Page 159 - Book Hosokawa Nanoparticle Technology Handbook

3.5 INTERACTIONS BETWEEN PARTICLES FUNDAMENTALS
2
where c[m/s] and D [m /s] are the mean thermal veloc- ⎛ p ⎞ ⎛ p ⎞
i
i
ity and diffusion coefficient of the ions, respectively,
k 1 ⎟
k 1 ⎟ 1. (3.5.37)
⎜
⎜
T[K] is the absolute temperature, and k 1.38 10 23 p 1⎜ ⎝ k 1 k ⎠ ⎟ p 1⎜ ⎝ k 1 k ⎠ ⎟ ⎠
[J/K] is the Boltzmann constant. The radius of the lim-
iting-sphere in Fuchs theory [m] is given as,
⎧ ⎛ ⎞ 5 ⎛ ⎞ ⎛ ⎞ 3 ⎫ References
2
3 ⎜
2 ⎟ ⎜

a ⎪ ⎪ ⎝ 1 a⎠ ⎟ ⎜ ⎝ 1 a ⎠ ⎝ 1 a⎠ ⎟ 2 ⎛ ⎞ 52 ⎪ [1] T. Matsuyama, H. Yamamoto: J. Soc. Powder Tech.,
⎪
2
⎨ ⎜ 1 2 ⎟ ⎟ ⎬ , Jpn., 34, 154–159 (1977).
2 ⎪ 5 3 15 ⎝ a ⎠ ⎪
⎪ ⎪ [2] M.M. Pauthenier, M.M. Moreau-Hanct: J. Phys.
⎩ ⎭ Radium, 3, 590–613 (1932).
(3.5.31) [3] S. Masuda, M. Washizu: J. Inst. Electros., Jpn., 3,
153–159 (1979).
where a is the radius of the particle and [m] is the [4] Editorial committee for basis of powder technology (eds):
mean free path of the ions.
Including the effect of the image force, the poten- Funtai-kougaku no kiso, Nikkankogyo, p. 86 (1992).
tial of an ion (r) [J] is calculated as [5] N.A. Fuchs: Pure Appl. Geophys., 56, 185–193 (1963).
[6] W.A. Hoppel, G.M. Frick: Aerosol Sci. Tech., 51, 1–21
2 ⎧ ⎫ (1986).
e ⎪ p p 0 a 3 ⎪
() ⎨
⎬ (3.5.32) [7] G.L. Natanson: Soviet Phys. Tech. Phys., 5, 538–551
r
4 0 ⎩ ⎪ r p 0 2 rr a 2 ) ⎭ ⎪ (1960).
2
(
2
3.5.1.3 Solid bridging (solution and precipitation,
2
2
is a correction coefficient given as
b / [6], sintering)
m
2
and b is a parameter given as the minimum of the
m
following function [7]: When fine particles are set at high temperature below
the melting point, atoms diffuse to reduce their total
surface energy. As a result, strong bonding between
⎧ 2 ⎫

2
b r ⎨ 1 (( ) r ( )) ⎬ (3.5.33) particles forms to be a sintered body. Depending on
2

⎩ 3 kT ⎭ diffusion path, pores are excreted to be densified.
Since specific surface area of particles increases with
decrease in particle size, nanoparticles have large
Bipolar diffusion charging is also called charge neu-
tralization. When the concentrations of both positive driving force for sintering, namely, their sinterability
and negative ions are sufficiently higher than that of is extremely high.
particles, the birth and death equations are In the contact region between two particles (neck
region), stress generates due to surface energy
(Fig. 3.5.4). The stress results in increase in excess
dn 0 nN n N nN n N ,

dt 1 1 0 0 1 1 0 0 vacancy concentration and decrease in vapor pres-
sure. These enhance mass transfer to advance sinter-
(3.5.34) ing. In solid state sintering, mass transfer occurs by
evaporation–condensation, surface diffusion, bound-
dn
p n N n N ary diffusion and bulk diffusion (Fig. 3.5.5).

dt p 1 p 1 p p Representative time dependences of neck size x

p 1 n p 1 N p p || 1 (3.5.35) assuming each mass transfer are listed in Table 3.5.4.
p
n N
In any case, neck size, x, is proportional to time and
the mth powers of grain size. The exponent, m,

Under the condition of N N , the equilibrium depends on the mass transfer route. In surface diffu-
charge distribution corresponding to the steady state sion and evaporation–condensation route, the distance
(dn /dt 0, dn /dt 0) can be calculated as between two particles does not change only causing
0
p
neck growth but not being densified [1]. On the other
hand, in case of bulk diffusion [2] and boundary dif-
n 0 1 ⎫ fusion [3] route, the distance between particles shrink
⎪
n T ⎪ during the neck growth. In these cases, shrinkage
⎪
p
⎪
n p
k p ⎬ (3.5.36) rates are obtained as shown in Table 3.5.4.
Furthermore, sintering theory in viscous flow [4] is
⎬
k 1
1
n T k ⎪ also listed in Table 3.5.4. Time-dependence of shrink-
1
⎪
p
n p
k p ⎪ age ratio is different in sintering mechanism.
Although grain-size effect also depends on sintering
k 1
1
⎭
n k ⎪ mechanism, smaller grain size has higher shrinkage
1
T
135