Comparison of normalized pore shrinkage versus reduced time for both GB cases with variable Dgb shown in (a), and vacancy dissociation and reattachment rates and net vacancy dissociation rates shown in (b ) and (c), respectively. Snapshots of the temporal evolution of the microstructures shown in (a1) to (a5) and comparison of the normalized shrinkage of a group of 4 pores in a grain boundary via grain boundary diffusion with the variable Dgb shown in (b).
Introduction
Pore shrinkage and microstructure densification were successfully simulated using this Monte Carlo algorithm. The influence of microstructural variables on densification kinetics is then examined.
Theoretical background
Driving force for solid-state sintering
Pressure due to curved surfaces
Changes in chemical potentials and solubilities at curved surfaces
Wetting angle as a function of interface energies
Sintering kinetics at the initial stage of sintering: two-particle model
Then, DP, L, and A are estimated as gs/r, the radius of the neck curvature, and 2px2ds, respectively, where ds is the surface diffusion thickness. Schematic representation of an equal-sized two-particle model for the initial stage of solid-state sintering without shrinkage (i.e. the distance between centers of mass of two particles is not reduced), where a and x are the radii of the particle and the neck, respectively, and r is the radius of the neck curvature [8].
Shrinkage behavior of pore arrays at the final stage sintering
Densification of a pore among three particles at the final stage of sintering
The pore disappears completely at the last time due to the shape changes (from concave to convex curvature). Changes in the rate of pore shrinkage by the pore shape [16] and fitted from the experimental results for the concave curvature of the pore and from the geometric approach for convex curvature.
Monte Carlo model
Definition of simulation domains
Periodic boundary conditions allow the simulation domain to be used as a representative volume of material without the influence of the external surface of the microstructure (in Figure 2.15, the red box indicates one simulation domain and is repeated in all directions to illustrate the periodic boundary conditions used during the simulation). Scheme of changes in lattice anisotropy depending on the number of nearest neighbors.
Preparation of simulation domains as inputs
As the third verification case, two spherical particles (qgrain) that initially come into contact with each other are surrounded by the outer vapor phase (q = -2) as shown in Fig. The particle radius in three-dimensional space is prescribed and varied to investigate the effect of the dissolution of the simulation domain on the coarsening behavior during solid-state sintering. This structure can be used to investigate three-dimensional characteristics of the development of the internal pore structure as well as the consolidation of the solid wires.
The entire domain is enveloped by barriers except for the XY planes (ie, periodicity along the z direction is used).
Description about the proposed Monte Carlo Potts model
In the proposed model, there are only 8 unique categories of the spin configuration for a pixel which has a non-zero interfacial energy, as shown in Fig. the proposed MC model for a pixel surrounded by 8 nearest interacting neighbors: A. Precipitation of a vacancy through a grain at the heterophase interface (B), grain-pore junction (C) or grain boundary or junction (D).
Decomposition of a vacancy in a grain from heterophase interface (B), grain-pore junction. C) or grain boundary or junction (D) Dvdiss.
Densification behavior of pore arrays lying on a grain boundary at final-stage sintering
3.1 (a), the variation of n#sn is a function of pore radius and the values of n#sn approach 8 as kT increases. 3.1 (b), the change in n#sn of the microstructures at equilibrium is plotted against kT for a pore radius of 6 in pixel as large as the Hassold system. Also, the picture shows the views of microstructures balanced at kTs of 0.7, 1.2 and 1.6.
The change in n#sn is a function of pore radius and kTs as shown in (a), and images of equilibrated microstructures with pore radii of 6 per pixel at kTs of 0.7, 1.2, and 1.6 are shown also in (b).
Initial configuration and conditions for pore densification simulation
By imposing these conditions, not only the compaction of the pores but also the simultaneous shrinkage of the microstructure is generically simulated through the grain boundary diffusion of vacancies and their escape from the microstructure in contrast to the previous explicit MC models [14- 24]. Validation of the model: pore compaction via grain boundary diffusion under variable grain boundary diffusion rates, Dgb, and initial interpore distance, l.
Validation of the model: pore densification via grain boundary diffusion under variable grain
Note here that the local roughness of the grain boundary near the junction is also increased for higher Dgb ((a), (b), (c) and (d)). It is known that the kinetics of the pore shrinkage by grain boundary diffusion the Eq. 2. Note that the data shows the linear dependence of the normalized pore shrinkage, (R0/l)3 – (Rt/l)3, on the reduced time, C(Dgb/l4)t.
Satsen for NvD og NvRA normaliseres af Dgb (dvs. GvD = NvD/(Dgbttotal) og GvRA = NvRA/(Dgbttotal)).
Influence of the motion of the fluctuating grain boundary on the pore densification
3.5 (c), (d), (g) and (h)) due to increased grain boundary oscillatory motion at zero temperature. Nevertheless, it is found that the normalized pore shrinkage (R0/l)3 – (Rt/l)3 with respect to the reduced time (Dgb/l4)t for the fluctuating GB case is slightly faster than that for the flat GB case for variable Dgb. Note that GvD, GvRA, and GvD - GvRA, when Dgb is fixed, are larger for the fluctuating GB case than for the flat GB case, resulting in a faster pore shrinkage rate for the fluctuating GB case.
From the noted simulation times, MCS, the pore shrinkage for the Fluctuating GB case is found to be slightly faster than that for the Flat GB case for variable Dgb (= 0.1 and 1.0) on the real simulation time scale.
Influence of the simulation temperature on the pore densification
As clearly shown in the figure, the normalized rate of pore shrinkage decreases with decreasing simulation temperature (dark red through red to orange). As expected, a higher kT results in a pore shrinkage rate that is higher than a lower kT for each GB case on the real time scale of the simulation. In (d), the difference in the two rates, GvD - GvRA, is plotted for each set of variables of the type kT, Dgb, and GB, showing that the net detachment rate is greater for lower Dgb, for higher kT, and for fluctuating GB, which has resulting in faster sintering kinetics.
Note that the degree of pore shrinkage decreases with decreasing simulation temperature (dark red through red to orange).
Influence of the grain boundary energy to surface energy ratio on the pore densification
The data of different GB types for each Egb and Dgb set are plotted with the same mark. It is found that higher Egb (= 1.25) results in much faster pore shrinkage kinetics in the actual simulation time scale than lower Egb (= 0.75) for each GB case. In (d), the difference in both rates, GvD - GvRA, is plotted for each set of variable Egb, Dgb and GB type, showing that net release rate is greater for smaller Dgb, for higher Egb, and for the Fluctuating GB, leading to faster sintering kinetics.
Note that the shrinkage rate of the pores decreases as the grain boundary energy decreases (dark blue through blue to light blue).
Influence of the interactions between the pores in an array on the densification behavior
In the figure, the change in the number of pores is marked with the corresponding data in the plot. It is worth noting that the rate decreases as the number of pores decreases, which in turn causes the pore space to increase, and thus the normalized pore shrinkage rate to decrease by definition. Note that the normalized shrinkage rate of the pore group is much smaller than that of a single pore via grain boundary diffusion for the Dgb variable.
As the number of remaining pores decreases, the normalized rate of pore shrinkage decreases but retains a linear dependence on reduced time.
Pore shrinkage rate as a function of the grain boundary diffusion rate and surface diffusion rate 68
Pore shrinkage rate as a function of the grain boundary diffusion rate and surface diffusion rate. The pore shrinkage kinetics were found to follow previous studies predicted, while the key features of the microstructural evolution were simultaneously realized. The model successfully simulated the equilibrium of the pore shape and the pore-grain junction with variable grain boundary energy to surface energy ratios.
Also, the more elongated, lenticular shape of the pore at the transition was developed as the ratio of grain boundary energy to surface energy increased.
Densification behavior of contacting three particles
The pore shrinkage calculated by the DA shows a linear dependence on time, as shown in (a). Initially, pore shrinkage is facilitated by the sharp morphology of the pore-grain junction. The pore shrinkage is calculated from the changes in pore area and radius, shown in (a) and (b), respectively.
The difference of the pore shrinkage rates is explained by the movement of vacancies on the grain boundaries as shown in (d) and (e).
Views of the time evolution of the microstructures with the highest (1) and lowest (second row) Egb/Ehi (= 1.5 and 0.5), respectively. The neck and pore structures for all cases stabilize rapidly within 500 MCS and their morphologies are a function of Egb/Ehi as shown in (a) and (e). Kinetics of pore shrinkage under conditions of kT = 0.7, Dgb/Dhi = 1.0, and flat GBs as a function of the interface energy ratio, Egb/Ehi.
The decoupled vacancy rates (GvD - GvRA, GvD_escape and GvD_moving (= (GvD - GvRA) - GvD_escape)) shown in (c), (d) and (e), respectively, are attributed to the local balanced form at pore-grain junctions varied with Egb/Ehi.
Influence of the simulation temperature, kT, on the densification kinetics
It is expected that the difference in pore shrinkage rates could decrease as a function of the total diffusion rate (the control of each diffusion rate is the main feature of this model). The kinetics of thickening were found to be a function of Egb/Ehi and kT. The discontinuous curvature changes to continuous surfaces with surface diffusions far from the neck, as shown in Fig. 1b.
At low kT (= 1.0), structural stabilization occurs by surface diffusion (points emitting microstructural features are indicated by orange colored circles and a, b, c and d in (a)).
Coarsening kinetics and changes in interfacial energies of two particles of equal size according to interfacial energy ratio. The neck growth rate also depends on Egb/Ehi due to energy competition at the vapor-grain junctions. The shape of the particles is transformed by surface diffusion from the sphere to the amorphous morphologies.
The surface diffusions break the particle size balance, leading to the continuous neck growth to reduce the collective interfacial energy.
The influence of the particle size anisotropy on the particle coarsening
Coarsening kinetics and changes in the interfacial energies of the two unequal-sized particles as a function of the interfacial energy ratio. It means that the total Eh of the necks increases due to the neck growth, but the total Eh of the surfaces decreases due to the structural changes, thus reducing the Esys of the materials. When the Egb exists between particles, the coarsening rates are a function of the size difference.
When integration is completed, the surface energy of the merged particle is stabilized by the surface diffusion, leading to shape changes of the isotropic faceted object.
Conclusions
Conclusions, issues, and future work
Conclusions
The compaction kinetics are expected to be enhanced by the mass diffusions for the convex pore shape due to the increase in the solubility of vacancies in solids at the high simulation temperature. In the case of the unequal sized particles, the particles were incorporated by the thermally activated surface diffusion and the grain boundary migration. By doing three validation cases, the proposed model was successfully verified to embody the solid state sintering through the combination of the diffusion mechanisms.
Densification behavior and microstructural evolution will be predicted by the proposed model during solid state sintering.
Issues and future work
Vagnon, Numerical simulation of microstructural evolution during mesoscale sintering in a 3D powder compact, Comput. Behrens, Comparison between different numerical models of densification during solid-state sintering of pure aluminum powder, Prod. Hodkin, The relationship between grain boundary energy and surface energy of nuclear ceramics as determined from pore geometries, J.
Sawbridge, The effect of fission products on the ratio of grain boundary energy to surface energy in irradiated uranium dioxide, J.
