Page 161 - Book Hosokawa Nanoparticle Technology Handbook

3.5 INTERACTIONS BETWEEN PARTICLES FUNDAMENTALS
Table 3.5.4
Typical equations of sintering rate for each sintering mechanism.
Sintering mechanism x/r vs. t L/L vs. t
0
Surface diffusion ⎛ x⎞ 7 ⎛ 56
D ⎞
(Kuczynski [1]) ⎜ ⎟ ⎜ ⎝ kTr 4 s ⎟ ⎠ t
⎝
r ⎠
Evaporating condensation ⎛ x⎞ 3 ⎛ 3

M 32 / p ⎞
0
r ⎟
(Kuczynski [1]) ⎜ ⎝ ⎠ ⎜ ⎝ 2 RT 32 2 2 ⎟ t
d r ⎠
/
32 /
Volume diffusion ⎛ x⎞ 5 ⎛ 80
D ⎞ ⎛ L ⎞ 52 / = ⎛ 20
D ⎞
V
V
r ⎟
3 ⎟
(Kingery [2]) ⎜ ⎝ ⎠ ⎜ ⎝ kTr 3 ⎟ ⎠ t ⎜ ⎝ L ⎠ ⎟ ⎜ ⎝ 2 kTr ⎠ t
0
D ⎞
Grain boundary diffusion ⎛ x⎞ 6 ⎛192
D ⎞ ⎛ L ⎞ 3 ⎛ 3
b
b
r ⎟
(Coble [3]) ⎜ ⎝ ⎠ ⎜ ⎝ kTr 4 ⎟ ⎠ t ⎜ ⎝ L ⎠ ⎟ ⎜ ⎝ kTr 4 ⎟ ⎠ t
0
9 ⎞
Viscous flow ⎛ x⎞ 2 ⎛ 3
⎞ ⎛ L ⎞ ⎛
r ⎟
(Frenkel [4]) ⎜ ⎝ ⎠ ⎜ ⎝ 2 r⎠ ⎟ t ⎜ ⎝ L ⎠ ⎟ ⎜ ⎝ 4 r⎠ ⎟ t
0
Solid phase
Liquid phase
Gas phase

Soild phase
Δ P = ( γ LV cos θ ) d
2
Figure 3.5.7
Capillary force P between two particles.

where is the thickness of liquid phase between
particles, the atomic volume,
the vapor/liquid
interfacial energy, D the diffusion constant of solid
in liquid, C the solid content in liquid, t the sinter-
ing time, k the Boltzmann constant, T the tempera-
ture, r the particle size and K the reaction constant.
r
The effect of time and grain size on shrinkage rate
is the same formula as solid phase sintering. The
exponent depends on the rate controlling process;
shrinkage ratio is proportional to 1/3 power and 1/2
power of time in diffusion controlling process and
interfacial reaction controlling process, respec-
tively. In the final stage, grain growth and shape
relaxation occur. Microstructure of sintered body by
Figure 3.5.6 liquid phase sintering includes secondary phase.
Densification behavior of Y O nano particles and Some of the secondary phase is glass, which
3
2
microstructure of the sintered body fired by adding 1 wt% influences the mechanical, chemical and thermal
MgO at 1,080 C to 76% and at 1,000 C for 20 h. properties.
137