Cluster-cluster aggregation of Ising dipolar particles under thermal noise

The cluster-cluster aggregation processes of Ising dipolar particles under thermal noise are investigated in the dilute condition. As the temperature increases, changes in the typical structures of clusters are observed from chainlike $\left(D\simeq 1\right)$ to crystalline $\left(D\simeq 2\right)$ through fractal structures $\left(D\simeq 1.45\right)$, where $D$ is the fractal dimension. By calculating the bending energy of the chainlike structure, it is found that the transition temperature is associated with the energy gap between the chainlike and crystalline configurations. The aggregation dynamics changes from being dominated by attraction to diffusion involving changes in the dynamic exponent $z=0.2$ to 0.5. In the region of temperature where the fractal clusters grow, different growth rates are observed between charged and neutral clusters. Using the Smoluchowski equation with a twofold kernel, this hetero-aggregation process is found to result from two types of dynamics: the diffusive motion of neutral clusters and the weak attractive motion between charged clusters. The fact that changes in structures and dynamics take place at the same time suggests that transitions in the structure of clusters involve marked changes in the dynamics of the aggregation processes.

Figure 1

(Color online) Typical snapshots of cluster-cluster aggregation in IDP systems at (a) $T=0.001$ and (b) $T=0.02$. Small parts of the system are shown for the clarity, which size is $250R\times 250R$ for each. One can see the chainlike structure for the system at low temperatures, while the particles form a more compact structure at high temperatures.

Figure 2

(Color online) The gyration radius $R_{\text{g}}$ versus cluster size $s$ for the system with $\varphi =0.01$. As the temperature increases, three types of structures appear. The dotted lines denote the relationship $s\propto R_{\text{g}}^D$ with $D=1$, $D=1.45$, and $D=2$. Decimal logarithms are taken for both axes.

Figure 3

(Color online) The temperature dependence of the crossover size $s_{\text{c}}$. The cross over size from $D\simeq 2.0$ to $D\simeq 1.45$ was estimated. The horizontal axis is $T_{\text{crystal}}-T$, with $T_{\text{crystal}}=0.04$. Decimal logarithms are taken for both axes. The profile of $s_{\text{c}}$ is well fitted by the dotted line denoting the relationship $s_{\text{c}}\propto {\left(T_{\text{crystal}}-T\right)}^{-3.0}$.

Figure 4

(Color online) The time evolution of average cluster size for (a) $\varphi =0.010$ and (b) $\varphi =0.020$. Decimal logarithms are taken for both axes. The asymptotic behavior of the cluster size obeys a power law, as fitted by the dotted lines, and the values of the dynamic exponent $z$ are obtained from their slope.

Figure 5

(Color online) The dynamic exponent $z$ for each temperature and density. The dotted line represents the value of $z$ at $T=0$ in [12]. The value of $z$ at finite temperatures converges to the value of about 0.45 to 0.5 in the dilute limit.

Figure 6

(Color online) The time evolution of the relative cluster number $N_{\text{c}}^{\text{rel}}$ for $\varphi =0.010$. Decimal logarithms are taken for both axes. The curves at $T\le 0.03$ tend to vanish, while that at $T=0.04$ converges to a constant value.

Figure 7

(Color online) A schematic diagram of the fractal structures of IDP corresponding to Fig. 2. Chainlike clusters $\left(D=1.0\right)$ appear at low temperatures, while a crossover phenomenon from $D=2$ to $D=1.45$ is observed at higher temperatures.

Figure 8

(Color online) The energy of a four-particle chain as a function of bending angle $\theta$. There is a small energy barrier to the formation of the tetragonal configuration.

Figure 9

(Color online) The time evolution of the relative number of neutral to charged clusters $N_{\text{c}}^{\text{rel}}$ against the average cluster size $\overline{s}$. Plots represent the simulation result and curves represent analytical values of the hetero-DLCA Smoluchowski equation for $\left({\gamma }_0,{\gamma }_1\right)=\left(-1,-1\right)$ and $\left({\gamma }_0,{\gamma }_1\right)=\left(-1,-1.44\right)$.