Aggregation kinetic equations

ISSN 2224–5227                                                                                                                               
№ 3. 2016  
 
 
83 
So we submit the following non-local modification of the Smoluchowski equation for aggregation in 
the uniform system:  
 
2
1
1 0 0
2
1
,
2
1
1 0 0
1
2
1
,
)
(
)
(
)
(
)
(
2
1
dt
dt
t
C
t
C
N
dt
dt
t
C
t
C
N
dt
dC
j
t t
j
i
j
i
i
j
t t
j
i
j
j
i
j
i
∑∫∫
∑∫∫
∞
=
−
=
−
−
−
=
    
  (34) 
i
C
 denotes the concentration of 
i
-mer.  
In our case the characteristic times 
j
i,
τ
of the aggregation of 
i
 and 
−
j
mers play a role of relaxation 
times. The simplest model equation for elements of the aggregation matrix can be constructed by analogy 
with model equation (6) for transfer kernels. We submit this equation as follows:  
 
 
0
,
,
0
,
,
,
=
+
∂
∂
+
∂
∂
j
i
j
i
j
i
j
j
i
j
i
j
i
i
N
f
s
N
r
s
N
r
τ
,     
  (35) 
where 
1
t
t
s
i
−
=
  
2
t
t
s
j
−
=
 
In equation (35) the coefficients 
i
r
 on a level with relaxation time 
ij
τ
 play a part of control 
parameters of globules “inertness”, the parameter
f
answers for media and particles characteristics. 
Independent integrals of equation (35) read 
j
j
i
i
r
s
r
s −
=
Ψ
1
;       
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
Ψ
i
j
i
i
j
i
j
i
I
s
r
f
N
,
0
,
,
2
exp
τ
    or  
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
Ψ
j
j
i
j
j
i
j
i
II
s
r
f
N
,
0
,
,
2
exp
τ
 
Thus the aggregation matrix, satisfying equation (35) and coming up to the condition of fast 
relaxation in time
j
i
t
,
τ
>>
, can be written as 
 
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
=
j
j
i
i
j
i
j
i
j
i
j
i
r
s
r
s
f
N
,
0
,
0
,
,
2
exp
τ
η
     
   (36) 
Let’s assume at the beginning 
1
=
=
j
i
r
r
 and
const
a
a
f
j
i
j
i
j
i
=
=
≡
,
,
0
,
τ
.  
Thus we have 
  
∑
∑
−
−
−
=
−
2
2
,
3
1
2
1
,
)
exp(
)
exp(
2
1
I
I
at
I
I
at
dt
dC
j
i
j
i
j
i
η
η
.  
(37) 
Here 
∑
1
means
∑
−
=
1
1
i
j
;   
∑
2
means
∑
∞
=1
j
;  
ds
s
C
as
I
j
i
t
)
(
)
2
exp(
0
1
−
∫
=
; 
ds
s
C
as
I
t
j
)
(
)
2
exp(
0
2
∫
=
; 
ds
s
C
as
I
t
i
)
(
)
2
exp(
0
3
∫
=
 
We didn’t find way to rigorous reducing equation (37) to an ODE form even in that case.  However, 
we try to simplify the problem by using asymptotic behaviour of integrals in (37). Namely, it is supposed 
that for small relaxation times we can use Laplace method in the neighbourhood of the time point
t
. But 
immediate substitution of the integrals expansions into equation (37) requires multiplying asymptotic 
sequences. Such procedure is dangerous, as it may lead to utter loss of checking orders of approximation.  
Therefore we rearrange the equations to the form which is free from a product of integrals:  
⎥
⎦
⎤
⎢
⎣
⎡
+
⎟
⎠
⎞
⎜
⎝
⎛−
−
+
⎟
⎠
⎞
⎜
⎝
⎛−
=
+
∑
∑
∑
−
−
2
2
,
3
2
,
1
2
1
,
2
2
2
exp
)
(
2
exp
2
1
j
j
i
j
i
i
j
i
j
j
i
j
i
i
C
I
I
C
at
I
C
I
C
at
dt
dC
a
dt
C
d
η
η
η
.  
(38)                     
 
Using then Laplace method we obtain the understandable asymptotic relations in which the orders of 
equations and approximations are concerted 

Доклады Национальной академии наук Республики Казахстан  
 
 
   
84  
 
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
=
−
−
−
−
dt
dC
dt
dC
at
a
C
t
C
at
a
I
j
i
j
i
j
i
j
i
)
0
(
2
exp
4
)
0
(
)
(
2
exp
2
2
)
1
(
1
,  
(39) 
 
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
=
dt
dC
dt
dC
at
a
C
t
C
at
a
I
j
j
j
j
)
0
(
2
exp
4
)
0
(
)
(
2
exp
2
2
)
1
(
2
,     
  (40) 
 
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎥
⎦
⎤
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
=
dt
dC
dt
dC
at
a
C
t
C
at
a
I
i
i
i
i
)
0
(
2
exp
4
)
0
(
)
(
2
exp
2
2
)
1
(
3
.   
 (41) 
As a result we get 
 
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛−
+
+
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛−
−
−
⎥⎦
⎤
⎢⎣
⎡
−
−
⎥⎦
⎤
⎢⎣
⎡
−
=
+
∑
∑
∑
∑
−
−
−
−
−
−
−
dt
dC
a
C
C
dt
dC
a
C
C
at
a
dt
dC
a
C
C
dt
dC
a
C
C
at
a
C
C
dt
d
a
C
C
a
C
C
dt
d
a
C
C
a
dt
dC
a
dt
C
d
i
i
j
j
j
i
j
i
j
j
j
i
j
i
j
i
j
j
i
j
j
i
j
i
j
i
j
i
j
j
i
j
j
i
j
i
i
)
0
(
2
)
0
(
)
0
(
2
)
0
(
2
exp
2
)
0
(
2
)
0
(
)
0
(
2
)
0
(
2
exp
1
)
(
1
4
)
(
1
2
2
,
1
,
2
,
1
,
2
2
η
η
η
η
  
(42) 
Let’s consider now the general case.  
The evolution equation reads   
 
4
3
2
)
(
,
)
(
,
,
2
1
1
)
(
,
)
(
,
,
)
)
(
exp(
)
)
(
exp(
2
1
I
I
t
g
g
I
I
t
g
g
dt
dC
j
j
i
i
j
i
j
i
j
i
j
i
j
j
j
i
j
j
i
j
i
∑
∑
+
−
−
+
−
=
−
−
−
−
η
η
,    
 (43) 
Here
i
n
m
i
n
m
r
a
g
2
,
)
(
,
=
;
j
n
m
j
n
m
r
a
g
2
,
)
(
,
=
; 
ds
s
C
s
g
I
j
i
t
j
i
j
i
j
)
(
)
exp(
0
)
(
,
1
−
−
−
∫
=
;
ds
s
C
s
g
I
j
t
j
j
i
j
)
(
)
exp(
0
)
(
,
2
∫
−
=
; 
ds
s
C
s
g
I
j
t
j
j
i
)
(
)
exp(
0
)
(
,
3
∫
=
; 
ds
s
C
s
g
I
i
t
i
j
i
)
(
)
exp(
0
)
(
,
4
∫
=
. 
By time-differentiating the evolution equation we obtain 
 
]}
)
exp(
)
)[exp(
)
(
exp(
)
)
(
exp(
)
(
{
]}
)
exp(
)
)[exp(
)
(
exp(
)
)
(
exp(
)
(
{
2
1
3
)
(
,
4
)
(
,
)
(
,
)
(
,
4
3
)
(
,
)
(
,
)
(
,
)
(
,
2
,
1
)
(
,
2
)
(
,
)
(
,
)
(
,
2
1
)
(
,
)
(
,
)
(
,
)
(
,
1
,
2
2
I
C
t
g
I
C
t
g
t
g
g
I
I
t
g
g
g
g
I
C
t
g
I
C
t
g
t
g
g
I
I
t
g
g
g
g
dt
C
d
i
i
j
i
j
j
j
i
j
j
i
i
j
i
j
j
i
i
j
i
j
j
i
i
j
i
j
i
j
j
j
i
j
j
i
j
i
j
i
j
j
i
j
i
j
j
j
i
j
j
i
j
i
j
j
j
i
j
j
i
j
i
j
j
j
i
j
j
i
j
i
+
+
−
+
+
+
−
+
−
−
−
+
+
−
+
+
+
−
+
−
=
∑
∑
−
−
−
−
−
−
−
−
−
−
−
−
−
−
η
η
   
(44) 
 
Unlike the first case we can’t now get rid of the products of integrals with the help of equation (34). 
That is why we are forced to resort to separate averaging of sums containing 
2
1
I
I
 and
4
3
I
I
 [10].  
There are not indisputable grounds for such procedure but we assume that (44) can be rewritten in the 
following form using coefficients 
i
A
 and 
i
B
 as functions of time
t
: 
 
.
)
)
(
exp(
)
)
(
exp(
2
1
4
3
2
)
(
,
)
(
,
,
2
1
1
)
(
,
)
(
,
,
2
2
Φ
+
+
−
+
+
+
−
−
=
∑
∑
−
−
−
−
I
I
t
g
g
B
I
I
t
g
g
A
dt
C
d
j
j
i
i
j
i
j
i
i
j
i
j
i
j
j
j
i
j
j
i
j
i
i
η
η
        
(45) 
Here  

ISSN 2224–5227                                                                                                                               
№ 3. 2016  
 
 
85 
 
].
)
exp(
)
[exp(
)
)
(
exp(
]
)
exp(
)
)[exp(
)
(
exp(
2
1
3
)
(
,
4
)
(
,
2
)
(
,
)
(
,
1
)
(
,
1
2
)
(
,
)
(
,
)
(
,
,
I
C
t
g
I
C
t
g
t
g
g
I
C
t
g
I
C
t
g
t
g
g
i
j
j
i
j
i
j
i
j
j
i
i
j
i
j
j
j
i
j
j
i
j
i
j
i
j
j
i
j
i
j
j
j
i
j
j
i
j
+
+
−
+
+
+
+
−
=
Φ
∑
∑
−
−
−
−
−
−
−
−
η
 (46) 
By repeated time-differentiating we get 
  
.
)
)
(
exp(
)
)
(
exp(
2
1
4
3
2
)
(
,
)
(
,
,
2
2
1
1
)
(
,
)
(
,
,
2
3
3
dt
d
I
I
t
g
g
dt
dB
B
I
I
t
g
g
dt
dA
A
dt
C
d
j
j
i
i
j
i
j
i
i
i
j
i
j
i
j
j
j
i
j
j
i
j
i
i
i
Φ
+
+
−
⎟
⎠
⎞
⎜
⎝
⎛
−
−
−
+
−
⎟
⎠
⎞
⎜
⎝
⎛
−
=
∑
∑
−
−
−
−
η
η
   
(47) 
 
The scheme of subsequent transformations is like that.  
1. From (43) and (45) we infer the expressions for sums containing 
2
1
I
I
 and
4
3
I
I
.   
2. Then we substitute these expressions to equation (47) and use asymptotic relations for integrals 
once again.  
By realizing this clear scheme we obtain the three-order ODE of rather unwieldy form, and it is no 
need to presenting this equation here. 
In any case, it is possible to conclude at once that account of an interference of non-simultaneous 
perturbations of 
i-mers concentration field may be important on close examination of aggregation 
processes. This shade was missed in our work [10].   
 

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