Numerical study of cluster formation in binary charged colloids

Cluster formation of oppositely charged colloidal particles is studied numerically. A simple Brownian dynamics method with a screened-Coulomb (Yukawa) potential is employed for numerical simulations. An equilibrium phase which consists of clusters and unassociated particles is obtained. It is shown that the equilibrium association number of clusters and their shapes are determined by charge numbers and charge ratio of the binary particles. The phase diagram of cluster formation for various charge numbers and their ratios is obtained. A simple relation between the association number and the charge ratio is found. It is demonstrated that in the case of high charge ratio the cluster takes a multilayer structure which is highly symmetric. It is also pointed out that the cluster-particle interaction changes dynamically in the cluster formation process, which is involved in the selection of final cluster structure.

Figure 1

Snapshots of the system obtained by the numerical simulation at $t=0$ (random initial state, top) and $t=300$ (cluster state, bottom), for $N_A=100,N_B=2400,Q=11,Z_B=100,R_s=1$, and $\varphi =0.001$.

Figure 2

Plot of the potential energy per pair of particles as a function of time, for $N_A=100,N_B=2400,Q=11,Z_B=100,R_s=1$, and $\varphi =0.001$.

Figure 3

(a) Pair-correlation functions $g_{AA}\left(r\right),g_{AB}\left(r\right)$, and $g_{BB}\left(r\right)$, for $N_A=100,N_B=2400,Q=11,Z_B=100,R_s=1$, and $\varphi =0.001$. Thin vertical lines in the figure of $g_{BB}$ indicate the expected peak positions for the cluster having icosahedral symmetry. (b) Normalized distribution $f_n$ of association number $n_a$.

Figure 4

Pair-correlation functions of the single cluster system $g_{AB}\left(r\right)$, and $g_{BB}\left(r\right)$ for (a) $Z_B=10$ and (b) $Z_B=100$ with $Q=11$ fixed.

Figure 5

Phase diagram of cluster formation in $(Q,Z_B)$ parameter space for single cluster systems. Symbols indicate the parameters for which the well-characterized clusters are formed (closed squares) or not (open squares) in the numerical simulations. All of the clusters (closed squares) have the association number $n_a=Q+1$.

Figure 6

Plot of the association number $n_a$ as a function of $Q$ for $Z_B=100$. Snapshots of clusters obtained by the simulations for corresponding $Q$ values are also shown.

Figure 7

Pair-correlation functions of the single cluster system $g_{AB}\left(r\right)$ for (a) $Q=31$ and (b) $Q=43$ with $Z_B=100$ fixed. Snapshots of clusters obtained by the simulations are shown together.

Figure 8

Association number $n_a$ as a function of $Q$ for large $Q$. Each column of the histogram is divided into a few parts showing $n_a^{\left(i\right)}$ for $i=1,2,...$ from bottom to top of each column, where $n_a^{\left(i\right)}$ is the number of particles belonging to the $i\mathrm{th}$ layer of a cluster.

Figure 9

Bond orientational order parameters $q_l$ of layer particles are shown (a) for the first layer (closed squares) of the cluster with $Q=31$, (b) for the second layer (closed squares) of the cluster with $Q=31$, (c) for the first layer (closed squares) and the third layer (closed circles) of the cluster with $Q=43$, and (d) for the second layer (closed squares) of the cluster with $Q=43$. The histograms in these figures indicate $q_l$ for (a, c) regular icosahedron and (b, d) regular dodecahedron. The error bars attached to the symbols show the standard deviations.

Figure 10

Total energy $\mathrm{\Delta }U$ of the cluster-particle system as a function of cluster-particle distance $\left|\mathit{R}\right|$ for (a) neutral clusters $\left(n_a=Q\right)$ with $Z_B=40$, 60, and 80 (from top to bottom) and (b) overcharged clusters $\left(n_a=Q+1\right)$ with $Z_B=40$, 60, and 80 (from bottom to top).

Figure 11

Dipole moment $\left|\mathit{d}\right|$ divided by $Z_{B}e$ of the overcharged cluster $\left(n_a=Q+1\right)$ as a function of cluster-particle distance $\left|\mathit{R}\right|$ for (a) $Z_B=50$ and $Q=1$, 2, …, 5 from top to bottom, (b) $Q=3,n_a=4$, and $Z_B=20$, 40, …, 100. In (a) two curves for $Q=3$ and 4 mostly overlap, and in (b) all curves coincide.

Figure 12

Decay of unstable cluster with $Q=3,n_a=5$, and $Z_B=70$. Initial configuration of the cluster (top) and that after emission of a particle (bottom) are shown.

Figure 13

Stability diagram of overcharged clusters in parameter space $(Q,Z_B)$. Circles (crosses) indicate parameters for stable (unstable) clusters with $n_a=Q+1$ and squares (plus) for stable (unstable) clusters with $n_a=Q+2$. No stable clusters with $n_a=Q+2$ for $Q=1$ and $Z_B\le 120$ were obtained.