3. Densification behavior of pore arrays lying on a grain boundary at final-stage sintering
3.8. Pore shrinkage rate as a function of the grain boundary diffusion rate and surface diffusion rate 68
3.8. Pore shrinkage rate as a function of the grain boundary diffusion rate and
as shown in Fig. 3.16 (d). It is found that the linear fitted have slopes, 1/n, from 0.17 to 0.25, which corresponds to n ~ 4 to 6. It is considered that the fluctuation in the value exists because of the small number of simulation attempts (i.e., 3 independent simulations for each case).
The observation above signifies that the normalized pore shrinkage rate, CDgb/l4, is proportional to Dhi1/nDgb. Based on the result, the normalized pore shrinkage rate can be defined to be C'Dhi1/nDgb/l4, where C' is a constant. By approximating n = 5, (C'/l4) values for all 36 different cases are calculated. It is apparent from Fig. 3.16 (e) that the values form a quite narrow bell-shaped distribution with only a couple of scattered points, which strongly suggests that the slope is proportional to Dhi1/nDgb. Using the average value of C'/l4(= 36.3´10-10) and replacing l = 45, C' is found to be ~ 0.015.
Fig. 3.14 The normalized pore shrinkages in the linear scale for 36 different combinations of Dgb
and Dhi: Each of Dgb and Dhi is varied with 0.1, 0.3, 0.5, 0.7, 0.9, and 1.0. It is found that all results show the linear relationship between the normalized pore shrinkage and the real simulation time, and for each constant Dhi, the normalized shrinkage of the pore occurs faster with higher Dgb in real
Fig. 3.15. Normalized pore shrinkages against the real simulation time in logarithmic scale. All of the shrinkage lines for all combinations of Dgb and Dhi in Fig. 3.14 cluster on one line with slope of unity in log-log scale.
Fig. 3.16. Normalized pore shrinkage rate against the real simulation time (t), defined as C(Dgb/l4)t (= (R0/l)3 – (Rt/l)3), as a function of (a) Dgb and (b) Dhi. Log-log scaled data in (a) and (b) are presented in (c) and (d), respectively. From the graphs, it is clear that the normalized pore shrinkage rate is proportional to Dhi1/nDgb and can be re-defined as C'Dhi1/nDgb/l4. By approximating n = 5 for all 36 different D and D sets, CD /l4 is found to form a bell-shaped curve (e) with an average
3.9. Conclusions
In this chapter, a new two-dimensional Monte Carlo (MC) model, incorporating the “spin- dynamics” with a spin-configuration-dependent rate, has been proposed for the evolution of a two-phase microstructure in the context of pore shrinkage and concurrent microstructural densification. The proposed model was explained in detail and the robustness of the model was assessed by simulating the case of the pore densification on a grain boundary in a bi-crystal via grain boundary diffusion. The pore shrinkage kinetics was found to follow what previous studies predicted, while the key features of the microstructural evolution were simultaneously realized. By examining the changes in spin configurations occurring during the “spin-dynamics” processes, it was possible to control the individual diffusion rates and to achieve the densification of the pore and the concurrent microstructural evolution under the variable parameters. Also, upon counting the number of successes for each category of the
“spin-dynamics” process, we were able to explain quantitatively why the sintering kinetics for smaller Dgb was faster than for higher Dgb with a fixed Dhi in a normalized, reduced time frame.
We also investigated the influence on sintering behaviors of the degree of grain boundary motion, of the grain boundary energy to surface energy ratio, of the simulation temperature and of the pore interactions in an array. For all variations in such parameters and conditions, the normalized pore shrinkage kinetics was found to be faster for smaller Dgb. The pore shrinkage rate was higher for the Fluctuating GB than for Flat GB, signifying that the enhanced grain boundary motion helps in attaining the local equilibrium at the junctions, resulting in the increased net outward flux of vacancies onto the grain boundary. The pore shrinkage rate increased as the temperature and grain boundary energy increased. The model successfully simulated the equilibrium of the pore shape and the pore-grain junction with variable grain boundary energy to surface energy ratios. The interactions of the pores, such as coarsening and coalescence, in an array was found to slow the pore densification. As the number of pores remained decreased, the normalized pore shrinkage rate decreased, but maintained the linear dependence on the reduced time, signifying that model successfully predicted both densification kinetics and the microstructural evolution at the same time.
The results on the normalized pore densification were consistent for all sets of variable parameters such that the more elongated the pore shape was and/or the shaper the pore tip region was, the faster pore shrinkage kinetics was obtained. It was found that such an equilibrated, sharp morphology of the pore-grain junction was achieved better when the grain boundary diffusion rate is smaller than the surface diffusion rate, and when the grain boundary mobility is higher. Also, the more elongated, lenticular shape of the pore at the junction was developed as the grain boundary energy to surface energy ratio increased. The results in the real and/or reduced time frame were confirmed by the increase in the detachment rate of the vacancy from the pore onto the grain boundary, the re-attachment rate of the vacancy from the grain boundary to the pore and the net detachment rate of the vacancies
from the pore onto the grain boundary for all sets of variable parameters. In addition, we investigated the effect of grain boundary diffusion rate, 𝐷'(, and surface diffusion rate, 𝐷71, on the normalized pore shrinkage rate, C(Dgb/l4)t (= (R0/l)3 – (Rt/l)3), using the simulation results. It was suggested that the normalized pore shrinkage rate is proportional to Dhi1/nDgb, where average n was found to be ~ 5.
Even though the proposed model has successfully simulated both the densification kinetics of the pore on a grain boundary in a bi-crystal via grain boundary diffusion and the concurrent microstructural densification, it is worthy of mentioning that the model realized those phenomena as a result of the stochastic minimization of interaction energy during the simulations. One may argue that the proposed MC model does not incorporate the physics that the diffusion rate is a function of both vacancy creation and vacancy migration, usually defined as Dtype= D0exp(-EcA/ kT)exp(-EmA / kT), where D0 is a constant, EcA is the activation energy for vacancy formation and EmA is the activation energy for vacancy migration. However, in the proposed MC model, the energy state of a vacancy is subject to change during the spin-dynamics process and the corresponding probability of accepting the process is expressed as exp(-DE /kT) as in the classical Metropolis algorithm of Eq. (2-28), where DE is the change in total energy of the system due to the spin-dynamics process. Then, the net flux of mutual diffusion of vacancy-matter is assumed to be proportional to D0exp(-EcA/ kT)exp(-EmA / kT)exp(-DE /kT). Here, in the proposed model, D0exp(-EcA/ kT)exp(-EmA / kT) is set to be constant as a function of diffusion path even though it may be rather drastic. One of the examples of such a simplification can be found in the MC studies for examining the effect of the anisotropy in grain boundary mobility or/and energy on the grain growth and recrystallization behavior, where grain boundary mobility or/and energy has constant values as a function of grain boundary misorientation, acting as a rate controlling barrier in the Metropolis algorithm [43, 45]. Therefore, even though the proposed MC model is short of expressing the exact physical mechanism for the diffusion of vacancies, the microstructural evolution generated by the driving force of the model, i.e., the minimization of total interaction energy of the system, is found to accompany inherently the kinetics of the shrinkage of the pore via grain boundary diffusion.
To current authors’ best knowledge, the relationship between the pore-microstructure densification behavior and the variable parameters we present here has rarely been clarified using a mesoscopic simulation method. Based on the results shown in this study, it is of immediate interest to apply the proposed model to other cases of solid-state sintering where multiple particles (or grains) and pores are consolidated via various vacancy diffusion modes. Eventually, it is of great interest to assess quantitatively the relative contribution of individual diffusion mechanisms to the densification behavior and kinetics of a given two-phase microstructure with many pores and grains by modulating the