Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: Towards engineering applications
Graphical abstract
The packing algorithm enables the generation of random close packings of nonspherical particles, agglomerates, mixtures of particles, and sintered packings.
Introduction
Random packings of particles are widely considered in science and engineering applications: they have been suggested as models for liquid and glass structure [1], [2] and they are used to represent granular materials [3], packed beds, and cermets [4] as well as in many other applications. In the last decades, several packing algorithms have been developed to represent the packing microstructure and for property evaluation. Many algorithms have been developed in particular for spherical particles: Monte-Carlo [5], [6], [7], drop-and-roll and sequential deposition algorithms [8], [9], [10], [11], [12], [13], collective rearrangement [14], [15], [16], discrete element methods [17], [18], [19] and molecular dynamics [20], [21] to cite the most common ones.
Recently, in order to have a better representation of particulate systems, and mainly due to the availability of increased computational resources, attention has shifted to simulating random packings of nonspherical particles. The first problem arising with nonspherical particles is the more complex shape than the spherical form. Different methods have been proposed to account for nonspherical shapes. In several methods the analytical equation of the particles is considered in the algorithm: packings of ellipsoids [22], [23], [24], [25], spherocylinders [26], superballs [27], [28], superellipsoids [29] and general convex particles [30] have been investigated. The use of the analytical equation to represent the particle has the clear advantage that the exact shape of the particles is considered. Although such a method is elegant and rigorous, the detection of overlaps is non-trivial and may lead to detection errors in some critical situations [31]. In addition, the algorithm is tailored for the specific shape under consideration and, clearly, particles with a shape that cannot be represented analytically cannot be simulated.
Another approach consists of tessellating the container and the particle shape with a grid, digitizing both the domain and the particles [32], [33], [34], [35]. In this way, any particle shape can be approximated with a coherent collection of pixels (2D) or voxels (3D), and the collision and overlap detection are simply noting whether two objects occupy the same site in the grid. On the other hand, quantitative predictions of packing characteristics, such as packing density, are sensitive to the resolution used and increasing the resolution through a finer grid leads to too much higher memory requirements than other methods [32]. Furthermore, the movement of the particles is discretized, such that particle trajectories are affected by the resolution of the grid.
The third method is the so-called multi-sphere (or sphere-assembly) approach [36], [37], [38], [39], [40], in which particles are represented by an assembly of component spheres reproducing their shape. As the digitizing method, general particle shapes, analytical or otherwise, can be reproduced by varying the position and the size of the component spheres within the particle. The detection of particle overlaps is carried out by checking if two component spheres, belonging to different particles, overlap, which is much easier if compared with the first approach where the analytical particle equations are used. On the other hand, higher resolutions, obtained with a larger number of component spheres, slow down the algorithm and require more computer memory, though usually less than in the case of the digitizing method.
The packing procedure is the second important feature to consider and it can affect the resulting packing properties. Existing physical simulation models include the discrete element method (DEM) [29], [40], [39] and the molecular dynamics method (MD) [23], [24], [28]. In these algorithms the real interaction forces are taken into account, rigorously simulating the dynamics of the packing generation in time domain. While even jammed configurations can be obtained [22], [41], these methods are usually very complex and, although specific technical solutions can be used to speed up the simulations, they are less computationally efficient than many purpose designed packing algorithms, at least for spheres [42]. Furthermore, for some engineering applications such a highly detailed physical representation is not necessary.
In Monte-Carlo methods [31], each particle is added to the domain one by one by selecting a random position and orientation and checking the overlaps with previously placed particles: if there are no overlaps, the current particle is accepted, otherwise a new position is tried and, after a predefined number of trials, the particle is rejected if an acceptable placement has not been found. Though this algorithm is straightforward, it is very time consuming and no rearrangement of particles is permitted (i.e., the orientation is completely random). This usually results in loose packings, and when rigid particles are simulated the majority if not all the particles are not in contact with one another.
Another packing algorithm, which has been widely adapted for nonspherical random packings [36], [38], is the collective rearrangement method (CR). In this algorithm all the particles are randomly distributed and oriented in a domain which is smaller than the volume that all the particles may occupy. At the beginning particles experience large overlaps, which are individually removed by iteratively moving and rotating each particle under the action of a restoring force and a restoring moment generated in consideration of the overlaps. The rearrangement of particles is usually coupled with a process of particle to domain reduction, consisting of either reducing (i.e., scaling down) the particle size or increasing (i.e., scaling up) the domain volume. When simulating hard particles, the algorithm stops when all the overlaps have been removed. This technique is expected to be faster than DEM and MD, since the physics is only approximated in order to save computational time. While this method usually provides close packings, they are generally not strictly jammed. Moreover, there is no check about the contact information of particles during the rearrangement process: typically, in the final configuration particles are arranged in unstable positions or are isolated, feature which is pronounced by the particle-to-container shrinking procedure.
Finally, in principle each packing algorithm could be coupled with one of the three approaches described above to represent the particle shape, although some concerns have been recently arisen when the sphere-assembly approach is coupled with the discrete element method [39], [40].
In this study, a collective rearrangement method coupled with the sphere-assembly approach is used, sharing some features in common with the Nolan and Kavanagh work [36]. However, in the present algorithm both the particles and the container maintain their dimensions, avoiding the particle shrinking procedure. A constraint has been introduced and applied to each particle in order to provide packings in which all the particles are stable. The coupling of these two characteristics ensures that all the particles contact each other in the final configuration, characteristics that cannot be generally guaranteed by conventional CR algorithms [21]. The unambiguous detection of contacts is a desired feature in many engineering applications because percolation thresholds and conduction properties of the packing are strictly related to the number of contacts [43], [44], [45], [7]. In addition, the algorithm has been generalized to allow for multiple polydisperse phases and to a controlled degree of particle overlap in order to simulate deformable particles and sintered multi-phase packings.
The study focuses on the application to rigid and non-rigid ellipsoids, cylinders and agglomerates, though the algorithm is sufficiently general that any particle shape can in principle be used. The chosen sub-set of shapes is common in several engineering applications, such as in polymer electrolyte [46] and solid oxide [47], [48], [49] fuel cells which, in part, motivated the developments described herein. In these applications it is important that in the reconstructed microstructure the packing is representative of a stable configuration and the particles experience a desired degree of overlap, in order to ensure that charges can be transported and converted throughout the packing [48], [50], [47], [51], [52], [53].
The paper is organized as follows: in Section 2 the algorithm is presented in detail; in Section 3, the algorithm is first explored by assessing the effects of its internal parameters, then simulation results for rigid spheres, ellipsoids and cylinders are compared to those obtained by other algorithms, and finally some results regarding the broader possibilities of the algorithm are shown.
Section snippets
General aspects
The algorithm was written in C++ programming language with the use of some functions provided in an open source finite element deal.II library [54]. The code was generalized to account for both 2D and 3D packings with one or more polydisperse or monodisperse phases of rigid or deformable particles: in this study only the three dimensional problem is described.
The algorithm begins with the definition of the domain size and the number, types and sizes of particles. The domain consists of a box of
Exploring the algorithm
In this section the effects of the main internal parameters of the algorithm (such as the number of iterations before reposition and the number of repositions before reallocation) and of the constraints (i.e., the stability check) are assessed.
Random packings of monodisperse rigid cylinders with diameter to height ratio equal to 1 were simulated. Rigid means that the kernel scale factor approached 1, with a contact allowance of 1.5% as a maximum. The starting porosity used was 0.1,which was
Conclusions
A modified collective rearrangement algorithm for simulating random packings of nonspherical particles, represented as an assembly of component spheres, was presented and discussed in this paper. Unlike previous CR methods, a stability constraint was introduced while the particle-to-container shrinking procedure was removed. In this way, during the packing generation the rearrangement process concerned only the neighborhood of the particle taken in consideration. More importantly, this ensured
Acknowledgments
This research was supported by the NSERC Solid Oxide Fuel Cell Canada Strategic Research Network from the Natural Science and Engineering Research Council (NSERC) and other sponsors listed at www.sofccanada.com. Special thanks to Sandro Pintus, who helped us in configuring the machines used to run the simulations at the Department of Civil and Industrial Engineering of the University of Pisa. Thanks to Olivier Blake, Fuel Cell Research Centre of Kingston, who assisted us in the production of
Glossary
- coordinates of the ith particle center of mass
- center of the kth component sphere in the ith particle in the global frame
- center of the kth component sphere in the ith particle in the local frame
- displacement of the ith particle due to the force
- force created by the kth component sphere of the jth particle with the hth component sphere of the ith particle on the ith particle
- resultant force on the ith particle
- i
- particle index (ith particle), usually considered as the
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