Morphology and Kinetics of Random Sequential Adsorption of Superballs: From Hexapods to Cubes

TL;DR

This paper introduces a universal simulation method for the random sequential adsorption of superballs, providing precise estimates of jamming fractions and adsorption kinetics, with implications for various scientific applications involving particle packing.

Contribution

The paper presents a novel low-entropy algorithm for simulating superball adsorption, improving accuracy in estimating jamming fractions and kinetics compared to traditional methods.

Findings

Accurate estimates of jamming fraction (p) for different superball shapes.

Precise determination of the adsorption exponent ta(p) across shape parameters.

Validation of the new algorithm against existing analytical and numerical results.

Abstract

Superballs represent a class of particles whose shapes are defined by ${|x|}^{2p}+{|y|}^{2p}+{|z|}^{2p} \le R^{2p}$, with $p\in(0,\infty)$ being the "deformation parameter". $0<p<0.5$ represents a family of hexapodlike (concave octahedrallike) particles, while for $0.5\leq p<1$ and $p>1$ one has, respectively, families of convex octahedrallike and cubelike particles, with $p=1,\;0.5$ and $\infty$ representing spheres, octahedra, and cubes. Colloidal zeolite suspensions, catalysis, and adsorption, as well as biomedical magnetic nanoparticles are but a few of the applications of packing of superballs. We introduce a universal method for simulating random sequential adsorption of superballs, which we refer to as "low-entropy" algorithm, in contrast with the conventional algorithm that represents a "high-entropy" method. The two algorithms yield, respectively, precise estimates of the jamming fraction $\phi_\infty(p)$ and $\nu(p)$, the exponent that characterizes the kinetics of adsorption at long times $t$, $\phi(\infty)-\phi(t)\sim t^{-\nu(p)}$. Precise estimates of $\phi_\infty(p)$ and $\nu(p)$ are obtained and shown to be in agreement, in some special limits, with the existing analytical and numerical results.

Equations

$${|x|}^{2p}+{|y|}^{2p}+{|z|}^{2p}\le R^{2p},$$

$$\varphi (\infty )-\varphi (t)\sim t^{-\nu (p)},$$

$$\varphi (t)=\frac{n(t)V_{sb}}{V},$$

$$S=-\sum _i{P}_i\log P_i=\mathcal{E}[-\log P(X)],$$

$$S=-{\int }_0^L\frac{1}{L}\ln \left(\frac{1}{L}\right)dx=\ln L,$$

$${\varphi }_{\infty }(p)=C_1\exp (-C_{2}p)\cos (C_{3}p-C_4)+C_5,$$

Full text

Morphology and Kinetics of Random Sequential Adsorption of Superballs: From Hexapods to Cubes

Pooria Yousefi,1 Hessam Malmir,2,† and Muhammad Sahimi3,‡

1*Faculty of Engineering, Science and Research Branch, Azad University, Tehran 14515-775, Iran

2Department of Chemical and Environmental Engineering, Yale University, New Haven, Connecticut 06511, USA

3Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089, USA*

Superballs represent a class of particles whose shapes are defined by ${|x|}^{2p}+{|y|}^{2p}+{|z|}^{2p}\leq R^{2p}$, with $p\in(0,\infty)$ being the deformation parameter. $0<p<0.5$ represents a family of hexapodlike (concave octahedrallike) particles, while for $0.5\leq p<1$ and $p>1$ one has, respectively, families of convex octahedrallike and cubelike particles, with $p=1,\;0.5$ and $\infty$ representing spheres, octahedra, and cubes. Colloidal zeolite suspensions, catalysis, and adsorption, as well as biomedical magnetic nanoparticles are but a few of the applications of packing of superballs. We introduce a universal method for simulating random sequential adsorption of superballs, which we refer to as low-entropy algorithm, in contrast with the conventional algorithm that represents a high-entropy method. The two algorithms yield, respectively, precise estimates of the jamming fraction $\phi_{\infty}(p)$ and $\nu(p)$, the exponent that characterizes the kinetics of adsorption at long times $t$, $\phi(\infty)-\phi(t)\sim t^{-\nu(p)}$. Precise estimates of $\phi_{\infty}(p)$ and $\nu(p)$ are obtained and shown to be in agreement, in some special limits, with the existing analytical and numerical results.

Random sequential adsorption (RSA) is an irreversible process for generating nonequilibrium packings of nonoverlapping particles, and is considered a very useful model to study and understand the structure of low-temperature phases of matter, as well as particle aggregation and jamming in a wide variety of applications, from granular media [1,2], to heterogeneous materials [3,4] and biological systems [5,6]. The RSA and its kinetics are also among important problems in statistical physics, which have been studied analytically and numerically for various particles and systems [1,5,7-17]. An important property of the RSA is the kinetics of the adsorption that typically approaches a very slow asymptotic saturation limit (jamming) in which no more particles can be added to the packing.

In this Letter we focus on a special class of particles, the so-called superballs, whose possible shapes include a variety of three-dimensional (3D) concave and convex particles. Colloidal zeolite suspensions with applications in catalysis, adsorption, and separation [18,19], as well as packings of magnetic nanoparticles with biomedical applications [20-22] are but some of the better-known applications of the RSA of superballs. Although optimal (lattice) packings and maximally randomly jammed (MRJ) systems of superballs have been studied by Jiao et al. [23,24], packing of superballs by the RSA and its kinetics have not been investigated. The importance of modeling and analysis of the RSA of superballs is due to the fact that by tuning a shape parameter one obtains a wide variety of particle shapes, ranging from hexapod- to octahedral- and cubelike particles, as well as spherical ones. At the same time, determining the saturation coverage and kinetics of the RSA of various types of particles by a unified approach is a long-standing problem, which we address in this Letter by studying a large family of superballs. One of the main questions that we address is how changing of particles’ shapes from concave to convex affects the maximum saturation coverage $\phi_{\infty}$ (sometimes called the jamming limit) of their packings. From practical view point, the adsorption rate of the particles is also an important property, which we study and compare with the existing conjectures on the kinetics of the RSA [1,2,5,7,8,10,13,15].

The shape of superballs is described by the following general equation,

[TABLE]

where $p\geq 0$ is the deformation parameter that indicates the extent to which the particle’s shape deviates (deforms) from that of a sphere, the limit $p=1$, and $R$ is the superballs’ radius. Depending on $p$, a superball may possess two types of shape anisotropy, namely, cubelike and octahedrallike shapes. As $p$ increases from 1 to $\infty$, one obtains a family of convex superballs with cubelike shapes. The limit $p\to\infty$ represents a perfect cube. As $p$ decreases from 1 to 0.5, a family of convex superballs with octahedrallike shapes are obtained. In the limit $p=0.5$ the superballs represent regular octahedral particles. For $p<0.5$, they still possess an octahedrallike shape, but similar to hexapods are concave, and approach a 3D “cross” in the limit $p\to 0$.

The algorithm that we utilize for simulating the RSA of superballs is a generalization of the one that we recently developed for cubic particles [25-27]. We begin with a large, empty box of volume $V$ in ${\bf R}^{3}$, generate superballs with given deformation parameter $p$ and randomly-selected positions and orientations, and place them sequentially in the simulation box. The deposition is subject to the nonoverlapping constraint, so that no newly inserted particle can overlap with any existing ones. The overlap occurrence depends, however, on the spatial coordinates and orientation of the superballs. Thus, (i) we first generate at random the three coordinates $x\in[0,x_{\rm max}]$, $y\in[0,y_{\rm max}]$ and $z\in[0,z_{\rm max}]$ of the superball’s center of mass, where $(x_{\rm max},y_{\rm max},z_{\rm max})$ are the dimensions of the simulation box, and (ii) the superball is then rotated using a quaternion [28], ${\bf q}=a+b{\bf i}+c{\bf j}+d{\bf k}$, which represents the orientations and rotations of 3D objects. A quaternion is simpler to compose than the Euler’s angles that are used to describe the orientation of a rigid body; avoid losing one degree of freedom (the so-called gimbal lock problem [29]); are more compact than the rotation matrices and more stable numerically, and their use is more efficient. Henceforth, we need four independent random numbers $a,\;b,\;c$, and $d$, uniformly sampled from a normal distribution with 0 mean and unit standard deviation in order to uniformly and randomly rotate the superballs. (iii) Periodic boundary condition is then imposed in all three directions. (iv) The surface points of the randomly positioned and rotated superball are then checked via Eq. (1) against the centers of the previously inserted superballs to see if any overlap occurs. If so, the superball is rejected and a new one is generated starting from step (i). Otherwise, the superball is accepted and the deposition process continues until the saturation or jamming limit of the system is reached.

The RSA rate is mainly limited by the volume exclusion from previously adsorbed particles. Its long-time kinetics is described by [1,10],

[TABLE]

in which $\phi(t)$ is the packing fraction at dimensionless time $t$:

[TABLE]

and $\phi(\infty)$ denotes the RSA maximum saturated packing fraction, with $V$ being the simulation box’s volume. $n(t)$ is the number of superballs generated up to time $t$, and $V_{sb}$ refers to the volume of the superballs [23].

Since the positions of superballs in an RSA packing are equiprobable throughout the simulation box, random sampling of a superball’s position follows a uniform probability density function (PDF). Thus, the predictability of the existence of an empty space for inserting a superball, which is a random variable $X$ with a uniform distribution is controlled by the interval in which the PDF is nonzero, with the simulation box size being $L=\max[x_{\rm max},y_{\rm max},z_{\rm max}]$. For $L\to 0$, the PDF becomes a delta function and the predictability is maximal, i.e., the uncertainty is minimum. In this limit, $X$ takes on the value at which the delta function is nonzero. For $L\to\infty$, however, the predictability of the state of $X$ is minimal, i.e., the uncertainty is maximum, and the same is true for all the possible states. Thus, one requires a measure of the uncertainty for the state of the random variable $X$ [30]. Let $X$ be a discrete random variable with possible values $\{x_{1},\cdots,x_{m}\}$ and probabilities $P_{i}=P(X=x_{i})$. The entropy $S$, a measure of uncertainty, is defined as the expected value of the information gained from observing $X$ [31]:

[TABLE]

where ${\cal E}$ is the expected value operator. Thus, $S$ depends on the probability distribution of $P_{1},\cdots,P_{m}$, but not on $x_{1},\cdots,x_{m}$. Applying the definition to the RSA in a simulation box of length $L$ yields

[TABLE]

implying that by shrinking the size $L$ of the sampling domain, the uncertainty $S$ of finding empty spaces in the RSA also decreases. But, the questions are, how much can one possibly limit the domain of the sampling, and how does it affect the maximum RSA packing fraction $\phi_{\infty}$ and its kinetics, which should be independent of the simulation box’s dimensions?

To address the questions, we must consider an approach that satisfies the isotropy of the packings and does not alter the RSA constraints. To do so, we propose to first sample the entire simulation box in order to randomly distribute the superballs, and achieve a state in which the number of iterations to find an empty space for inserting a superball becomes very large. We refer to this step as phase I. We then divide the simulation box into uniform and equal grid cells, with their size selected such that they can accommodate a few superballs, at least 3 or 4. We then sweep the cells one by one, referred to as phase II, to more accurately identify any possible empty space that can accomodate new superballs. By decreasing the domain of the sampling in phase II, the predictability of finding empty space for particle insertion increases, ensuring that the true saturation is reached.

We refer to the combination of the two phases as the low-entropy RSA, which yields precise estimates of the maximum saturation coverage, $\phi_{\infty}$. This approach cannot, however, capture accurately and efficiently the kinetics of the RSA because phase II that explores the cells sequentially is a very slow process. Thus, it makes reaching the true asymptotic kinetics of the RSA difficult. Hence, we still need to use the conventional RSA, which we refer to as a high-entropy RSA process, in which phase II is not considered, or has a negligible effect in the simulations and, therefore, almost all the superballs are generated and inserted by phase I. The use of low- and high-entropy terminology is motivated by Eq. (5), according to which by limiting the size $L$ of the domain of sampling, entropy $S$ is kept very low, and the absence of this effect in the high-entropy case.

Figure 1 shows three RSA packings. The densest packing in Fig. 1 is that of cubelike particles with $p=5/4$ whose shape is still close to spherical particles, $p=1$. Using the algorithms and setting $R=1$ for the superballs, we computed precise estimates for $\phi_{\infty}$ and the kinetics of adsorption for ten types of superballs as a function of the deformation parameter $p$, ranging from $p=0.25$ (concave octahedrallike particles), to 0.5 (octahedra), 0.75 (convex octahedrallike particles), and 1 (spheres), as well as 6 values of $p$ for the cubelike superballs, using the low-entropy RSA algorithm. The results are shown in Fig. 2. The maximum value of $\phi_{\infty}(p)$ corresponds to spherical particles, for which we obtained, $\phi_{\infty}\approx 0.384457\pm 0.003991$, which is in agreement with the previous estimates, $\phi_{\infty}\approx 0.38278$ [12], 0.3841307 [13], 0.38 [32], and 0.382 [33]. As $p\to 0$, $\phi_{\infty}(p)$ decreases, with the RSA packing of hexapodlike superballs with $p=0.25$ having the lowest $\phi_{\infty}$ that we computed, $\phi_{\infty}\approx 0.151749\pm 0.001553$. Beyond spheres we have cubelike particles with $p>1$ for which $\phi_{\infty}$ also decreases, and for $p>10$ reaches the saturation limit, $\phi_{\infty}\approx 0.333=1/3$ for the cubic particles [34]. This is presumably the most accurate estimate of $\phi_{\infty}$ for the RSA packing of cubes, since preventing overlaps between the particles is based on Eq. (1) as $p\to\infty$, and not based on approximations in terms of edge-edge, edge-face, and corner-face intersections [16].

The plot of $\phi_{\infty}$ versus $p$ shown in Fig. 2 has a shape distinctly different from that of the lattice and MRJ packings of superballs [23,24], for which $\phi_{\infty}$ has its lowest value for the spherical particles, $p=1$. This demonstrates that the RSA is a process completely different from those that generate dense equilibrated configurations of superballs. The list of estimates of $\phi_{\infty}(p)$ along with their standard errors, computed by using the low-entropy RSA, is presented in Table 1. The fitted curve in Fig. 2 may be approximated by

[TABLE]

with $C_{1}$, $C_{2}$, …, $C_{5}$ being, respectively, $\approx 1.661$, 2.089, 0.567, 1.901, and 0.336.

Table 1. Maximum packing fraction $\phi_{\infty}$ of superballs versus the deformation parameters $p$.

[TABLE]

To obtain accurate results for the kinetics of the RSA packing, Eq. (2), we use the high-entropy RSA algorithm. Figure 3 shows the dependence of $\phi(2t)-\phi(t)$ on the dimensionless time $t$ defined by, $t=n_{i}V_{sb}/V$ in which $n_{i}$ denotes the number of the RSA iterations, i.e. the number of successive addition of the particles. Since it is not practical to carry out the simulations for too long, we derive the asymptotic behavior by analyzing $\log[\phi(2t)-\phi(t)]$ that exhibits the same scaling as $\log[\phi(\infty)-\phi(t)]$, when plotted against $\log(t)$ [15]. As illustrated, the slope of $\log[\phi(2t)-\phi(t)]$, which corresponds to $-\nu(p)$ in Eq. (2), has its maximum and minimum values at, respectively, $p=1$ (spheres) and $p=0.25$ (hexapodlike superballs). For spherical particles, one has, $\nu\approx 0.33$, which was previously predicted [5,7,8,12,30]. For cubes and octahedra, however, $\nu\approx 0.23$ and 0.18, respectively. The plot of $\log[\phi(\infty)-\phi(t)]$ with respect to $\log(t)$ decays slowly for concave octahedrallike particles. For $p=0.25$ we obtain $\nu\approx 0.07$. In this limit, the relation $\phi(\infty)-\phi(t)\sim\log(t)$ also quantifies the asymptotic behavior extremely accurately, which is not surprising as $\nu$ is very small.

To further characterize the structure of the RSA packings, we computed the pair correlation function $g_{2}(r)$ at $\phi_{\infty}(p)$. Figure 4 presents the results. The peak of $g_{2}(r)$ occurs at $r=D$ for the spherical particles, where $D=2R$ is the superballs’ diameter, and moves from left to right with increasing $p$. Since for superballs with $p<1$ the diameter $D$ equals the diameter of their circumscribed sphere, Fig. 4 indicates that for $p=0.25$ and $p=0.5$ (octahedron) and $r<D$ we still have some neighboring superballs around the reference one, but the peak of $g_{2}(r)$ occurs at smaller $r$ for smaller $p$, and has lower values for such values of $p$. On the other hand, for superballs with $p>1$ the diameter $D$ is the diameter of their inscribed sphere, which is why the peak of $g_{2}(r)$ in Fig. 4 for $p=2.0$ (cubelike particles) occurs at $r>D$, although the value of the peak is still lower than that of spheres ($p=1$). Furthermore, the limiting values for $g_{2}(r)$ of spheres are similar to those of Refs. [7,8,12,13], namely, $\lim_{r\to D^{+}}g_{2}(r)\sim-\ln{(r/D-1)}$, which supports the conjecture on the logarithmic singularity of $g_{2}(r)$ at $r=D$ for spherical particles.

Summarizing, developing a new simulation algorithm, we presented the results of a comprehensive study of the RSA of superballs. The maximum packing fraction and the kinetics of the adsorption were studied and the relevant quantities were estimated. We proposed a universal approach, namely, the low-entropy process that leads to the precise estimates of the jamming. The highest saturated packing fraction among superballs belongs to spherical particles ($p=1$). Both the long-time kinetics and the pair-correlation function $g_{2}(r)$ of the RSA of superballs at $p=1$ support the previous conjectures on the RSA of spherical particles.

Work at USC was supported in part by the Petroleum Research Fund, administered by the American Chemical Society.

†hessam.malmir@yale.edu

‡moe@usc.edu

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