Page 158 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 3 CHARACTERISTICS AND BEHAVIOR OF NANOPARTICLES AND ITS DISPERSION SYSTEMS
Obviously in the above theory the charge on the par-
ticle is assumed to distribute uniformly on the surface.
Such an assumption can be satisfied only when a par-
ticle has either low electrical resistance or rotational
motion. For a particle with high resistance and no rota-
tion, especially when the permittivity of the particle is
low, the particle can capture only a small amount of
charge, less than half of the Pauthenier limit [3].
(ii) Diffusion charging [4]
Ions in the gas phase reach the particle surface by
thermal diffusion. For diffusion charging by unipolar
Figure 3.5.3 ions, the evolution of the number concentration of
3
Schematic illustration of electric flux lines around a particle. particles n [m ] with p elementary charges is gov-
p
erned by the following birth and death equation,
assuming that the ion concentration N apart from the
field, E . The electric field around the particle, particles is sufficiently high compared to the concen-
0
including the effect of polarization of the particle, is tration of the particles:
calculated as,
⎛ ⎞ 3 Q dn 0 nN, (3.5.26)
a

ˆˆ
E E p 0 ⎜ ⎟ 3 ( E rr E r, ˆ dt 00
)
0
0
0
p 2 ⎝ ⎠ 4 0 r 2
r
0
(3.5.22) dn p
( n n N p ,0 (3.5.27)
)
dt p 1 p 1 p p
where rˆ is the unit radius vector. Fig. 3.5.3 shows a
schematic illustration of the electric flux lines. Note
that the number of the lines in the schematic illustra- where n is the number concentration of neutral parti-
0
tion does not quantitatively correspond to the strength cles.
of the electric field. Because ions in the air migrate Letting n represent the total number concentration
T
along the electric field lines, the ion can reach the par- of all the particles, the solutions of the above equa-
ticle surface and attach itself if the flux line is termi- tions are given as,
nated by the particle. The component in the radial
direction, E , of the electric field, E, at the particle n 0 exp( Nt) (3.5.28)
r
surface (r a) is obtained from equation (3.5.22) as, n T 0
⎧ ⎪ ⎫
⎪
E E cos ⎨ 2 p 0 1 ⎬ Q (3.5.23) n ⎛ p 1 ⎞ p exp( Nt)
⎜
N⎟
r ra 0 2 4 a 2 p
k j
⎩ ⎪ p 0 ⎭ ⎪ 0 n T ⎜ ⎝ k 0 ⎟ ⎠ j 0
(3.5.29)
p 1
k 0 ( i j ) N
With the condition of E 0 for all (the angle ji

r
between E and r), no electric field can reach the parti-
3 1
cle surface, thus no more ions can impact the particle The rate constant [m s ] is called the combination
p
via this mechanism. Therefore, this condition gives the coefficient, which represents the probability of
maximum charge Q max for the impact (field) charging: impact between a single ion and a single particle with
p elementary charges per unit time and unit ion con-
3 centration. In order to determine the combination
2
Q max 4 0 a E 0 p (3.5.24) coefficient, a lot of theoretical and experimental
p 2 0 effort has been expended. Fuchs theory [5], demon-
strated to have the widest applicability in terms of
The time dependence of this charging is given as, particle size (including the nanoscale), gives the com-
bination coefficient as
t
Qt () Q
max
t 4 eBN (3.5.25)
0
⎛ ()⎞

c 2 exp ⎜ ⎟
where e ( 1.60 10 19 C) is elementary charge, B i ⎝ kT ⎠ ,
1
3
2 1
[m s V ] and N[m ] are the electric mobility of the p ⎛ ()⎞
2 ⎛ (()r ⎞ (3.5.30)
ion and the ion number density apart from the parti- 1 exp ⎜ ⎟ c i 1 2 ∫ exp ⎜ ⎟ d r
cles, respectively. This is called Pauthenier theory [2]. ⎝ k T ⎠ 4 D i r ⎝ kT ⎠

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