4. Densification behavior of contacting three particles
4.3. Influence of the simulation temperature, kT, on the densification kinetics
The influence of the simulation temperature, kT, on the pore shrinkage is examined in Chapter 3.5 for a pore on a grain boundary. It is found that the roughness of the heterophase interfaces increases and the vacancy movement is much activated as the kT increases, resulting that the pore shrinkage rate is much facilitated at the high kT. To study effects of the kT on the densification behavior of the three- particle systems, the solid-state sintering is simulated under variable kT at constant Dgb/Dhi (=1.0), Egb/Ehi (=1.0), Dgbm (=0.01) and kTgg (=0.0). The kT is varied from 0.7 to 1.0. In this temperature range, the solid and vapor mass is dissolved into a volume of the other phase at the high kT above 1.0.
Furthermore, the simulations are run until 3.0ร105 MCS for all conditions to examine the particle coarsening after the pore completely shrinks.
In Fig. 4.7, snapshots of the microstructural evolution at the initial, final and 0.005 and 0.010 pore shrinkage values are presented. Those pore shrinkages are marked by the gray arrows in Fig. 4.8 (a). The 1st and 2nd rows are representative conditions of the low (kT = 0.7) and high temperature(kT = 1.0), respectively. It is found that the heterophase interfaces as well as the grain boundaries become rough for all kTs. However, the grain boundaries are corrugated by the vacancy movement and the flat morphologies are formed by the grain boundary migration. The grain boundaries are pinned by the vapor-grain junctions, and the solid mass diffuses by the heterophase interface diffusions across the junctions, resulting that the area fraction of particles is non-identical. The pore shrinkage rate is increased at the high kT as shown in the marked MCS. The irregular pore shape is observed in the snapshots because the pore shrinkage time is much faster than time to establish equilibrated morphologies (in this case, the equilibrated shape is triangle).
In Fig. 4.8, the influence of the kTs on the densification kinetics is presented. It is found that the pore shrinkage is a linear function of the time in the linear scale, and its rate increases as the kT increases as presented in Figs. 4.8 (a) and (b), respectively. As presented in Fig. 4.8 (c), the GvD - GvRA
for the high kT is much higher than that for the low kT because the local equilibrium at the pore-grain junctions is established for the short time by the activated heterophase interface diffusions. In addition, in Fig. 4.8 (d), a number of vacancies at the high kT diffuse along the grain boundaries at the same time, leading to the high GvD_escape. As presented in Figs. 4.8 (e) and (f), the necks rapidly grow at the initial time up to X/D ~ 0.35 to reduce the collective interface energy. After that, the neck growth rate depends on the kTs while the pore shrinks. It is apparent that the grain boundary length for the vacancy movement is increased and a few of vacancies simultaneously diffuse. After the pore shrinkage is done, the necks become the almost equilibrated structure as a function of the Egb/Ehi. Therefore, the X/D at the final time shows similar value (X/D: 0.4 ~ 0.5). Note that the equilibrated neck size is determined by the Egb/Ehi
and the neck growth rate is a function of the kT.
Fig. 4.7. Snapshots of the changes in microstructures against the simulation time at the given shrinkage (= initial, 0.005, 0.010, and final presented at each column) for the relatively low (1st) and high (2nd row) kTs. After the simulation for solid-state sintering starts, the morphologies of the solid particles and inner pore are changed by the given Egb/Ehi (= 1.0) and kT (i.e., the triangular pore shape is established at initial time) as shown in (a) and (e). Roughness of the heterophase interfaces increases at the high kT, but the flat grain boundaries are attained (kTgg = 0.0). Note that the high kT makes it easy to change area fraction of particles because the solid mass transfers by the thermally activated diffusions across the vapor-grain junctions.
Fig. 4.8. Kinetics of pore shrinkage and neck growth at the variable kT (= 0.7~1.0 with 0.5 intervals), and fixed Dgb/Dhi (= 1.0), Egb/Ehi (= 1.0), and Dgbm (=0.01). The kTgg is set to be 0.0 to make the energetically stable flat boundary. The pore shrinkage rates represent the linear relationship against KT as shown in (a) and (b). Note that movement of vacancies detached onto the grain boundaries (GvD - GvRA, and GvD_escape) is increased as kT increases as presented in (c) and (d). During the pore shrinkage, necks are gradually grown at low kT, and equilibrated neck structures are formed by the grain growth and surface diffusions as shown in (e). The equilibrated X/D is maintained due to the fixed Egb/Ehi (= 1.0) as presented in (f).
4.4. Influence of the diffusion through lattice on the densification kinetics
The diffusion types are embodied by the categorized spin-configurational changes in the proposed model and imposed by the pre-defined diffusion rates. In this session, the densification kinetics done by the grain boundary diffusions under variable ๐๐s is compared with results from the lattice and grain boundary diffusion mode during solid-state sintering. The diffusions through the particle lattice are named the bulk diffusion (Dvb), and vacancy detachment from the interfaces to volume of particles is imposed by the dissolution process (Dvdiss). The simulations are performed under variable kTs (= 0.7~1.0) at the fixed Dgb/Dhi (=1.0), Egb/Ehi (=1.0), Dvb (=1.0), Dvdiss (=1.0), Dgbm (=0.01) and kTgg (=0.0).
In Fig. 4.9, the pore shrinkage difference between two different diffusion modes is presented.
It is noticeable that the vacancy diffusion paths in solids are categorized as both grain boundary and bulk diffusions only, but the volume diffusions, (i.e., vacancies diffuse from a grain boundary to particle volume or from a particle volume onto a grain boundary) are found inherently in this model. All mass transportation is much activated at the high kT as shown in Fig. 4.9 (a), leading to the increase in the pore shrinkage rates compared with the results from the grain boundary diffusion. It is expected that the net detachment rate, GvD - GvRA, for the bulk and grain boundary diffusion mode is higher than that for the grain boundary diffusion only. However, the results are different from the expectation (i.e., the GvD
- GvRA for the grain boundary diffusion is higher than that for bulk diffusion) as presented in Fig. 4.9 (b).
To understand the reason why the pore shrinkage is enhanced for the bulk and grain boundary diffusions, the GvD_escape is examined as presented in Fig. 4.9 (c). It is found that the GvD_escape for the bulk and grain boundary diffusion mode is higher than that for the grain boundary diffusion, and this difference increases as the kT increases. Note that, the equilibrated pore-grain junctions make it easier for vacancies to be detached than curved heterophase interfaces, but the detached vacancies escape from the materials by the bulk diffusions because the equivalent spin configurations in the particle volume have the zero-energy dissipation for the vacancy movement. It is expected that the difference in the pore shrinkage rates might be reduced as a function of the bulk diffusion rate (Controlling each diffusion rate is the main characteristics in this model). For example, ratio of the grain boundary diffusivity and the lattice diffusivity is ~106 order for the typical FCC metals [61], suggesting that the grain boundary diffusion for the pore shrinkage is the main mechanism, and the pore shrinkage is facilitated by the lattice diffusions at the high temperature. Furthermore, the influence of the bulk diffusion on the densification kinetics will be enhanced for the low Egb/Ehi with the convex pore shape at high kT [39]
due to the Gibbs-Thompson relation.
Fig. 4.9. Difference of the pore shrinkage kinetics as a function the diffusion modes (i.e., GB diffusion and all possible diffusions). Pore shrinkage rates for the bulk and grain boundary diffusion mode are higher than that for the grain boundary diffusion as shown in (a). The difference of the GvD
- GvRA, and GvD_escape as shown in (b) and (c), respectively, is decreased for the bulk and grain boundary diffusion mode, suggesting that the number of moving vacancies for the bulk diffusion is lower than that for the grain boundary diffusion (i.e., detached vacancies diffuse in a volume of a particle and escape from materials to free space for the short time).
4.5. Conclusions
In this chapter, the densification behavior of three contacting particles during solid-state sintering at the final stage was explored to validate the proposed model for the enlarged microstructures.
The microstructures were concurrently evolved by the pore shrinkage and neck growth, and the densification kinetics was elucidated by qualitatively observing the microstructural changes against time and quantitatively investigating the microstructural characteristics and the vacancy movement during simulations based on the 1st validation studies.
The densification of the three-particle systems was performed at the fixed interface energy ratio, Egb/Ehi (=1.0), and low kT (=0.7) to understand the stochastic phenomena occurring by mass transportation for 5 trials of simulations. The pore shape was rapidly transformed from concave curvature to triangular morphologies, and the flat grain boundary was constructed to reduce the collective interface energy at initial time. It was apparent that the pore shrinkage calculated by the pore area change was a linear function of time as reported in Abdeljawad et al. [39], and the shrinkage behavior was dependent on the establishment of energetically stable structures at the pore-grain junctions. It was apparent that not only the net detached vacancies but also the moving vacancies contributed to the pore shrinkage. To simulate the stochastic mass transport, the Monte Carlo model was utilized, and the densification kinetics using the proposed model to control the diffusion paths and rates was different from the results by the vacancy annihilation process [16].
It was found that the densification kinetics was a function of the Egb/Ehi and kT. Especially, the neck length for the vacancy diffusion along the grain boundaries was determined by the interface energy ratio (from pore-grain junction to vapor-grain junction), and the flat grain boundaries were attained by the grain boundary migration whenever the grain boundaries were disrupted by the vacancy movement, suggesting that the microstructures were completely evolved by the categorized diffusion modes in the proposed model. It was noteworthy that the densification behavior of the three-particle systems was consistent with the results of the 1st validation case by investigating the increase in the net detachment rate of the vacancies at the locally equilibrated pore-grain junctions as a function of the interface energy ratio and simulation temperatures.
From the results by grain boundary diffusions, the influence of the bulk diffusions on the densification kinetics was clarified and apparent at the relatively high temperature (above half of the critical temperature) at the fixed interface energy ratio. Furthermore, vacancies detached onto grain boundaries were much favorable than dissolved into volume of a grain because the hetero-interface
5. Coarsening behavior of contacting two particles in three dimensions
In this chapter, coarsening behavior of two-particle systems is explored by the proposed model in three-dimensional space because results from the three-dimensional model are much accurate than that from the two-dimensional model. This model is validated by comparison with the previous coarsening theories. As mentioned in Chapter 2.5, the two-particle coarsening is considered at the initial stage sintering except for pore shrinkage (i.e., a pore is not isolated in particles, and material systems consist of the particles and vapor channels). It is known that the coarsening behavior of particles is characterized by the neck size ratio, X/D, where X and D are the temporal neck diameter and the initial particle diameter, respectively. If two spherical equal-size particles are incorporated to one sphere, the final X/D is equal to 1.2 by the geometrical approximation. The coarsening kinetics, expressed as (X/D)n ยต t, is a function of the diffusion mechanisms and the kinetic exponent, n, (1/n = slope in the log-log scale against time) is equal to 7 if two particles are consolidated by the surface diffusion. A method to measure the X as presented in Fig. 5.1 is considered for the discretized three-dimensional domains and the D is approximated from the initial particle volume. The neck-surface solid voxels (placed at the vapor-grain junctions) are found by checking the neighboring spin of all solid voxels whether a solid site has a different solid spin (qsolid โ qneighbor, the solid site is placed on a neck) and an outer vapor spin (qneighbor = -2, the solid site is placed on a heterophase interface) at once. For the neck-surface solid members, the mass center is calculated and a distance between the member and mass center is approximated to a circular neck radius, Y. Finally, the neck diameter, X, is obtained by two times of the neck radius (i.e., X = 2รY).
Fig. 5.1. Schematic of the method to calculate the neck size ratio from the discretized elements in three dimensions. The red and blue colors represent the solid particles. The green X is the mass center of the surface-neck solid members. The neck radius, Y, marked by the purple arrows is equal to the average distance between the member and mass center.
5.1. Coarsening behavior of contacting two particles as a function of particle size and simulation temperature
To examine the ideal coarsening behavior of contacting two spherical particles, the input microstructure, where the spherical particles with equal size initially contact with each other (X/D ~ 0.2 at the initial time) embedded by the outer vapor phase, is prepared. A line connecting the particle mass centers is parallel to the X axis. The particle size and kT are varied, and the kTgg is set to be 0.0 for the flat grain boundary. Rates of the heterophase interface diffusion and grain boundary migration (Dhi and Dgbm) are imposed to be 1.0 and 0.01, respectively.
As the 1st case, the ideal particle consolidation is considered. In the ideal condition, the crystallographic orientations of particles are identical so that the grain boundary energy is not imposed between particles (Egb = 0.0). The particle coarsening occurs by the only surface diffusions (Dhi = 1.0) in this work. The variable particle size is utilized (r = r1 = r2 = 5 ~ 20 in voxel) and the kT is varied with 1.0 to 4.0. All individual simulations were run for 5 times and their average was recorded as a result with an error bar.
In Fig. 5.2, linear dependence of particle size on the X/D (solid lines) against the time in the logarithmic scale is presented. It is found that the linear relationship is changed to a step function as the particle size increases (i.e., the X/D changes for large particles become almost 0. this is named the stagnation of the particle coarsening) at the low kT (=1.0) as shown in Fig. 5.2 (a), signifying that the small size particles are completely coarsened (= X/D reaches to the theoretical value of 1.2), but the large particles are not consolidated at the X/D ~ 0.95. However, the stagnant time is apparently reduced, and the theoretical X/D (=1.2) is obtained regardless of the particle size as the kTs increase from Figs.
5.2 (a) to (g).
To find the reason of the coarsening stagnation, microstructural features are examined for radius of 20 in voxel at the kT of 1.0 as presented in Fig. 5.3. The colors generally represent the different grains.
However, they have the identical crystallographic orientations (Egb = 0.0). At the initial time, small neck is rapidly expanded by supplement of solid mass via the surface diffusions near the neck, leading to the formation of step geometries as shown in Fig. 5.3 (a). The step length is a function of the particle size as shown in Fig. 5.2 (a) (i.e., the starting X/D and time at the stagnation are varied with particle size).
Also, it is found that the steps between the necks with the negative curvature and the particle surface with the positive curvature become discontinuous. The discontinuous curvature is changed to the continuous surfaces by the surface diffusions far from the neck as presented in Fig. 5.3 (b). Time to
scale. As presented in Fig. 5.3 (c), flat surface structures are constructed because the Ehi is imposed to the particles only (i.e., Egb = 0.0). These morphologies are found in the experimental images [62] and the snapshots of simulational results at low temperature [63] during solid-state sintering. If the X/D reaches ~0.95, the particle coarsening stops because the neck size of the enlarged terrace become equivalent with the particle radius as shown in Fig. 5.3 (d) and the wide terrace structures make it hard for solid mass to overcome the structural energy barrier through the surface diffusions because the flat surfaces are energetically stable. In the smallest particle size (r = 5 in voxel), the particle structure shows that the solid mass overcomes the structural energy barrier because structures of the positive particle curvature are relatively unstable. Note that the stagnant time is reduced and the neck growth rate in log- log scale becomes more linear as the kTs increase as shown in Figs. 5.2 (b) ~ (g) because the high kT makes it easy for solid mass to overcome the structural energy barrier by the thermally activated movement. Finally, two particles are completely merged with each other (X/D = 1.2).
The slope of the X/D and time in the log-log scale is equal to the 1/n, where n is the kinetic exponent of the particle coarsening, and it is known as the value of 0.143 (=1/7 by the surface diffusions) as presented in Fig. 5.3. Note that the particle coarsening occurs by the surface diffusions to reduce the collective interface energy presented by the normalized surface energy profiles marked by the dash lines in Fig. 5.2. The surface energy of particles is defined as the total number of vapor voxels for a solid voxel on surfaces (equivalent to the number of broken neighbors) and normalized by the initial value of the input microstructure as given by
๐ธ&= _h๐1,(*ร ๐ธ71j
* 18#
Eq. (5-1)
, where n is the number of solid voxels on surfaces, and Ni,bn is the number of the broken neighbors, respectively. In this case, total surface energy is equal to total interface energy because the Egb is not imposed. At the high kT above 1.0, the normalized surface energy is increased at initial time, suggesting that the initial spherical structure is not equilibrated shape at the given kT. It is found that the collective interface energy is decreased against time. In addition, the normalized surface energy is drastically increased like a hump at high kT due to the evaporation of solid mass into vapor phase (this is the inherent artifact of the proposed model).
Fig. 5.2. Kinetics of neck growth for variable particle size against time in log-log scale at the variable kT (= 1.0 ~ 4.0). At the low kT (= 1.0), the structural stabilization occurs by the surface diffusion (points extracting microstructural features are marked by the orange color circles and a, b, c, and d in (a)). At the relatively high kTs, the neck growth shows a linear function of the time regardless of the particle size in the log-log scale and the theoretical X/D is achieved while the surface energy of particles is reduced.