3. Densification behavior of pore arrays lying on a grain boundary at final-stage sintering
3.3. Validation of the model: pore densification via grain boundary diffusion under variable grain
For the validation of the proposed model, the same variable parameters are used for the simulations of the pore densification on a grain boundary in a bi-crystal through grain boundary diffusion as previously reported [14]. Both the grain boundary energy, Egb, and the surface energy, Ehi, are defined to be equal to 1.0 (with an arbitrary unit), the surface diffusion rate, Dhi, is equal to 1.0, variable initial interpore distances, l, are equal to 45 or 90 (in pixel), and variable grain boundary diffusion rates, Dgb, are 1.0 or 0.1. Grain growth is simulated with the solid diffusion rate across the grain boundary, Dgbm, of 0.1 and the simulation temperature for the grain growth, kT, of 0.0 so that the new configuration from the “spin-flip” process is accepted with a success probability of 0.1 whenever it lowers the energy of the system. For “spin-exchange” processes, the simulation temperature, 𝑘𝑇, is set to be 0.8, equivalent to ~ 0.5 kTc disordering the microstructure.
In Fig. 3.2., the temporal evolution of the pore on a grain boundary in a bi-crystal during the densification is presented when Dgb = 1.0 ((a) to (d)) or Dgb = 0.1 ((e) to (h)), respectively. Again, a circular pore with a radius of 6 (in pixel) is placed on grain boundary of a bi-crystal system, realized on the simulation domain with a size of 90 ´ 90 (in pixel). Therefore, the pore distance, l, is 45 in pixel in the figures. The microstructures are chosen at 10% ((a) and (e)), 40% ((b) and (f)), 60% ((c) and (g)) and 90% ((d) and (h)) of the pore shrinkage defined as [1.0 - (R03 – Rt3)/R03]´100, for each Dgb case, where Rt and R0 are the radii of the pore at time t and zero, respectively. As expected, the smaller grain boundary diffusion rate is (Dgb = 0.1), the much slower pore densification is obtained in the real simulation time scale, MCS. Note that it takes ~ 9 times longer for the simulation with smaller Dgb to accomplish 90% of the pore shrinkage (please refer to the annotated MCS numbers in the figures (d) and (h)). From the figures, it is apparent that both shrinkage and densification of the microstructure (manifested by the increase in green area and by the decrease in red area) occur by the grain boundary diffusion of vacancies. It is also found that, during the densification, the pore constantly moves along
and the grain boundary migration whenever the area of the pore and the local morphology of the junction changes due to the densification process. Note that, when Dgb = 0.1 (from (e) to (h)), the pore attains the equilibrium, lenticular shape while the local equilibrium at the junctions is established with the dihedral angle close to ~ 120 º, as obvious from the imposed condition that Egb = Ehi = 1.0. Also, note that the equilibrated “cusps” between the sink pores (i.e., free space) and the grain boundary are formed, as predicted by Burke [3]. However, when Dgb = 1.0, the centered pore has slightly rougher surface and less elongated shape than that obtained from the simulation with Dgb = 0.1. This was suggested by Hassold et al. [14], noticing that a pore might have less complete equilibrated shape during the densification with higher Dgb since the microstructures were found to sinter more slowly in a normalized time scale (not in a real time scale) with higher Dgb, with a fixed Dhi of 1.0. Unfortunately, the mechanism of the enhanced sinterability due to the equilibrium of the pore and the junction has not been clearly elucidated in the work [14] (this issue will be explored later in Chapter 3.8). Nevertheless, it is qualitatively shown in Fig. 3.4 that relatively slower surface diffusion rate with higher Dgb (= 1.0) makes it more difficult for the pore and the junction to re-establish the equilibrium than with smaller Dgb (= 0.1). Note here that the local roughness of the grain boundary near the junction is also increased for higher Dgb ((a), (b), (c) and (d)). This signifies that the grain boundary is subject to migrate relatively farther to attain the local equilibrium at junctions. Such long-range motion of the grain boundary compensates relatively slower shape adjustment of the pore near the junction through the surface diffusion, resulting in rougher morphology of the grain boundary near the junction.
The kinetics of the pore shrinkage through grain boundary diffusion is known to obey the Eq.
(2-23), R03-Rt3 = C(Dgb/l)t [14, 25], where Rt and R0 are the radii of the pore at time t and zero, respectively. C is a linear constant. In Figs. 3.3 (a) and (b), the data up to 90 % of pore shrinkage, averaged from 3 independent simulations for each set of variable grain boundary diffusion rates (Dgb = 1.0 and 0.1) and initial pore distances (l = 45 and 90), are plotted by normalizing Eq. (2-23) by l3 for direct comparison of the simulation results with the prediction from the Eq. (2-24), (R0/l)3 – (Rt/l)3 = C(Dgb/l4)t. Fig. 3.3 (b) is the magnified image of the area marked by a square in Fig. 3.3 (a). For all subsequent figures error bars are constructed using the maximum and minimum values of 3 independent simulations for each case. Note that the data show the linear dependence of the normalized pore shrinkage, (R0/l)3 – (Rt/l)3, on the reduced time, C(Dgb/l4)t. Also, as is apparent from Figs. 3.3 (a) and (b), the normalized shrinkage rate (i.e., slope of linearly fitted lines of the data) for Dgb = 0.1 is higher than that for Dgb = 1.0 for each fixed l. This is consistent with the result that the simulation with higher Dgb (= 1.0) completes 90% of the pore shrinkage only 9 times faster (not 10 times) than that with smaller Dgb (= 0.1) in the real simulation time scale, as shown in Fig. 3.2. The slopes of the linearly fitted lines in Figs. 3.2 (a) and (b) are plotted in Fig. 3.3 (c) for each set of Dgb and l, reassuring that, with a fixed l, microstructures sinter slightly faster with lower Dgb in the normalized units, as previously reported
[14]. Also, the data fall onto each other along a straight line in log-log scale (Fig. 3.3 (d)), as previously reported [14]. Note that the current simulation (solid and open marks) completes faster than the previous model (dashed lines), while maintaining the slope of the line close to ~ 1.0 [14]. In addition, the effect of the spatial resolution of the domain on simulation results is also examined to validate the results. It is found that the current simulation domain size gives converged, stable prediction of the universal log- log linearity between normalized pore shrinkage vs. normalized time.
As mentioned before, clear explanation for faster normalized sintering kinetics with smaller Dgb
was not given in the previous study [14]. To analyze quantitatively the dependence of the pore shrinkage rate on Dgb with a given l, the numbers of successful processes in which a vacancy detaches from the tip region of the centered pore onto the grain boundary, NvD (i.e., number of the successful spin- exchange processes of (B or C) à D in Fig. 2.17, categorized as “the other cases” in Table 2.2), and a randomly walking vacancy on the grain boundary re-attaches to the tip region of the pore, NvRA (i.e., number of the successful spin-exchange processes of D à (B or C) in Fig. 2.17, categorized as “the other cases” in Table 2.2), are counted for the entire simulations. The inner vapor phase identically has -1 spin, but classifying the pore and vacancies is available by the “Burn algorithm” which is method to inspect spins of neighboring sites for -1 spin and list up the consecutive neighboring sites as independent area. As results of the burn algorithm, the number of pore pixels is considered as pore area during 1 MCS. When some pore pixels are detached, the change in the pore area is positive and its difference is exactly equal to the number of detached vacancies (Apore_before - Apore_after > 0 and DA = NvD). In other case, when the re-attachment of the detached vacancies onto heterophase interfaces of a pore occurs, the change in the pore area is negative and exactly equal to the number of re-attached vacancies (Apore_before
- Apore_after < 0 and DA = NvRA). Net detachment is defined as deviation in the NvD and NvRA. Rate of the NvD and NvRA is normalized by Dgb (i.e., GvD = NvD/(Dgbttotal) and GvRA = NvRA/(Dgbttotal)). The deviation in the densification rates of the high and low Dgb is clearly explicated by the net detachment rate (GvD - GvRA).
A schematic of such processes is shown in Fig. 3.4 (a), in which a vacancy detaches from the tip region of the pore onto the grain boundary (marked by a white arrow) and a vacancy on the grain boundary re-attaches to the tip region of the pore (marked by a black arrow). From Fig. 3.4 (b), it is found that both GvD and GvRA are higher when Dgb is smaller (= 0.1) for each variable l. Together with the conclusions obtained from the examination of the morphological evolution in Fig. 3.2, the result suggests that more sharply defined and rapidly equilibrated shape of the pore at the pore-grain junction attained from smaller D in the reduced time scale enhances both the detachment and the re-attachment
tip and the junction is sharper, resulting in smaller energy difference accompanied by the corresponding spin configurational change. Since the local equilibrium is attained relatively faster for smaller Dgb case, the result that the more vacancy detachments occur in the reduced time frame is reasonable. As the number of detachment increases, it is also reasonable to expect that the number of re-attachments increases in proportion. However, when a randomly walking vacancy on a grain boundary attempts to coarsen the pore by the re-attachment onto the region of the rapidly equilibrated pore, the corresponding rate will be decreased due to the higher curvature of the pore surface at the junction established with smaller Dgb. This will lead to the enhanced net rate of detachment of the vacancy from the pore, leading to faster sintering kinetics. The differences between both rates, GvD and GvRA, for all sets of Dgb and l are plotted in Fig. 3.4 (c), confirming that net detachment rate is bigger for smaller Dgb (= 0.1), resulting in faster sintering kinetics than Dgb = 1.0 for each case that variable l = 45 or 90 under the constant surface diffusion rate, Dhi, of 1.0. If diffusivity through grain boundaries, Dgb, is a simple function of densification time, the final time when the densification is complete might be reversely multiple proportion of l/Dgb. However, it is consistent with the results that the densification kinetics with Dgb of 0.1 is almost 9 times of that with Dgb of 1.0 in the real simulation time scale. In addition, all slopes of the normalized pore shrinkage close to unity in the logarithmic scale as the results of previous study by Hassold et al. [14].
Fig. 3.2. Temporal evolution of microstructures with l = 45 for the cases of ((a) to (d)) Dgb = 1.0 and ((e) to (h)) Dgb = 0.1 presented at 10%, 40%, 60% and 90% of the pore shrinkage, respectively. It is found that the pore densification kinetics is slower for Dgb = 0.1 than for Dgb = 1.0 about 9 times in real simulation time scale (MCS). During the densification, the pore constantly moves along the grain boundary and changes the shape due to the surface diffusion and the interaction with the grain boundary to maintain the equilibrium in shape and at junction. Note that the simulation with higher Dgb features less equilibrated pore shape and rougher grain boundary morphology than that with smaller Dgb.
Fig. 3.3. (a) and (b) normalized pore shrinkage kinetics obtained from simulations using different grain boundary diffusion rates (Dgb = 1.0 and 0.1) and initial pore distances (l = 45 and 90). (b) is the magnified image of the area marked by a square in (a). The data are averaged from the results of 3 independent simulations for each case, with error bars constructed using the maximum and minimum values. Note the normalized pore shrinkage, (R0/l)3 – (Rt/l)3, shows the linear dependence on the reduced time, C(Dgb/l4)t, as predicted by Eq. (2-24), when normalized by l4. As is apparent from (a), (b) and (c), the slope of the linear shrinkage line for Dgb = 0.1 is higher than that for Dgb = 1.0 for each case of variable l, reassuring that microstructures sinter slightly faster with lower 𝐷'(
in the normalized units, as reported by Hassold et al. [14]. The data from different cases cluster on a line with a slope of unity in log-log scale as shown in (d).
Fig. 3.4. (a) A schematic of the “spin-exchange” processes in which a vacancy detaches from the tip region of the pore onto the grain boundary (marked by a white arrow) and a vacancy on the grain boundary re-attaches to the tip region of the pore (marked by a black arrow). (b) Both detachment and re-attachment rates of vacancies, GvD and GvRA, defined as the number of successful processes, NvD and NvRA, divided by a reduced total simulation time, Dgbttotal, are higher for Dgb = 0.1 than Dgb= 1.0. In (c), the difference in both rates is plotted for each case, showing that net detachment rate is bigger for smaller Dgb, resulting in faster sintering kinetics.