Radical Voronoï tessellation from random pack of polydisperse spheres: Prediction of the cells’ size distribution

Highlights

• Random packs of polydisperse spheres have been computed with LAMMPS.

• Radical Vorono tessellation have been performed on each pack.

• Cell sizes have been statistically characterized in each case.

• Relationships between spheres’ population and cell sizes are derived.

• The bias introduced by cross sectioning (2D apparent sizes) is investigated.

Abstract

This paper investigates the relevance of the representation of polycrystalline aggregates using Radical Voronoï (RV) tessellation, computed from Random Close Packs (RCP) of spheres with radius distribution following a lognormal distribution. A continuous relationship between the distribution of sphere radii with that of RV cell volumes is proposed. The stereology problem (deriving the 3D grain size distributions from 2D sections) is also investigated: two statistical methods are proposed, giving analytical continuous relationships between the apparent grain size distribution and the sphere radius distribution. In order to assess the proposed methods, a 3D aggregate has been generated based on a EBSD map of a real polycrystalline microstructure.

Introduction

Polycrystalline materials, such as metals or minerals, are usually composed of numerous individual grains, connected to each other by their grain boundaries. As a result, their thermomechanical properties are strongly influenced by the morphology and the crystalline orientation of each grain. The numerical generation of random geometries, representative of real microstructures, is still challenging because it requires to statistically reproduce the physical properties of the modelled grains in order to model the behaviour of polycrystalline aggregates [[1], [2], [3]].

The Poisson–Voronoï (PV) tessellation [4] has been extensively used over the last decades to numerically generate microstructure similar to real ones [[1], [2], [5], [6], [7], [8], [9]]. Consider a set of $n$ separate points (called seeds) $\{{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n\}$ in a given volume $\mathrm{\Omega }$. In PV geometry, the $i$th cell is defined such that:

$${\mathrm{C}}_i=\left\{\mathit{x}\in \mathrm{\Omega }|\Vert \mathit{x}-{\mathit{p}}_i\Vert =\underset{k=1}{\overset{n}{min}}\Vert \mathit{x}-{\mathit{p}}_k\Vert \right\}$$

where $\Vert \cdot \Vert$ denotes the euclidean distance. The PV tessellation results in a series of convex and non-overlapping cells bounded by planar surfaces. Nevertheless, considering the definition given in (1), size and shape of each Voronoï cell are only driven by the distance from its centre to those of its neighbours. As a result, cell sizes cannot easily be constrained. For instance, Groeber et al. [7] have developed the Constrained Voronoi Tessellation Method (CVTM), consisting in recursively adding new seeds in order to result in realistic cell morphologies.

The Radical Voronoï (RV) tessellation, also known as Laguerre–Voronoï [10], helps to impose the size of each cell through the introduction of a local radius, denoted $r$. Consider a set of $n$ seeds $\{{\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n\}$ and a set of $n$ radii $\{r_1,r_2,\dots ,r_n\}$. For the RV tessellation, (1) becomes:

$${\mathrm{C}}_i=\left\{\mathit{x}\in \mathrm{\Omega }|d_{\mathrm{L}}\left(\mathit{x},{\mathit{p}}_i,r_i\right)=\underset{k=1}{\overset{n}{min}}\left\{d_{\mathrm{L}}\left(\mathit{x},{\mathit{p}}_k,r_k\right)\right\}\right\}$$

where $d_{\mathrm{L}}$ denotes the euclidean distance in Laguerre geometry, so that:

$$d_{\mathrm{L}}\left(\mathit{x},{\mathit{p}}_k,r_k\right)=\Vert \mathit{x}-{\mathit{p}}_k{\Vert }^2-r_k^2.$$

Consider two cells $i$ and $j$; according to (2), if

$$\Vert {\mathit{p}}_i-{\mathit{p}}_j\Vert =r_i+r_j$$

then the boundary between the two cells is located at the distance $r_i$ (resp. $r_j$) from ${\mathit{p}}_i$ (resp. ${\mathit{p}}_j$). Eq. (4) is satisfied if ${\mathit{p}}_i$ and ${\mathit{p}}_j$ are the centres of two spheres, in contact with each other, with respective radii $r_i$ and $r_j$. As a result, the dimensions of the cells resulting from the RV tessellation computed from Random Close Packing (RCP) of spheres are strongly inherited from their radii.

It is worth mentioning that the numerical generation of RCP of spheres is not straightforward. The two main methods used in the literature are the collective rearrangement method [[11], [12], [13], [14], [15]] and the sequential generation. The latter usually consists in numerically pouring a closed volume with the spheres, by simulating a gravitational effect [[16], [17], [18]]. Yang et al. [11] have investigated the evolution of the packing factor of RCPs computed from sphere radii following a lognormal distribution. They have shown that the packing factor was an increasing function of the so-called shape parameter of the distribution (denoted $\sigma$ below, see Section 2.1) starting from 0.637. Falco et al. [19] have adapted the “3D-clew” technique [20] to generate the RCPs of sphere. This method consists in adding the spheres one by one into the pack, ensuring the tangency with other spheres. Rodrigues et al. [9] use a virtual container with moving walls instead of gravity. Fan et al. [14] have generated RCPs of spheres by collective rearrangement, in order to investigate the geometry of the RV tessellation thanks to the Qhull software package [21]. They have shown that if the volumes of the spheres follow a lognormal distribution, then those of the resulting RV cells follow a lognormal distribution too. In addition, they have focused on the geometrical characterization of each RV cell. In particular, they have shown that the mean number of faces per RV cell was a decreasing function of the Coefficient of Variation (CV) of sphere volumes, ranging from 13.04 to 14.11. Finally, Fan et al. [14] have investigated the relationship between the CV of the sphere volumes with that of the RV cell volumes.

It is well known that sequential algorithms usually result in anisotropic RCPs [[16], [22], [23]]. For instance, Tory et al. [16] have reported that the length of projection along the vertical direction (i.e. direction of applied gravitational force) of monodisperse spheres was 11  % greater than those computed along the horizontal directions.

The geometry of a real polycrystalline structure is usually described by the morphology and size of its grains. In order to numerically reproduce those structures using a RV tessellation, it is necessary to predict the adequate distribution for the RCP spheres. Grain sizes can be measured by 3D imaging techniques (such as X-ray tomography or neutron diffraction) but they are more frequently evaluated from 2D cross sections, for instance by microscopy or Electron Backscatter Diffraction (EBSD) mapping. Those 2D observations induce a bias in grain sizes, for they only provide information about the intersection of the 3D grains and the cutting plane. For instance, it can be shown that the mean apparent radius of a sphere with unit radius cut at random latitudes is $\pi ∕4$ [24]. For polydisperse particles, deriving their 3D sizes from 2D observations is called stereology problem. It is usually solved using an iterative routine applied on finite histograms [[25], [26], [27]]. Lopez-Sanchez and Llana-Fúnez [27] have proposed a routine, namely the two-step method, for the resolution of stereology problems. It consists in:

• deriving the 3D histogram from 2D sections, thanks to the so-called Saltykov method [25];

• fitting a lognormal distribution on the 3D histograms.

The Saltykov method (also known as Scheil–Schwartz–Saltykov method) is an iterative procedure, consisting in evaluating the probability of cutting each grain (considered as spherical) at a certain latitude, resulting in a reduced apparent size. This method is based on the finite histogram (finite number of classes) of apparent sizes. The fundamental assumption is that the upper class (i.e. larger apparent sizes) is given by the largest grains cut near their equatorial planes. When cutting a particle of radius $R$ at a random latitude, the probability of finding an apparent radius $r$ comprised in between $r_1$ and $r_2$ is given by:

$$P\left(r_1<r<r_2\right)=\frac{1}{R}\left(\sqrt{R^2-r_1^2}-\sqrt{R^2-r_2^2}\right).$$

Eq. (5) is first used for evaluating the number of grains with radius $R=(r_1+r_2)∕2$, $r_1$ and $r_2$ being the limits of the upper class. Then, (5) can be used for each other class to subtract the contribution of grains with size $R$ in the overall histogram. Finally, the previous steps are reiterated, starting from the (remaining) second upper class, and so on.

The aim of this work is to establish continuous inverse functions for predicting the radius distribution parameters (denoted $E$ and $\sigma$ below) based on 3D or 2D observations. Fig. 1 schematically illustrates the method used in this paper for this purpose: RCPs have been computed with different distributions for the sphere radii, then RV tessellations have been carried out on each RCP. Finally, the geometries of the RV cells have been statistically evaluated. The procedure used in this work for generating the geometries is detailed in Section 2. Section 3 investigates the geometry of each RCP. A continuous relationship between the radius distribution of the spheres and that of the cell volumes is proposed in this section. In Section 4, the bias introduced when one attempts to characterize the grain size from 2D sections is studied and methods for evaluating the 3D distribution based on 2D sections are proposed. In order to assess those methods, a numerical microstructure, representative of a real one, has been generated; those microstructures are compared in Section 5.