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Results

Figure 2 shows a series of steps in a simulated sintering process. Each picture shows four 100 pixel by 100 pixel slices from the original 100³ model. The slices are taken at depths of 45, 46, 56, and 55 pixels going clockwise from the upper left slice. Figure 2a shows slices from the original monosize sphere packing. The spheres are nine pixels in diameter, and are centered on pixels. The counting sphere used to compute local curvature is five pixels in diameter and 400 pixels are moved each cycle. Fig. 2b shows the same slices after 100 sintering cycles. Many inter-particle necks have begun to form. Fig. 2c shows the effect of 200 cycles, with thickening of the necks and partial agglomeration of groups of particles. Fig. 2d shows the four slices after 400 cycles, where many of the closest packed particles have almost totally agglomerated, leaving some large pores where the random particle packing was originally looser. After 575 cycles, all internal pores have filled-in, as can be seen in Fig. 2e, with only external roughness remaining. Finally, in Fig. 2f, after 800 cycles, the external shape is approaching its equilibrium spherical shape of uniform curvature.

Figure 2a: Showing four 100 pixel by 100 pixel slices from the original 100³ pixel model after 0 sintering cycles. The original porosity was 53.4%.

Figure 2b: 100 sintering cycles.

Figure 2c: 200 sintering cycles.

Figure 2d: 400 sintering cycles.

Figure 2e: 575 sintering cycles.

Figure 2f: 800 sintering cycles.

The percolation threshold of the pore space in porous materials that are changing in time can be an important property. The percolation threshold is defined as the value of the porosity at which the pore space changes from being connected (through-sample paths exist) to disconnected (no through- sample paths exist) [6]. The percolation threshold during sintering for powdered metal and ceramic compacts has been rarely studied, although some experimental results are available [22,23,24].

In the model presented above, since all phases are represented by pixels on a cubic lattice, it becomes possible to assess the degree of connectivity of any material phase, including pore space, using well-known lattice percolation techniques [6]. In Ref. [6], the cement paste model had periodic boundary conditions and maintained a constant volume at all times, so the percolation threshold could be defined as that point at which connectivity was lost between opposite faces of the periodic cell. In the present case, as the material shrinks to its final spherical form, a shell of empty space forms between the solid and the original boundaries of the packing box, so it becomes difficult to define appropriate starting and ending points for determining bulk connectivity. Thus, we have chosen to use an alternate approach, where we calculate the site-weighted average cluster size [25] of the pore space, and graph this as a function of overall internal porosity in Fig. 3. Internal porosity has been defined as any porosity within a fixed distance (diameter) of the center of mass of the solid phase. For the results shown in Fig. 3, this distance was set at 42 pixels. If all the particles were consolidated into one sphere, this sphere would have a radius of 48 pixels. Setting the limit at 42 pixels excludes extraneous surface pores. Percolation theory predicts a sharp break in the site- weighted average cluster size at the percolation threshold [25]. In Fig. 3, a sharp break occurs at about 15% porosity, for both of the two different initial particle packings used. We then take the pore space percolation threshold to be about 15 ± 3%. This value is not too much bigger than the experimental value of about 9% found for Cu, Ni, and UO₂ [22,23]. Our threshold is probably systematically higher due to the fact that other processes, involving elastic forces and grain boundaries, are operating in real sintering situations, which are not included in our model. It should be noted that the way that the sintering algorithm redistributes pixels is not changed after the pore space becomes disconnected. In reality, after the pore space becomes discontinuous, material could only be transported from external surfaces to internal pores via volume or grain boundary diffusion, and not by evaporation/condensation or surface diffusion mechanisms.

Figure 3: Showing the site-weighted average cluster size for the pore space of the model as a function of porosity. The sharp break in the data indicates the percolation threshold.