42 4 Gas-Phase Synthesis of Nanoparticles
Figure 4.2 Geometrical model to estimate the probability of the collision of two particles
with diameters d 1 and d 2 . The lines give the limitation of the cylinders circumscribing the
trajectories of the particles.
d 1
d 2
c 1 D t
c 2D
t
passes the volume v of the cylinder in the short time interval Δt:
v
d c t
d
kT
m
t
d
kT
d
=
=

 

 
=










π
π
π
π
πρ
π
4
4
2
2
4
2
2
6
2
2
0 5
2
3
0
∆
∆
.
.5 5
0 5
3
∆
∆
t
kT d
t
=

 

 
ρ
.
(4.4)
(ρ is the density of the particle).
To obtain the collision probability of two particles with the diameters d 1 and
d 2 , it is necessary to estimate the probability p 1 to find, for example, the particle
with the number 1 in a well-defined volume element.
p
v
V
1
1
=
total
(4.5)
(V total is the volume of the device under consideration). To find the particles
with the numbers 1 and 2 in the time interval Δt in the same volume element
is given by the product:
p
p p
1 2
1 2
− =
.
(4.6)
To relate the probability for collision with the particle diameters one obtains
p
V
kT d
t V
kT d
t
1 2
1
0 5
2
0 5
1 3
1 3
− =

 

 

 

 
=
total
total
cons
ρ
ρ
.
.
∆
∆
t t d d
T
1 2
0 5
( )
.
.
(4.7)
Equation (4.7) shows that, at constant temperature, the probability for the collision of two particles is proportional to the square root of the product of the
two diameters.
Therefore, one may call this parameter, depending only on the geometry
of the particles, “collision parameter”. Furthermore, Eq. (4.7) shows that the
collision probability increases linearly with temperature. Hence, if particle
growth by coagulation should be minimized, one has to reduce the temperature
as far as possible.