In a search for a shape maximizing packing fraction for two-dimensional random sequential adsorption

Cieśla, Michał; Paja̧k, Grzegorz; Ziff, Robert M.

doi:10.1063/1.4959584

Cited by 45 publications

(55 citation statements)

References 33 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The exact geometry of target sites thus intimately affects the kinetics, which should also be true in higher dimensions. It would be very interesting to find out if the same or similar universality classes govern also other jamming properties for non-spherical shapes, e.g., the observed peak in the packing density at specific aspect ratios of elongated shapes (see, e.g., [9,[42][43][44]), which allows the identification of optimally dense granular packings that are highly relevant for developing new functional granular materials [3]. Since the model Eq.…”

Section: β)(12)mentioning

confidence: 99%

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Baule

2017

Phys. Rev. Lett.

View full text Add to dashboard Cite

Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage ∼ t −ν as t → ∞. However, the exact value of the exponent ν is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi's seminal 'car parking problem'), where ν = 1 can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general non-spherical particles ν = 1/(d +d), where d denotes the dimension of the domain andd the number of orientational degrees of freedom of a particle. Here, we solve this long standing problem analytically for the d = 1 case -the 'Paris car parking problem'. We prove that the scaling exponent depends on particle shape, contrary to the original conjecture, and, remarkably, falls into two universality classes: (i) ν = 1/(1 +d/2) for shapes with a smooth contact distance, e.g., ellipsoids; (ii) ν = 1/(1 +d) for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains in particular why many empirically observed scalings fall in between these two limits.The question of how particle shape affects the dynamical and structural properties of particle aggregates is one of the outstanding problems in statistical mechanics with profound technological implications [1][2][3]. Jammed systems are particularly challenging, since they are dominated by the geometry of the particles and are not described by conventional equilibrium statistical mechanics [4]. Exploring the effect of shape variation thus relies on extensive computer simulations [5][6][7][8] or mean-field theories whose solutions require similar computational efforts [9,10]. From a theoretical perspective it is striking that so far there has been hardly any insight from exactly solvable analytical models, even though these are most suitable to identify and classify shapes in the infinite shape space.In this letter, we consider the probably simplest nontrivial packing model that takes into account excluded volume effects due to shape anisotropies: random sequential adsorption (RSA). Since Renyi's seminal work on the 'car parking problem' (the RSA of monodisperse spheres on a line) [11,12], RSA models have been widely used to model particle aggregation and jamming in physical, chemical and biological systems [13][14][15]. Their great appeal is the paradigmatic nature of the adsorption mechanism: the particles' positions and orientations are selected with uniform probability and then placed sequentially into the domain if there is no overlap with any previously placed particles. Particles are not able to move or reorient once being placed.Two key features of RSA models are: (i) the existence of a finite jamming density φ in the infinite time limit φ(∞) = lim t→∞ φ(t) and (ii) the algebraic time dependence of the approach to jamming, which has been conj...

show abstract

Section: β)(12)mentioning

confidence: 99%

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Baule

2017

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…It corresponds to one-dimensional RSA, the saturated packing fraction of which was analytically calculated by Renyi's as a solution to the so-called car parking problem [10]. RSA is most commonly utilized in two dimensions [11][12][13], because it models monolayers obtained in irreversible adsorption process [11].…”

Section: Introductionmentioning

confidence: 99%

Random sequential adsorption of Platonic and Archimedean solids

Kubala

2019

Phys. Rev. E

View full text Add to dashboard Cite

The aim of the study presented here was the analysis of packings generated according to random sequential adsorption protocol consisting of identical Platonic and Archimedean solids. The computer simulations performed showed, that the highest saturated packing fraction θ = 0.402 10(68) is reached by packings build of truncated tetrahedra and the smallest one θ = 0.356 35(67) by packings composed of regular tetrahedra. The propagation of translational and orientational order exhibited microstructural propertied typically seen in RSA packings and the kinetics of 3 dimensional packings growth were again observed not to be strictly connected with the dimenstion of the configuration space. Moreover, a number of optimizations for the RSA algorithm were described allowing generation of significantly larger packings, which translated directly to a lower statistical error of the results obtained. Additionally, the polyhedral order parameters provided can be utilized in other studies regarding particles of polyhedral symmetry.

show abstract

“…Vertical axis is the logarithm of all three quantities and ranges from 0 to 6. From top to bottom: { [8,8], [10,5], [4,4], [6,3] For very large ρ (left on the horizontal axis) all parkings are 1-parkings: they yield exactly 1 disc. As ρ decreases, other numbers become possible and typically we see a distribution of k-values: some samples yield more parked discs than others.…”

Section: Large Discs and Critical Radiimentioning

confidence: 99%

“…The only true critical radius we have found is on the sphere, although on the plane [4,4] there is a radius which is almost critical, where the fraction of 4-parkings reaches 99.3%. Critical intervals appear to occur for the hyperboloid [8,8] (2-parkings), plane [4,4] (2-parkings), plane [6,3] (3-parkings), projective plane (3-parkings), and sphere (2-parkings). Except for the hyperboloid [8,8], these intervals typically occur directly after the transition from 1-parkings.…”

Section: Large Discs and Critical Radiimentioning

confidence: 99%

“…Because the one-dimensional model resembles parking a car on the curb in a busy city, this process is often called random parking; note that it is distinct from random packing. Variants of RSA can include non-uniform disc sizes or shapes [15,3], modelling the motion of objects before they find a parking spot [14,7], or non-uniform sticking distributions, as in cooperative sequential adsorption in which the sticking probability depends on the locations of the objects already attached [22].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Random Sequential Adsorption of Discs on Surfaces of Constant Curvature: Plane, Sphere, Hyperboloid, and Projective Plane

Chen

Holmes-Cerfon

2017

J Nonlinear Sci

View full text Add to dashboard Cite

We present an algorithm to simulate random sequential adsorption (random "parking") of discs on constant-curvature surfaces: the plane, sphere, hyperboloid, and projective plane, all embedded in three-dimensional space. We simulate complete parkings by explicitly calculating the boundary of the available area in which discs can park and concentrating new points in this area. This makes our algorithm efficient and also provides a diagnostic to determine when each parking is complete, so there is no need to extrapolate data from incomplete parkings to study questions of physical interest.We use our algorithm to study the number distribution and density of discs parked in each space, where for the plane and hyperboloid we consider two different periodic tilings each. We make several notable observations: (i) On the sphere, there is a critical disc radius such the number of discs parked is always exactly four: the random parking is actually deterministic. We prove this statement rigorously, and also show that random parking on the surface of a d-dimensional sphere would have deterministic behaviour at the same critical radius. (ii) The average number of parked discs does not always monotonically increase as the disc radius decreases: on the plane (square with periodic boundary conditions), there is an interval of decreasing radius over which the average decreases. We give a heuristic explanation for this counterintuitive finding. (iii) As the disc radius shrinks to zero, the density (average fraction of area covered by parked discs) appears to converge to the same constant for all spaces, though it is always slightly larger for a sphere and slightly smaller for a hyperboloid. Therefore, for parkings on a general curved surface we would expect higher local densities in regions of positive curvature and lower local densities in regions of negative curvature.

show abstract

In a search for a shape maximizing packing fraction for two-dimensional random sequential adsorption

Cited by 45 publications

References 33 publications

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Random sequential adsorption of Platonic and Archimedean solids

Random Sequential Adsorption of Discs on Surfaces of Constant Curvature: Plane, Sphere, Hyperboloid, and Projective Plane

Contact Info

Product

Resources

About