small fractal dimension, d f ~ 1, in polyelectrolytes in the limit of high electrostatic
interactions (l B ~ 0.1 nm) [94]. Formation of linear structures was observed also in
Monte Carlo simulations of a short polyelectrolyte with various numbers of beads,
N ¼ 8–80 [95]. The value of d f was noticeably increasing as Bjerrum length l B
decreased.
The effect of the range of interactions on the structural and kinetic properties of a
computer-simulated two-dimensional aggregating colloidal system was studied
using the repulsive Yukawa potential [96] (Eq. 9). The increase in l D or u 0 provokes
arrangement of aggregates into linear structures. The repulsive interactions also
have a strong influence on the kinetic behavior of the coagulation process. The
structure of small clusters composed of up to 80 particles interacting simultaneously via attractive and repulsive forces was recently simulated [97]. A shortrange attraction was simulated by generalization of the Lennard–Jones potential
and a long-range repulsion was simulated by the screened electrostatic Yukawa
potential. The competition between attraction and repulsion resulted in formation of
stable clusters, and the ground-state clusters were preferentially growing almost in
one dimension. The extensive numerical simulations were done for suspended
charged colloidal particles at a screening length comparable to the particle radius
[98]. It was shown that at low temperature, particles organize into quasi onedimensional aggregates connecting via branching mechanism into a percolating
gel structure. Note that experimental data indicate a more elongated and open
morphology of the fractal-like aerosol agglomerates at larger charge [99].
3.2.1 DLA-Like Model
The impact of long-range interactions on cluster morphology has been intensively
studied for different variants of the DLA model [100–103]. It was shown using a
quasi-deterministic two-dimensional (2D) particle–cluster growth model with
attractive cluster–particle forces proportional to R
Àa that the growing cluster was
dendritic. Moreover, strong reduction of the fractal dimension, d f , with increase in
a was observed [100]. The 2D DLA model with the power-like potential, uðRÞ ¼
u 0 R
Àa , was also studied [101, 104]. Here, u 0 and a are parameters. Note that this
model is equivalent to the ordinary DLA in the limits of short range interactions,
a ! 1, or u 0 ¼ 0. In this problem, the aggregates were grown on a triangular
lattice and the effects of anisotropy were important.
The detailed structure of clusters was strongly dependent on a, or u 0 , e.g., in the
case of attractive interactions at large u 0
j jand long-range interaction (i.e., for small a),
the clusters grew with stable tips and were of dendritic shape. The observed effects
reflected the anisotropy induced by the underlying lattice that stabilized the tips of the
growing arms. In the limit of small u 0
j j and large a, the transition from dendritic to
tip-split aggregates was observed. At fixed value of u 0 , the estimated fractal dimension d f increased with increase in a, which reflected tip destabilization [101].
The similar cluster–cluster variants of DLA aggregation with either attractive or
repulsive interactions were also studied [102, 103]. These simulations were done
Aggregation of Charged Colloidal Particles
73
interactions (l B ~ 0.1 nm) [94]. Formation of linear structures was observed also in
Monte Carlo simulations of a short polyelectrolyte with various numbers of beads,
N ¼ 8–80 [95]. The value of d f was noticeably increasing as Bjerrum length l B
decreased.
The effect of the range of interactions on the structural and kinetic properties of a
computer-simulated two-dimensional aggregating colloidal system was studied
using the repulsive Yukawa potential [96] (Eq. 9). The increase in l D or u 0 provokes
arrangement of aggregates into linear structures. The repulsive interactions also
have a strong influence on the kinetic behavior of the coagulation process. The
structure of small clusters composed of up to 80 particles interacting simultaneously via attractive and repulsive forces was recently simulated [97]. A shortrange attraction was simulated by generalization of the Lennard–Jones potential
and a long-range repulsion was simulated by the screened electrostatic Yukawa
potential. The competition between attraction and repulsion resulted in formation of
stable clusters, and the ground-state clusters were preferentially growing almost in
one dimension. The extensive numerical simulations were done for suspended
charged colloidal particles at a screening length comparable to the particle radius
[98]. It was shown that at low temperature, particles organize into quasi onedimensional aggregates connecting via branching mechanism into a percolating
gel structure. Note that experimental data indicate a more elongated and open
morphology of the fractal-like aerosol agglomerates at larger charge [99].
3.2.1 DLA-Like Model
The impact of long-range interactions on cluster morphology has been intensively
studied for different variants of the DLA model [100–103]. It was shown using a
quasi-deterministic two-dimensional (2D) particle–cluster growth model with
attractive cluster–particle forces proportional to R
Àa that the growing cluster was
dendritic. Moreover, strong reduction of the fractal dimension, d f , with increase in
a was observed [100]. The 2D DLA model with the power-like potential, uðRÞ ¼
u 0 R
Àa , was also studied [101, 104]. Here, u 0 and a are parameters. Note that this
model is equivalent to the ordinary DLA in the limits of short range interactions,
a ! 1, or u 0 ¼ 0. In this problem, the aggregates were grown on a triangular
lattice and the effects of anisotropy were important.
The detailed structure of clusters was strongly dependent on a, or u 0 , e.g., in the
case of attractive interactions at large u 0
j jand long-range interaction (i.e., for small a),
the clusters grew with stable tips and were of dendritic shape. The observed effects
reflected the anisotropy induced by the underlying lattice that stabilized the tips of the
growing arms. In the limit of small u 0
j j and large a, the transition from dendritic to
tip-split aggregates was observed. At fixed value of u 0 , the estimated fractal dimension d f increased with increase in a, which reflected tip destabilization [101].
The similar cluster–cluster variants of DLA aggregation with either attractive or
repulsive interactions were also studied [102, 103]. These simulations were done
Aggregation of Charged Colloidal Particles
73