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# Simulations: Qualitative results

We turn now to more qualitative simulations, where we show model results for configurations of surfaces that are important in real experimental situations. Figure 10a shows the case of two isolated circles, with the diameter of the larger being 21 pixels, and the diameter of the smaller being 17 pixels, so that the initial curvature difference is 1/17 - 1/21 = 0.011. Since the circles are isolated, mass transfer can only take place through the vapor phase. This case is often referred to as coarsening [1]. Figure 10b shows the same system after 150 pixels have been moved, with n=15 being moved each cycle. A b=11 curvature box was used. The model has moved material from the smaller diameter, higher curvature circle to the larger diameter, lower curvature circle, increasing the curvature difference. Note that local equilibrium is maintained for each particle, as they individually remain quite circular. Figure 10c shows the system after N=300 pixels have been moved. The smaller circle is now almost gone, and will remain only a few more cycles.

Figure 10: Showing two isolated circles, of original diameter 17 and 21 pixels.

(a) Before (n=15,b=11, N=) mass transfer

(b) During (N=150) mass transfer

(d) Mass transfer is complete.

Figure 11a shows the case of two 21-pixel diameter circles that are just touching at a point. Because of the digital nature of the circles, they actually touch at a several-pixel face. This is an important case in sintering, because it simulates a two-particle contact in a random particle compact. It should be noted that there are no grain boundaries in the model at present. However, a grain boundary energy could easily be incorporated by including a site-site mismatch term in the counting [13,14].

After N=20 pixels have been moved (b=7, n=1), the system now looks like the picture shown in Fig. 11b. The negative curvature neck region has begun to fill in using material transported from the positive curvature outer particle surfaces. Figure 11c shows the system after N=100 pixels have been moved. An ellipsoidal shape has resulted. We have checked that this system's equilibrium shape is a circle, with uniform curvature (within the resolution of the counting box), which implies that the particle centers have been moved by the model.

Figure 11: Showing two 21-pixel digital circles that are originally just touching.

A small random packing of 15-pixel diameter circles is shown in Fig. 12a, before any material transport. There are a number of inter-particle necks and internal pores. The model was run for this packing with b=7 and n=1. Figure 12b shows the system after N=35. Five out of the original seven internal pores have closed up, due to material being transported into the interior from the exterior. The necks have noticeably thickened. In Fig. 12c, there is only one internal pore left after N=150, and the rest of the particles have densified considerably. Finally, in Fig. 12d, all internal pores have been filled in, and the external surface is much smoother. This system will also achieve the equilibrium shape of one circular particle.

Figure 12: Showing a collection of randomly packed 15-pixel diameter digital circular particles (b=7, n=1).

(d) N=250

It is interesting to note that vapor phase transport is not usually considered to contribute to densification during sintering [25]. In the case of vapor-transport-limited diffusion, all local curvature gradients would be relaxed on a short time scale and then material transport would proceed very slowly owing to small differences in curvature separated by distances large compared to the characteristic diffusion length. However, in this model of interface-limited coarsening, local gradients in curvature are relaxed at the same rate as curvature differences between different particles. The critical diffusion length is infinite and densification can and does proceed.

As our final qualitative illustration of the model results, we show in Fig. 13 the effect of alternating hot and cold zones on an initially smooth planar solid surface. In Fig. 13, the zones are arranged, from left to right, as hot-cold-hot-cold. Figure 13a shows the flat equilibrium surface before temperature differences were applied. Figure 13b shows the results after N=1000 pixels had been moved (n=10, b=7) from the originally flat surface. Material from the hot zones has been removed, and deposited in the cold zones. The surface in each zone has obtained uniform "curvature", so as to be in local equilibrium within the zone.

Figure 13: Showing the effect of hot-cold temperature zones on an originally planar solid-air interface. The termperature zones are arranged from left to right as hot-cold-hot-cold.

The effect of different temperatures was simulated by adding a fixed number to the calculated hot zone curvatures. In this way we can simulate the Gibbs-Thomson effect, since the chemical potential at the surface will be µ = µ(T) + γΩ K, where µ(T) is the potential at the flat interface at temperature T, γ is the surface tension of an isotropic interface, Ω is the volume of the diffusing species, and K is the curvature. Adding a constant to the counting is equivalent to changing the temperature via the relation µ(T). The equilibrium structure will be governed by

where the constant C is fixed by the assigned difference in counting in the simulation.

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