CHEMICAL ENGINEERINGTRANSACTIONS 
 

VOL. 57, 2017 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Sauro Pierucci, Jiří JaromírKlemeš, Laura Piazza, SerafimBakalis 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN978-88-95608- 48-8; ISSN 2283-9216 

Modelling  the  Formation  of  Particulate  Composition  in 
Aggregating  Disperse  Media  with  Account  of  the  Clusters 

Age  Aspect 
Aikerim Kazenova, Arnold Brener*, Vladimir Golubev, Gulmira Kenzhalieva   
State University of South Kazakhstan, Tauke Khan, 5, Shymkent, Kazakhstan  
amb_52@mail.ru  

The work deals with the simplified approach to develop modelling of the integrated effect of geometric and 
thermodynamic factors on the stability of the dispersion fractional composition. The influence of the age factor 
is described in the form of the ultimate reduction of the aggregation activity of colliding clusters when 
increasing their orders. Some experimental data evidencing  the possibility of such simplified approach are 
also resulted. The novelty of the submitted models is determined by the fact that the kinetics of internal 
transformations of clusters structure is included in the description of the cluster aggregation process as whole.  

1. Introduction 

The contemporary view of the mechanism of formatting the new (insoluble) phase in the system lies in the fact 
that the intensive transition of the component to insoluble state is thermodynamically similar to the formation of 
crystals nuclei in supersaturated solutions, although solid particles may be only partially crystallized (Kaschiev, 
2000). An intensive transition to the insoluble state starts when the critical concentration of the component 
forming insoluble phase is reached (De Groot and Mazur, 2013 ). New phase of nuclei may have different 
sizes and masses, but the probabilities of arising the nuclei of different masses are different (Leyvraz, 2003). 
Therefore the particles of the smallest possible size of the insoluble phase, corresponding to the  negative 
work of nucleate formation, can be regarded as  1-mers  with the most probable size (Sonntag et al., 1987). 
Taking this hypothesis, it is reasonable to assume that the equilibrium distribution of the dispersed phase in 
the volume of the medium, i.e. the formation of the fractional composition, is determined mainly by two factors. 
The first factor is the rate of coagulation (Sonntag et al., 1987) and the second one is the fragmentation of the 
solid  clusters phase (Zahnov et al., 2011).   
The  rate  of coagulation depends on the sign and magnitude of the total interaction energy, which, in 
accordance with the theory of Deryagin-Verwey-Overbeek (DFO) is determined by the superposition of ionic-
electrostatic repulsion forces and the van der Waals-London attraction forces (Voloschuk and Sedunov, 
1975).The relationship between the components of the total energy of clusters interaction is determined by 
two geometric factors: the effective size of the cluster, which depends on its order, and the characteristic 
distance between the clusters (Aldous, 1999). At the same time, the cluster has a complex structure, and its 
fractal dimension is also dependent on cluster order (Conoglio and Stanley, 1984). The aggregation activity of 
the cluster  depends on the number of active centers on its surface (Ernst, 1986). When the process of 
capturing the new particles by the cluster surface slows down significantly, then after a while the inner 
structure of the cluster can be changed  under the influence of internal transformations. Thus the cluster  
acquires a stable structure which characterizes by an absolute or local minimum of the free energy of its 
surface. It is correct unless, of course, the system has no interactions that can overcome the energy barrier of 
the local extreme of the surface energy (Mirlin, 2000). Thus, a quasi-stable equilibrium distribution of the 
dispersed phase is achieved at a certain aggregation activity of its surface, which in turn depends on the 
cluster order and the age (Brener and Dil'man, 2016). 
The purpose of this work is to offer a simplified approach to modeling the integrated effect of geometric and 
thermodynamic factors on the stability of the dispersion fractional composition. The influence of the age factor 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1757140

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Kazenova A., Brener A., Golubev V., Kenzhalieva G., 2017, Modelling the formation of particulate composition in 
aggregating disperse media with account of the clusters age aspect, Chemical Engineering Transactions, 57, 835-840  
DOI: 10.3303/CET1757140 

835



is described in the form of the ultimate reduction of the aggregation activity of colliding clusters when 
increasing their orders.  Some of experimental results confirming the prospect of such a simplified approach 
are presented here in this work.  

2. Theoretical considerations and heuristic models  

In terms of formal kinetic models the two main mechanisms controlling stabilization of the dispersion 
composition can be distinguished. Firstly, it is the balance between the rates of disaggregation and 
aggregation and, secondly, it is reducing of the aggregation activity with increasing the order of clusters and 
with growth of the cluster age (Dairabay et al., 2016).  
In some cases,  the first mechanism is, apparently, dominant (Alexander and Orbach, 1982). In other cases, 
such as phase transitions and aggregation processes in supersaturated systems, a second mechanism of 
stabilizing the fractional composition becomes prevalent after the transition of the process to the diffusion-
limited stage (Blackman and Marshall, 1994). 
It can be also considered systems with comparable kinetics of both mechanisms. If due to certain 
prerequisites a highly concentrated dispersion of the monomer-embryos arises, the fast many-body 
aggregation (Brener, 2014) can begin as the mobility of the monomers is large enough, and the diffusion 
resistance is small (Doering and ben-Avraham, 1988). 
Then, when after a certain transition period the clusters having high orders arise, and the diffusion resistance 
increases due to the decrease in the concentration of particles in the dispersion, the process proceeds to the 
diffusion-limited type (Happel, Brenner, 2012), and at the same time, the large clusters start structuring with 
reducing their mobility and aggregation activity (Brener and Dil'man, 2016). Apparently, the frequency of 
many-body collisions on this period reduced (Brener, 2014).    
Thus, the process of adding new particles to a  high-order cluster can be significantly slowed down, and after 
a while it can acquire a pseudo-stable structure of the cluster surface (Barabasi and Vicsek, 1991). Formation 
of these stable (screened) structures can be described as a process of reducing  local fluctuations (Coniglio 
and Stanley, 1984). Each reduction means a transition into a new state with its own free energy level of the 
cluster surface (Mirlin, 2000).  
The characteristic times of local relaxation processes is significantly less than the global stabilization time 
(Brener, Dil'man, 2016). Moreover, it is  most likely  the transition to a stable global state for higher order 
clusters will move through a series of local stabilization, while the total stabilization time can exceed the 
characteristic time for the process.  
Based on these considerations, let us consider the process of stabilizing  the fractional composition by the 
way of  the second of  above mentioned mechanisms using the modified Smoluchowski model of binary 
coagulation (Brener and Dil'man, 2016). 

21

1 0 0

211,1
1 )()( dtdttCtC

dt

dC

J

t t

JJ




  (1) 

21

1 0 0

21,2

1

1 0 0

121, )()()()(
2

1
dtdttCtCdtdttCtC

dt

dC

J

t t

JIJI

I

J

t t

JIJJIJ
I










   (2) 

The above heuristic arguments can be interpreted as reducing the probability of mutual particles capture with 
an increase in the total order and  ages of the colliding clusters. In terms of the formal kinetic model it can be 
described as follows  




ji

ji 0lim ,  (3) 

At the same time the requirement of the spatial homogeneity of the distribution of the clusters is consistent 
with the decrease of elements of the aggregation matrix with increasing orders of the colliding particles. This 
approach offers the simplified model instead of an inner kinetic equation for describing the transformation of 
clusters structure with their ages (Brener, Dil'man, 2016). The proposed approach allows passing from the 
infinite chain of coagulation equations to a closed finite system of equations that can be effectively 
investigated  with the help of  qualitative methods and subjected to computational experiments.  
On the other hand, regarding to the small orders of clusters it can play a main role the increase in the effective 
capture cross section with increasing the particle radius, and decrease in the mobility of the particles with 
increasing their order (Duprat and Stone, 2016). 

836



Here we propose to use the model elements of the matrix in the form of an even function of the following  
parameter (Brener et al., 2009). 

ji

ji




  (4) 

The expansion of this parameter to the Taylor series  reads  

 44
2

20,  aaaji  (5) 

Here 0sa  when s . 

It is obvious that under condition consta 0  expression (5) contradicts to hypothesis (3). To resolve this 

contradiction it is possible to set the coefficient 0a  in the form of  decreasing function of  ji  , such as: 

 ji
k

a


0  (6) 

Confining ourselves to two terms of the expansion in a Taylor series in the expression (5), the following  
approximate model of the aggregation matrix can be obtained: 

 

2

2, 
















ji

ji
a

ji

k
ji 

 (7) 

The derivative of the matrix elements by an order of one of the components is: 

   2
21

,
4

ji

j

ji

ji
a

ji

k

i

ji

























 (8) 

Hence, for the reason of the symmetry of the aggregation process for each component (Brener et al., 2009), it 
is sufficient to consider the stabilization of the disperse composition for the aggregation of globules of the 
same orders, i.e. at ji  . 
Since the intensity of the process according to our model and in accordance with the conclusions of the  DFO 
theory (Voloschuk and Sedunov, 1975) decreases with increasing the orders of the particles, the consideration 
can be limited by two terms on the right hand in the usual Smoluchowski equation for aggregating  the two  i-
mers of great orders (Wattis, 2006): 

0
2

1
22,

2
,

2  iiiiiii
i CCC

dt

dC
 (9) 

From the condition of the process stationarity we obtain the ratio between concentrations of 2i -mers  and i-
mers, what value will be called the instability factor: 

 
 



ziC

C
W

ii

ii

i

i









1

23

2

~

2,

,2  (10) 

Here  


1

2

9
3 










k

a
z  (11) 

Figure 1 depicts some results of the numerical experiments for the instability factor as a function of the orders 
of collided clusters.  Figure 1 shows the feature of the behaviour of curves, namely, all the curves intersect in 
a small neighbourhood of the point 2zi  .  
This allows counting on being able to relate the two parameters  controlling the stability of the disperse 
systems  with two parameters that  control the scale of the intermolecular interaction energy in the theory of 
van der Waals-Hamaker (Dairabay, 2016 b): 

837



2
12 h

A
Vm





 , (12) 

where mV is the energy of interaction,
  

A
* is the  van der Waals-Hamaker constant, h is the distance between 

particles (Sonntag et al., 1987).   

 

1- 2,0 ; 2- 5,0 ; 3- 1 ; 4- 2,1 ; 5- 2  

Figure 1: The instability factor W
~

 as a function of the parameter   (look formula (6)) 

 
Thus, the heuristic model  of stabilization of disperse systems in the  aggregation  process  provided in this 
section,  acquires some physical justification, since it is possible to calculate the control parameters of the 
model for the known physical media.   

3. Some experimental data 

In this section, the brief review of  some experimental data obtained by research group  led by Professor V. 
Golubev in the State University of South Kazakhstan has been submitted. These data are  described in detail 
in the PhD  thesis by Dairabai D.D. (Dairabay, 2016 b), and here we will try to give them an interpretation in 
the spirit of the considerations outlined in the previous section.  
The main impetus for this experiment was to test the conclusion that under the high initial density of primary 
nuclei dispersion  the conditions of the aggregation process for creating the dispersions with high level of the 
uniformity of the cluster distribution by sizes and orders can be chosen. 
Experimental studies of the nucleation and subsequent coagulation of the solid particles of SiO2 which are 
arisen when  desublimation from the gas phase mixture (NH4)2SiF2 + NH3 + H2O  were carried out  on the 
special laboratory-scale plant (Dairabay, 2016 b). The above gas mixture has been given off  by the help of 
heat treatment of Karatau phosphorite by the ammonium fluoride. The dispersion characteristics were studied 
with the help of electron microscopy. The scanning electron microscope JSM-6490LV (SEM) was used. 
Figure 2 shows some representative photographs of the dispersions obtained through experiments. 
 

838



 
А 

 
В 

 
С 

 
D 

Figure 2: Photomicrographs of the cluster dispersions in the processes of silica vapour desublimation. 

Figure 2A is a photomicrograph of a dispersion which arose under the relative supersaturation of 4.5 as a 
result of a sharp discharge of the vapour pressure. The average cluster size is  380 µm (micrometers), the 
standard deviation is  of the order of 25  µm. 
Figure 2B is a photomicrograph of a dispersion with the initial supersaturation of 1.4 which arose while the 
sudden depressurization. It is clearly seen that the degree of the homogeneity of  the dispersion is significantly 
lower than in the first case (4A). The average size of the clusters is less than 160 µm, at the same time the 
standard deviation exceeds 90 µm. 
Figure 2C corresponds to the first case, i.e. the relative supersaturation is 4.5, but the picture was taken in the 
zoom of  25,000, which allows considering the single nucleates. It is clearly seen  a high proportion of almost 
spherical particles that arise during the intense many-particle collisions (Brener, 2014). Particles of more 
complex shapes correspond to clusters arisen for a more long time by the way of  mostly binary collisions. 
Figure 2D corresponds to supersaturation of 1.4, but the picture was taken with the zoom in 5000. It is evident 
that  even at the micro level the degree of heterogeneity of the dispersion  is large. 
Studies clearly indicate that provided a large initial supersaturation when the rapid formation of a large number 
of nuclei (monomers) of the disperse phase per unit volume of the apparatus is observed, the contribution of 
many-body collisions is also large (Brener, 2014). This is clearly seen in the figures  taken at high 
magnification. Namely, large clusters of the particles are composed of many small nuclei, moreover the large 
globular clusters grow quite round, which is only possible with a simultaneous merger of many small particles 
across the surface of the larger clusters. Then, after a sharp decrease in the vapour concentration the 
aggregation process becomes limited by diffusion resistance in the gas phase (Happel and Brenner, 2012). 
Because of this the intensity of the aggregation process drops sharply, and, in terms of our model,  this leads 
to a sharp decrease in the values of coagulation kernels. As a result the dispersion with sufficiently 
homogeneous fractional composition can be obtained. Under the low initial supersaturation the aggregation 
process is "blurred" over time, since its rate is determined mainly by binary collisions. Such regime of the 

839



process is accompanied by competition between diffusion  rate and the aggregation kinetics (Kaschiev, 2000). 
Therefore, the fractional composition of the dispersion becomes very heterogeneous. Of course, we must bear 
in mind that in the described experiments a solid phase formation occurred on the solid substrate. The kinetics 
of this process has certain differences from the kinetics of the desublimation in the volume of an apparatus 
(Sonntag et al., 2012).  Nevertheless, it can be supposed that the proposed interpretation is still valid as these 
differences are  relevant only  to the short stage of initial nucleation (Voloschuk and Sedunov, 1975). 

4. Conclusions 

The submitted simplified approach for accounting the influence of the clusters orders and their ages on the 
aggregation activity allows offering the control parameters which are likely to be useful for defining optimal 
technological regimes of producing the dispersions with needed characteristics of fractional composition. The 
results of the study demonstrate a possibility to get the satisfactory interpretation of the experimentally 
observed regularities of nucleation-coagulation processes with the help of the submitted model approach. 
However, this problem should be investigated more deeply. 

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