J. Am. Ceram. Soc., 98 [11] 3496–3502 (2015) DOI: 10.1111/jace.13684 © 2015 The American Ceramic Society
Journal
Sintering Simulation of Periodic Macro Porous Alumina
Robert Besler,‡ Marcel Rossetti da Silva,‡ Jefferson J. do Rosario,‡ Maksym Dosta,§ Stefan Heinrich,§ and Rolf Janssen‡,† ‡
Institute of Advanced Ceramics, Hamburg University of Technology, Hamburg, Germany
§
Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, Hamburg, Germany evolution for constrained and unconstrained sintering,14 the effect of substrate on film sintering15 and sintering of copper/ alumina composites.16 In addition, Henrich et al. investigated particle rearrangement during free and preassisted solid-state sintering,17 stress-induced anisotropy in sintering of alumina,18 the effect of particle size distribution on solid-state sintering,19 and the influence of initial coordination number.20 In general, according to Zipse21 numerical simulations like DEM have three advantages: (1) they can provide results when testing is impossible, (2) can test complex theories, and (3) can be used to evaluate reoccurring engineering problems. Typically, the initial starting configuration for sintering is particles of different sizes packed to a compact of 55%–65% relative density. Therefore, pores within these powder compacts are smaller than the grain size and located as interstices between grain clusters. However, inverse 3D-ordered macro porous (3DOM) ceramics exhibit an initial structure, where pores are larger than the grains and are structured in an ordered array. This special morphology provides attractive characteristics, such as high surface area, low heat conductivity, and photonic band gap. Studies have been carried out on the fabrication and application of inverse opals as catalyst carriers22 and thermal barrier coatings.23,24 Particularly, the latter operates at temperatures where sintering may destroy the initial well-ordered pore structure. There are few experimental literature publications available concerning the behavior of periodic porosity at high temperature. Sokolov et al. analyzed the changes of an a-alumina 3DOM structure during sintering and observed experimentally a preferential densification of nodes.25 The review from Rudisill et al.26 on the stability of templated materials gives an overview on the experimental state-of-the-art in this field. However, a detailed understanding of this problem still needs support by respective simulations, which allows changing one parameter at a time. Therefore, in this work, we compare the sintering behavior of 3DOM alumina structures with different pore size to filling particle ratios at a temperature of 1473 K using the DEM based simulation system MUSEN-DEM.27 With this simulation system it is possible to adjust the simulated structure close to experimental packing densities and pore distributions. As the processing route chosen determines essentially the size ratio of the spherical voids to the particles filling the interstices, one focus was to vary this size ratio k in our simulations. In addition, the initial structures were densified using both grain-boundary diffusion as well as surface diffusion as the dominant mass transport mechanism. Densification evolution curves and optical analysis of the simulated structures were used to evaluate their stability.
Three-dimensionally ordered macroporous (3DOM) ceramic materials are considered for a variety of applications. One of its many subclasses, inverse opals, is constituted by the ordered arrangement of the pores, resulting in the functionality of a photonic crystal and leading to strong reflection of incident electromagnetic radiation. Exposing these porous structures to high temperatures, however, can lead to sintering of the desired structure and loss of functionality. Therefore, discrete element method (DEM) simulations are performed on inverse opal structures with random homogenous distributed alumina particles forming the struts and nodes. Grain-boundary diffusion as well as surface diffusion are modeled via respective parameters of a contact model applied in MUSEN-DEM. Furthermore, the void to particle size ratio is varied to simulate fine and coarse grained 3DOM ceramics. Results indicate that nodes densify at higher rates and to a larger extent when compared to struts. An increase in the void to particle size ratio results in similar trends but with lower densification rates. This behavior is observed regardless whether surface or grain-boundary diffusion is considered as the dominant transport mechanism, with the latter giving higher densification rates. Variations in particle coordination due to the initial random packing favor local desintering, thereby causing the formation of defects/crack nuclei.
I.
Introduction
I
N ceramic processing one of the key steps is sintering, which assigns the quality of the finished product, for example, final shape, microstructure, and strength. Therefore, numerous studies have been performed in the past to analyze and predict the outcome of sintering processes. The most applied sintering models use a continuums mechanical approach. This approach was reviewed by Olevsky1 and by Bordia and Scherer.2–4 Another important approach is based on the Monte Carlo method, which can be used to describe the evolution of micro- and meso-structures during sintering.5–7 The latest and most promising approach is the discrete element method (DEM) developed by Cundall and Strack.8 The special feature of this method is the possibility to treat each particle as a discrete element with all degrees of freedom. This ability to analyze the motion of particles in large systems has attracted intense research in the past decade in various areas, particularly in process engineering. Recently, DEM was used also in the field of material science to model processes like compaction of ceramic powders,9 compression tests,10,11 bending tests12 and sintering.13–20 For example, Martin et al. simulated sintering of copper powder at various temperatures.13 His group also investigated crack
II.
R. Bordia—contributing editor
Simulation Method and Sample Preparation
(1) Simulation Method The computation is based on the solution of Newton’s and Euler’s laws of motion in each time step. The particle motion is calculated with the balance of forces and momentums as
Manuscript No. 36067. Received December 12, 2014; approved May 10, 2015. † Author to whom correspondence should be addressed. e-mail:
[email protected]
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Fig. 1.
Sintering Simulation of 3DOM Alumina
Schematic drawing of two particles in contact.
shown in Eqs. (1) and (2), where ri is the position vector to the particle i with the mass mi. Forces acting on each particle can be drag, gravity, pressure gradients, contact forces between particles, and adhesive forces. mi
Ii
o2 ri ¼ RFi ot2
(1)
oxi ¼ Mi ot
(2)
(2) Contact Model One approach to model the sintering of a material until a relative density up to 90% is by using a contact model, describing the contact between two particles as shown in Fig. 1. Thereby, the Hertz model denotes a completely elastic contact.28 Particles are modeled as spheres with angular xi and translational velocity ui with a soft sphere approach. This model allows the particles to have an overlap dn, as an intersecting part between two particles in contact. For the sintering of alumina the main diffusion mechanisms are considered to be surface and grain-boundary diffusion. Using the model developed by Parhami and McMeeking,29 the normal force consists of two terms: FN ¼ FV þ FS ¼
pa4s a urel;n pRi cs 2bDb b
(3)
The first one describes the viscous force FV depending on the relative velocity in normal direction urel,n, the contact radius as, the parameter b and the diffusion parameter Db. With an increasing overlap dn, the contact radius grows and the viscous force intensifies. The second term is the sintering force FS dependent on the surface energy cs, the particle radius Ri and the parameters a and b. These parameters depend on the ratio of the grain-boundary diffusion dbDb to the surface diffusion dsDs in the following relation: 15 u¼
d b Db ds Ds
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Fig. 2.
Schematic drawing of two overlapping particles.
diffusion mechanism having a two times higher effect on the densification than surface diffusion, for example, φ = 2. Surface diffusion as the dominant densification path can be simulated by a = 5/2, therefore φ = 5/2. With the approximation that the simulation is not dependent on the dihedral angle, the equation for the normal force can be simplified to [Eq. (3)]. The diffusion parameter Db is dependent on the atomic volume Ω, the Bolzmann constant k, the absolute temperature T, and the grain-boundary thickness db with Db being the diffusion coefficient for vacancy transport. Db ¼
X db Db kT Qb
Db ¼ D0b e Rg T
(5)
(6)
The activation energy for grain-boundary diffusion Qb and the universal gas constant Rg are constants for the calculation of the diffusion coefficient. To obtain the contact radius as, a geometric model derived by Coble30 was used. In Fig. 2 is a visualization given for the neck formation of two particles approaching each other. The material localized in the overlapping parts of the particles is redistributed to form sintering necks. as ¼
pffiffiffiffiffiffiffiffiffiffiffiffi 2Ri dn
(7)
The tangential force FT [Eq. (8)] depends mainly on the relative velocity in tangential direction urel,t and a dimensionless pseudo coefficient of frictions gPart between two particles in contact. This coefficient describes the resistance to slip and has been set here for all simulations to 0.01 = gPart. These parameters exhibited the best agreement with obtained experimental results.15 FT ¼ gPart
pa2s Ri urel;t 2bDb
(8)
(4)
For two particles in contact, b = 4 can be used for all values of φ, whereas a has to be set specifically. This approach allows changing the dominant sintering mechanism between grain boundary and surface diffusion. By using a = 9/2,15 sintering can be simulated with grain-boundary
(3) Generation of Virtual Samples For all simulations that were carried out, a limitation of about 40 000 particles was imposed in the simulation system MUSEN-DEM27 to avoid prohibitive simulation time. The numerical samples consisted of homogeneous random
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Journal of the American Ceramic Society—Besler et al. Table I.
Properties of Numerical Sample
DP,T (nm)
k
10 20 30 40
Table II.
dP (nm)
1000 2000 3000 4000
nP
686 4883 16 184 41 521
100
Simulation Parameters Used for All Simulations
Parameter
Symbol
Value
Density Particle diameter Atomic volume Surface energy Grain-boundary thickness times diffusion coefficient Activation energy Sliding friction coefficient (tangential force) Initial relative density Temperature Time step Saving time step
q dP Ω cs dbD0b
3950 100 8.47e-30 1.1 1.3e-8
Qb gPart
475 0.01
qrel T Dt tSave
0.62 1473 1e-4 10
Unit
kg/m³ Nm m³ J/m² m³/s kJ/mol — — K s s
packages with mono-sized particles representing alumina particles with diameters of dP = 100 nm. To investigate the influence of different node and strut sizes on the sintering of inverse opal structures, four different ratios between the sizes of template to the size of the filling particles k have been used, defining the position and size of pores. The larger the template particle diameter DP,T the larger are the interstitial spaces between them and, therefore, a higher number of filling particles in the node and the strut is present. The ratio k was varied between 10 and 40 as shown in Table I. Without changing the particle size of the alumina particles it is possible to keep the same simulation parameters for all simulations. Table II presents the simulation parameters.14 To generate the inverse opal structure [see Fig. 3(c)] the following three steps were carried out: At first the template particles were assembled into an ordered FCC unit cell array, which represent the colloidal crystal template, that is, polymer spheres. As described by Rudisill et al.,26 there are two different shapes of interstitials, tetrahedral, and octahedral, in the FCC colloidal crystal template. In the second step, the size and coordinates of the filling region were set using the template spheres’ coordinates and radius. The cube vertexes must cross the corner of particle centers, as it is shown with the dashed lined box in Fig. 3(b). This cube is filled with the filling particles (red)
(a)
(b)
Fig. 3.
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with a random homogeneous packing, up to a relative density of 0.62. This value of relative density was chosen as it is a typical upper value for green compacts. The algorithm to fill the simulation box is a dynamic method, which is performed as follows: (i) the number of particles is calculated from the porosity required and the volume of the simulation box, (ii) this number of particles is filled into the box occupying random sites, thereby an overlap of 0.1% between particles is allowed, and (iii) the particles are rearranged by the force-biased algorithm,31,32 where sets of particles are shifted repeatedly to reduce the overlap. The resulting particle setting is according to the generation algorithm homogeneous and isotropic, with no gradients in packing density. The third and final step to produce the inverse opal structure is to delete all overlapping template and alumina particles dT,F according to the following condition [Eq. (9)]: jri rT;j j\ðRi RT;j Þ ð1 nÞ
(9)
where ri is the position vector of the filling particle i and rT,j the position vector of the template particle j. Ri and RT,j are, respectively, the radii of the filling and template particles. The result is the desired inverse opal structure [Fig. 3(c)]. To ensure stable structures for k = 10 and 20, the coefficient ξ was introduced in order to reduce RT,j. This value was set to ξ10 = 7.5% for k = 10 and ξ20 = 2% for k = 20. The sample consists of one complete and eight quarter of the octapodal nodes. Every complete octapodal node is surrounded with eight tetrapodal nodes, where each tetrapodal node is connected to an eighth of one octapodal node with a strut. All simulations were performed with the use of free boundary conditions due to the fact that the unit cell consists of 28 structural parts, which represent the smallest units of the structure. Each of these structural parts consists of three
Fig. 4. Schematic 2-D visualization of strut and node configuration. The areas used for calculating the packing density during the simulations are indicated by spheres.
(c)
Schematic visualization of sample generation.
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Table III. Average Number of Particle Midpoints in the Measurement Spheres for the Four Analyzed Structures
Strut Tetrapodal node Octapodal node full Octapodal node quarter
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Sintering Simulation of 3DOM Alumina
k = 10
k = 20
†
†
0 1† 22 4
1 5 156 7
k = 30
k = 40
7 43 377 43
38 120 976 136
† In the table, only the midpoints of particles are considered for simplification. In the calculation, however, also fractions of particles are taken into account giving higher reliability of the computation results.
different subsections, for example, 1/8 octapodal node, strut and 1/4 tetrapodal node. To confirm this assumption, simulations were conducted with four unit cells, which yielded similar results. Therefore, the effect of the particles at the extremities was negligible. The tetrapodal node can be approximated as a pyramid and the octapodal as a cube. The strut is assumed to be a cylinder in between. The inverse structures were used as a starting situation for all simulations. Therefore, all effects during sintering are only due to a mass transport by grain boundary and surface diffusion. For analyzing the densification during sintering, a sphere is situated in the cube, the pyramid, and in the cylinder (Fig. 4) and the density evaluated. The density evolution was measured throughout the whole simulation time until a relative density of 90%, to avoid additional effects of grain growth in final stage sintering. Four struts, four tetrapodal
nodes, one complete octapodal node, and 3 quarter of an octapodal node were taken into account for the analysis of the density evolution. In Table III, the average numbers of particles in the measurement volumes are listed. Using an approximation algorithm for dividing the particles in subvolumes, it is possible to gain reliable results for measurements with more than six particles in the measurement volume, Table III. Thus, only measurements of the full octapodal and the quarter octapodal nodes will be presented for k = 10 and 20, (Fig. 6). For other k the densification of all structural parts can be discussed. The values for the full octapodal and quarter octapodal node will be averaged and shown in one curve.
III.
Results
In Fig. 5, the coordination numbers of the inverse opals are plotted for the initial stage, as well as for grain boundary and surface diffusion as dominant transport mechanisms. After the simulation, a clear increase in the coordination number is visible for all ratios. The cross section in the middle of the octapodal nodes demonstrates that the coordination number is always higher in the nodes when compared to the struts. In addition, the strut thickness and the coordination number inside the strut increases with the pore to particle size ratio. Densification via surface diffusion is, as expected, slower than the densification with grain-boundary diffusion as dominant mass transport mechanism. The observed desinterings (indicated by black arrows in Fig. 5) occurs independent of
(a)
(b)
(c)
Fig. 5. Cross sections in Y-plane directly through the centered octapodal node, colored with the coordination number, (a) initial stage, (b) after simulation with grain-boundary diffusion and (c) after simulation with surface diffusion.
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Journal of the American Ceramic Society—Besler et al. (a)
(b)
Fig. 6. Relative density evolution of the centered octapodal node over the sintering time for grain boundary and surface diffusion; (a) k = 10 and (b) k = 20.
the dominant densification mechanism. Due to the initial particle coordination, the particles can only rearrange in one specific way. Therefore, the particle positions after the simulation are determined by the initial coordination. Only a change in particle positions in the initial structure would result in a change in desintering positions. In the case of k = 10 there are only few particles in the struts; hence, there are not many possibilities to rearrange in a manner that initial small defects can heal during sintering. Therefore, local desintering occurs, which can be considered as the formation of crack nuclei. In Figs. 5(b) and (c) it can be observed that the structure becomes more stable with increasing k. The sample with k = 40 does not show any completely developed cracking. The red arrows in Fig. 5(b) indicate the beginning of desintering at the struts. The densification via grain boundary and surface diffusion for k = 10 and 20 is given in Fig. 6. Densification evolutions are calculated as a function of simulation time. The simulations were performed up to 90% of relative density, to avoid additional effects of grain growth in final stage sintering. The k = 10 sample reaches the final relative density of 90% after a simulation time of 200 s with grain-boundary diffusion being the dominant mass transport mechanism [Fig. 6(a)]. The behavior for surface diffusion as dominant mechanism is the same; only the simulated time needed to reach 90% relative density is about 50% enhanced. This result fits very well to the observations Bouvard and McMeeking showed in,33 where surface diffusion-controlled sintering exhibited lower shrinkage if compared to grainboundary diffusion-controlled sintering. Figure 6(b) reveal the densification behavior of the k = 20 sample, exhibiting almost the same behavior. The necessary densification time increases 10% for grain-boundary diffusion and 20% for surface diffusion. In addition, less desintering is observed for k = 20 than for k = 10, Fig. 5(b). The densification of the central nodes in Fig. 6 follows the expected behavior, for example, grain-boundary diffusion results in faster densification when compared to surface diffusion as dominant transport mechanism, even for low k. In Figs. 7 and 8, the densification behavior is analyzed for the different parts of the opal structure for k = 30 and 40, respectively. The calculation indicate an almost identical densification of all areas of the inverse opal in the first stage, for example, between 60% and 70% of theoretical density. Then, the densification in the struts slows down and at the end of the calculation densities between 70% and 75% are calculated. However, in the nodes densification continues until the set limit of 90%, which is obtained always for the octopodal nodes and, for high k, also in case of tetrapodal nodes. For k = 30 octapodal nodes densify 5% more than the tetrapodal node for both simulations. Still, there is a difference in the densification time needed to achieve a comparable densification for dominant grain boundary or dominant surface diffusion.
The comparison of all samples shows that the densification takes longer for samples with a higher k. The variation in time is shown in Table IV. The densification times are normalized using the densification time for k = 10 and grainboundary diffusion as reference (=1). The difference in densification time for dominant grain boundary and dominant surface diffusion for all k is in the range from 50% for k = 10% and 72% for k = 30.
IV.
Discussion
The DEM simulations performed demonstrate an almost identical behavior for grain boundary and surface diffusion as dominant transport mechanism—the only difference being the simulation time. This is due to the fact that even in case of dominant surface diffusion our simulations still consider a low but effective mass transport via grain-boundary diffusion. The densification curves for the nodes are in agreement with previous work using respective parameters in the contact model for sintering structures without large or even ordered pores14 and the work of Rudisill et al.26 confirmed the applicability of our DEM code for 3DOMs. However, different to homogenous powder compacts, the sintering of 3DOM is inhomogeneous—nodes sinter to a higher density than struts for situations where the matrix—the filling particles—is randomly distributed. Interestingly, sintering of struts and nodes starts generally with the same rate and slows down only for struts during intermediate sintering. This behavior is almost independent on k and occurs even at k = 40, for example, the model representative for 3DOM materials with high number of grains in the strut cross sections. Although no steady state was achieved in our simulations, it is most likely that even prolonged sintering will not result in a full densification of struts by sintering inverse opal structures. Using the DEM simulation it is also possible to analyze the forces acting on single particles. It was observed that the forces and the particle movement inside the strut are directed to the nodes. In particular, there is a tendency that forces are higher in direction of the octapodal node than to the tetrapodal node. The higher forces correspond with higher resulting velocities of the particles in the struts (see Fig. 9). Looking at the global velocity distribution, one can observe that the whole structure moves toward the centered octapodal node. This global particle movement superimposes the movement of the single particles. Considering the absolute value of the particles velocity, higher velocities of the particles are observed in the region close to the octapodal node, whereas the lowest particle velocities are at the center of the nodes (Fig. 9). In Fig. 10, the velocity distribution at struts close to the nodes is exemplified showing the formation of defects, which can be considered as crack nuclei. At these nuclei, particle movements towards the nodes can be observed. Particles in
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Sintering Simulation of 3DOM Alumina (a)
Fig. 7.
(b)
Relative density evolution of all structural parts over the sintering time for (a) grain boundary and (b) surface diffusion k = 30.
(a)
Fig. 8.
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(b)
Relative density evolution of all structural parts over the sintering time for (a) grain boundary and (b) surface diffusion k = 40.
Table IV. Normalized Densification Time for 90% Relative Density for All k with Grain boundary and Surface Diffusion Being the Dominant Mass Transport Mechanism
Grain-boundary diffusion Surface diffusion
k = 10
k = 20
k = 30
k = 40
1.0 1.5
1.1 1.7
1.3 1.8
1.7 2.5
Fig. 10. Crack development during sintering in struts between octopodal and tetrapodal nodes for k = 20 at a simulation time of 145 s with grain-boundary diffusion as densification mechanism.
Fig. 9. Particle velocity in a strut for k = 20 at the end of simulation with grain-boundary diffusion as densification mechanism.
the strut on the right-hand side lost their contact some time steps ago, resulting in high velocities close to the octapodal strut. The variation in k indicates that nodes with higher ratios need more time to sinter to the desired final density, for example, structures with larger voids densify with slower speed. This behavior is in good agreement with the experimental work carried out by Sokolov et al., where they state that structures with bigger pores and the same size of alumina grains are more stable than structures with small pore sizes.25 In addition to the larger size of pores, structures of higher ratio exhibit more particles in the nodes and struts that are involved in the sintering
process. Therefore, more forces are acting on a single particle in opposite directions. Furthermore, low ratios exhibit a situation where a considerable fraction of particles are only attached partly to the particle network, for example, more particles are located at the void surface where no forces are acting in the direction of this free surface. Therefore, particles at the void surface are attracted more to the center, where more particles are located. In addition to packing density evolution, the DEM simulation also allows for the analysis of the average densification homogeneity of structures with nonideal particle distribution. In our simulations, the particles have been filled in the nodes and struts using a random homogenous model. This results in an initial particle configuration with local agglomerates and areas where particles are less coordinated. During sintering, the lower coordinated area can exhibit a further reduction in coordination, for example, local desintering. Only if new contacts are developed during the simulation these areas densify. Otherwise, defects are formed which can be considered as a kind of crack nuclei (as indicated in the 2D representation in Fig. 5). It is very likely that these defects are responsible for the crack formation typically observed for
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heat-treated 3DOM structures.26,34 Our results indicate that sintering of structures with higher number of particles in the struts and nodes offer a high probability for rearrangement processes, which can lead to the healing of initial packing defects. In summary, DEM sintering simulations using MUSENDEM proved to be a promising tool to analyze sintering of complex structures. In particular, the option to generate initial configuration variations, which are close to sample variations in experimental research, offers a route for a detailed analysis of the sintering behavior of complex structures.
V.
Conclusions
It is shown that a numerical investigation concerning the high-temperature stability of 3DOM structures is possible with DEM, yielding complimentary results to more timeintensive experimental investigations. Simulation results did show a limited densification of struts whereas nodes can sinter to high final densities. The observed behavior is almost identical for grain boundary and for surface diffusion as the dominant transport mechanism—the only difference being the densification rate. Furthermore, the ratio of void to particle size k does not alter the general behavior, although structures with larger k exhibit slower densification rates. As the initial particle packing in the struts and nodes is simulated as random homogenous packing and, therefore, with slight variations in local particle coordination, some particles experienced a reduction in contacts indicating local desintering. As densification proceeds, this desintering results in the formation of defects/crack nuclei. These defect formations are balanced by rearrangement processes, and structures with higher k, for example, with higher number of particles in the cross section of nodes and struts, show therefore less defect formation. Consequently, homogeneous fillings of fine particles are recommended for 3DOMs serving at temperatures at which sintering phenomena are not negligible. In addition, the material should be doped in a way that grain-boundary diffusion is inhibited.
Acknowledgments We gratefully acknowledge financial support from the German Research Foundation (DFG) via SFB 986 “M3”, project A3, C4, and C5, as well as Brazilian agencies CNPq and CAPES for supporting Marcel Rossetti da Silva via project BRAGECRIM and Science without Borders.
References 1
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