Note that Brownian kernel k
B
ij has a constant value of about 2 when i % j. The
kernels for Brownian ðk ij
B
f Þ or ballistic ðk ij
b
f Þ aggregation of fractal particles are [147]:
k ij
B
f ’
1
2
i
À1=d f þ j
À1=d f
i
1=d f þ j
1=d f
;
(49)
k ij
b
f ’
1
2
i
À1
þ j
À1
À
Á 1=2 i
1=d f þ j
1=d f
;
(50)
where d f is the fractal dimension of the cluster.
The exact analytical solutions of PBE for Brownian and ballistic kernels have not
yet been obtained. In slow aggregation regime, these kernel may be estimated as:
k
s;BðbÞ
ij
¼ k
BðbÞ
ij =W ij :
(51)
Note that the constant kernel kinetics is the slowest as compared to kinetics of
the sum and product kernels. It is interesting to note that formation of the infinite
10
0
10
0
10
0
10
-1
10
-2
10
0
10
-1
10
-2
10
1
10
1
10
2
10
2
10
0
10
1
10
2
0
0.5
1
0.1
2.0
10.0
0.1
0.2
0.49
0.1
0.4
10.0
t
* =
t * =
t * =
k ij =2
k ij =i+j
k ij =2ij
m
m
m
f
f
f
Fig. 8 Normalized cluster distribution functions f(m) ¼ mn m /(mn m ) max for constant (k ij ¼ 2),
sum (k ij ¼ i + j), and product (k ij ¼ 2ij) kernels at different dimensionless times t* ¼ t/t a
Aggregation of Charged Colloidal Particles
83