Connectedness percolation in the random sequential adsorption packings of elongated particles

Connectedness percolation phenomena in the two-dimensional packing of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption off-lattice models with preferential orientations of the particles along a given direction. The partial ordering was characterized by the order parameter $S$, with $S=0$ for completely disordered films (random orientation of particles) and $S=1$ for completely aligned particles along the horizontal direction $x$. The aspect ratio (length-to-width ratio) of the particles was varied within the range $\varepsilon \in [1;100]$. Analysis of connectivity was performed assuming a core-shell structure of the particles. The value of $S$ affected the structure of the packings, the formation of long-range connectivity, and the behavior of the electrical conductivity. The effects can be explained by taking accounting of the competition between the particles' orientational degrees of freedom and excluded volume effects. For aligned deposition, anisotropy in the electrical conductivity was observed with the values along the alignment direction ${\sigma }_x$ being larger than the values in the perpendicular direction ${\sigma }_y$. Anisotropy in the localization of the percolation threshold was also observed in finite-sized packings, but it disappeared in the limit of infinitely large systems.

Figure 1

Examples of RSA packings in the jamming state for discorectangles with aspect ratios $\varepsilon =2$ (a); $\varepsilon =5$ (b); and at different values of the order parameters: $S=0$ (random orientation), $S=0.5$ (partial orientation), and $S=1$ (complete alignment along the horizontal direction $x$).

Figure 2

Coverage concentration $\phi$ versus the deposition time $t$ for the RSA packing of random ($S=0$) and perfectly aligned ($S=1$) discorectangles with aspect ratio $\varepsilon =4$ at different values of $L/l$. Here, ${\phi }_{\text{j}}$ is the jamming coverage. Inset shows an enlarged portion of the $\phi \left(t\right)$ plot near the saturation concentration.

Figure 3

Approaches to description of the connectivity analysis (a) and calculation of electrical conductivity (b) of the RSA packing of discorectangles on a 2D substrate. A core-shell structure of the particles was assumed. Intersections of the particle cores were forbidden. For the connectivity analysis, each particle was assumed to be covered by a soft (penetrable) shell of thickness $\delta d$. To calculate the electrical conductivity $\sigma$, a discretization approach with a supporting mesh was used. The mesh cells with centers located at the cores, shells, or pores parts were assumed to have electrical conductivity of ${\sigma }_{\text{c}}, {\sigma }_{\text{s}}$, and ${\sigma }_{\text{m}}$, respectively. (c) For larger values of the aspect ratio (slender-rod limit), discorectangles were treated as zero-width rods with the electrical conductivity ${\sigma }_{\text{c}}$. (d) Example of a transformation of slender rods into a RRN.

Figure 4

Scaling dependencies of the critical shell thickness ${\delta }_{\text{c}}$ at different values of particle coverage ${\phi }_c$ (a) and of the critical particle coverage ${\phi }_{\text{c}}$ at different fixed values of shell thickness $\delta$ (b). The data are presented for an aspect ratio of $\varepsilon =4$ for completely disordered ($S=0$, dashed lines) and completely aligned ($S=1$, solid lines) packings. For $S=0$ the data along the $x$ and $y$ directions almost coincide. Here, $L(=16l,32l,64l,128l)$ is the size of the system. $\nu =\frac{4}{3}$ is the 2D correlation length percolation exponent [64].

Figure 5

Critical shell thickness ${\delta }_{\text{c}}$ versus the aspect ratio $\varepsilon$ at different coverages $\phi$ for completely disordered $S=0$ (a) and completely aligned $S=1$ (b) packings.

Figure 6

Critical coverage ${\phi }_{\text{c}}$ versus the aspect ratio $\varepsilon$ at different shell thickness $\delta$ for completely disordered $S=0$ (a) and completely aligned $S=1$ (b) packings.

Figure 7

Examples of intrinsic conductivities $\left[\sigma \right]$ versus the order parameter $S$. The data are presented along the $x$ and $y$ directions for discorectangles with aspect ratios $\varepsilon =2,4,10$ (a). Dependencies of the parameters ${\left[\sigma \right]}_0, \kappa$ [see Eq. (5)] versus $\varepsilon$ (b).

Figure 8

Normalized intrinsic conductivity $\left[\sigma \right]/{\left[\sigma \right]}_{\infty }$ in the $x$ and $y$ directions versus the inverse mesh size $1/m$ at different aspect ratios $\varepsilon =4,20$, and $S=1$. Here, the value of ${\left[\sigma \right]}_{\infty }$ corresponds to the intrinsic conductivity in the limit of $m\to \infty$.

Figure 9

Electrical conductivity $\sigma$ versus the difference $d\phi =\left|\phi -{\phi }_{\sigma }\right|$ for RSA packings of disks ($\varepsilon =1$) at different shell thicknesses $\delta$. Here, the value of ${\phi }_{\sigma }$ was identified from the concentration at the percolation jump for each independent run, with calculations being carried out using a mesh size of $m=1024$. Dashed lines corresponds to the classical exponents $s=t\approx \frac{4}{3}$ [64].

Figure 10

Electrical conductivity $\sigma$ versus the difference $\left|\phi -{\phi }_{\sigma }\right|$ for RSA packings of discorectangles with different values of aspect ratios $\varepsilon$ for a fixed shell thickness of $\delta =0.2$ for completely disordered $S=0$ (a) and completely aligned $S=1$ (b) packings. Here, the value of ${\phi }_{\sigma }$ was identified from the concentration at the percolation jump for each independent run, with the calculations being performed using a mesh size of $m=1024$. Dashed lines correspond to the classical exponents $s=t\approx \frac{4}{3}$ [64].

Figure 11

Electrical conductivity $\sigma$ versus the difference $|\phi -{\phi }_{\sigma }|$. The data are presented for discorectangles with an aspect ratio $\varepsilon =10$ and a shell thickness of $\delta =0.3$, for completely aligned $S=1$ RSA packings. Here, the value of ${\phi }_{\sigma }$ was identified from the concentration at the percolation jump for each independent run, with the calculations being performed using mesh sizes of $m=1024$ and 2048. Moreover, above the percolation threshold, the electrical conductivity obtained within the t model is also presented. Dashed lines correspond to the classical exponents $s=t\approx \frac{4}{3}$ [64].

Figure 12

Electrical conductivity $\sigma$ versus the difference $|\phi -{\phi }_{\sigma }|$. The data are presented for discorectangles with the aspect ratios $\varepsilon =50,100$ and a shell thickness of $\delta =0.8$ for completely disordered $S=0$ and completely aligned $S=1$ RSA packings. Here, the value of ${\phi }_{\sigma }$ was identified from the concentration at the percolation jump for each independent run. Dashed lines correspond to the exponents $t\approx 2.3$ and $t\approx \frac{4}{3}$ [64].

Figure 13

Representation of the mesh square lattice with deposited discorectangle. The centers of the mesh cells were located at the cores, shells, or pores parts. Each cell was associated with a set of four resistors. The electrical conductivities of the whole bonds between two similar sites were ${\sigma }_{\text{c}}, {\sigma }_{\text{s}}$, and ${\sigma }_{\text{m}}$. For bonds located between different sites, there are only three possible combinations of the electrical conductivities of the entire bonds between core and shell sites ${\sigma }_{\text{cs}}$, pore and shell sites ${\sigma }_{\text{ms}}$, and core and pore sites ${\sigma }_{\text{cm}}$.

Figure 14

Explanation of the notation used. $(x_1,y_1)\in {\mathbb{R}}^2, {\alpha }_1\in [-\pi /2,\pi /2), (x_2,y_2)\in {\mathbb{R}}^2, {\alpha }_2\in [-\pi /2,\pi /2) [d=2r, l=2(r+a)]$.