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164 Chung-Shin J. Yuan and Thomas T. Shen
In the case of ion bombardment, combining Eqs. (19)–(21) gives
t
q = 4π K Pa E
2
0
f
0
t + 4 K q m N
0
i
i
0
=4π (8.85 ×10 −12 )( )( −12 )(10 6 )
310
0.1
×
0.1+ ( 4 8.85 ×10 −12 ) (1 602 ×10 −19 )(4 ×10 −4 )(10 14 )
.
= 3.162 ×10 −16 ( C or 1,974 electronic charges )
Note that q for this particular case happens to be 2082 electronic charges. Thus, in 0.1
max
s, the particle has already acquired 95% of its maximum charge.
For the case of diffusion charging, Eq. (26) provides the answer:
π
( 85.
48 × 10 −12 )(10 −6 )(1 38. × 10 −23 )(313 )
q =
d
1 602 × 10 −19
.
( 10 −8 )(10 14 )(2 566. × 10 −38 )(500 )(0 1. )
× ln + 1
( 48 85. × 10 −12 )(1 38. × 10 −23 )(313 )
= 2.02 × 10 −17 C (or 126 electronic charges )
Comparing the charges from two mechanisms illustrates the significance of field charging
as opposed to diffusion charging for a particle of 1 mm radius. Diffusion charging con-
tributes less than 6% of the total charge.
Repeating the above calculations for a particle of 0.1 µm radius and charging time of 0.1
s gives field charging of 20 electronic charges and diffusion charging of 8 electronic
charges. Being a contributor of almost 30% of the total charge, it obviously shows that the
diffusion-charging mechanism cannot be neglected for smaller particles.
Previously, we made use of particle space charge density σ without actually defin-
p
ing it; see Eqs. (9), (11), and (15). We are now in a position to do so. Assuming that field
charging is the more dominant charging mechanism, Eq. (19) gives the maximum
charge acquired by a particle:
2
q max =πK 0 Pa E
4
Assuming that the particle concentration is N per unit volume, the total charge acquired
p
by all particles (i.e., particle space-charge density) is
2
σ = 4πΚ 0 Pa EN p
p
which may be also be expressed as
σ = K 0 PES (27)
p
where S is the total particle surface per unit volume of gas,
S = 4πa 2 N p
