using off-lattice models both for 2D [103] and three dimensional (3D) [102]
systems. The interaction energy u ij of two non-overlapping clusters including
i and j particles was approximated by the power law:
u ij ¼ u 0
X i
k¼1
X j
i¼1
R
Àa
kl ;
(32)
where R kl is the distance between the kth particle in the cluster i and lth particle in
the cluster j.
Both the theory and simulation results gave the following estimates for fractal
dimensionality d f versus parameter a dependence:
d f ¼
d 0 ; a > 2d 0
d 0 ða þ 2Þ= 2 d 0 þ 1
ð
Þ
ð
Þ ; 1 8
> <
> :
(33)
where d 0 is the fractal dimensionality in the cluster–cluster aggregation model
without interactions. It equals d 0 ~ 1.6 and d 0 ~ 2.1 for 2D and 3D systems,
respectively.
Figure 5 presents d f versus a dependencies for 2D and 3D systems. The figure
shows that short range interactions (a > 2d 0 ) do not change the value of d f , but long
range (1 < a ; 2d 0 ) interactions lead to substantial changes in d f . Moreover, long
range interactions can have an important effect on the local structure of clusters
[102, 103]. The shape and size of the aggregates may be sensitively dependent on
the balance between attraction and repulsion interactions [98].
3.2.2 Eden-Like Model
The stochastic Eden-like model of aggregation of the charged particles in 2D
systems was developed [105, 106]. In this model, the particles overcome the
electrostatic repulsive barrier created by the aggregate and stick to it due to the
existence of short-range attractions.
By variation of two model parameters, the screening length, l, and the attractive
binding energy per particle, u 0 , formation of the aggregates with the following
morphologies was studied: linear or near-linear, linear with bending, worm-like,
dense-branching or dense-branching with a core, and compact Eden-like aggregates.
Figure 6 presents an example of a cluster with a charged core and external branches.
The regions of finite and infinite growth of clusters with different morphologies are
presented in the form of a diagram as l versus u 0 . The structure and fractal properties
of the ramified clusters were studied. It was found that the clusters did not reveal the
fractal properties at any values of l and u 0 and a sharp transition between linear
(d f ¼ 1) and dense-branching (d f ¼ 2) morphologies was observed.
74
N.I. Lebovka
systems. The interaction energy u ij of two non-overlapping clusters including
i and j particles was approximated by the power law:
u ij ¼ u 0
X i
k¼1
X j
i¼1
R
Àa
kl ;
(32)
where R kl is the distance between the kth particle in the cluster i and lth particle in
the cluster j.
Both the theory and simulation results gave the following estimates for fractal
dimensionality d f versus parameter a dependence:
d f ¼
d 0 ; a > 2d 0
d 0 ða þ 2Þ= 2 d 0 þ 1
ð
Þ
ð
Þ ; 1 8
> <
> :
(33)
where d 0 is the fractal dimensionality in the cluster–cluster aggregation model
without interactions. It equals d 0 ~ 1.6 and d 0 ~ 2.1 for 2D and 3D systems,
respectively.
Figure 5 presents d f versus a dependencies for 2D and 3D systems. The figure
shows that short range interactions (a > 2d 0 ) do not change the value of d f , but long
range (1 < a ; 2d 0 ) interactions lead to substantial changes in d f . Moreover, long
range interactions can have an important effect on the local structure of clusters
[102, 103]. The shape and size of the aggregates may be sensitively dependent on
the balance between attraction and repulsion interactions [98].
3.2.2 Eden-Like Model
The stochastic Eden-like model of aggregation of the charged particles in 2D
systems was developed [105, 106]. In this model, the particles overcome the
electrostatic repulsive barrier created by the aggregate and stick to it due to the
existence of short-range attractions.
By variation of two model parameters, the screening length, l, and the attractive
binding energy per particle, u 0 , formation of the aggregates with the following
morphologies was studied: linear or near-linear, linear with bending, worm-like,
dense-branching or dense-branching with a core, and compact Eden-like aggregates.
Figure 6 presents an example of a cluster with a charged core and external branches.
The regions of finite and infinite growth of clusters with different morphologies are
presented in the form of a diagram as l versus u 0 . The structure and fractal properties
of the ramified clusters were studied. It was found that the clusters did not reveal the
fractal properties at any values of l and u 0 and a sharp transition between linear
(d f ¼ 1) and dense-branching (d f ¼ 2) morphologies was observed.
74
N.I. Lebovka