Biomedical Engineering Reference
In-Depth Information
aF/2kT. However, when particle sizes are in the range of nanometers, the
linear approximation is no longer reasonable. Then, the diffusion equation
becomes a nonlinear equation that can only be solved via numerical
methods. Applying this approach to sintering two particles, the ratio of the
neck-to-particle radius as a function of the length of time at a given
temperature was calculated and shown by Fig. 13.7. The differences between
prediction by linear solutions and nonlinear solutions are significant during
the initial stage of sintering and diminish as sintering time increases. The
distinction between linear and nonlinear solutions also diminishes as particle
size increases.
The rapid rate of sintering due to the rapid rate of diffusion is also
supported by recent studies which indicate that the coefficient of diffusion,
D, is size dependent as shown in equation 13.10,
44
D
0
exp
E
ðÞ
RT
exp
2S
vib
ðÞ
3R
1
Dr
ð
;
T
Þ¼
½
13
:
10
r
=
r
0
1
where D
0
is pre-exponential constant, E(
)
is bulk melting entropy, r is particle radius, r
0
= 3h (h is atomic diameter)
for nano particles, R is ideal gas constant, and T is absolute temperature.
The dependence of the coefficient of diffusion on particle size is attributed
to the theory that, as the size of the nanocrystals decreases, the activation
energy of diffusion decreases and the corresponding coefficient of diffusion
increases based on the Arrhenius relationship between them. Together, these
theories, based on nonlinear diffusion law and the increase of the coefficient
of diffusion with decreasing particle size, convincingly explain the rapid
formation of necks bonding with neighboring particles.
The rapid kinetics of the sintering of nano particles were also
demonstrated by using molecular dynamic simulations (MD).
45-52
The
basic approach for simulating sintering using the MD method involves
tracking the motion of atoms under stress caused by surface or interfacial
energy. The kinetics of sintering is given as the rate of decreasing distance
between two atoms in the middle of two particles in contact. It was shown
that sintering of nano particles at the atomic level can be accomplished by
dislocation motion and grain boundary rotation, as well as other
mechanisms. It was further predicted that the sintering time of nano
particles would be in the range of a few hundred picoseconds. Although the
predicted sintering time is far from engineering reality, the results of the
simulation can be used as a basis for understanding the initial bonding or
formation of the necks between nano particles.
?
) is bulk activation energy, S
vib
(
?
