Voronoi tessellations

The most common models of cellular solids are generated by Voronoi tessellation of distributions of 'seed-points' in space. Around each seed there is a region of space that is closer to that seed than any other. This region defines the cell of a Voronoi (or Dirichlet) tessellation [Stoyan et al., 1995]. Placing a solid wall at each face of these cells results in a closed-cell Voronoi tessellation. An open-cell Voronoi tessellation results if only the edges where two cell-walls intersect are defined as solid. For several different random (e.g. Poisson) distributions of seed-points, the average number of faces per cell falls in the range 13.7-15.5 [Oger et al., 1996].

The Voronoi tessellation can also be obtained [Stoyan et al., 1995] by allowing spherical bubbles to grow with uniform velocity from each of the seed points. Where two bubbles touch, growth is halted at the contact point, but allowed to continue elsewhere. In this respect the tessellation is similar to the actual process of liquid foam formation [Van der Burg et al., 1997]. Of course, physical constraints such as minimisation of surface energy will also play an important role. Depending on the properties of the liquid and the processing conditions, the resultant solid foam will be comprised of open and/or closed cells.

The amount of order in the Voronoi tessellation depends on the order in the seed points. If regular arrays are used, ordered anisotropic foams will result. Indeed the open-cell models used by Warren and Kraynik [Warren & Kraynik, 1997], Zhu et al  [Zhu et al., 1997], and Ko [Ko, 1965] turn out to be equivalent to Voronoi tessellations of the BCC (Figure 1d), FCC (Figure 1c), and HCP lattices. If a purely random (Poisson) distribution of points is used, highly irregular isotropic foams containing a wide size distribution of large and small cells will result.

It is worth noting that the tessellation of the BCC array (the tetrakaidecahedral cell model discussed above) is a reasonable approximation to the foam introduced by Lord Kelvin [Weaire & Fortes, 1994,Warren & Kraynik, 1997,Grenestedt, 1999]. The cells of the Kelvin foam are uniformly shaped, fill space, and satisfy Plateau's law of foam equilibrium (three faces meet at angles of 120°, and four struts join at 109.5°). In order for this to be true, the faces and edges are slightly curved [Weaire & Fortes, 1994], unlike those of the tetrakaidecahedral cell model.

In this study, we wish to examine foams that have a roughly uniform cell size, but which are still random and isotropic. A non-periodic, evenly spaced, and isotropic arrangement of seed-points is therefore necessary. Such a distribution is provided by the centre points of equi-sized hard spheres in thermal equilibrium [Torquato, 1991]. If the spheres are quite closely packed, the Voronoi cell size will be approximately equal to the sphere diameter d ₀. We generate distributions by placing 122 spheres in a simple cubic array within a periodic cube of size 125d₀ ³ and allow the spheres to move by a Monte Carlo algorithm until an isotropic distribution is reached.

A pixel in the digital model is defined as belonging to a face (edge) if it is approximately equidistant from at least two (three) sphere centres. The density of the model is changed by varying the thickness of the cell edges. An illustration of the open-cell model (with only 63 cells) is shown in Figure 4(a).

Figure 4: Random models of open-cell solids. (a) The open-cell Voronoi tesselation with ρ / ρ_s =0.15. The model shown has 63 cells, whereas the computations were performed on samples with 125 cells. (b) The low-coordination number ($\bar z$=5.5) node-bond model with ρ / ρ_s=0.13. (c) A digital model of the high coordination number foam ( $\bar z\approx12$ ≈ 12) with ρ / ρ_s = 0.075. (d) The open-cell Gaussian random field model with reduced density ρ / ρ _s ≈> 0.1.

(a)

(b)

(c)

(d)

In the low density limit, the Young's modulus of the open-cell tessellation can be fitted by (equation 1) to within a maximum of 5 % relative error with the parameters C=0.930 and n=2.04 or

$$\frac EE_s = 0.930 \left(\frac{\rho}{\rho_s} \right)^{2.04} \rm {for}\;\; 0.04 < \frac\rho\rho_s < 0.5.$$ (8)

The FEM data and (equation 8) (solid line) are shown in Figure 5. This simple scaling relation cannot reproduce the high density behaviour (E  → E_s as ρ → ρ _s unless C is fortuitously equal to one). Rather than choosing a three- or four-parameter relation to describe the full density range, we instead use the equation

$$\frac E{E_s}= \left( \frac{ p-p_0 }{p - p_0}\right)^m\;\; (p=\frac \rho\rho_s),$$ (9)

which has found to be useful for describing the properties at high densities. With m=2.12 and p₀ =-0.0056, the formula describes the FEM data to within 5 % for 0.04 < ρ / ρ_s  < 1. The fit is shown on Fig 5 as a dashed line. Note that the fitting parameters p ₀ and m are not the conventional percolation threshold and exponent. However, since the actual percolation threshold of the Voronoi tessellation is expected to be zero, it is interesting, but perhaps fortuitous, that the value of p ₀ is quite small.

Figure 5: The Young's ($\circ$), bulk ($\Box$) and shear ($\triangle$) moduli of the open cell Voronoi tesselations. The lines are the empirical fits to the data given in the text (see (equation 8)).

Interestingly, just like the periodic models, the bulk modulus shows a near linear decrease with density. The low density limit of the bulk modulus can be described by ( equation 1) with C=0.209 and n=1.22. For the shear modulus C=0.404 and n=2.12. At the lowest density ( ρ / ρ _s = 0.05) the Poisson's ratio is relatively high (ν =  0.44), and the trend indicates that (ν → 0.5) as ρ / ρ_s → 0.

It is possible that we have retained some type of order in our models that is responsible for this behaviour. Specifically, the hard sphere equilibrium distribution, which underlies our tessellation, undergoes a 'liquid-solid' transition around a concentration of c=0.5 [Torquato, 1991], and our simulations were performed at a concentration of $c=122/125 \times \frac{\pi}{6}=0.511$. For comparison, the value $c=\frac{\pi}{6}=0.524$ corresponds to a close-packed simple cubic array of spheres. The solid side of the transition is characterised by regions (grains) of ordered spheres. Therefore, our models may have some long-range order and corresponding anisotropy. It is likely that the range of the order is actually quite small so close to the transition, but to check the result we performed several equilibrium simulations using 122 hard spheres of decreasing diameter. The lowest concentration was c=0.05, which corresponds to spheres of about half the size used in the original simulations. The open-cell Voronoi tessellations of these distributions showed a polydisperse size distribution since the seed points may now be bunched in regions (see Figure 7). We computed the Poisson's ratio of the open-cell tessellations at a solid density of ρ / ρ =0.05 and found only a 10 % decrease (from 0.44 to 0.4) at the lowest concentration. This indicates that the behaviour is not an artifact of our seed-point distributions, and seems to hold approximately for all isotropic Voronoi tessellations. The implications of this finding are discussed later.

Figure 7: The effect of sphere packing fraction on polydispersity in Voronoi tesselations (cross-sections of 3-D models). (a) monodisperse cells result from a relatively close-packed (fraction c=0.51) distibution of hard spheres (b) polydisperse cells based on the same number of hard spheres (with smaller radii) at a lower packing fraction (c=0.05). The increased polydispersity only decreased the Poisson's ratio by 10%.