e.g. curve-fitting experimental data to
densification equations, equation 13.8 can be used to estimate the sintering
temperature of different particle-sized powders.
