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 [15]. The Voronoi tessellation can also be obtained [15] 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 [25]. Of course physical constraints, such as minimization 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.

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 [26,7]. 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 120º, and four struts join at 109.5º). In order for this to be true, the faces and edges are slightly curved [26], unlike those of the tetrakaidecahedral cell model.

To generate foams with a roughly uniform cell size we use 122 seed points corresponding to the center's of close-packed (fraction 0.511) hard spheres in thermal equilibrium [18]. A pixel in the digital model is defined as belonging to a face if it is approximately equidistant from at least two sphere centers. The density of the model is changed by varying the thickness of the cell faces. An illustration of the model (with only 63 cells) is shown in Fig. 2.

Figure 2: The Voronoi tessellation model of a closed cell foam. The reduced density is / _s 0.18. The model shown has 63 cells, whereas the computations were performed on samples with 122 cells.

Using M=128 pixels in each direction to resolve the structure, and a wall thickness of two pixels, the minimum density obtainable (using 122 cells) was / _s = 0.16. In order to examine the stiffness at lower densities we also generated samples with 26 cells. We found that foams of 26 and 122 cells had the same stiffness (within 1 %), indicating that finite size effects are very small for the model. The results are given in Table 1 and plotted in Fig. 3. In the low density limit the Young's modulus of the closed-cell model can be described to within a 1.5 % relative error by,

$$\frac EE_s = 0.563 \left(\frac{\rho}{\rho_s} \right)^{1.19} \rm {for}\;\; 0.1 < \frac\rho\rho_s < 0.3.$$ (9)

This simple scaling relation cannot reproduce the high density behavior (E E_s as _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

$$\frac E{E_s}= \left( \frac{ p-p_0 }{p - p_0}\right)^m\;\; (p=\frac \rho\rho_s),$$ (10)

which has been found to be useful for describing the properties at high densities. With m=2.09 and p₀=-0.140 Eq. (10) describes the data to within 4 % for 0.15 < / _s < 1.

Table 1: Properties of the closed cell Voronoi tessellation model. Data at the two lowest densities were obtained for 3 realizations of the 26 cell model, and the remainder were obtained for 5 realizations of the 122 cell model. The relative density of the cell-edges ( / _s )_o was obtained by deleting all the faces in the closed-cell models.

/ s	E/Es	( / s )o	$\phi$	M
0.104	0.038	0.016	0.15	128
0.137	0.052	0.028	0.21	96
0.165	0.065	0.045	0.27	128
0.216	0.090	0.077	0.36	96
0.255	0.11	0.l08	0.42	80
0.312	0.15	0.16	0.52	64
0.400	0.21	0.24	0.60	48
0.553	0.34	0.35	0.62	64
0.655	0.45	0.46	0.71	64
0.744	0.56	0.56	0.76	64
0.895	0.80	0.80	0.89	64

Figure 3: The Young's ($\circ$) modulus of the Voronoi tessellation
($\circ$) and Gaussian random field model ($\triangle$). The solid lines are the empirical fits to the data given in Eqs. (9) and (11), respectively. The dashed lines are the fits to Eq. (10).