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Sintering kinetics of large pores
Mechanics of Materials 37 (2005) 705–721 www.elsevier.com/locate/mechmat
Sintering kinetics of large pores J. Pan b
a,*
, H.N. ChÕng a, A.C.F. Cocks
b
a School of Engineering, University of Surrey, Guildford GU2 7XH, UK Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
Received 29 September 2003
Abstract The sintering kinetics of large pores in a dense polycrystalline solid is studied using computer simulation. Coupled grain-boundary diffusion, surface diffusion and grain-boundary migration are assumed as the underlying mechanisms for the microstructural evolution. A numerical method developed by ChÕng and Pan [ChÕng, H.N., Pan, J., 2004. Cubic spline elements for modelling microstructural evolution of materials controlled by solid state diffusion and grain-boundary migration. Journal of Computational Physics 196, 724], which is an improved version of that developed by Pan et al. [Pan, J., Cocks, A.C.F., Kucherenko, S., 1997. Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration. Proceedings of the Royal Society, London A 453, 2161], is used for the computer simulation. A well-known textbook theory due to Kingery and Francois [Kingery, W.D., Francois, B., 1967. The sintering of crystalline oxides. I. Interaction between grain boundaries and pores. In: Kuczynski, G.C., et al. (Eds.), Sintering Related Phenomena. Goedon and Breach, New York, p. 471] predicts that a critical coordination number exists above which a pore does not shrink. The computer simulation demonstrates that this is not a general rule and that the critical coordination number only exists if one assumes that the pore is surrounded by identical grains. This numerical finding explains the contradiction between existing experimental observations and the critical coordination number theory. The numerical results also support the analytical expression developed by Ma [Ma, J., 1997. Constitutive modelling of the densification of porous ceramic components. PhD thesis, Engineering Department, Cambridge University] and Cocks [Cocks, A.C.F., 2001. Constitutive modelling of powder compaction and sintering. Progress in Materials Science 46, 201] for the sintering potential of powder containing large pores. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Sintering; Computer simulation; Large pore; Modelling; Solid-state diffusion; Grain-growth; Densification; Finite element method
*
Corresponding author. Tel.: +44 1483 689671; fax: +44 1483 306039. E-mail address: [email protected] (J. Pan).
0167-6636/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2004.06.002
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J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
1. Introduction Most ceramic components and an increasing number of metal components are manufactured by sintering powder compacts, a process during which pores are eliminated at elevated temperatures (Pan, 2003). A commonly believed textbook theory is that a large pore may grow instead of shrink during sintering (Kingery and Francois, 1967). Here the number of grains surrounding a pore, i.e. the coordination number of a pore, defines ‘‘large’’. It is predicted that a critical coordination number exists, which depends on the dihedral angle of the material, and that a pore will shrink only if its coordination number is less than the critical value. It follows that a large pore can only be eliminated after sufficient grain-growth, which reduces the coordination number to that below the critical value. However, there is increasing experimental evidence contradicting with the theory. In a well-designed experiment, Slamovich and Lange (1992) demonstrated that it is the characteristic diffusion distance, instead of the coordination number, that controls the behaviour of a large pore. They artificially introduced large pores into two Zr(Y)O2 powders, exhibiting either sluggish (ZrO2— 3 mol% Y2O3) or rapid (ZrO2—8 mol% Y2O3) grain-growth kinetics. The two powder compacts containing the artificial large pores of the same size were sintered under identical conditions during which their microscopic morphologies and densities were monitored. Grain growth was much faster in the Zr(8Y)O2 matrix than in the Zr(3Y)O2 matrix, so that the average coordination number of the pores in the Zr(8Y)O2 compact quickly became much smaller than that in the Zr(3Y)O2 compact. If the coordination number controls the densification behaviour, then the Zr(8Y)O2 powder compact would densify faster than the Zr(3Y)O2 compact. The experimental results showed exactly the opposite: the Zr(8Y)O2 powder compact densified significantly slower than the Zr(3Y)O2 compact. After any common period of sintering, the grain size in the Zr(8Y)O2 matrix was much larger than that in the Zr(3Y)O2 matrix; the former, therefore, had a much longer characteristic diffusion distance than the latter. Earlier
work by Chen and Xue (1990) on the creep behaviour of the two materials indicated that the diffusivities of the two materials are similar regardless of their yttria contents. Hence the experiment showed that it is the characteristic diffusion distance instead of the coordination number that controls the densification behaviour. More recently, Flinn et al. (2000) carried out a carefully designed experiment using Al2O3 powder. Their experiment was designed to study micromechanical failure from pores, and to rationalise the fracture strength of ceramics as a function of grain size and density. They also introduced artificial large pores and found that pores as large as 125 lm in diameter in their initially submicron sized powders (which reached an average grain-size of about 1 lm by the end of the sintering) shrunk continuously throughout the sintering process. Flinn et al. (2000) realized the contradiction between their experimental results and the critical coordination number theory and explained the contradiction by arguing that the pore shrinkage is strain-controlled. However, the Ôstrain-controlledÕ deformation does not explain why the thermodynamic argument leading to the critical coordination number theory is not applicable in these experiments. Recently the issue of large pores has attracted more attention because of the effort to sinter nanostructured ceramics. It is very easy for nano-sized particles to form agglomerates and a nano-powder compact contains large inter-agglomerate pores as well as small intra-agglomerate pores. During sintering, the small intra-agglomerate pores disappear quickly leaving the large inter-agglomerate pores behind. Most of the sintering time is therefore spent in eliminating the large pores. This process has to be carefully controlled since excessive grain-growth can easily destroy the desired nanostructure (Mayo, 1996). Understanding the sintering kinetics of large pores is therefore crucial for sintering nano-structured ceramics. Ma (1997) and Cocks (2001) proposed that the sintering potential is independent of the coordination number and only depends on the size of the pore. This idea was supported by a kinetic model for the sintering of large pores developed by Pan et al. (1999). The kinetic model was however over-simplistic. The purpose of this paper is to pre-
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
sent a set of more realistic kinetic models for the sintering of large pores and to demonstrate that the coordination number does not necessarily control whether a pore shrinks or grows. Firstly, we provide an appropriate thermodynamic argument showing that there is not necessarily a thermodynamic barrier preventing a pore of any coordination number from being eliminated from a dense solid. We then support the thermodynamic argument by carrying out a series of computer simulations for microstructural evolution in a dense solid containing very large pores. This is a set of fully kinetic models which are solved using the numerical technique developed by ChÕng and Pan (2004), which is an improved version of the numerical technique initially developed by Pan et al. (1997). Coupled surface and grain-boundary diffusion is assumed as the underlying mechanism for densification. Grain-boundary migration (leading to grain-growth) is also included in the model. Finally, we compare the sintering potentials and the densification rates which are obtained analytically using the expressions proposed by Ma (1997) and Cocks (2001), and numerically using the computer simulations. The numerical results support the analytical expressions.
lution of a powder compact is the excess free ergy associated with the free surfaces of powder particles. The upper limit of the free ergy reduction in the sintering process can be pressed as
enthe enex-
DEtotal ¼ As cs :
ð1Þ
Here As is the total free surface area of the powder and cs is the specific energy for the free surface. This free energy reduction drives two processes during sintering, i.e. densification and graingrowth. When a powder compact densifies, the free surfaces are replaced by grain-boundaries as shown in Fig. 1(b). If a full densification is achieved then the change of free energy associated with the densification is DEdensification ¼ ðAs cs Agb cgb Þ:
First, let us consider a powder compact without agglomeration and large pores as shown in Fig. 1(a). The driving force for the microstructural evo-
ð2Þ
Here Agb is the total grain-boundary area in the dense solid and cgb is the specific energy for the grain-boundary. During densification approximately two free surfaces coalesce into one grainboundary as shown schematically in Fig. 1, i.e. As 2Agb and the total change of free energy related to densification can be estimated as DEdensification As ðcs 0:5cgb Þ:
2. Driving force for elimination of large pores
707
ð3Þ
This part of the free energy reduction drives the densification. The remaining part of the free energy reduction, DEgrain-growth 0:5As cgb ;
Fig. 1. Schematic illustration of the sintering process of a powder compact without agglomeration and large pores.
ð4Þ
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drives the grain-growth. For most materials cs is larger than cgb, DEdensification is therefore normally negative which means that there is always a thermodynamic force driving the powder compact to eliminate the pores. Next let us consider a two-dimensional, polycrystalline solid containing periodically arranged large pores. A theory due to Kingery and Francois (1967) predicts that there exists a thermodynamic barrier for the shrinkage of the large pore. This barrier can be easily explained using the example shown in Fig. 2. A pore of radius R is surrounded by n identical grains. If the pore shrinks a little as indicated by the dashed lines, the radius of the pore is reduced by dR and the total change of free energy can be calculated as dE ¼ ndRcgb 2pdRcs :
ð5Þ
Fig. 2. Large pore surrounded by four identical grains.
Since the pore shrinks only if dE < 0, this leads to the following condition for the large pore to shrink: n < 2pcs =cgb :
ð6Þ
If the pore is surrounded by too many grains, then the free energy gain due to the extension of the grain-boundaries into the pore outbalances the free energy loss due to the elimination of the pore surface, so the pore should grow instead of shrink. This is the simplest version of the coordination number theory. It can be refined by considering the three-dimensional case, the effect of the outer surface of the compact and mass conservation in the variational argument (Kellett and Lange, 1989). There are two problems with the critical coordination number theory. The first one is that the above argument assumes that the large pore is surrounded by identical grains and that the grains move simultaneously into the large pore as it shrinks. However one rarely finds elongated grains centred around a large pore in specimens sintered from powder compacts containing large pores. The second problem is that increasing experimental evidence contradicts the theory as discussed in the introduction. In fact the thermodynamic barrier predicted by Eq. (6) is a result of the imposed kinetic route (that the grains move simultaneously into the pore). Instead of considering a small incremental variation of the pore size, let us consider the initial and final states of the material as shown in Fig. 3(a) and (b) respectively. Imagine the large
Fig. 3. Schematic illustration of elimination of a large pore from a dense solid.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
pore in Fig. 3(a) is somehow eliminated from the solid producing the dense solid shown in Fig. 3(b). If the grains remain roughly equiaxed throughout the elimination process, then it is obvious that the total free energy of the material shown in Fig. 3(b) is less than that shown in Fig. 3(a) simply because the free energy associated with the pore surface diminished and the free energy of the grain-boundaries does not increase because the grains remain equiaxed. Eqs. (3) and (4) can now also be used to estimate the free energy reduction associated with densification and graingrowth for a powder compact containing large pores. In this imaginary process, the total free energy is reduced no matter how large the coordination number is. This shows that eliminating a large pore from a dense solid does not necessarily have a free energy barrier contrary to that predicted by Eq. (6). The key condition here is that the grains remain equiaxed in the sintering process, which is almost universally met. It remains to show how the grains can rearrange themselves into the large pore while still remaining roughly equiaxed. In the following sections, we use a two-dimensional kinetic model to demonstrate that a combination of grain-boundary sliding and neighbour-switching can indeed help a large pore to by-pass the thermodynamic barrier predicted by Eq. (6).
3. A kinetic model for sintering To develop a kinetic model for the large pore, we consider solid-state sintering and further assume that matter redistribution is controlled by coupled grain-boundary and surface diffusion. Grain-boundary migration (leading to grain-growth) is also considered such that grainboundary diffusion occurs along a migrating grainboundary network. Following linear kinetic laws are assumed for the three processes (Herring, 1955; Hillert, 1965) although non-linear kinetic laws can also be accommodated by the variational method (Pan, 2004): jgb ¼ M gb
or ; os
ð7Þ
709
oðcs js Þ ; os
ð8Þ
vm ¼ M m cgb jgb ;
ð9Þ
js ¼ M s and
where jgb and js are the diffusive fluxes along the grain-boundaries and free surface respectively, vm is the migration velocity of the grain-boundary, Mgb, Ms and Mm are the mobilities for grainboundary diffusion, surface diffusion and grainboundary migration respectively, r is the stress normal to a grain-boundary, cs and cgb are the specific surface and grain-boundary energies respectively, js and jgb are the principal curvatures of the free surface and grain-boundary respectively, and s is a local coordinate along either a grainboundary or a free surface. Apart from the three kinetic laws, the normal stress r has to satisfy mechanical equilibrium, the diffusive fluxes have to satisfy matter conservation, the chemical potential has to be continuous everywhere, and finally the dihedral angles have to be such that the equilibrium between the interfacial tensions is satisfied. Existing data on superplastic deformation of several fine-grained ceramics indicate that grainboundary sliding is not a rate limiting mechanism (Chen and Xue, 1990). In the current model, we allow the grain-boundaries to slide freely (connected with no energy dissipation). It can be shown that the problem described above can be re-stated using the following variational principle (Needleman and Rice, 1980; Suo, 1997): among all the virtual migration velocities of the interfaces and the virtual diffusive fluxes that satisfy matter conservation, the actual velocity and diffusive flux fields minimise the functional Z Z 1 2 1 2 P¼ jgb dC þ js dC 2M 2M gb s Cgb Cs Z 1 2 dE ð10Þ vm dC þ þ 2M dt m Cgb in which Cgb represents the grain-boundary network, Cs represents the free surface in the pore and E is the total free energy of the system as discussed in the previous section:
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E¼
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
Z Cgb
cgb dC þ
Z cs dC:
ð11Þ
Cs
The microstructure evolves to reduce E. The variational principle (i.e. dP = 0) was derived from the fact that the reduction rate of E has to be balanced by the rates of the three energy dissipation mechanisms: grain-boundary diffusion, surface diffusion and grain-boundary migration. A robust finite element technique was developed by Pan et al. (1997) to solve the above variational problem. Recently ChÕng and Pan (2004) further improved the numerical technique by introducing a set of cubic spline finite element formulations. The numerical technique has been verified using a wide range of testing cases. Here we present one more test case using the cubic spline elements developed by ChÕng and Pan (2004) to demonstrate the validity of the numerical method. We consider the free sintering of a close packed array of circular cylinders as shown in Fig. 4. By symmetry we can isolate the dashed triangular region as the repeating cell and use the cell in the computer simulation. This problem has been solved by Svoboda and Riedel (1995) and is the simplest case of the type of situations considered in this paper. We solve this problem using the cubic spline finite element method. All the initial conditions and material parameters were taken as the same as those used by Svoboda and Riedel (1995). Twelve cubic spline elements were used to represent each cylinder surface; three elements were used to represent half of the neck (grain-
Fig. 4. A two-dimensional regular array of monosized cylinders. The dashed triangular region shows the repeating cell used in the computer simulation.
Fig. 5. Computer simulated evolution of a pore between three circular cylinders. The dihedral angle is set as 120°, diffusivity ratio Ms/Mgb = 1 and the initial radius of cylinder R = 1. t0 ¼ 0:0, t1 ¼ 7:0152 107 , t2 ¼ 2:602 105 , t3 ¼ 3:7033 104 , t4 ¼ 6:0978 104 and t5 ¼ 6:7742 104 .
boundary). Fig. 5 shows the computer simulated sintering process of the pore obtained using the cubic spline finite element method. Frequent remeshing had to be undertaken during the computer simulation. In Fig. 6, the numerical result of
Fig. 6. Comparison of porosity as a function of time obtained by Svoboda and Riedel (1995) and by finite element method. The equilibrium dihedral angle is 120° and the diffusivity ratio is Ms/Mgb = 1.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
porosity as a function of time obtained by Svoboda and Riedel (1995) is compared to that obtained using the cubic spline finite element method. It can be seen that the two numerical results agree very well with each other.
4. Sintering kinetics of large pores 4.1. Three microstructural models for dense material containing periodically arranged pores Our purpose here is to demonstrate that the critical coordination number theory is a direct consequence of the assumption that a large pore is surrounded by uniform and identical grains, and in general a pore with its coordination number larger than the critical value can still shrink without the help of grain-growth. We construct three microstructural models for a material containing large pores. In the first model, we assume that a large pore is surrounded by identical grains as shown in Fig. 7. The proposition that the pore is surrounded by identical grains makes it difficult to connect the grains at the pore surface with the rest of the microstructure in a realistic manner. Here
Fig. 7. A numerical model of a large pore surrounded by identical grains.
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we simply ignore the interaction between these grains and the rest of the material. The outside boundaries of the grains are treated as neither grain-boundaries nor free surfaces and the outside ends of the grain-boundaries are sealed (i.e. no diffusive flux is allowed in or out of these ends). To maintain symmetry, all the grains can only move toward or away from the centre of the pore simultaneously at the same speed. The relative motion between the grains has to be accommodated by diffusion along the grain-boundaries, which in turn has to be accommodated by diffusion along the pore surface. This deposits matter onto either the grain-boundaries as the grains move away from the centre of the pore, or the pore surface as the grains move towards the centre of the pore. Three cubic spline elements were used to model each grain-boundary and facet of the pore surface respectively. In the numerical study we vary the number of grains surrounding the pore, the ratio of specific surface energy cs to specific grainboundary energy cgb, and the ratio of grainboundary length lgb to the pore radius Rpore. In most fine powders, especially nano-sized powders, the particles are often bounded together to form agglomerates. Consequently, the powder compact contains both small intra-agglomerate pores and large inter-agglomerate pores. In the sintering process, the small intra-agglomerate pores disappear quickly leaving the large interagglomerate pores behind. Most of the sintering time is spent in eliminating the large pores which are embedded in an almost fully dense solid matrix. In our second model, we assume that a large pore is embedded in a matrix of hexagonal grains as shown in Fig. 8. The matrix grains are assumed to be initially hexagonal because this is the simplest possible model away from the identical grain model. The difference between this model and the identical grain model in the characteristics of the microstructural evolution can be clearly observed. The pore was created by removing a few grains from a uniform hexagonal polycrystalline structure. Only a quarter of the model is shown because of symmetry. It can be taken as a representative unit of a two-dimensional infinite body containing a periodic array of pores. No exchange of matter is allowed between the unit and the surrounding
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Fig. 8. A numerical model of a large pore in a hexagonal grain matrix.
Fig. 9. A numerical model of a large pore in an irregular grain matrix.
body. The pore has a coordination number of 24. According to Eq. (6), this pore should grow if cs < 3.8cgb. In the finite element model each facet on the pore surface is represented by three surface diffusion elements. Each grain-boundary is represented by three grain-boundary diffusion elements and another three grain-boundary migration elements. The model consists of 163 grains which are discretised into 2715 cubic spline elements with 9815 degrees of freedom. The third model is shown in Fig. 9, which is more realistic and assumes that the large pore is embedded in an irregular grain matrix. The irregular grain structure was in fact taken from a scanning electronic micrograph of alumina with some modifications to accommodate the pore. Symmetry is assumed about the horizontal and vertical axes, hence only a quarter of the pore is shown. Again no matter exchange is allowed between the representative unit and its surrounding material. Each grain-boundary is represented by three grain-boundary diffusion elements and another three grain-boundary migration elements, and each facet of the pore surface is represented by three surface diffusion elements. The model consists of 354 grains which are discretised using 6144 cubic spline elements with a total number of 22,120 degrees of freedom. The pore has a coor-
dination number of 52. According to Eq. (6), this pore should grow if cs < 8.3cgb. When presenting the numerical results, all the length scales are non-dimensionalised by a characteristic length scale, lchar, which is taken as the initial length of one facet of the pore surface in the first model, the initial grain size d in the second model and the initial average grain-size in the third model respectively. The material parameters are non-dimensionalised as following: Ms ¼
Ms ; M gb
Mm ¼
M m l2char ; M gb
cs ¼
cs ; cgb
ð12Þ
and the time is non-dimensionalised as t ¼
M gb cgb l4char
t:
ð13Þ
4.2. Behaviour of a large pore surrounded by identical grains Using the model shown in Fig. 7, a series of computer simulations were undertaken and it was found that the kinetic model supports the critical coordination number theory. The model predicts that the pore shrinks if its coordination number is less than a critical value and grows if its coordination number is larger than the critical
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
value. Fig. 5 in Section 3 shows an example for the pore to shrink where its coordination number is 3. Figs. 10 and 11 show two examples for the pore to grow where the coordination number is 10 and 24 respectively. In the case of Fig. 10 cs/cgb = 1 and in the case of Fig. 11, cs/cgb = 3. lgb/Rpore was taken as 4 in both cases. The computer simulations show that the critical coordination number depends on not only cs/cgb but also lgb/Rpore. Fig. 12 presents the critical coordination numbers obtained from the computer simulation for different ratios of specific surface energy to specific grain-boundary energy, cs/cgb, and for different ratios of the grain-boundary length to the pore radius, lgb/Rpore. Also presented in the figure is the prediction of Eq. (6). It can be seen that as lgb/Rpore increases, the numerical results approach those predicted by Eq. (6). When deriving Eq. (6), the relative motion between the grains caused by redistributing matter away from the grain-boundaries and onto the pore surface was not considered. Eq. (6) is therefore only precise when the grain-boundaries are much longer than the pore radius.
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Fig. 11. Microstructural evolution of large pore surrounded by 24 identical grains at various time steps: t ¼ 0:0, 11.3 and 34.3. Material parameters are taken as Ms/Mgb = 1, cs/cgb = 3 and lgb/Rpore = 4.
Fig. 12. Comparison between critical coordination numbers obtained numerically and Eq. (6).
4.3. Behaviour of a large pore surrounded by hexagonal grains Fig. 10. Computer simulated microstructural evolution of a large pore surrounded by 10 identical grains. t0 ¼ 0:0, t1 ¼ 2:2 and t2 ¼ 208:2 for material parameters Ms/Mgb = 1, cs/cgb = 1 and lgb/Rpore = 4.
Using the model shown in Fig. 8, a series of computer simulations were undertaken. For many materials cs/cgb 3 (Cottrell, 1995), which was
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used in the simulation. Ashby (1990) provides a large collection of data for the surface and grainboundary diffusion coefficients and the grainboundary mobility which vary over a wide range and depends on the sintering temperature. In our numerical study, the ratio of Ms/Mgb was varied between 0.01 and 1000, and the ratio of (Mmd2)/ Mgb was varied between 0.001 and 1000 to cover most part of these data. A total number of 14 simulations were carried out and in all the cases the computer model predicts that the pore shrinks.
Fig. 13 presents an example of the simulations for which Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/ cgb = 3. This case can be compared to that shown in Fig. 11. The two cases share a common coordination number of 24. The material parameters are the same except that the grain-boundary mobility Mm is irrelevant in the identical grain model. The initial pore areas in the two cases were set to be as close as possible. Furthermore, the characteristic diffusion distances (i.e. the distance over which matter has to be transported in order for the pore
Fig. 13. Computer simulated microstructural evolution of a large pore in a hexagonal grain matrix. (a) t ¼ 1:6764 103 , (b) t ¼ 3:449 103 and (c) t ¼ 5:2517 103 for material parameters Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/cgb = 3.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
to grow or shrink) in the two cases were also set to be as close as possible. In the identical grain model shown in Fig. 11, the characteristic diffusion distance is simply the length of the grain-boundary, which is set as 12. In the hexagonal grain model, the characteristic diffusion distance can be taken as the average of lgb1, lgb2 and lgb3 shown in Fig. 8, which was calculated as 11.93. The contrast between the predictions of the two models is dramatic. While the identical grain model predicts that the large pore grows, the hexagonal grain model predicts that the large pore shrinks, in direct contradiction to the critical coordination number theory. As can be seen from Fig. 13, the pore is surrounded by three types of grains, i.e. full hexagonal grains, half grains and grains which occupy 5/6 of the full hexagon. Because each of the smallest grains at the pore surface is surrounded by three larger grains, the smallest grains have a tendency to shrink leading to grain-growth. On the other hand, the two triple junctions marked by a and a 0 in Fig. 8 tend to approach and finally meet each other. This disappearance of the shorter facets of the pore surface leads to a change in the coordination number and causes the smallest grains to meet each other. New grain-boundaries are formed between them which are oriented in the radial direction. This is very similar to the mechanism described by Ashby and Verrall (1973) as ‘‘neighbour-switching’’ which maintains the equiaxed structure during superplastic deformation. The simulations were terminated before the shorter facets of the pore surface disappeared. In a real powder compact, such a change of morphological relation between the grains occurs frequently allowing the pore to shrink continuously. Fig. 14 compares the pore area as functions of time obtained from the two different models respectively. Noticing that the log scale is used for the time axis in the figure, it can be seen that while the identical grain model predicts a slow growth, the hexagonal grain model predicts a much faster shrinkage. The total amount of mass in the model should be conserved in the computer simulations which can be used as a measure of accuracy of the numerical analysis. The total mass variation was 0.00006% in the identical grain model and 0.018% in the hexagonal grain model.
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Fig. 14. Comparison of numerical calculated pore area as a function of time for a pore surrounded by hexagonal grains and by identical grains respectively. Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/cgb = 3.
Fig. 15 shows another example in which the grain-boundary mobility Mm is increased by an order of magnitude from the case shown in Fig. 13. All the other parameters remain the same. It can be observed from the figure that the faster grain-boundary migration leads to the shrinkage of the small grains in the vicinity of the large pore. The first grain switching event now occurs away from the pore surface and is rapidly followed by elimination of the small grains and a reduction of the coordination number. Fig. 16 compares the pore areas as a function of time for the two cases shown in Figs. 13 and 15. It is interesting to see that fast grain-boundary migration actually leads to a slightly slower shrinkage rate of the pore, which suggests that grain-growth does not accelerate the pore elimination process; instead, it retards the process. Figs. 13 and 15 show two different scenarios for the microstructure evolution. The pore shrinkage is accompanied by a neighbour-switching event in Fig. 13 and by grain-growth in Fig. 15 respectively. In most of the practical applications, the first scenario is preferable since grain-growth destroys the fine grain structure, which leads to poor mechanical properties. Similar computer simulations were undertaken using cs/cgb = 1 and a coordination number of 10
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Fig. 15. Computer simulated microstructural evolution of a large pore in a hexagonal grain matrix. (a) t ¼ 1:1334 103 , (b) t ¼ 2:3489 103 and (c) t ¼ 3:3609 103 for material parameters Ms/Mgb = 1, (Mmd2)/Mgb = 100 and cs/cgb = 3.
so that they can be compared with the identical grain model shown in Fig. 10. The smaller coordination number was used because Eq. (6) predicts that the critical coordination number is 6 for cs/ cgb = 1. The ratio of Ms/Mgb was varied between 0.000001 and 1000 and the ratio of (Mmd2)/Mgb was varied between 0.000001 and 1000. A total number of 26 computer simulations were carried out. Once again, in all the cases the hexagonal grain model predicts that the pore shrinks. Fig. 17 presents an example of these simulations for which Ms/Mgb = 1 and (Mmd2)/Mgb = 1. It can be
seen that this is a case that is dominated by neighbour-switching. If one increases the grain-boundary mobility significantly, then grain-growth occurs as the pore shrinks in a similar way to that shown in Fig. 15. 4.4. Behaviour of a large pore surrounded by irregular grains Fig. 18 presents a computer simulation for the microstructural evolution of a large pore embedded in an irregular grain matrix. Fig. 19 shows
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was repeated using another set of material parameters, Ms/Mgb = 10, (Mmd2)/Mgb = 0.1242 and cs/ cgb = 3. The numerical results also predict that the pore shrinks for this case, and the details are not shown here.
5. Sintering potential and densification rate Ma (1997) and Cocks (2001) proposed an expression of sintering potential, refereed to as rs, for powder compacts containing larger pores: rs ¼ Fig. 16. Pore area as a function of time obtained using model in Fig. 10 for two different values of the normalized grainboundary migration mobility (Mmd2)/Mgb.
the numerically obtained pore area and free energy as functions of time. The material parameters used in the simulation are cs/cgb = 3, Ms/Mgb = 1 and (Mmd2)/Mgb = 0.124. For cs/cgb = 3, Eq. (6) predicts that the critical coordination number is 19. However it can be observed from Figs. 18 and 19 that the numerical model predicts that the pore shrinks despite a much larger coordination number of 52. In this computer simulation the grain-boundary mobility, Mm, was deliberately set to be small to make the point that the pore can shrink without the help of grain-growth. The strange value of (Mmd2)/Mgb is due to the average grain-size used in the non-dimensionalisation (the initial grain size d was calculated as 11.1 lm). Nevertheless some grain-growth did occur and the simulation was terminated before the smallest grain disappeared. It is possible to continue the computer simulation and follow the entire process of grain-growth and pore-shrinkage. However our purpose here is to establish that a large pore can shrink despite the large coordination number. The computer simulation shows the large pore shrinks in a matrix of irregular grains. The disappearance of the smaller grains from the irregular grain structure simply leads to another irregular grain structure and does not alter this numerical finding. This simulation
cs 0:5cgb dE ¼2 dV pore Rpore
ð14Þ
in which E is the total free energy of the material defined by Eq. (11), Vpore is the total volume of the pores, and Rpore is the radius of the pores. A two dimensional version of the sintering potential can be derived as rs ¼
cs 0:5cgb 1 dE ¼ l dApore Rpore
ð15Þ
in which l is the third dimension of the Ôtwo dimensionalÕ material model and Apore is the total area of the pores. In the computer simulation, both the free energy (Eq. (11)) and the pore area were calculated by numerical integrations at each time step. Figs. 16 and 20 show examples of the calculated pore area and free energy as functions of time. Therefore the sintering potential can be calculated from the numerical analysis using numerical differentiation: rs
1 DE : l DApore
ð16Þ
The densification rate of the large pore, A_ pore ¼ dApore =dt, can also be calculated by numerical differentiation from the data shown in Fig. 16. On the other hand, A_ pore can be estimated from an equivalent continuum model as shown in Fig. 21 using the sintering potential determined from Eq. (16). The problem shown in Fig. 21(b) is basically the viscous sintering model studied by Mackenzie and Shuttleworth (1949). If matter redistribution is controlled by grainboundary diffusion, i.e. if Ms Mgb, then the
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J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
Fig. 17. Computer simulated microstructural evolution of large pore in a hexagonal grain matrix. (a) t ¼ 0:0, (b) t ¼ 7:4804 103 and (c) t ¼ 1:8356 102 . Ms/Mgb = 1, (Mmd2)/Mgb = 1 and cs/cgb = 1.
matrix material deforms by Coble creep and its unidirectional strain rate is given by
e_ ¼
r M gb ¼b 3 r g d
ð17Þ
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
719
Fig. 19. Free energy and pore area as functions of time for the computer simulation shown in Fig. 21. Ms/Mgb = 1, (Mmd2)/ Mgb = 0.1242 and cs/cgb = 3.
Fig. 20. Numerically obtained free energy as functions of time for the cases shown in Figs. 13 and 15. Fig. 18. Computer simulated microstructural evolution of large pore in an irregular grain matrix. (a) t ¼ 1:0946 104 , (b) t ¼ 3:6668 103 . Ms/Mgb = 1, (Mmd2)/Mgb = 0.1242 and cs/ cgb = 3. The initial grain size d was calculated as 11.1 lm.
in which r is the remote stress, Mgb the grainboundary diffusion mobility, d the grain size, b a numerical number which is 36 for two dimensional hexagonal material ( Cocks and Searle, 1990) and 148 for three dimensional material (Coble, 1963), and g = d3/(bMgb) is the apparent viscosity of the matrix material. From the classical viscous solution for a hollow cylinder under internal pressure (for example, see Rekach, 1979), the densification rate of the pore area can be obtained as
A_ ¼ 4pR2pore rs =g ¼ 4pbM gb ðcs 0:5cgb ÞRpore =d 3 :
ð18Þ
Tables 1 and 2 compares the numerically obtained sintering potentials and densification rates to those obtained using Eqs. (15) and (18). The following non-dimensionalisation is used in the comparison: s ¼ r
d rs cgb
_ ¼ and A
d2 _ A: cgb M gb
ð19Þ
Table 1 presents the comparison for the hexagonal grain model as shown in Fig. 8 while Table 2 for the irregular grain model as shown in Fig. 9.
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J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
Fig. 21. Analogue of the micromechanical model (a) to a plane strain viscous-elastic problem of a hollow cylinder (b).
Table 1 Comparison of the sintering potential and shrinking rate of the pore area obtained from the computer simulation using the kinetic model shown in Fig. 8 and using the analytical expressions (Eqs. (15) and (18)) Cases
Material parameters
(a) (b) (c) (d) (e) (f)
Mm Mm Mm Mm Mm Mm
Analytical models
Mm = 0, Ms Mgb
¼ 0:1, M s ¼ 1000 ¼ 1, M s ¼ 100 ¼ 10, M s ¼ 1 ¼ 10, M s ¼ 10 ¼ 100, M s ¼ 1 ¼ 100, M s ¼ 100
r s obtained numerically
A_ obtained numerically
0.8428 0.7692 0.7998 0.8243 1.0342 1.1351
3313.65 3415.38 3361.58 3607.72 3264.46 3878.64
0.8049 (Eq. (15))
3512.99 (Eq. (18))
The comparison was made at the initial pore size.
Table 2 Comparison of the sintering potential and shrinking rate of the pore area obtained from the computer simulation using the kinetic model shown in Fig. 9 and using the analytical expressions (Eqs. (15) and (18)) Cases
Material parameters
r s obtained numerically
A_ obtained numerically
(g) Analytical models
M m ¼ 0:1242, M s ¼ 1 Mm = 0, Ms Mgb
0.3357 0.3206 (Eq. (15))
8899.69 8819.31 (Eq. (18))
The comparison was made at the initial pore size.
The comparison is made at the initial pore size because both the sintering potential and the densification rate depend on the pore size and the time. From the two tables it can be seen that the numerical and analytical results agree with each other fairly well. As expected the agreement is less good when Mm is large (see cases (e) and (f) in Table 1).
However the expected best agreement at the boundary diffusion controlled extreme (Ms Mgb) is not observed. In fact the numerical results scatter around the analytical values and the densification rate is not very sensitive to M s and M m . It is useful to point out that the experimental values of the kinetic mobilities (Ms, Mgb and Mm) are
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
often only accurate to an order of magnitude. In such a context, it can be concluded from the tables that the analytical expression (Eq. (18)) provides a good estimation for the densification rate.
6. Concluding remarks In a real powder compact, no critical coordination number exists above which a pore will not shrink and there is always a thermodynamic driving force to eliminate the pores regardless of their coordination number. When grain-growth is prohibited, a large pore shrinks by a neighbourswitching mechanism similar to the superplastic deformation. It is therefore not necessary and even counter productive to promote grain-growth during sintering in order to eliminate large pores. A kinetic model assuming that the pore is surrounded by identical grains seriously under-predicts the densification rate and the continuum viscous sintering model proposed by Ma (1997) and Cocks (2001) can be used to calculate the densification rate for powder compact containing large pores.
Acknowledgement H. ChÕng acknowledges gratefully a PhD studentship from the University of Surrey. The finite element mesh shown in Fig. 18(a) was generated by Michael Harding as part of his Mater Degree thesis.
References Ashby, M.F., 1990. HIP 6.0, Background Reading. Engineering Department Report, Cambridge University. Ashby, M.F., Verrall, A.A., 1973. Diffusion accommodated flow and superplasticity. Acta Metall. 21, 149–163. Chen, I.-W., Xue, L.A., 1990. J. Am. Ceram. Soc. 73, 2585–2609. ChÕng, H.N., Pan, J., 2004. Cubic spline elements for modelling microstructural evolution of materials controlled by solid state diffusion and grain-boundary migration. J. Comput. Phys. 196, 724–750. Coble, R.L., 1963. A model for boundary diffusion controlled creep in polycrystalline materials. J. Appl. Phys. 34 (6), 1679–1682.
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Cocks, A.C.F., 2001. Constitutive modelling of powder compaction and sintering. Prog. Mater. Sci. 46, 201–229. Cocks, A.C.F., Searle, A.A., 1990. Cavity growth in ceramic materials under multiaxial stress states. Acta. Metall. Mater. 38, 2493–2505. Cottrell, A., 1995. An Introduction to Metallurgy. The Institute of Materials, pp. 340–345. Flinn, B.D., Bordia, R.K., Zimmermann, A., Rodel, J., 2000. Evolution of defect size and strength of porous alumina during sintering. J. Eur. Ceram. Soc. 20, 2561–2568. Herring, C., 1955. Surface tension as a motivation for sintering. In: Kingston, W.E. (Ed.), The Physics of Powder Metallurgy. McGraw-Hill, New York. Hillert, M., 1965. Acta Metall. 13, 227. Kellett, B.J., Lange, F.F., 1989. Thermodynamics of densification: I, Sintering of simple particle arrays, equilibrium configurations and pore stability and shrinkage. J. Am. Ceram. Soc. 72, 725–734. Kingery, W.D., Francois, B., 1967. The sintering of crystalline oxides. I. Interaction between grain boundaries and pores. In: Kuczynski, G.C. et al. (Eds.), Sintering Related Phenomena. Goedon and Breach, New York, pp. 471– 496. Ma, J., 1997. Constitutive modelling of the densification of porous ceramic components. PhD thesis, Engineering Department, Cambridge University. Mackenzie, J.K., Shuttleworth, R., 1949. A phenomenological theory of sintering. Proc. Phys. Soc. Lond. Sect. B 62, 833– 852. Mayo, M.J., 1996. Processing of nanocrystalline ceramics from ultrafine particles. Int. Mater. Rev. 41, 85–115. Needleman, A., Rice, J.R., 1980. Plastic creep flow effects in the diffusive cavitation of grain boundaries. Acta Metall. 28, 1315–1332. Pan, J., 2003. Modelling sintering at different length scales. Int. Mater. Rev. 48, 69–85. Pan, J., 2004. Solid state diffusion under a large driving force and sintering of nano-sized particles. Philos. Mag. Lett. 84, 303–310. Pan, J., Cocks, A.C.F., Kucherenko, S., 1997. Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration. Proc. R. Soc. London, Ser. A 453, 2161–2184. Pan, J., Kucherenko, S., Yeomans, J.A., 1999. Computer simulation of solid state sintering. In: The Sixth Conference and Exhibition of the European Ceramic Society, vol. 2, pp. 469–470. Rekach, V.G., 1979. Manual of the Theory of Elasticity. Mir Publishers, Moscow, pp. 151–159. Slamovich, E.B., Lange, F.F., 1992. Densification of large pore: I. Experiment. J. Am. Ceram. Soc. 75, 2498–2508. Suo, Z., 1997. Motion of microscopic surfaces in materials. Adv. Appl. Mech. 33, 193–294. Svoboda, J., Riedel, H., 1995. Quasi-equilibrium sintering for coupled grain-boundary and surface diffusion. Acta. Metall. Mater. 43, 499–506.
Sintering kinetics of large pores J. Pan b
a,*
, H.N. ChÕng a, A.C.F. Cocks
b
a School of Engineering, University of Surrey, Guildford GU2 7XH, UK Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
Received 29 September 2003
Abstract The sintering kinetics of large pores in a dense polycrystalline solid is studied using computer simulation. Coupled grain-boundary diffusion, surface diffusion and grain-boundary migration are assumed as the underlying mechanisms for the microstructural evolution. A numerical method developed by ChÕng and Pan [ChÕng, H.N., Pan, J., 2004. Cubic spline elements for modelling microstructural evolution of materials controlled by solid state diffusion and grain-boundary migration. Journal of Computational Physics 196, 724], which is an improved version of that developed by Pan et al. [Pan, J., Cocks, A.C.F., Kucherenko, S., 1997. Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration. Proceedings of the Royal Society, London A 453, 2161], is used for the computer simulation. A well-known textbook theory due to Kingery and Francois [Kingery, W.D., Francois, B., 1967. The sintering of crystalline oxides. I. Interaction between grain boundaries and pores. In: Kuczynski, G.C., et al. (Eds.), Sintering Related Phenomena. Goedon and Breach, New York, p. 471] predicts that a critical coordination number exists above which a pore does not shrink. The computer simulation demonstrates that this is not a general rule and that the critical coordination number only exists if one assumes that the pore is surrounded by identical grains. This numerical finding explains the contradiction between existing experimental observations and the critical coordination number theory. The numerical results also support the analytical expression developed by Ma [Ma, J., 1997. Constitutive modelling of the densification of porous ceramic components. PhD thesis, Engineering Department, Cambridge University] and Cocks [Cocks, A.C.F., 2001. Constitutive modelling of powder compaction and sintering. Progress in Materials Science 46, 201] for the sintering potential of powder containing large pores. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Sintering; Computer simulation; Large pore; Modelling; Solid-state diffusion; Grain-growth; Densification; Finite element method
*
Corresponding author. Tel.: +44 1483 689671; fax: +44 1483 306039. E-mail address: [email protected] (J. Pan).
0167-6636/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2004.06.002
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J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
1. Introduction Most ceramic components and an increasing number of metal components are manufactured by sintering powder compacts, a process during which pores are eliminated at elevated temperatures (Pan, 2003). A commonly believed textbook theory is that a large pore may grow instead of shrink during sintering (Kingery and Francois, 1967). Here the number of grains surrounding a pore, i.e. the coordination number of a pore, defines ‘‘large’’. It is predicted that a critical coordination number exists, which depends on the dihedral angle of the material, and that a pore will shrink only if its coordination number is less than the critical value. It follows that a large pore can only be eliminated after sufficient grain-growth, which reduces the coordination number to that below the critical value. However, there is increasing experimental evidence contradicting with the theory. In a well-designed experiment, Slamovich and Lange (1992) demonstrated that it is the characteristic diffusion distance, instead of the coordination number, that controls the behaviour of a large pore. They artificially introduced large pores into two Zr(Y)O2 powders, exhibiting either sluggish (ZrO2— 3 mol% Y2O3) or rapid (ZrO2—8 mol% Y2O3) grain-growth kinetics. The two powder compacts containing the artificial large pores of the same size were sintered under identical conditions during which their microscopic morphologies and densities were monitored. Grain growth was much faster in the Zr(8Y)O2 matrix than in the Zr(3Y)O2 matrix, so that the average coordination number of the pores in the Zr(8Y)O2 compact quickly became much smaller than that in the Zr(3Y)O2 compact. If the coordination number controls the densification behaviour, then the Zr(8Y)O2 powder compact would densify faster than the Zr(3Y)O2 compact. The experimental results showed exactly the opposite: the Zr(8Y)O2 powder compact densified significantly slower than the Zr(3Y)O2 compact. After any common period of sintering, the grain size in the Zr(8Y)O2 matrix was much larger than that in the Zr(3Y)O2 matrix; the former, therefore, had a much longer characteristic diffusion distance than the latter. Earlier
work by Chen and Xue (1990) on the creep behaviour of the two materials indicated that the diffusivities of the two materials are similar regardless of their yttria contents. Hence the experiment showed that it is the characteristic diffusion distance instead of the coordination number that controls the densification behaviour. More recently, Flinn et al. (2000) carried out a carefully designed experiment using Al2O3 powder. Their experiment was designed to study micromechanical failure from pores, and to rationalise the fracture strength of ceramics as a function of grain size and density. They also introduced artificial large pores and found that pores as large as 125 lm in diameter in their initially submicron sized powders (which reached an average grain-size of about 1 lm by the end of the sintering) shrunk continuously throughout the sintering process. Flinn et al. (2000) realized the contradiction between their experimental results and the critical coordination number theory and explained the contradiction by arguing that the pore shrinkage is strain-controlled. However, the Ôstrain-controlledÕ deformation does not explain why the thermodynamic argument leading to the critical coordination number theory is not applicable in these experiments. Recently the issue of large pores has attracted more attention because of the effort to sinter nanostructured ceramics. It is very easy for nano-sized particles to form agglomerates and a nano-powder compact contains large inter-agglomerate pores as well as small intra-agglomerate pores. During sintering, the small intra-agglomerate pores disappear quickly leaving the large inter-agglomerate pores behind. Most of the sintering time is therefore spent in eliminating the large pores. This process has to be carefully controlled since excessive grain-growth can easily destroy the desired nanostructure (Mayo, 1996). Understanding the sintering kinetics of large pores is therefore crucial for sintering nano-structured ceramics. Ma (1997) and Cocks (2001) proposed that the sintering potential is independent of the coordination number and only depends on the size of the pore. This idea was supported by a kinetic model for the sintering of large pores developed by Pan et al. (1999). The kinetic model was however over-simplistic. The purpose of this paper is to pre-
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
sent a set of more realistic kinetic models for the sintering of large pores and to demonstrate that the coordination number does not necessarily control whether a pore shrinks or grows. Firstly, we provide an appropriate thermodynamic argument showing that there is not necessarily a thermodynamic barrier preventing a pore of any coordination number from being eliminated from a dense solid. We then support the thermodynamic argument by carrying out a series of computer simulations for microstructural evolution in a dense solid containing very large pores. This is a set of fully kinetic models which are solved using the numerical technique developed by ChÕng and Pan (2004), which is an improved version of the numerical technique initially developed by Pan et al. (1997). Coupled surface and grain-boundary diffusion is assumed as the underlying mechanism for densification. Grain-boundary migration (leading to grain-growth) is also included in the model. Finally, we compare the sintering potentials and the densification rates which are obtained analytically using the expressions proposed by Ma (1997) and Cocks (2001), and numerically using the computer simulations. The numerical results support the analytical expressions.
lution of a powder compact is the excess free ergy associated with the free surfaces of powder particles. The upper limit of the free ergy reduction in the sintering process can be pressed as
enthe enex-
DEtotal ¼ As cs :
ð1Þ
Here As is the total free surface area of the powder and cs is the specific energy for the free surface. This free energy reduction drives two processes during sintering, i.e. densification and graingrowth. When a powder compact densifies, the free surfaces are replaced by grain-boundaries as shown in Fig. 1(b). If a full densification is achieved then the change of free energy associated with the densification is DEdensification ¼ ðAs cs Agb cgb Þ:
First, let us consider a powder compact without agglomeration and large pores as shown in Fig. 1(a). The driving force for the microstructural evo-
ð2Þ
Here Agb is the total grain-boundary area in the dense solid and cgb is the specific energy for the grain-boundary. During densification approximately two free surfaces coalesce into one grainboundary as shown schematically in Fig. 1, i.e. As 2Agb and the total change of free energy related to densification can be estimated as DEdensification As ðcs 0:5cgb Þ:
2. Driving force for elimination of large pores
707
ð3Þ
This part of the free energy reduction drives the densification. The remaining part of the free energy reduction, DEgrain-growth 0:5As cgb ;
Fig. 1. Schematic illustration of the sintering process of a powder compact without agglomeration and large pores.
ð4Þ
708
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
drives the grain-growth. For most materials cs is larger than cgb, DEdensification is therefore normally negative which means that there is always a thermodynamic force driving the powder compact to eliminate the pores. Next let us consider a two-dimensional, polycrystalline solid containing periodically arranged large pores. A theory due to Kingery and Francois (1967) predicts that there exists a thermodynamic barrier for the shrinkage of the large pore. This barrier can be easily explained using the example shown in Fig. 2. A pore of radius R is surrounded by n identical grains. If the pore shrinks a little as indicated by the dashed lines, the radius of the pore is reduced by dR and the total change of free energy can be calculated as dE ¼ ndRcgb 2pdRcs :
ð5Þ
Fig. 2. Large pore surrounded by four identical grains.
Since the pore shrinks only if dE < 0, this leads to the following condition for the large pore to shrink: n < 2pcs =cgb :
ð6Þ
If the pore is surrounded by too many grains, then the free energy gain due to the extension of the grain-boundaries into the pore outbalances the free energy loss due to the elimination of the pore surface, so the pore should grow instead of shrink. This is the simplest version of the coordination number theory. It can be refined by considering the three-dimensional case, the effect of the outer surface of the compact and mass conservation in the variational argument (Kellett and Lange, 1989). There are two problems with the critical coordination number theory. The first one is that the above argument assumes that the large pore is surrounded by identical grains and that the grains move simultaneously into the large pore as it shrinks. However one rarely finds elongated grains centred around a large pore in specimens sintered from powder compacts containing large pores. The second problem is that increasing experimental evidence contradicts the theory as discussed in the introduction. In fact the thermodynamic barrier predicted by Eq. (6) is a result of the imposed kinetic route (that the grains move simultaneously into the pore). Instead of considering a small incremental variation of the pore size, let us consider the initial and final states of the material as shown in Fig. 3(a) and (b) respectively. Imagine the large
Fig. 3. Schematic illustration of elimination of a large pore from a dense solid.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
pore in Fig. 3(a) is somehow eliminated from the solid producing the dense solid shown in Fig. 3(b). If the grains remain roughly equiaxed throughout the elimination process, then it is obvious that the total free energy of the material shown in Fig. 3(b) is less than that shown in Fig. 3(a) simply because the free energy associated with the pore surface diminished and the free energy of the grain-boundaries does not increase because the grains remain equiaxed. Eqs. (3) and (4) can now also be used to estimate the free energy reduction associated with densification and graingrowth for a powder compact containing large pores. In this imaginary process, the total free energy is reduced no matter how large the coordination number is. This shows that eliminating a large pore from a dense solid does not necessarily have a free energy barrier contrary to that predicted by Eq. (6). The key condition here is that the grains remain equiaxed in the sintering process, which is almost universally met. It remains to show how the grains can rearrange themselves into the large pore while still remaining roughly equiaxed. In the following sections, we use a two-dimensional kinetic model to demonstrate that a combination of grain-boundary sliding and neighbour-switching can indeed help a large pore to by-pass the thermodynamic barrier predicted by Eq. (6).
3. A kinetic model for sintering To develop a kinetic model for the large pore, we consider solid-state sintering and further assume that matter redistribution is controlled by coupled grain-boundary and surface diffusion. Grain-boundary migration (leading to grain-growth) is also considered such that grainboundary diffusion occurs along a migrating grainboundary network. Following linear kinetic laws are assumed for the three processes (Herring, 1955; Hillert, 1965) although non-linear kinetic laws can also be accommodated by the variational method (Pan, 2004): jgb ¼ M gb
or ; os
ð7Þ
709
oðcs js Þ ; os
ð8Þ
vm ¼ M m cgb jgb ;
ð9Þ
js ¼ M s and
where jgb and js are the diffusive fluxes along the grain-boundaries and free surface respectively, vm is the migration velocity of the grain-boundary, Mgb, Ms and Mm are the mobilities for grainboundary diffusion, surface diffusion and grainboundary migration respectively, r is the stress normal to a grain-boundary, cs and cgb are the specific surface and grain-boundary energies respectively, js and jgb are the principal curvatures of the free surface and grain-boundary respectively, and s is a local coordinate along either a grainboundary or a free surface. Apart from the three kinetic laws, the normal stress r has to satisfy mechanical equilibrium, the diffusive fluxes have to satisfy matter conservation, the chemical potential has to be continuous everywhere, and finally the dihedral angles have to be such that the equilibrium between the interfacial tensions is satisfied. Existing data on superplastic deformation of several fine-grained ceramics indicate that grainboundary sliding is not a rate limiting mechanism (Chen and Xue, 1990). In the current model, we allow the grain-boundaries to slide freely (connected with no energy dissipation). It can be shown that the problem described above can be re-stated using the following variational principle (Needleman and Rice, 1980; Suo, 1997): among all the virtual migration velocities of the interfaces and the virtual diffusive fluxes that satisfy matter conservation, the actual velocity and diffusive flux fields minimise the functional Z Z 1 2 1 2 P¼ jgb dC þ js dC 2M 2M gb s Cgb Cs Z 1 2 dE ð10Þ vm dC þ þ 2M dt m Cgb in which Cgb represents the grain-boundary network, Cs represents the free surface in the pore and E is the total free energy of the system as discussed in the previous section:
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E¼
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
Z Cgb
cgb dC þ
Z cs dC:
ð11Þ
Cs
The microstructure evolves to reduce E. The variational principle (i.e. dP = 0) was derived from the fact that the reduction rate of E has to be balanced by the rates of the three energy dissipation mechanisms: grain-boundary diffusion, surface diffusion and grain-boundary migration. A robust finite element technique was developed by Pan et al. (1997) to solve the above variational problem. Recently ChÕng and Pan (2004) further improved the numerical technique by introducing a set of cubic spline finite element formulations. The numerical technique has been verified using a wide range of testing cases. Here we present one more test case using the cubic spline elements developed by ChÕng and Pan (2004) to demonstrate the validity of the numerical method. We consider the free sintering of a close packed array of circular cylinders as shown in Fig. 4. By symmetry we can isolate the dashed triangular region as the repeating cell and use the cell in the computer simulation. This problem has been solved by Svoboda and Riedel (1995) and is the simplest case of the type of situations considered in this paper. We solve this problem using the cubic spline finite element method. All the initial conditions and material parameters were taken as the same as those used by Svoboda and Riedel (1995). Twelve cubic spline elements were used to represent each cylinder surface; three elements were used to represent half of the neck (grain-
Fig. 4. A two-dimensional regular array of monosized cylinders. The dashed triangular region shows the repeating cell used in the computer simulation.
Fig. 5. Computer simulated evolution of a pore between three circular cylinders. The dihedral angle is set as 120°, diffusivity ratio Ms/Mgb = 1 and the initial radius of cylinder R = 1. t0 ¼ 0:0, t1 ¼ 7:0152 107 , t2 ¼ 2:602 105 , t3 ¼ 3:7033 104 , t4 ¼ 6:0978 104 and t5 ¼ 6:7742 104 .
boundary). Fig. 5 shows the computer simulated sintering process of the pore obtained using the cubic spline finite element method. Frequent remeshing had to be undertaken during the computer simulation. In Fig. 6, the numerical result of
Fig. 6. Comparison of porosity as a function of time obtained by Svoboda and Riedel (1995) and by finite element method. The equilibrium dihedral angle is 120° and the diffusivity ratio is Ms/Mgb = 1.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
porosity as a function of time obtained by Svoboda and Riedel (1995) is compared to that obtained using the cubic spline finite element method. It can be seen that the two numerical results agree very well with each other.
4. Sintering kinetics of large pores 4.1. Three microstructural models for dense material containing periodically arranged pores Our purpose here is to demonstrate that the critical coordination number theory is a direct consequence of the assumption that a large pore is surrounded by uniform and identical grains, and in general a pore with its coordination number larger than the critical value can still shrink without the help of grain-growth. We construct three microstructural models for a material containing large pores. In the first model, we assume that a large pore is surrounded by identical grains as shown in Fig. 7. The proposition that the pore is surrounded by identical grains makes it difficult to connect the grains at the pore surface with the rest of the microstructure in a realistic manner. Here
Fig. 7. A numerical model of a large pore surrounded by identical grains.
711
we simply ignore the interaction between these grains and the rest of the material. The outside boundaries of the grains are treated as neither grain-boundaries nor free surfaces and the outside ends of the grain-boundaries are sealed (i.e. no diffusive flux is allowed in or out of these ends). To maintain symmetry, all the grains can only move toward or away from the centre of the pore simultaneously at the same speed. The relative motion between the grains has to be accommodated by diffusion along the grain-boundaries, which in turn has to be accommodated by diffusion along the pore surface. This deposits matter onto either the grain-boundaries as the grains move away from the centre of the pore, or the pore surface as the grains move towards the centre of the pore. Three cubic spline elements were used to model each grain-boundary and facet of the pore surface respectively. In the numerical study we vary the number of grains surrounding the pore, the ratio of specific surface energy cs to specific grainboundary energy cgb, and the ratio of grainboundary length lgb to the pore radius Rpore. In most fine powders, especially nano-sized powders, the particles are often bounded together to form agglomerates. Consequently, the powder compact contains both small intra-agglomerate pores and large inter-agglomerate pores. In the sintering process, the small intra-agglomerate pores disappear quickly leaving the large interagglomerate pores behind. Most of the sintering time is spent in eliminating the large pores which are embedded in an almost fully dense solid matrix. In our second model, we assume that a large pore is embedded in a matrix of hexagonal grains as shown in Fig. 8. The matrix grains are assumed to be initially hexagonal because this is the simplest possible model away from the identical grain model. The difference between this model and the identical grain model in the characteristics of the microstructural evolution can be clearly observed. The pore was created by removing a few grains from a uniform hexagonal polycrystalline structure. Only a quarter of the model is shown because of symmetry. It can be taken as a representative unit of a two-dimensional infinite body containing a periodic array of pores. No exchange of matter is allowed between the unit and the surrounding
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J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
Fig. 8. A numerical model of a large pore in a hexagonal grain matrix.
Fig. 9. A numerical model of a large pore in an irregular grain matrix.
body. The pore has a coordination number of 24. According to Eq. (6), this pore should grow if cs < 3.8cgb. In the finite element model each facet on the pore surface is represented by three surface diffusion elements. Each grain-boundary is represented by three grain-boundary diffusion elements and another three grain-boundary migration elements. The model consists of 163 grains which are discretised into 2715 cubic spline elements with 9815 degrees of freedom. The third model is shown in Fig. 9, which is more realistic and assumes that the large pore is embedded in an irregular grain matrix. The irregular grain structure was in fact taken from a scanning electronic micrograph of alumina with some modifications to accommodate the pore. Symmetry is assumed about the horizontal and vertical axes, hence only a quarter of the pore is shown. Again no matter exchange is allowed between the representative unit and its surrounding material. Each grain-boundary is represented by three grain-boundary diffusion elements and another three grain-boundary migration elements, and each facet of the pore surface is represented by three surface diffusion elements. The model consists of 354 grains which are discretised using 6144 cubic spline elements with a total number of 22,120 degrees of freedom. The pore has a coor-
dination number of 52. According to Eq. (6), this pore should grow if cs < 8.3cgb. When presenting the numerical results, all the length scales are non-dimensionalised by a characteristic length scale, lchar, which is taken as the initial length of one facet of the pore surface in the first model, the initial grain size d in the second model and the initial average grain-size in the third model respectively. The material parameters are non-dimensionalised as following: Ms ¼
Ms ; M gb
Mm ¼
M m l2char ; M gb
cs ¼
cs ; cgb
ð12Þ
and the time is non-dimensionalised as t ¼
M gb cgb l4char
t:
ð13Þ
4.2. Behaviour of a large pore surrounded by identical grains Using the model shown in Fig. 7, a series of computer simulations were undertaken and it was found that the kinetic model supports the critical coordination number theory. The model predicts that the pore shrinks if its coordination number is less than a critical value and grows if its coordination number is larger than the critical
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
value. Fig. 5 in Section 3 shows an example for the pore to shrink where its coordination number is 3. Figs. 10 and 11 show two examples for the pore to grow where the coordination number is 10 and 24 respectively. In the case of Fig. 10 cs/cgb = 1 and in the case of Fig. 11, cs/cgb = 3. lgb/Rpore was taken as 4 in both cases. The computer simulations show that the critical coordination number depends on not only cs/cgb but also lgb/Rpore. Fig. 12 presents the critical coordination numbers obtained from the computer simulation for different ratios of specific surface energy to specific grain-boundary energy, cs/cgb, and for different ratios of the grain-boundary length to the pore radius, lgb/Rpore. Also presented in the figure is the prediction of Eq. (6). It can be seen that as lgb/Rpore increases, the numerical results approach those predicted by Eq. (6). When deriving Eq. (6), the relative motion between the grains caused by redistributing matter away from the grain-boundaries and onto the pore surface was not considered. Eq. (6) is therefore only precise when the grain-boundaries are much longer than the pore radius.
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Fig. 11. Microstructural evolution of large pore surrounded by 24 identical grains at various time steps: t ¼ 0:0, 11.3 and 34.3. Material parameters are taken as Ms/Mgb = 1, cs/cgb = 3 and lgb/Rpore = 4.
Fig. 12. Comparison between critical coordination numbers obtained numerically and Eq. (6).
4.3. Behaviour of a large pore surrounded by hexagonal grains Fig. 10. Computer simulated microstructural evolution of a large pore surrounded by 10 identical grains. t0 ¼ 0:0, t1 ¼ 2:2 and t2 ¼ 208:2 for material parameters Ms/Mgb = 1, cs/cgb = 1 and lgb/Rpore = 4.
Using the model shown in Fig. 8, a series of computer simulations were undertaken. For many materials cs/cgb 3 (Cottrell, 1995), which was
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used in the simulation. Ashby (1990) provides a large collection of data for the surface and grainboundary diffusion coefficients and the grainboundary mobility which vary over a wide range and depends on the sintering temperature. In our numerical study, the ratio of Ms/Mgb was varied between 0.01 and 1000, and the ratio of (Mmd2)/ Mgb was varied between 0.001 and 1000 to cover most part of these data. A total number of 14 simulations were carried out and in all the cases the computer model predicts that the pore shrinks.
Fig. 13 presents an example of the simulations for which Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/ cgb = 3. This case can be compared to that shown in Fig. 11. The two cases share a common coordination number of 24. The material parameters are the same except that the grain-boundary mobility Mm is irrelevant in the identical grain model. The initial pore areas in the two cases were set to be as close as possible. Furthermore, the characteristic diffusion distances (i.e. the distance over which matter has to be transported in order for the pore
Fig. 13. Computer simulated microstructural evolution of a large pore in a hexagonal grain matrix. (a) t ¼ 1:6764 103 , (b) t ¼ 3:449 103 and (c) t ¼ 5:2517 103 for material parameters Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/cgb = 3.
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
to grow or shrink) in the two cases were also set to be as close as possible. In the identical grain model shown in Fig. 11, the characteristic diffusion distance is simply the length of the grain-boundary, which is set as 12. In the hexagonal grain model, the characteristic diffusion distance can be taken as the average of lgb1, lgb2 and lgb3 shown in Fig. 8, which was calculated as 11.93. The contrast between the predictions of the two models is dramatic. While the identical grain model predicts that the large pore grows, the hexagonal grain model predicts that the large pore shrinks, in direct contradiction to the critical coordination number theory. As can be seen from Fig. 13, the pore is surrounded by three types of grains, i.e. full hexagonal grains, half grains and grains which occupy 5/6 of the full hexagon. Because each of the smallest grains at the pore surface is surrounded by three larger grains, the smallest grains have a tendency to shrink leading to grain-growth. On the other hand, the two triple junctions marked by a and a 0 in Fig. 8 tend to approach and finally meet each other. This disappearance of the shorter facets of the pore surface leads to a change in the coordination number and causes the smallest grains to meet each other. New grain-boundaries are formed between them which are oriented in the radial direction. This is very similar to the mechanism described by Ashby and Verrall (1973) as ‘‘neighbour-switching’’ which maintains the equiaxed structure during superplastic deformation. The simulations were terminated before the shorter facets of the pore surface disappeared. In a real powder compact, such a change of morphological relation between the grains occurs frequently allowing the pore to shrink continuously. Fig. 14 compares the pore area as functions of time obtained from the two different models respectively. Noticing that the log scale is used for the time axis in the figure, it can be seen that while the identical grain model predicts a slow growth, the hexagonal grain model predicts a much faster shrinkage. The total amount of mass in the model should be conserved in the computer simulations which can be used as a measure of accuracy of the numerical analysis. The total mass variation was 0.00006% in the identical grain model and 0.018% in the hexagonal grain model.
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Fig. 14. Comparison of numerical calculated pore area as a function of time for a pore surrounded by hexagonal grains and by identical grains respectively. Ms/Mgb = 1, (Mmd2)/Mgb = 10 and cs/cgb = 3.
Fig. 15 shows another example in which the grain-boundary mobility Mm is increased by an order of magnitude from the case shown in Fig. 13. All the other parameters remain the same. It can be observed from the figure that the faster grain-boundary migration leads to the shrinkage of the small grains in the vicinity of the large pore. The first grain switching event now occurs away from the pore surface and is rapidly followed by elimination of the small grains and a reduction of the coordination number. Fig. 16 compares the pore areas as a function of time for the two cases shown in Figs. 13 and 15. It is interesting to see that fast grain-boundary migration actually leads to a slightly slower shrinkage rate of the pore, which suggests that grain-growth does not accelerate the pore elimination process; instead, it retards the process. Figs. 13 and 15 show two different scenarios for the microstructure evolution. The pore shrinkage is accompanied by a neighbour-switching event in Fig. 13 and by grain-growth in Fig. 15 respectively. In most of the practical applications, the first scenario is preferable since grain-growth destroys the fine grain structure, which leads to poor mechanical properties. Similar computer simulations were undertaken using cs/cgb = 1 and a coordination number of 10
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Fig. 15. Computer simulated microstructural evolution of a large pore in a hexagonal grain matrix. (a) t ¼ 1:1334 103 , (b) t ¼ 2:3489 103 and (c) t ¼ 3:3609 103 for material parameters Ms/Mgb = 1, (Mmd2)/Mgb = 100 and cs/cgb = 3.
so that they can be compared with the identical grain model shown in Fig. 10. The smaller coordination number was used because Eq. (6) predicts that the critical coordination number is 6 for cs/ cgb = 1. The ratio of Ms/Mgb was varied between 0.000001 and 1000 and the ratio of (Mmd2)/Mgb was varied between 0.000001 and 1000. A total number of 26 computer simulations were carried out. Once again, in all the cases the hexagonal grain model predicts that the pore shrinks. Fig. 17 presents an example of these simulations for which Ms/Mgb = 1 and (Mmd2)/Mgb = 1. It can be
seen that this is a case that is dominated by neighbour-switching. If one increases the grain-boundary mobility significantly, then grain-growth occurs as the pore shrinks in a similar way to that shown in Fig. 15. 4.4. Behaviour of a large pore surrounded by irregular grains Fig. 18 presents a computer simulation for the microstructural evolution of a large pore embedded in an irregular grain matrix. Fig. 19 shows
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
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was repeated using another set of material parameters, Ms/Mgb = 10, (Mmd2)/Mgb = 0.1242 and cs/ cgb = 3. The numerical results also predict that the pore shrinks for this case, and the details are not shown here.
5. Sintering potential and densification rate Ma (1997) and Cocks (2001) proposed an expression of sintering potential, refereed to as rs, for powder compacts containing larger pores: rs ¼ Fig. 16. Pore area as a function of time obtained using model in Fig. 10 for two different values of the normalized grainboundary migration mobility (Mmd2)/Mgb.
the numerically obtained pore area and free energy as functions of time. The material parameters used in the simulation are cs/cgb = 3, Ms/Mgb = 1 and (Mmd2)/Mgb = 0.124. For cs/cgb = 3, Eq. (6) predicts that the critical coordination number is 19. However it can be observed from Figs. 18 and 19 that the numerical model predicts that the pore shrinks despite a much larger coordination number of 52. In this computer simulation the grain-boundary mobility, Mm, was deliberately set to be small to make the point that the pore can shrink without the help of grain-growth. The strange value of (Mmd2)/Mgb is due to the average grain-size used in the non-dimensionalisation (the initial grain size d was calculated as 11.1 lm). Nevertheless some grain-growth did occur and the simulation was terminated before the smallest grain disappeared. It is possible to continue the computer simulation and follow the entire process of grain-growth and pore-shrinkage. However our purpose here is to establish that a large pore can shrink despite the large coordination number. The computer simulation shows the large pore shrinks in a matrix of irregular grains. The disappearance of the smaller grains from the irregular grain structure simply leads to another irregular grain structure and does not alter this numerical finding. This simulation
cs 0:5cgb dE ¼2 dV pore Rpore
ð14Þ
in which E is the total free energy of the material defined by Eq. (11), Vpore is the total volume of the pores, and Rpore is the radius of the pores. A two dimensional version of the sintering potential can be derived as rs ¼
cs 0:5cgb 1 dE ¼ l dApore Rpore
ð15Þ
in which l is the third dimension of the Ôtwo dimensionalÕ material model and Apore is the total area of the pores. In the computer simulation, both the free energy (Eq. (11)) and the pore area were calculated by numerical integrations at each time step. Figs. 16 and 20 show examples of the calculated pore area and free energy as functions of time. Therefore the sintering potential can be calculated from the numerical analysis using numerical differentiation: rs
1 DE : l DApore
ð16Þ
The densification rate of the large pore, A_ pore ¼ dApore =dt, can also be calculated by numerical differentiation from the data shown in Fig. 16. On the other hand, A_ pore can be estimated from an equivalent continuum model as shown in Fig. 21 using the sintering potential determined from Eq. (16). The problem shown in Fig. 21(b) is basically the viscous sintering model studied by Mackenzie and Shuttleworth (1949). If matter redistribution is controlled by grainboundary diffusion, i.e. if Ms Mgb, then the
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Fig. 17. Computer simulated microstructural evolution of large pore in a hexagonal grain matrix. (a) t ¼ 0:0, (b) t ¼ 7:4804 103 and (c) t ¼ 1:8356 102 . Ms/Mgb = 1, (Mmd2)/Mgb = 1 and cs/cgb = 1.
matrix material deforms by Coble creep and its unidirectional strain rate is given by
e_ ¼
r M gb ¼b 3 r g d
ð17Þ
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
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Fig. 19. Free energy and pore area as functions of time for the computer simulation shown in Fig. 21. Ms/Mgb = 1, (Mmd2)/ Mgb = 0.1242 and cs/cgb = 3.
Fig. 20. Numerically obtained free energy as functions of time for the cases shown in Figs. 13 and 15. Fig. 18. Computer simulated microstructural evolution of large pore in an irregular grain matrix. (a) t ¼ 1:0946 104 , (b) t ¼ 3:6668 103 . Ms/Mgb = 1, (Mmd2)/Mgb = 0.1242 and cs/ cgb = 3. The initial grain size d was calculated as 11.1 lm.
in which r is the remote stress, Mgb the grainboundary diffusion mobility, d the grain size, b a numerical number which is 36 for two dimensional hexagonal material ( Cocks and Searle, 1990) and 148 for three dimensional material (Coble, 1963), and g = d3/(bMgb) is the apparent viscosity of the matrix material. From the classical viscous solution for a hollow cylinder under internal pressure (for example, see Rekach, 1979), the densification rate of the pore area can be obtained as
A_ ¼ 4pR2pore rs =g ¼ 4pbM gb ðcs 0:5cgb ÞRpore =d 3 :
ð18Þ
Tables 1 and 2 compares the numerically obtained sintering potentials and densification rates to those obtained using Eqs. (15) and (18). The following non-dimensionalisation is used in the comparison: s ¼ r
d rs cgb
_ ¼ and A
d2 _ A: cgb M gb
ð19Þ
Table 1 presents the comparison for the hexagonal grain model as shown in Fig. 8 while Table 2 for the irregular grain model as shown in Fig. 9.
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Fig. 21. Analogue of the micromechanical model (a) to a plane strain viscous-elastic problem of a hollow cylinder (b).
Table 1 Comparison of the sintering potential and shrinking rate of the pore area obtained from the computer simulation using the kinetic model shown in Fig. 8 and using the analytical expressions (Eqs. (15) and (18)) Cases
Material parameters
(a) (b) (c) (d) (e) (f)
Mm Mm Mm Mm Mm Mm
Analytical models
Mm = 0, Ms Mgb
¼ 0:1, M s ¼ 1000 ¼ 1, M s ¼ 100 ¼ 10, M s ¼ 1 ¼ 10, M s ¼ 10 ¼ 100, M s ¼ 1 ¼ 100, M s ¼ 100
r s obtained numerically
A_ obtained numerically
0.8428 0.7692 0.7998 0.8243 1.0342 1.1351
3313.65 3415.38 3361.58 3607.72 3264.46 3878.64
0.8049 (Eq. (15))
3512.99 (Eq. (18))
The comparison was made at the initial pore size.
Table 2 Comparison of the sintering potential and shrinking rate of the pore area obtained from the computer simulation using the kinetic model shown in Fig. 9 and using the analytical expressions (Eqs. (15) and (18)) Cases
Material parameters
r s obtained numerically
A_ obtained numerically
(g) Analytical models
M m ¼ 0:1242, M s ¼ 1 Mm = 0, Ms Mgb
0.3357 0.3206 (Eq. (15))
8899.69 8819.31 (Eq. (18))
The comparison was made at the initial pore size.
The comparison is made at the initial pore size because both the sintering potential and the densification rate depend on the pore size and the time. From the two tables it can be seen that the numerical and analytical results agree with each other fairly well. As expected the agreement is less good when Mm is large (see cases (e) and (f) in Table 1).
However the expected best agreement at the boundary diffusion controlled extreme (Ms Mgb) is not observed. In fact the numerical results scatter around the analytical values and the densification rate is not very sensitive to M s and M m . It is useful to point out that the experimental values of the kinetic mobilities (Ms, Mgb and Mm) are
J. Pan et al. / Mechanics of Materials 37 (2005) 705–721
often only accurate to an order of magnitude. In such a context, it can be concluded from the tables that the analytical expression (Eq. (18)) provides a good estimation for the densification rate.
6. Concluding remarks In a real powder compact, no critical coordination number exists above which a pore will not shrink and there is always a thermodynamic driving force to eliminate the pores regardless of their coordination number. When grain-growth is prohibited, a large pore shrinks by a neighbourswitching mechanism similar to the superplastic deformation. It is therefore not necessary and even counter productive to promote grain-growth during sintering in order to eliminate large pores. A kinetic model assuming that the pore is surrounded by identical grains seriously under-predicts the densification rate and the continuum viscous sintering model proposed by Ma (1997) and Cocks (2001) can be used to calculate the densification rate for powder compact containing large pores.
Acknowledgement H. ChÕng acknowledges gratefully a PhD studentship from the University of Surrey. The finite element mesh shown in Fig. 18(a) was generated by Michael Harding as part of his Mater Degree thesis.
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