Particle morphology effects in random sequential adsorption

The properties of the random sequential adsorption of objects of various shapes on a two-dimensional triangular lattice are studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps, whereby the size of the objects is gradually increased by wrapping the walks in several different ways. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density ${\theta }_{\text{J}}$ and on the temporal evolution of the coverage fraction $\theta \left(t\right)$. Our results suggest that the order of symmetry axis of a shape exerts a decisive influence on adsorption kinetics near the jamming limit ${\theta }_{\text{J}}$. The decay of probability for the insertion of a new particle onto a lattice is described in a broad range of the coverage $\theta$ by the product between the linear and the stretched exponential function for all examined objects. The corresponding fitting parameters are discussed within the context of the shape descriptors, such as rotational symmetry and the shape factor (parameter of nonsphericity) of the objects. Predictions following from our calculations suggest that the proposed fitting function for the insertion probability is consistent with the exponential approach of the coverage fraction $\theta \left(t\right)$ to the jamming limit ${\theta }_{\text{J}}$.

Figure 1

Plots of $ln({\theta }_{\text{J}}-\theta \left(t\right))$ vs $t$ for wrapping triangles $T_3,T_4,T_6$, rhombuses $R_3,R_6$, and hexagons $H_4,H_7,H_{19}$ from Tables 1, 2, 3. The curves correspond to various values of the order of symmetry axis of the shape, $n_s$, as indicated in the legend. Additionally, the slanted straight lines with the slope $-1/\sigma =-1,-1/2,-1/3,-1/6$ are shown, indicating the late-time RSA behavior and are guides for the eye.

Figure 2

Jamming coverages ${\theta }_{\text{J}}$ for all wrapping (a) triangles $T_j$, (b) rhombuses $R_j$, and (c) hexagons $H_j$, from Tables 1, 2, 3. Here $j=2,...,30$ denotes the number of sites covered by an object. Numerical values of the symmetry order $n_s$ of the shapes are given in the square brackets above the corresponding plot symbols.

Figure 3

The coefficients $c_j$ [Eq. (3)] as a function (a) of the number $n$ of deposited objects, and (b) of the coverage $\theta$, in the late stage of the deposition process, for all wrapping triangles $T_j$ (Table 1). The horizontal line represents the final value $N=L^2=57600$ of the coefficients $c_j$ that is reached when the coverage $\theta$ approaches to the jamming limit ${\theta }_J$. The solid black line has slope 1 and is a guide for the eye.

Figure 4

Shown here is the insertion probability $p_j$ $vs$ the coverage $\theta$ for triangles $T_3,T_4,T_6$, rhombuses $R_3,R_6$, and hexagons $H_4,H_7,H_{19}$ from Tables 1, 2, 3.

Figure 5

The decay of the insertion probability $p_j\left(\theta \right)$ for the same objects as in Fig. 4 in the range of coverage $\theta$ where the corresponding probability $p_j$ is lower than $p_j^c=0.15$ (thin horizontal line). The continuous superimposed lines are the fits according to Eq. (4). The fitting parameters ${\lambda }_1$ and ${\lambda }_2$ are reported in Fig. 6.

Figure 6

Parameters (a) ${\lambda }_1$ and (b) ${\lambda }_2$ of the fit (4) for all wrapping triangles from Table 1. Numerical values of the symmetry order $n_s$ of the shape $j$ are given in the square brackets above the corresponding plot symbols.

Figure 7

Characteristic density ${\theta }^{\left(c\right)}$ (Eq. (5)) for all wrapping triangles $T_j$ from Table 1 (solid symbols, left-hand axis). The opened symbols are plotted against the right-hand axis and give the values of the shape factor $\zeta \left(j\right)$ for all wrapping triangles $T_j$. The characteristic density ${\theta }^{\left(c\right)}\left(j\right)$ is anticorrelated with the shape factor $\zeta \left(j\right)$.

Figure 8

Construction of the polygon determined by the first neighboring sites on the lattice for the wrapping triangles $T_6$ and $T_8$, rhombus $R_8$, and hexagon $H_9$. The shape factor of the extended object is equal to the shape factor (6) of the polygon of the first neighboring sites on the lattice.

Figure 9

Characteristic density ${\theta }^{\left(c\right)}$ [Eq. (5)] for all wrapping rhombuses $R_j$ from Table 2 (solid symbols, left-hand axis). The opened symbols are plotted against the right-hand axis and give the values of the shape factor $\zeta \left(j\right)$ for all wrapping rhombuses $R_j$. The characteristic density ${\theta }^{\left(c\right)}\left(j\right)$ is anticorrelated with the shape factor $\zeta \left(j\right)$.

Figure 10

Characteristic density ${\theta }^{\left(c\right)}$ [Eq. (5)] for all wrapping hexagons $H_j$ from Table 3 (solid symbols, left-hand axis). The opened symbols are plotted against the right-hand axis and give the values of the shape factor $\zeta \left(j\right)$ for all wrapping hexagons $H_j$. The characteristic density ${\theta }^{\left(c\right)}\left(j\right)$ is anticorrelated with the shape factor $\zeta \left(j\right)$.

Figure 11

The product between the shape factor and characteristic density, $\zeta \left(j\right)\times {\theta }^{\left(c\right)}\left(j\right)$, as a function of the object size $j$ for all wrapping triangles $T_j$, rhombuses $R_j$, and hexagons $H_j$ from Tables 1, 2, 3. The black solid curve is the stretched exponential fit of Eq. (7).

Figure 12

Characteristic density ${\theta }^{\left(c\right)}$ [Eq. (5)] as a function of the shape factor $\zeta$ for various triplets of triangles, rhombuses and hexagons $\left(T_j,R_j,H_j\right)$ of the same size, $j=14,17,19,22,25,28,30$. The letter above the plot symbol indicates the object type. Sizes of the objects for the corresponding triplets are given in the legend.

Figure 13

Functions ${\mathrm{\Phi }}_1\left(\theta \right)$ and ${\mathrm{\Phi }}_2\left(\theta \right)$ [Eqs. (9) and (10)] obtained by fitting Eq. (4) to the insertion probability data for wrapping triangles $T_2$ and $T_{27}$ in the case of $p_j^c=0.15$. The solid lines give the fitting function $p_j\left(\theta \right)={\mathrm{\Phi }}_1\left(\theta \right){\mathrm{\Phi }}_2\left(\theta \right)$ [Eq. (4)]. The dot-dashed lines give the approximation $p_j\left(\theta \right)={\mathrm{\Phi }}_1\left(\theta \right)exp\left[-{\left({\theta }_{\text{J}}/{\theta }^{\left(c\right)}\right)}^{{\lambda }_2}\right]$, as indicated in the legend. Approximation (11) is applicable in the narrower density range in the case of the large object $T_{27}$ than in the case of the dimer $T_2$. The parameters of the fit (4) are ${\theta }^{\left(c\right)}=0.8313,{\lambda }_2=3.6964$ for shape $T_2$, and ${\theta }^{\left(c\right)}=0.3619,{\lambda }_2=2.5266$ for shape $T_{27}$. The thin vertical lines indicate the values of jamming coverage for shapes $T_2$ $\left({\theta }_{\text{J}}=0.9141\right)$ and $T_{27}$ $\left({\theta }_{\text{J}}=0.6498\right)$.

Figure 14

Relaxation time $\sigma$ calculated from the expression (13) for all wrapping triangles (Table 1). Results are given for three values of threshold $p_j^c=0.12,0.02,0.002$, as indicated in the legend. The full circles correspond to values of $\sigma$ in Table 1. Open circles correspond to values $6/n_s\left(j\right)$ [Eq. (14)] for $j=2,...,30$. Numerical values of the symmetry order $n_s\left(j\right)$ of the shape are given in the square brackets above the $x$ axis.