Diffusion-driven self-assembly of rodlike particles: Monte Carlo simulation on a square lattice

The diffusion-driven self-assembly of rodlike particles was studied by means of Monte Carlo simulation. The rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size $L\times L$ up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, $L$, was varied from 128 to 2048, and periodic boundary conditions were applied along both $x$ and $y$ axes, while the length of the $k$-mers (determining the aspect ratio) was varied from 2 to 12. The $k$-mers oriented along the $x$ and $y$ directions ($k_x$-mers and $k_y$-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of $k$-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the $k$-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long $k$-mers. For long $k$-mers ($k\ge 6$), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state; the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation; and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of $k$-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.

Figure 1

Examples of patterns at different times, $t_{\text{MC}}$, from the videok6 in the Supplemental Material [50]. Here, $L=256, k=6$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state, $p_j\approx 0.769$ [45].

Figure 2

Fragment of a square lattice with two deposited 3-mers of different orientations. The conductivities of the bonds for (a) conventional and (b) insulating ends models are indicated.

Figure 3

Number of intraspecific, $n$, and interspecific, $n_{xy}$, contacts, the shift ratio, $R$, and electrical conductivity for conventional model, $\sigma$, versus the MC time $t_{\text{MC}}$. The data correspond to the same MC run as for the patterns presented in Fig. 1, $L=256, k=6$, with use of the initial $r$ configuration. The concentration of the $k$-mers corresponds to the jamming state, $p_j\approx 0.769$ [45]. Here, all vertical axes are linear.

Figure 4

Examples of patterns at $t_{\text{MC}}={10}^7$ for different values of length of the $k$-mers. Here, $L=256$ and the initial r configuration were used. The concentration of $k$-mers corresponds to the jamming state [45].

Figure 5

Relationship between the initial number (i.e., at $t_{\text{MC}}=1$) of intraspecific, $n^i$, and interspecific, $n_{xy}^i$, contacts versus the length of the $k$-mers. The normalized numbers of intraspecific contacts, $n_f^{*}$ versus $k$ at $t_{\text{MC}}={10}^7$ are also presented. Here, $L=256$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state [45].

Figure 6

Normalized number of (a) intraspecific, $n^{*}=n/n^i$, and (b) interspecific, $n_{xy}^{*}=n_{xy}/n_{xy}^i$, contacts versus the time $t_{\text{MC}}$ for different values of length of the $k$-mers. Here, $n^i$ and $n_{xy}^i$ are the corresponding initial values at $t_{\text{MC}}=1$, while $L=256$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state [45]. The values of (a) $n_f^{*}$ and (b) $t_t$ correspond to the final values at $t_{\text{MC}}={10}^7$ and to the transition period, respectively.

Figure 7

Shift ratio, $R$, versus the time $t_{\text{MC}}$ for different values of length of the $k$-mers. Here, $L=256$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state [45].

Figure 8

Electrical conductivity, $\sigma$, versus the time $t_{\text{MC}}$ for different values of length of the $k$-mers for (a) conventional and (b) insulating ends models. Here, $L=256$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state [45].

Figure 9

Electrical conductivity, $\sigma$, versus the length of $k$-mers for different values of $t_{\text{MC}}$ for conventional and insulating ends models. Here, $L=256$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state [45].

Figure 10

Examples of patterns at $t_{\text{MC}}={10}^7$ for $k=12$ and different sizes of the lattice $L=128–2048$. Here the initial $r$ configuration was used. The concentration of $k$-mers corresponds to the jamming state.

Figure 11

Examples of patterns at different times, $t_{\text{MC}}$, from the video videok12L2048 in the Supplemental Material [50]. Here, $L=2048, k=12$, and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state.

Figure 12

Normalized number of (a) interspecific contacts, $n_{xy}^{*}=n_{xy}/n_{xy}^i$, and (b) the shift ratio, $R$, versus the time $t_{\text{MC}}$ for $k=12$ and different sizes of the system, $L$. Here, $n_{xy}^i$ is the initial value at $t_{\text{MC}}=1$ and the initial $r$ configuration were used. The concentration of $k$-mers corresponds to the jamming state.

Figure 13

Examples of patterns at different times, $t_{\text{MC}}$, from the video videok12vd in the Supplemental Material [50]. Here, $L=256, k=12$, initial $v$ configuration (upper row) and $d$ configuration (bottom row) were used. Concentration of $k$-mers corresponds to the jamming state of $r$ configuration [45].

Figure 14

Number of (a) intraspecific contacts, $n$ (b) interspecific contacts, $n_{xy}$, (c) shift ratio, $R$, and (d) electrical conductivity, $\sigma$, versus time $t_{\text{MC}}$ for $k=12$. Here, $L=256$ and different initial $r, v$, and $d$ configurations were used. The concentration of $k$-mers corresponds to the jamming state of the $r$ configuration [45].

Figure 15

Examples of patterns at $t_{\text{MC}}={10}^7$ for different values of length of the $k$-mers. Here, $L=256$ and the initial $d$ configuration were used. The concentration of $k$-mers corresponds to the jamming state for the initial $r$ configuration [45].

Figure 16

Number of interspecific, $n_{xy}$, contacts versus time $t_{\text{MC}}$ for different values of $k$. Here, $L=256$, and the initial $d$ configuration were used. The concentration of $k$-mers corresponds to the jamming state for the initial $r$ configuration [45]. Inset shows the half-time $t_{1/2}$ required to attain the level $0.5n_{xy}^s$.

Figure 17

Number of interspecific, $n_{xy}$, contacts versus time $t_{\text{MC}}$ for initial $r, v$, and $d$ configurations, $L=2048$, for (a) $k=6$ and (b) $k=12$. The concentration of $k$-mers corresponds to the jamming state for the initial $r$ configuration [45]. The corresponding patterns at $t_{\text{MC}}={10}^7$ are also shown.