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Three-dimensional computer modeling of grain growth and pore shrinkage during sintering
Materials Chemistry and Physics 67 (2001) 12–16
Three-dimensional computer modeling of grain growth and pore shrinkage during sintering G. Tomandl∗ , P. Varkoly1 Institute for Ceramic Materials, University of Mining and Technology, Freiberg 09596, Germany
Abstract This paper deals with modeling of grain growth and pore shrinkage during sintering. The simulation takes as the theoretical basis the curvature of the grain boundaries and the pore surfaces and their mobilities. The movement of the grain boundaries is calculated for each grain separately using the flux equation for grain growth j = ck (1/r2 −1/r1 ), ck being a rate constant, r1 and r2 effective radii for each grain boundary. The modification of these boundaries in a micro structure is calculated in two and three-dimensional space starting with a Voronoy mosaic at the beginning of sintering. Two kinds of pores are distinguished: sphere like pores situated between two grains and non spherical pores between at least three grains. The results of several computer runs are shown and discussed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Sintering; Crystal growth; Computer modeling and simulation
1. Introduction Modeling all effects of sintering ceramic materials is of great importance because the comparison of simulation and experiment can improve understanding of sintering process. Moreover simulation of the influence of the powder characteristics of additives, the effects of various temperature-time regimes could save much experimental work which has to be performed for optimizing a special sintering task. In literature many attempts are made for mathematical treatment of sintering. In this treatment the approach of Oel [1] and Hillert [2] is applied, later also used in [3] and [4]. Our earlier investigation (Tomandl and co-workers [5–8]) continued these ideas, including also pore shrinkage and the coupling between grain growth and pore shrinkage. However, the methods used there did not use computer modeling but tried to solve the problem by numerical solution of the integro-differential equations. This implied a compromise, because several approximations had to be assumed. The present model of grain growth and pore shrinkage is based on the following assumptions. • No additives, no liquid phase. • Isothermal sintering.
∗ Corresponding author. E-mail address: [email protected] (G. Tomandl). 1 Present address.: Gesellschaft für Software- und Systementwicklung, Nürnberg, Germany.
• The grains are polyhedras in three-dimensional space (3d) or polygons in two-dimensional space, however with curved grain boundary areas or lines, respectively. • The driving force for the movement of the grain boundaries is the curvature. • The kinetic is described by the mobility of the grain boundaries. • Pores may be situated within grains, they show almost no shrinkage and are therefore neglected. • Pores may be situated at the grain boundaries and are treated as spheres, driving force is the energy surface. • Pores surrounded by at least three grains (2d) or four grains (3d) are treated similar to grains, however with different mobilities. • Pores reduce the movement of grain boundaries. • Pores do not contain slowly diffusing gases. • Pores may change their places in the case of a fast moving grain boundary compared to the mobility of the pore.
2. Grain growth The movement of the grainboundaries is calculated similar to the theory of Oel [1] and Hillert [2] and to Tomandl and co-workers [5–8]. In this theory the grains are treated as spherical like bodies. The driving force for grain growth between each two grains is assumed as proportional to the difference of the reciprocal radii. The materials flux is
0254-0584/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 0 ) 0 0 4 1 3 - 2
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Fig. 1. Calculation of the movement vectors for 2d.
described by the equation 1 dr 1 − = ck j= dt r2 r1 with ck = 2uk γ Vm , j = mass flux, uk = mobility of the atoms perpendicular to the grain boundary (i.e. mobility of the grain boundary), 1G = free enthalphy, ck = effective rate constant for grain growth, γ = grain boundary energy, V m = mol. volume. In this present approach each grain side is curved. This curvature is calculated by using the individual distances to the points of gravity of the adjacent grains. A motion vector is calculated according to Fig. 1 1 1 − bw0 = ck r1 r2 however, this motion vector bw0 is not directly used. Grains tend to become spherical. This means grain boundaries have the tendency to align perpendicularly to the straight line joining the two points of gravity of the neighbouring grains. So two different motion vectors (2d) for the corners of each grain side are calculated according to Fig. 1. Effectively a curved grain boundary as shown in Fig. 2 results. This function can only be calculated by numerical methods, therefore
13
Fig. 3. Calculation of the movement vectors for 3d.
in the graphic representations of the grain structures only the circular approximation is shown. The three-dimensional case cannot be solved in exactly the same way as in the 2d case, because the edge points of one grain boundary are not situated on one plane. In Fig. 3 it is shown how the two vectors bw1 and bw2 are calculated. In the present computer model not the movement of the curved grain boundaries is used directly but instead the movement of the edges using the movement vectors bw1 and bw2 is evaluated. For this purpose the average of the movement vectors belonging to one edge is used. The most difficult problem is the disappearing of grains when they are small enough. In this case only three-corner grains in 2d or four-corner grains in 3d are allowed to disappear. In order to guarantee this a special treatment in the program is necessary. Within an individual time scale first these particles must be allowed to shrink to three-corner (four-corner) grains.
3. Pore shrinkage 3.1. Pores situated at grain boundaries between two grains or within one grain According to Oel [1] or Tomandl and co-workers [5–8] the pore shrinkage is described approximately as cp ds = − 2 with cp = 2(σ + γ )up Vm dt s
Fig. 2. View of the curved grain boundary.
s = pore radius, σ = surface energy of the pore, γ = grainboundary energy, up = mobility of the atoms filling the pore, V m = mol. volume. Pores sitting on the grain boundaries occupy some part of the common area of grain boundary of two grains. Therefore, this part of the area is not effective for the grain boundary movement. For pores within grains cp is very small. In these simulations these pores are neglected.
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G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Several other equations related to pore shrinkage can be found in [9–15]. 3.2. Pores sitting at grain corners Pores may be surrounded by more than two grains either as a result of the voids from the initial state of sintering or if the length of the grain boundary where the pore is sitting, becomes smaller than the pore diameter. In the latter case the program rearranges the pores from case 3.1 to 3.2. These pores are treated like grains as polyhedras (3d) or polygons (2d), however with different effective mobility constants ck . The motion vector is assumed always in the direction towards the pore center. 3.3. Specialities The pores within grain boundaries have a limited velocity of movement. If the motion of the grain boundary belonging to the pore moves faster, then the pore is released from the grain boundary and positioned within the grain. The pore mobility is assumed as M p = csep /s according to [11]. If a grain disappears, then several pores may come together. Therefore, pore coalescence is taken into account.
Starting with a regular point lattice and a statistical variation of the point positions it is possible within some limitations to create a micro structure with a particle size distribution which is close to a predefined one. Because of the limited size of data storage in a computer the size of the simulated specimen is rather small. Therefore, a special trick has been used to perform the simulation with virtually an infinite number of grains. The initial Voronoy mosaic is repeated in all four (2d) or six directions (3d) and also the subsequent calculations of the boundary movements. Therefore, no surface of the simulated body has to be taken in account. In addition to the features of the program discribed above several extensions have been implemented which will not be shown in this paper. The grain growth in a multi phase system can be calculated using different mobility constants ck . The influence of additives can be taken in account in that way that due to concentrating the additive at the grain boundaries the constants ck are reduced with time. If the additive is not diffusing fast enough along the grain boundaries, then this can be considered in assuming the ck ’s neighbouring large grains being reduced to a higher extent than the ck ’s in the vicinity of smaller grains. 4. Results
3.4. Implementation in a computer program
4.1. 2d-Simulations
Two programs have been written, one for 2d-simulation, another for 3d-simulation. The first step is the calculation of an initial micro structure using a Voronoy mosaic.
In Fig. 4 a series of synthesized microstructures can be seen showing crystal growth and shrinkage of the two kinds of pores. The largest grain at the end of this series is marked by hatching. The pores disappear after a relatively short time.
Fig. 4. Series of synthesized microstructures. The hatched particle becomes the largest one. The image numbers are proportional to the square of time.
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Fig. 5. Average diameters of grains and the two kinds of pores.
The picture number is roughly proportional to the square of the sintering time. The average particle size dependent on time is shown in Fig. 5. After a slow initial growth an approximately linear increase follows. A square root law cannot be recognized. It would take a much longer time to reach this regime. The shrinkage of the two types of pores is also shown in Fig. 5. The spherical pores at the grain boundaries do not shrink but disappear almost abruptly. 4.2. 3d-Simulations It was not yet possible to calculate cross-sections through the 3d microstructure. Therefore, in Fig. 6 only a per-
15
Fig. 7. Cumulative grain diameter distribution functions in 3d.
spective image of five grains and their change with time is shown. The series shows the disappearing of the left most grain. Fig. 7 shows as an example how to evaluate the generated microstructures a series of the normalized cumulative volume distribution functions of the effective grain diameters. An increase of about factor 10 can be recognized within the simulated time range. About 50 000 grains in the initial Voronoy mosaic have been calculated. In addition the distribution functions and average values for shrinkage of the two kinds of pores are determined. However, the statistical error of these is rather large, because of the limited number of treated pores.
Fig. 6. Change with time of five grains in 3d.
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G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
5. Conclusions It has been shown that it is possible using the above mentioned assumptions, to simulate ceramic microstructures in a rather realistic way. The advantage of this method is the need of a relatively small computer memory because only the edge points of the particles must be stored. Further on it is possible to implement much more features into this model taking into account multiple phases, influence of additives, impediment of pores to grain boundary movement, pores filled with gases, anisotropic effects etc. In future it is essential to compare these simulations with real experiments. For this purpose it is necessary to convert the three-dimensional quantities into two-dimensional ones in order to enable the comparison with experimental data obtained on polished sections of sintered specimens using image analysis.
Acknowledgements This investigation was supported by the German Research Foundation, DFG. References [1] H.J. Oel, Crystal growth in ceramic powders, Mater. Sci. Res. 4 (1969) 249.
[2] M. Hillert, On the theory of normal and abnormal grain growth, Acta metall. 13 (1965) 227. [3] L-Q. Chen, D. Fan, Computer simulation model for coupled grain growth and ostwald ripening — application to Al2 O3 two phase systems, J. Am. Ceram. Soc. 79 (1996) 1163. [4] D. Fan, L-Q. Chen, Computer simulation of grain growth using a continuum field model, Acta mater. 45 (1997) 611. [5] G. Tomandl, Computer calculations of sintering of polycrystalline ceramics, Sci. Ceram. 9 (1977) 158. [6] G. Tomandl, A. Stiegelschmitt, Mechanismus des Sinterns bei Ein- und Mehrstoffsystemen unter Berücksichtigung der wichtigsten technologischen Größen, Ber. Deutsch. Keram. Ges. 55 (1978) 169. [7] G.Tomandl, in: G.C. Kuczynski (Ed.), Statistical Theory of Crystalline Ceramics without Liquid Phase, Sintering Processes, Vol. 13, Plenum Press, New York, 1980, p. 61. [8] G. Tomandl, A. Stiegelschmitt, Comparison between theory and experiment in sintering of undoped MgO, in: Fiftth International Round Table Conference On Sintering 1981 Portoroz, Jugoslavia, J. Mat. Sci. Monogr. 14 (1982) 101. [9] E.B. Slamovich, F.F. Lange, Densification of large pores: II, driving potential and kinetics, J. Am. Ceram. Soc. 76 (6) (1993) 1584. [10] R.L. Coble, Sintering crystalline solids. 1. Intermediate and final stage diffusion model, J. Appl. Phys. 32 (5) (1961) 787. [11] R.J. Brook, The impurity-drag effect and grain growth kinetics, Scr. Metal 2 (1968) 375. [12] R.J. Brook, Pore-grain boundary interactions and grain growth, J. Am. Ceram. Soc. 52 (1969) 56. [13] F.M.A. Carpay, The effect of pore drag on ceramic microstructures, in: R.M. Fulrath, J.A. Pask (Eds.), Ceramic Microstructures ’76, 1967, p. 261. [14] J. Svoboda, H. Riedel, Pore-boundary interactions and evolution equations for the porosity and the grain size during sintering, Acta Metall. Mater. 40 (1992) 2829. [15] H. Riedel, J. Svoboda, A theoretical study of grain growth in porous solids during sintering, Acta Metall. Mater. 41 (1993) 1929.
Three-dimensional computer modeling of grain growth and pore shrinkage during sintering G. Tomandl∗ , P. Varkoly1 Institute for Ceramic Materials, University of Mining and Technology, Freiberg 09596, Germany
Abstract This paper deals with modeling of grain growth and pore shrinkage during sintering. The simulation takes as the theoretical basis the curvature of the grain boundaries and the pore surfaces and their mobilities. The movement of the grain boundaries is calculated for each grain separately using the flux equation for grain growth j = ck (1/r2 −1/r1 ), ck being a rate constant, r1 and r2 effective radii for each grain boundary. The modification of these boundaries in a micro structure is calculated in two and three-dimensional space starting with a Voronoy mosaic at the beginning of sintering. Two kinds of pores are distinguished: sphere like pores situated between two grains and non spherical pores between at least three grains. The results of several computer runs are shown and discussed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Sintering; Crystal growth; Computer modeling and simulation
1. Introduction Modeling all effects of sintering ceramic materials is of great importance because the comparison of simulation and experiment can improve understanding of sintering process. Moreover simulation of the influence of the powder characteristics of additives, the effects of various temperature-time regimes could save much experimental work which has to be performed for optimizing a special sintering task. In literature many attempts are made for mathematical treatment of sintering. In this treatment the approach of Oel [1] and Hillert [2] is applied, later also used in [3] and [4]. Our earlier investigation (Tomandl and co-workers [5–8]) continued these ideas, including also pore shrinkage and the coupling between grain growth and pore shrinkage. However, the methods used there did not use computer modeling but tried to solve the problem by numerical solution of the integro-differential equations. This implied a compromise, because several approximations had to be assumed. The present model of grain growth and pore shrinkage is based on the following assumptions. • No additives, no liquid phase. • Isothermal sintering.
∗ Corresponding author. E-mail address: [email protected] (G. Tomandl). 1 Present address.: Gesellschaft für Software- und Systementwicklung, Nürnberg, Germany.
• The grains are polyhedras in three-dimensional space (3d) or polygons in two-dimensional space, however with curved grain boundary areas or lines, respectively. • The driving force for the movement of the grain boundaries is the curvature. • The kinetic is described by the mobility of the grain boundaries. • Pores may be situated within grains, they show almost no shrinkage and are therefore neglected. • Pores may be situated at the grain boundaries and are treated as spheres, driving force is the energy surface. • Pores surrounded by at least three grains (2d) or four grains (3d) are treated similar to grains, however with different mobilities. • Pores reduce the movement of grain boundaries. • Pores do not contain slowly diffusing gases. • Pores may change their places in the case of a fast moving grain boundary compared to the mobility of the pore.
2. Grain growth The movement of the grainboundaries is calculated similar to the theory of Oel [1] and Hillert [2] and to Tomandl and co-workers [5–8]. In this theory the grains are treated as spherical like bodies. The driving force for grain growth between each two grains is assumed as proportional to the difference of the reciprocal radii. The materials flux is
0254-0584/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 0 ) 0 0 4 1 3 - 2
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Fig. 1. Calculation of the movement vectors for 2d.
described by the equation 1 dr 1 − = ck j= dt r2 r1 with ck = 2uk γ Vm , j = mass flux, uk = mobility of the atoms perpendicular to the grain boundary (i.e. mobility of the grain boundary), 1G = free enthalphy, ck = effective rate constant for grain growth, γ = grain boundary energy, V m = mol. volume. In this present approach each grain side is curved. This curvature is calculated by using the individual distances to the points of gravity of the adjacent grains. A motion vector is calculated according to Fig. 1 1 1 − bw0 = ck r1 r2 however, this motion vector bw0 is not directly used. Grains tend to become spherical. This means grain boundaries have the tendency to align perpendicularly to the straight line joining the two points of gravity of the neighbouring grains. So two different motion vectors (2d) for the corners of each grain side are calculated according to Fig. 1. Effectively a curved grain boundary as shown in Fig. 2 results. This function can only be calculated by numerical methods, therefore
13
Fig. 3. Calculation of the movement vectors for 3d.
in the graphic representations of the grain structures only the circular approximation is shown. The three-dimensional case cannot be solved in exactly the same way as in the 2d case, because the edge points of one grain boundary are not situated on one plane. In Fig. 3 it is shown how the two vectors bw1 and bw2 are calculated. In the present computer model not the movement of the curved grain boundaries is used directly but instead the movement of the edges using the movement vectors bw1 and bw2 is evaluated. For this purpose the average of the movement vectors belonging to one edge is used. The most difficult problem is the disappearing of grains when they are small enough. In this case only three-corner grains in 2d or four-corner grains in 3d are allowed to disappear. In order to guarantee this a special treatment in the program is necessary. Within an individual time scale first these particles must be allowed to shrink to three-corner (four-corner) grains.
3. Pore shrinkage 3.1. Pores situated at grain boundaries between two grains or within one grain According to Oel [1] or Tomandl and co-workers [5–8] the pore shrinkage is described approximately as cp ds = − 2 with cp = 2(σ + γ )up Vm dt s
Fig. 2. View of the curved grain boundary.
s = pore radius, σ = surface energy of the pore, γ = grainboundary energy, up = mobility of the atoms filling the pore, V m = mol. volume. Pores sitting on the grain boundaries occupy some part of the common area of grain boundary of two grains. Therefore, this part of the area is not effective for the grain boundary movement. For pores within grains cp is very small. In these simulations these pores are neglected.
14
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Several other equations related to pore shrinkage can be found in [9–15]. 3.2. Pores sitting at grain corners Pores may be surrounded by more than two grains either as a result of the voids from the initial state of sintering or if the length of the grain boundary where the pore is sitting, becomes smaller than the pore diameter. In the latter case the program rearranges the pores from case 3.1 to 3.2. These pores are treated like grains as polyhedras (3d) or polygons (2d), however with different effective mobility constants ck . The motion vector is assumed always in the direction towards the pore center. 3.3. Specialities The pores within grain boundaries have a limited velocity of movement. If the motion of the grain boundary belonging to the pore moves faster, then the pore is released from the grain boundary and positioned within the grain. The pore mobility is assumed as M p = csep /s according to [11]. If a grain disappears, then several pores may come together. Therefore, pore coalescence is taken into account.
Starting with a regular point lattice and a statistical variation of the point positions it is possible within some limitations to create a micro structure with a particle size distribution which is close to a predefined one. Because of the limited size of data storage in a computer the size of the simulated specimen is rather small. Therefore, a special trick has been used to perform the simulation with virtually an infinite number of grains. The initial Voronoy mosaic is repeated in all four (2d) or six directions (3d) and also the subsequent calculations of the boundary movements. Therefore, no surface of the simulated body has to be taken in account. In addition to the features of the program discribed above several extensions have been implemented which will not be shown in this paper. The grain growth in a multi phase system can be calculated using different mobility constants ck . The influence of additives can be taken in account in that way that due to concentrating the additive at the grain boundaries the constants ck are reduced with time. If the additive is not diffusing fast enough along the grain boundaries, then this can be considered in assuming the ck ’s neighbouring large grains being reduced to a higher extent than the ck ’s in the vicinity of smaller grains. 4. Results
3.4. Implementation in a computer program
4.1. 2d-Simulations
Two programs have been written, one for 2d-simulation, another for 3d-simulation. The first step is the calculation of an initial micro structure using a Voronoy mosaic.
In Fig. 4 a series of synthesized microstructures can be seen showing crystal growth and shrinkage of the two kinds of pores. The largest grain at the end of this series is marked by hatching. The pores disappear after a relatively short time.
Fig. 4. Series of synthesized microstructures. The hatched particle becomes the largest one. The image numbers are proportional to the square of time.
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
Fig. 5. Average diameters of grains and the two kinds of pores.
The picture number is roughly proportional to the square of the sintering time. The average particle size dependent on time is shown in Fig. 5. After a slow initial growth an approximately linear increase follows. A square root law cannot be recognized. It would take a much longer time to reach this regime. The shrinkage of the two types of pores is also shown in Fig. 5. The spherical pores at the grain boundaries do not shrink but disappear almost abruptly. 4.2. 3d-Simulations It was not yet possible to calculate cross-sections through the 3d microstructure. Therefore, in Fig. 6 only a per-
15
Fig. 7. Cumulative grain diameter distribution functions in 3d.
spective image of five grains and their change with time is shown. The series shows the disappearing of the left most grain. Fig. 7 shows as an example how to evaluate the generated microstructures a series of the normalized cumulative volume distribution functions of the effective grain diameters. An increase of about factor 10 can be recognized within the simulated time range. About 50 000 grains in the initial Voronoy mosaic have been calculated. In addition the distribution functions and average values for shrinkage of the two kinds of pores are determined. However, the statistical error of these is rather large, because of the limited number of treated pores.
Fig. 6. Change with time of five grains in 3d.
16
G. Tomandl, P. Varkoly / Materials Chemistry and Physics 67 (2001) 12–16
5. Conclusions It has been shown that it is possible using the above mentioned assumptions, to simulate ceramic microstructures in a rather realistic way. The advantage of this method is the need of a relatively small computer memory because only the edge points of the particles must be stored. Further on it is possible to implement much more features into this model taking into account multiple phases, influence of additives, impediment of pores to grain boundary movement, pores filled with gases, anisotropic effects etc. In future it is essential to compare these simulations with real experiments. For this purpose it is necessary to convert the three-dimensional quantities into two-dimensional ones in order to enable the comparison with experimental data obtained on polished sections of sintered specimens using image analysis.
Acknowledgements This investigation was supported by the German Research Foundation, DFG. References [1] H.J. Oel, Crystal growth in ceramic powders, Mater. Sci. Res. 4 (1969) 249.
[2] M. Hillert, On the theory of normal and abnormal grain growth, Acta metall. 13 (1965) 227. [3] L-Q. Chen, D. Fan, Computer simulation model for coupled grain growth and ostwald ripening — application to Al2 O3 two phase systems, J. Am. Ceram. Soc. 79 (1996) 1163. [4] D. Fan, L-Q. Chen, Computer simulation of grain growth using a continuum field model, Acta mater. 45 (1997) 611. [5] G. Tomandl, Computer calculations of sintering of polycrystalline ceramics, Sci. Ceram. 9 (1977) 158. [6] G. Tomandl, A. Stiegelschmitt, Mechanismus des Sinterns bei Ein- und Mehrstoffsystemen unter Berücksichtigung der wichtigsten technologischen Größen, Ber. Deutsch. Keram. Ges. 55 (1978) 169. [7] G.Tomandl, in: G.C. Kuczynski (Ed.), Statistical Theory of Crystalline Ceramics without Liquid Phase, Sintering Processes, Vol. 13, Plenum Press, New York, 1980, p. 61. [8] G. Tomandl, A. Stiegelschmitt, Comparison between theory and experiment in sintering of undoped MgO, in: Fiftth International Round Table Conference On Sintering 1981 Portoroz, Jugoslavia, J. Mat. Sci. Monogr. 14 (1982) 101. [9] E.B. Slamovich, F.F. Lange, Densification of large pores: II, driving potential and kinetics, J. Am. Ceram. Soc. 76 (6) (1993) 1584. [10] R.L. Coble, Sintering crystalline solids. 1. Intermediate and final stage diffusion model, J. Appl. Phys. 32 (5) (1961) 787. [11] R.J. Brook, The impurity-drag effect and grain growth kinetics, Scr. Metal 2 (1968) 375. [12] R.J. Brook, Pore-grain boundary interactions and grain growth, J. Am. Ceram. Soc. 52 (1969) 56. [13] F.M.A. Carpay, The effect of pore drag on ceramic microstructures, in: R.M. Fulrath, J.A. Pask (Eds.), Ceramic Microstructures ’76, 1967, p. 261. [14] J. Svoboda, H. Riedel, Pore-boundary interactions and evolution equations for the porosity and the grain size during sintering, Acta Metall. Mater. 40 (1992) 2829. [15] H. Riedel, J. Svoboda, A theoretical study of grain growth in porous solids during sintering, Acta Metall. Mater. 41 (1993) 1929.