Gaussian random fields

The Voronoi tessellation has regular cells with perfectly flat walls. Since some solids have irregularly shaped cells we consider a model based on Gaussian random fields (GRF), which shows a large variation in cell shapes and sizes. To generate the model, one starts with a GRF field y(r) which assigns a (spatially correlated) random number to each point in space. A two-phase solid-pore model [16,27] can be defined by letting the region in space where − β < y(r) < β be solid, while the remainder [ $\vert y({\bf r})\vert \geq \beta$] corresponds to the pore-space. A closed-cell model can be obtained from the model by forming the union set of two statistically independent level cut GRF models [28]. An example is shown in Fig. 8. Details for generating the models have been previously described [29].

Figure 8: The closed-cell Gaussian random field model with reduced density ρ / ρ _s ≈ 0.2.

The closed cell GRF model has very irregular cells with strongly curved walls, which are known to have a strong influence of the overall properties of foams [12]. The Young's modulus of the model can be described to within a 2 % relative error by,

$$\frac EE_s = 0.694 \left(\frac{\rho}{\rho_s} \right)^{1.54} \rm {for}\;\; 0.15 < \frac\rho\rho_s < 0.4$$ (11)

in the low density regime, and Eq. (10) with m=2.30 and p₀ = −0.121 for 0.15 < ρ / ρ_s < 1 (relative error 3 %). As in the case of the closed-cell Voronoi tessellation, the difficulty of resolving the very thin cell walls prohibits lower densities from being studied at present.