This force results in an acceleration Q 1 Q 2 = mr
2
ð
Þ, which reduces the speed and
changes the direction of the path of the particles. As a result, the collision volume
passed during time interval Dt is reduced and consequently the collision probability
is also reduced. An exact solution for this problem is extremely complex, as care
must be taken to include all possible directions of the particle flight. However, for
qualitative purposes, in a first approximation, the reduction of the collision volume
can be described by the factor 1= Q 1 Q 2
À
Á
11= d 1 d 2
ð
Þ. Inserting this factor into
Eq. (4.5) leads to the following modified collision parameter:
p 1À2 ¼ p 1 p 2 ¼ k 3 Tðd 1 d 2 Þ
0:5 1
d 1 d 2
¼ k 3 T
1
d 1 d 2
ð
Þ
0:5
ð4:9Þ
As a consequence of introducing a repelling term, Eq. (4.9) now describes a
reduced collision probability with increasing particle size. The particle growth by
coagulation and agglomeration is limited. In analogy to Eq. (4.5), the term d 1 d 2
ð
Þ
À0:5
is now called the “collision parameter,” while the temperature dependence remains
unchanged linearly.
The consequence of particle charging on the “collision parameter” is visible in
Figure 4.4a and b, where this modified “collision parameter” is plotted against particle
size for different collision partners. In contrast to Figure 4.3, where the collision
parameter increases with increasing particle size, in Figure 4.4a (calculated for d 0 ¼ 0)
a continuous decrease of the collision parameter with increasing size of the collision
partner is also observed. When considering particles of 5 nm and more, in the case of
charged particles the collision parameter is more than one order of magnitude smaller
as compared to neutral particles. This is entirely different in Figure 4.4b, where d 0 ¼ 3
was assumed. Here, a maximum of the collision parameter is realized for the particle
size d 1 ¼ d 2 ¼ d 0 ¼ 3. Clearly, it will be very difficult to obtain particles with sizes
significantly larger than d 0 . Both Figure 4.4a and b show that charging the particles
limits the size of the particles, and the long tail of the particle size distribution on the
side of the large particles in the size distribution may be avoided. The experimental
results have confirmed these considerations.
Now, one may want to estimate the temporal evolution of the particle size
distribution as a function of the collision parameters. In a first approximation,
which neglects the kinetic processes in detail this is possible. As boundary condition, one has to assume a constant number of atoms N tot of the species that will form
the particles. The process of coagulation reduces the number of particles in the
system; however, as the concentration of these atoms in the carrier gas atmosphere
is extremely low, the change in the total number of particles (hence, the concentration) during coagulation is negligible.
Based on these basic assumptions, it is possible to develop a model based on a
Markov chain [3]. To do this, the number of particles is calculated as a function of
time t, using discrete time steps Dt ¼ 1. At t ¼ 0, the beginning of the process, there
are only particles of size (volume) 1 in the system. At the start of the reactions, t ¼ 0,
the starting value is Nð0; 1Þ ¼ N tot . At the time, reaction step t, there are N t; i
ð Þ
particles of volume i and N t; j
ð Þ particles of volume j present in the system. In the
case where a particle with volume i and a second one with volume j collide and
50j 4 Gas-Phase Synthesis of Nanoparticles
2
ð
Þ, which reduces the speed and
changes the direction of the path of the particles. As a result, the collision volume
passed during time interval Dt is reduced and consequently the collision probability
is also reduced. An exact solution for this problem is extremely complex, as care
must be taken to include all possible directions of the particle flight. However, for
qualitative purposes, in a first approximation, the reduction of the collision volume
can be described by the factor 1= Q 1 Q 2
À
Á
11= d 1 d 2
ð
Þ. Inserting this factor into
Eq. (4.5) leads to the following modified collision parameter:
p 1À2 ¼ p 1 p 2 ¼ k 3 Tðd 1 d 2 Þ
0:5 1
d 1 d 2
¼ k 3 T
1
d 1 d 2
ð
Þ
0:5
ð4:9Þ
As a consequence of introducing a repelling term, Eq. (4.9) now describes a
reduced collision probability with increasing particle size. The particle growth by
coagulation and agglomeration is limited. In analogy to Eq. (4.5), the term d 1 d 2
ð
Þ
À0:5
is now called the “collision parameter,” while the temperature dependence remains
unchanged linearly.
The consequence of particle charging on the “collision parameter” is visible in
Figure 4.4a and b, where this modified “collision parameter” is plotted against particle
size for different collision partners. In contrast to Figure 4.3, where the collision
parameter increases with increasing particle size, in Figure 4.4a (calculated for d 0 ¼ 0)
a continuous decrease of the collision parameter with increasing size of the collision
partner is also observed. When considering particles of 5 nm and more, in the case of
charged particles the collision parameter is more than one order of magnitude smaller
as compared to neutral particles. This is entirely different in Figure 4.4b, where d 0 ¼ 3
was assumed. Here, a maximum of the collision parameter is realized for the particle
size d 1 ¼ d 2 ¼ d 0 ¼ 3. Clearly, it will be very difficult to obtain particles with sizes
significantly larger than d 0 . Both Figure 4.4a and b show that charging the particles
limits the size of the particles, and the long tail of the particle size distribution on the
side of the large particles in the size distribution may be avoided. The experimental
results have confirmed these considerations.
Now, one may want to estimate the temporal evolution of the particle size
distribution as a function of the collision parameters. In a first approximation,
which neglects the kinetic processes in detail this is possible. As boundary condition, one has to assume a constant number of atoms N tot of the species that will form
the particles. The process of coagulation reduces the number of particles in the
system; however, as the concentration of these atoms in the carrier gas atmosphere
is extremely low, the change in the total number of particles (hence, the concentration) during coagulation is negligible.
Based on these basic assumptions, it is possible to develop a model based on a
Markov chain [3]. To do this, the number of particles is calculated as a function of
time t, using discrete time steps Dt ¼ 1. At t ¼ 0, the beginning of the process, there
are only particles of size (volume) 1 in the system. At the start of the reactions, t ¼ 0,
the starting value is Nð0; 1Þ ¼ N tot . At the time, reaction step t, there are N t; i
ð Þ
particles of volume i and N t; j
ð Þ particles of volume j present in the system. In the
case where a particle with volume i and a second one with volume j collide and
50j 4 Gas-Phase Synthesis of Nanoparticles