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A model for the sintering of spherical particles of different sizes by solid state diffusion
PII:
Acta mater. Vol. 46, No. 13, pp. 4671±4690, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00144-X 1359-6454/98 $19.00 + 0.00
A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES OF DIFFERENT SIZES BY SOLID STATE DIFFUSION J. PAN{, H. LE{, S. KUCHERENKO and J. A. YEOMANS School of Mechanical and Materials Engineering, University of Surrey, Guildford GU2 5XH, U.K. (Received 30 September 1997; accepted 29 March 1998) AbstractÐIn this paper the numerical scheme developed by Pan and Cocks (Acta metall. 43, 1395±1406, 1995) is used to simulate the co-sintering process of two spherical particles of dierent sizes by coupled grain-boundary and surface diusion. The numerical analysis reveals many interesting features of the cosintering process. For example, it is found that the shrinkage between the two particles is not aected signi®cantly by the size dierence of the two particles as long as the dierence is less than 50%. Based on the numerical results, empirical formulae for the characteristic time of the co-sintering process and for the shrinkage rate between the two particles are established. The empirical formulae can be used to develop constitutive laws for early-stage sintering of powder compacts which take into account the eect of particle size distribution. To demonstrate this, a densi®cation rate equation for compacts with bimodal particle size distributions is derived. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Sintering is a crucial step for the fabrication of a ceramic component from a powder compact. Two things occur during sintering: grain growth and densi®cation. The driving force for the microstructural changes is the excess free energy associated with the large free-surface area of the ®ne powders. The actual mechanism of matter redistribution depends on the material system, the particle size, the sintering temperature and the level of external pressure, if such pressure is applied. Coupled grainboundary and surface diusion is often the dominant mechanism for the sintering of a ®ne particle compact which is subjected to a moderate pressure [1, 2]. This is a mechanism where grainboundary diusion transports matter to the junctions between grain-boundaries and pore surfaces, and surface diusion redistributes that matter onto the pore surfaces. A typical example of sintering by this mechanism is the pressureless sintering of alumina powder with a particle size of about 3± 8 mm at 14008C. In this paper we concentrate on this coupled diusion mechanism only. Considerable eorts have been made to understand the sintering process. Constitutive laws have been developed that enable ®nite element analysis to be performed for the sintering process [3]. The analysis can predict the history of stress, strain, relative density and grain size at any location of a {To whom all correspondence should be addressed. {Current address: Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K.
component during the sintering process [4]. However, the current generation of densi®cation laws assume uniform particle size and ignore the fact that most of the commercial powders consist of particles with a wide range of sizes. There have been relatively few studies examining the eect of particle size distribution on sintering. Patterson and Benson [5] did an experimental study on the eect of powder size distribution on sintering. Ting and Lin [6] derived a shrinkage rate equation for powder compacts taking into account the eect of particle size distributions. Grain growth was also considered in their model. Ting and Lin did not investigate the detailed kinetics of the co-sintering process between particles of dierent sizes. It was simply assumed that the sintering rate equation between two particles of dierent sizes takes exactly the same form as that between two particles of single size. For bimodal powder mixtures, German [7] proposed a simple model for prediction of the sintered density vs mixture composition from the knowledge of the densi®cation of the large and small powders. Probably the ®rst theoretical study of the co-sintering process between a pair of spherical particles of dierent sizes was made by Coble [8] who used the approximiate geometric relationships for identical particles to derive an approach rate equation for the non-identical particles. Recently Tanaka [9] re-investigated this problem. Assuming that the two particles maintain their truncated spherical shape during the co-sintering process, Tanaka obtained the rate equations for the sintering and coarsening of the two-particle system. More
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recently Parhami et al. [10] investigated a similar problem using a variational approach. Three degrees of freedom were used to de®ne the geometry of a representative unit of a row of particles of two dierent sizes. Using the classical Rayleigh± Ritz method, a numerical solution was obtained for the co-sintering process. It is important to realize that the selected degrees of freedom in these models limit the co-sintering process to a speci®c kinetic route which can be very dierent from the actual one. For example, it was assumed that the main parts of the two particles remain spherical during the sintering process. This is not quite correct since surface diusion is often not fast enough for the two particles to maintain the near-equilibrium shape. The accuracy of these models can only be found when they are compared with a full solution. The main purpose of this paper is to investigate the sintering kinetics of two spherical particles of
dierent sizes by means of computer simulation. Such a computer simulation was made possible by a numerical scheme developed recently by Pan and Cocks [11]. Ignoring the interaction between the pair of particles and the particles surrounding them, the two particles can be considered as a representative unit of a powder compact. The numerical analysis is shown to reveal many interesting features of the co-sintering process. Based on the numerical results, empirical formulae are established in order to describe the various aspects of the co-sintering process analytically. These empirical formulae can be used to develop constitutive laws for early-stage sintering taking into account the eects of particle size distributions. To demonstrate this, a densi®cation rate equation is derived for powder compacts with bimodal particle size distributions based on the empirical formula.
Fig. 1. The co-sintering process of two particles in contact with each other (schematic drawings).
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Experimental work is underway to verify the major ®ndings of this numerical study. 2. DRIVING FORCE FOR SINTERING
Figure 1(a) shows the representative unit that is considered in this work. The total free energy of the system is 1 E gs dAs ggb dAgb s
gb
where gs and ggb are the speci®c energies of the free surface and the inter-particle boundary (the grainboundary), respectively. The tendency to reduce E is the driving force for a shape change of the system. For the two-particle system, it is obvious that E reaches its minimum value when the two particles become one perfect sphere. There are several mechanisms by which matter redistribution can be achieved. These include viscous ¯ow, lattice diusion, evaporation and condensation, and coupled grain-boundary and surface diusion. As mentioned in the Introduction (Section 1), only the last mechanism is considered here. The shape evolution is controlled by the kinetic law as well as by the driving force. A comprehensive discussion about the roles played by driving forces and kinetic laws, respectively, in the microstructral evolution has been given by Sun et al. [12]. Where the grain-boundary meets the free surface, the equilibrium between the surface tensions of the two particles and the grain-boundary tension has to be maintained. As shown in Fig. 1(b), this requirement of equilibrium tends to ``bend'' the grainboundary towards the small particle with ends pinned at the junction. The grain-boundary then moves towards the smaller particle to ¯atten itself. The combination of these two mechanisms results in migration of the boundary towards the smaller particle as long as the junction itself moves. The ¯attening of the grain-boundary is assumed to be much faster than the movement of the junction. 3. THE KINETIC LAW
The usual linear kinetic law (Fick's law) is assumed for grain-boundary and free-surface diusion. The diusive ¯ux j, de®ned as volume of matter ¯owing across unit area perpendicular to the ¯ux direction per unit time, is assumed to depend linearly on the gradient of the chemical potential m of the diusing species: jÿ
Dd rm kT
2
where D is diusivity, d is the thickness of the layer through which the material diuses, k is Boltzmann's constant and T is the absolute temperature.
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Along the grain-boundary, the gradient of the chemical potential is directly related to the gradient of stress, s, acting normal to the grainboundary [13], i.e. m ÿOs 3 where O is the atomic volume. Along the free surface, the gradient of the atomic chemical potential is related to the gradient of the free-surface curvature [13], i.e m ÿOgs k
4
Here gs is the surface tension and k is the principal curvature of the surface. In equation (2), Dd should be replaced by Dgbdgb for grain-boundary diusion and by Dsds for free-surface diusion. The subscripts ``gb'' and ``s'' represent grain-boundary and free surface, respectively. 4. NUMERICAL SCHEME AND NONDIMENSIONALIZATION
The coupled grain-boundary and surface diusion problem is generally too dicult to solve analytically. Cavity growth and sintering are two opposite phenomena. Under many practical circumstances both processes can be controlled by the coupled diusion mechanism. For cavity growth, a steady state solution was obtained by Chuang and Rice [14] and later a self-similar solution was obtained by Chuang et al. [15]. For the sintering problem of uniform particles, Svoboda and Riedel obtained an analytical solution for the so-called ``small scale'' diusion problem [16]. For more general situations, numerical methods have to be used. Pharr and Nix [17] studied the cavity growth problem while Bross and Exner [18] studied the sintering problem using similar numerical methods at almost the same time. More recent studies of sintering using numerical analysis include the work by Bouvard and McMeeking [19] and Zhang and Schneibel [20]. These eorts have considerably improved our understanding of sintering kinetics. However, there is a common problem to all these previous studies: surface diusion was assumed to be symmetric about the grain-boundary and consequently the sintering of particles of dierent sizes cannot be studied using these numerical schemes since matter diuses from the smaller particle to the larger one when the two particles in contact are of dierent sizes. Recently, a general numerical method has been developed by Pan and coworkers [11, 21], which can be used to simulate microstructural evolution controlled by solid state diusion and grain-boundary migration. Using this numerical method, the evolution history of a prescribed network of grainboundaries with internal and external free surfaces can be followed. The grain-boundaries and the free
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surfaces can take any arbitrary shape. As a demonstrating example of the numerical scheme, the co-sintering process of two cylinders of dierent radii was simulated [11]. In this work, the numerical scheme described in Ref. [11] is used with some straightforward modi®cations to consider the axisymmetric problem of the two spherical particles instead of the two cylinders considered previously. The underlying principle used in this study is exactly the same as that described in Ref. [11], i.e. the chemical potential at the junction between the grain-boundary and the free surface is taken as an unknown. Matter conservation in the vicinity of the junction is used to determine the chemical potential. As soon as this chemical potential is known, the central ®nite dierence scheme is used to determine the migration velocity of the free surface. The approach velocity between the two particles is obtained analytically for the diusion problem in the circular disk of contact between the two particles. The pro®le of the system is then updated using the direct Euler integration scheme and the entire procedure is repeated for a required number of time steps. Rather than repeat the details of the numerical scheme here, emphasis is placed on the physical merits of the numerical analysis. It proves convenient to discuss the numerical results in terms of non-dimensionalized groups of material properties that control the sintering kinetics. A reference ``strain rate'' is de®ned as: e_ gb
Dgb dgb O gs kT r42
5
where r2 is the initial radius of the large particle. In this paper, all the lengths are scaled by r2 and the various physical variables are non-dimesionalized in the following way: k kr2 , s
sr2 gs
j
j e_ gb r22
_ W_ W e_ gb r2 t e_ gb t
6
where W_ is the approaching velocity between the two particles and t is the time. The free energy E can be non-dimensionalized as E 7 E 2 dA s g gb dA gb r2 gs s gb
and dA dA 2 r2 Then equation (2) becomes jgb rs for the grain-boundary diusion and js D s rk
g gb
ggb gs
8
10
for the free-surface diusion, in which Ds ds D s Dgb dgb
11
Two non-dimensionalized groups of the material properties have emerged from the above analysis: D s , which represents the relative importance of the two diusion processes, and g gb , which determines the dihedral angle, C, through the equation cos C
g gb 2
12
The numerical results are presented using D s and C as the input material properties. To apply the results to any real material system, data required by equation (5) as well as D s and g gb need to be available so that equation (6) can be used to transform the non-dimensionalized results into those for the material system concerned.
5. OVERVIEW OF THE COMPUTER SIMULATION
The computer simulations presented cover a wide range of dierent combinations of r1/r2, D s , C, and s 1 , which represent the ratio of particle radii, the ratio of surface diusivity over grain-boundary diffusivity, the dihedral angle and the normalized average stress applied on the grain-boundary, respectively. The ratio of particle radii, r1/r2, is varied between 0.1 and 1.0 and D s is varied between 0.01 and 100. Three dierent values of the dihedral angle, i.e. C = 458, C = 608 and C = 758, and three levels of the applied stress are used. Table 1 summarizes all the dierent cases. In total about 150 simulations were performed covering the various dierent cases. Ideally, the numerical analysis should start from zero contact area between the two particles. Numerically, however, the analysis has to start from a small initial contact area. In the numerical
Table 1. An overview of the computer simulation r1/r2
in which
9
0.1, 0.2, 0.3, 0.5, 0.7, 0.9, 1.0
D s
C
s 1
0.01, 0.1, 1.0, 100.0
458, 608, 758
0.0, 5.0, 50.0
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
simulations, the radius of the initial contact area has been taken as 0.1r1. The total volume of the two particles should remain constant throughout the co-sintering process but numerical errors cause volume ¯uctuations during the simulation. In all the simulations performed, the maximum ¯uctuation of the total volume was within 4% of the initial volume of the small particle. This indicates the high accuracy of the numerical scheme. The co-sintering process can be divided into two distinct stages, i.e. the stages before and after the grain-boundary disappears. Once the grain-boundary disappears, the system becomes one particle
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which evolves eventually into a perfect sphere as shown in Fig. 1. The second stage, which is completely controlled by surface diusion, is of little practical interest since at this stage the interaction between the system and the surrounding particles becomes signi®cant. All the simulations were therefore terminated as soon as the grain-boundary migrates out of the system. 6. CHARACTERISTICS OF SHAPE EVOLUTION OF THE TWO-PARTICLE SYSTEM
One of the purposes of the numerical study is to understand how the two-particle system evolves.
Fig. 2. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =1.0 and C = 608. (a) t 0, (b) t 1:153 10ÿ2 , (c) t 21:95 10ÿ2 , (d) t 28:19 10ÿ2 .
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Based on such an understanding, a simpler model of the system can be proposed and approximate solutions can be obtained using the variational approach described by Parhami et al. [10]. It is obvious that the closer the sizes of the two particles, the greater is the in¯uence of the smaller particle on the shape evolution of the entire system. This can be seen by comparing the numerical result from the case of r1/r2=0.5 with that of r1/r2=0.1 as shown in Figs 2 and 3, respectively. In these examples C = 608 and D s =1. For r1/r2=0.5 the large particle changes its shape rapidly from a sphere into a bulb. For r1/r2 =0.1, however, the large particle manages to maintain its original shape.
Varying D s (the ratio of surface diusivity to grain-boundary diusivity) changes whichever of the two diusion processes dominates the co-sintering process. It is the slower process that controls the overall rate, for example, when D s =100 grainboundary diusion dominates and when D s =0.01 surface diusion dominates. Figures 4 and 5 show the cases for D s =0.01 and 100, respectively, which can be compared with Fig. 2 where D s =1. In these examples, C = 608 and r1/r2=0.5. It can be observed that although D s varies over a range of ®ve orders of magnitude, the pattern of the shape evolution remains similar. The grain-boundary diffusion-controlled case shows a slightly smaller grain-boundary during the entire process. These ob-
Fig. 3. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.1, D s =1.0 and C = 608. (a) t 0, (b) t 1:227 10ÿ4 and (c) t 2:9 10ÿ4 .
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Fig. 4. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =0.01 and C = 608. (a) t 0, (b) t 0:8078, (c) t 10:56 and (d) t 20:05.
servations are also true for other ratios of particle radii. The numerical results suggest that D s mainly in¯uences the rate of the process not the pattern of evolution. The dihedral angle has a dramatic eect on the pattern of the shape evolution. This is clearly demonstrated by comparing Fig. 6, where C = 458, with Fig. 2, where C = 608, for r1/r2=0.5 and by comparing Fig. 7, where C = 458, with Fig. 8,
where C = 608, for r1/r2=0.9. In these examples, D s =1.0. It can be seen that a smaller dihedral angle helps the small particle to maintain a more rounded shape throughout the co-sintering process. As mentioned in the Introduction (Section 1), in eort to model the co-sintering process, Tanaka [9] assumed that the two particles maintain their truncated spherical shape as matter is transferred from the small particle to the large one while Parhami et
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Fig. 5. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =100 and C = 608. (a) t 0, (b) t 0:6 10ÿ3 , (c) t 7:171 10ÿ3 and (d) t 9:687 10ÿ3 .
al. [10] modi®ed Tanaka's model by introducing a cylindrical disk between the two spheres. From the computer simulations presented above it can be seen that the actual shape evolution of the two-particle system is very dierent from those assumed by Tanaka and Parhami et al. In fact the two-particle system can be better approximated by a system that consists of two truncated spheres connected by a truncated cone, as shown in Fig. 9(a). The approxi-
mate system is completely determined by ®ve geometric parameters: the radii of the large and small truncated spheres, r1 and r2, the radii of the top and bottom sections of the truncated cone, r1 and r2, and the height of the truncated cone, h. Matter conservation requires that only four of the ®ve parameters are independent; the system therefore has four degrees of freedom. The system starts from h = 0 and r1=r2=r0, where r0 is the initial neck
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Fig. 6. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =1.0 and C = 458. (a) t 0, (b) t 7:948 10ÿ2 , (c) t 65:49 10ÿ2 and (d) t 80:77 10ÿ2 .
size. By comparing the numerical results for all the dierent cases, it is found that for r1/r2 larger than 0.5, the junction between the truncated cone and the large truncated sphere can be regarded as a smooth one. As a consequence, the degrees of freedom of the approximate system can be reduced to three as shown by Fig. 9(b). For r1/r2 less than 0.5, the approximate model with four degrees of freedom, shown by Fig. 9(a), is more appropriate.
7. TIME TO DISAPPEARANCE OF THE GRAINBOUNDARY
A characteristic time describing the co-sintering process is the time taken for the inter-particle boundary (i.e. the grain-boundary) to migrate out of the system. In the following discussions, this characteristic time is referred as to td (which is nondimesionalized by e_ gb ). Once the grain-boundary has disappeared, grain growth is complete as far as
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Fig. 7. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9, D s =1.0 and C = 458. (a) t 0, (b) t 12:68 (c) t 15:25 and (d) t 16:39.
the two particles are concerned and it is experimentally dicult to distinguish the system from its surrounding particles in a powder compact. The dependence of td on relative diusivity D s and the ratio of particle radii r1/r2 is shown in Figs 10±12, for C = 458, C = 608 and C = 758, respectively. The interesting feature of the numerical results is that td depends linearly on r1/r2 on the
log±log plots. The numerical results can be best ®tted using the following empirical formula 4:63 ggb r1 13 td 0:15 D sÿ0:85 gs r2 which is plotted in Figs 10±12 using solid and dashed lines to compare with the numerical results. Using equations (5) and (6) we obtain
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Fig. 8. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9, D s =1.0 and C = 608. (a) t 0, (b) t 0:7869, (c) t 5:242 and (d) t 5:84.
Dgb dgb 0:85 ggb kT Ogs Dgb dgb Ds ds gs 4:63 r1 r42 r2
td 0:15
14
Equation (14) suggests that the in¯uence of grainboundary diusivity on the characteristic time is much weaker than that of surface diusivity. This is expected since it is the local surface diusion of matter from the small particle to the large one that controls the grain-boundary migration. The grainboundary is pinned at the junction between the grain-boundary and the free surface, and can only
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Fig. 9. An approximate model of the two-particle system (a) using four independent degrees of freedom and (b) using three independent degrees of freedom.
migrate when the junction moves by surface diusion. Grain-boundary diusion has little in¯uence on this process. It does, however, control the neck growth and shrinkage between the two particles which will be discussed in the Section 8. Equation (14) also suggests that the in¯uence of the size of
the large particle on the characteristic time is much weaker than that of the small one. This is simply because the grain-boundary always migrates through the small particle. Equation (14) breaks down when r1/r2 approaches unity for which td should be in®nity in
Fig. 10. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines represent the empirical formula of equation (13). C = 458.
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Fig. 11. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines represent the empirical formula of equation (13). C = 608.
theory. In reality the symmetric con®guration of two identical particles is unstable and a small perturbation of the particle geometry or material prop-
erty can destroy the symmetry. This was observed in the numerical simulation for identical particles during which numerical errors destroy the sym-
Fig. 12. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results. The solid and dashed lines represent the empirical formula of equation (13). C = 758.
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metry slightly and cause the grain-boundary to migrate towards one of the particles.
8. SHRINKAGE AND CONTACT SIZE
Based on their ®nite dierence studies, Bouvard and McMeeking [19] suggested the following empirical formula to describe the relationship _ and the radius between the approach velocity, W, of the contact area, x, between two identical particles W_ a b s1 r 15 r_egb x=r4 x=r2 gs where r is the radius of the particles, s1 is the average stress transmitted onto the contact area, a is a pure number which depends on D s and b is a pure number which is insensitive to D s and s1. Figures 13±15 present the numerical results from _ to x for various values of r / this study relating W 1 r2 and for s 1 =0 (Fig. 13), s 1 =5 (Fig. 14), and s 1 =50 (Fig. 15), respectively, where _ W_ W r2 e_ gb x and
x r2
s 1
s1 r2 gs
In these examples, D s =1 and C = 608. The numerical results can be ®tted using a modi®ed version of equation (15) r1 a b s 1 r2 W_ 0:5 1 r2 gs r2 e_ gb x=r2 4 x=r2 2 16 which is plotted in the ®gures using solid and dashed lines. Here a and b have been taken as 9 and 8, respectively, which is consistent with the values used by Bouvard and McMeeking [19]. It can be seen that the numerical results con®rm equation (15) for the case of uniform particles. As the size of the small particle decreases, the approach rate decreases. This is incorporated in the empirical formula given by equation (16) by introducing a factor 0.5(1 + r1/r2). The interesting feature of the numerical results is that the particle ratio does not have a signi®cant eect on the approach rate provided that r1/r2 is larger than 0.5. This is especially the case when stress is applied as shown by Figs 14 and 15. As the matter redistribution from the grainboundary to the neck is a local event in the early stages of the co-sintering process, the relative size of the two particles only comes into play at a later stage. The numerical results indeed indicate that W_ depends on r1/r2 in the later stages. When stress is
Fig. 13. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =0.
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Fig. 14. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =5.
applied, it tends to dominate the driving force for the co-sintering process and further weakens the in¯uence of r1/r2. From Figs 13±15 it can be seen that
equation (16) breaks down for r1/r2 less than 0.5. The numerical results can be better ®tted by introducing a dierent factor, 0.5(1 + (r1/r2))x for
Fig. 15. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =50.
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example, to equation (16) where x is an empirical exponent. No attempt to do so has been made here since such ®tting does not give any further insight into the process. Figure 16 shows the numerical results relating the shrinkage W to the radius of the contact area x between the two particles. Coble used a simple relationship between the two variables [8] w 1 x 2 17 r 2 r in which r is the initial radius of the particles. For two identical particles, the numerical results con®rm equation (17) when the neck radius is larger than 0.2r. For non-identical particles, the numerical relationship can be ®tted using a modi®ed version of equation (17) ÿz 2 W r1 1 x 0:5 1 18 r2 r2 d r2 in which d and z are empirical parameters. Equation (18) is plotted on Fig. 16 using z = 1.5 and d = 2.4 for comparision with the full numerical results. The empirical ®tting breaks down when the neck size is very small or when r1/r2 is less than 0.5. Figure 17 presents the numerical results relating the normalized shrinkage, W/r2, to the normalized time, t_egb , for three dierent values of r1/r2 and for D s =1 and C = 608. Again it can be seen that the size ratio of the two particles has almost no eect on the shrinkage. This observation is true for the
entire range of values of D s and C covered in Table 1. From Fig. 17, it can also be observed that W/r2 is linearly dependent on t_egb on the log±log scale which suggests the following empirical relationship 1=n W ÿ lt_egb r2
19
where l and n are empirical parameters. It is found that n is insensitive to D s and C and is within the range between 3 and 4. The empirical parameter l is found to be dependent on D s and C. For example, l = 0.006316, 0.4453 and 1.1874 for D s =0.01, 0.1, and 1.0, respectively when C = 608. At any ®xed normalized shrinkage, the shrinkage rate is linearly dependent on l. The numerical results suggest that as surface diusivity increases over three orders of magnitude relative to grainboundary diusivity, the shrinkage rate increases over the same orders of magnitude. Figure 18 presents the numerical relationship between W/r2 and t_egb covering D s =0.01±10, C = 458±758 and r1/r2=0.5±1.0. Equation (19) is plotted on top of the numerical results using n = 4 and l = 1.1874. Equation (19) is invalid for D s larger than 10, i.e. the linear relationship between W/ r2 and t_egb on the log±log scale breaks down when D s is larger than 10. This is shown in Fig. 19 for an extreme case of D s =100. Probably the most interesting numerical result presented in this section is that for a wide range of
Fig. 16. Relationship between the shrinkage and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (18). C = 608, D s =l and s 1 =0.
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Fig. 17. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608 and D s =1.
material parameters, the shrinkage and shrinkage rate between the two particles are not signi®cantly aected by the size dierence between the two particles. This numerical ®nding simpli®es the task of constructing densi®cation laws for powder compacts when taking into account the eect of size distributions. It means that the size distribution only in¯uences densi®cation (in the early stages) by in¯uencing the initial density and the number of contacts of a powder compact. The diusion kinetics between the particles are not aected at least in the early stages of the sintering process. This
conclusion is not valid if the size dierence between the particles is larger than 50%. 9. A DENSIFICATION RATE EQUATION FOR POWDER COMPACTS WITH A BIMODAL PARTICLE SIZE DISTRIBUTION
For a powder compact which consists of particles of only two dierent radii, Rl and Rs, for large and small particles, respectively, nl and ns are the number fractions of the large and small particles. There are three dierent types of contacts between the particles; nss, nll and nls represent the number frac-
Fig. 18. The shrinkage as a function of time. The shaded band represents the numerical results covering D s =0.01±10, C = 458±758 and r1/r2=0.5±1.0. The solid line represents the empirical formula of equation (19) using n = 4.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Fig. 19. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608 and D s =100.
tions of contacts between small and small, large and large, and large and small particles, respectively. Turner [22] demonstrated that the number fractions of the dierent contacts can be expressed as 2 n l Rl nll nl Rl ns Rs ls
ÿ
2nl Rl ns Rs nl Rl ns Rs
nss
2
ns Rs nl Rl ns Rs
2
e_ ÿ
nsll nsls nsss W_ 2nsll Rl nsls Rl Rs 2nsss Rs
in which W_ is the shrinkage rate of a contact. Using equation (20) and equations (21)±(23), it can be shown that nsls V s 1 R s s nll 1 ÿ V s R 2s
25
2 1 V s 1 ÿ V s R 3s
26
and nsss nsll
20
Furthermore Turner found that the number fractions of the contacts in a string of two sizes of particle can be given as nsll
2nll Rl 2nll Rl nls Rl Rs 2nss Rs
21
nsls
nls Rl Rs 2nll Rl nls Rl Rs 2nss Rs
22
nsss
2nss Rs 2nll Rl nls Rl Rs 2nss Rs
23
where the superscript ``s'' denotes a string. In order to obtain the densi®cation rate equation, either equation (16) together with equation (18) or equation (19) can be used. For simplicity, the latter is used here, i.e. it is assumed that the normalized shrinkage rates for the three dierent types of contacts are the same. The line strain rate of the powder compact is simply
24
where R s =Rs/Rl and V s is the volume fraction of the small particles in the compact. Using equations (25 and 26), equation (24) can be rewritten as W_ 1 ÿ e_ ÿ w R s ,V s 2 Rl
27
in which ÿ w R s ,V s ÿ 2 ÿ ÿ R 3s 1 ÿ V s V s 1 ÿ V s 1 R s R s V s2 ÿ 2 ÿ ÿ 2 R 3s 1 ÿ V s 0:5V s 1 ÿ V s 1 R s R s V s2 R s 28 Identifying r2 in equation (19) as Rl and substituting equation (19) into equation (27) gives e_ ÿ
1 w R s ,V s l_egb 1=n t 1ÿn=n 2n
29
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Referring to the relative density of the compact as D and the densi®cation rate as D_ then 1 D_ D 3 ÿ3_e D D0
30
in which D0 is the initial density of the compact. Combining equation (29) with equation (30) gives 1 1=n 1ÿn=n 3 ÿ D 3ÿ D_ l_egb t w R s ,V s D0 D 2n
31
which can be integrated with respect to time giving 1ÿ
1 1=n 1=n D0 3 1 ÿ ÿ w Rs ,Vs l_egb t 2 D
32
Eliminating t from equations (31) and (32) and noticing equation (5) results in 0 11ÿn 1 1 ÿ 1 D D D_ 0 3C 3 B1 ÿ 33 Awn R s ,V s 4 @ A D D Rl D0 in which A is the only parameter that depends on material properties and temperature: Dgb dgb Ogs 3 21ÿn l 34 A 2n kT As discussed in Section 8, l is an empirical parameter which depends on Dsds/Dgbdgb and gs/ggb, and increases as Dsds/Dgbdgb increases. In practice A can be determined experimentally as a single par-
4689
ameter of the material which is advantageous as the constituent properties, especially the thermal diusivity, are dicult to measure accurately. It must be pointed out that equation (33) is only valid for the early stage of sintering during which the interaction between dierent contacts is insignificant and the contact number is not aected by density changes of the compact although it would be possible to release the second limitation. The conclusion that size dierence does not aect the shrinkage between two particles can be used to develop densi®cation rate equations for powder compact with a known size distribution. This has not been pursued here but rather equation (33) has been constructed to establish a macroscopic consequence of the numerical ®ndings so that experiments can be performed to verify the numerical results. ÿ Function w R s ,V s represents the dierence in densi®cation rate between a uniform powder com ÿ pact and a bimodal powder compact. w R s ,V s is plotted in Fig. 20 where the eects of volume fraction and size of the small particles can be seen clearly. 10. CONCLUSIONS
Particle size distribution can in¯uence the sintering process in two respects, i.e. grain growth and densi®cation. The two processes become strongly coupled in the intermediate stage of the sintering process. In the early stage, however, they can be treated separately. For early stage sintering, equation (14) can be used to formulate a rate
Fig. 20. Eect of the volume fraction V s and relative size R s of the small particles on the initial densi®cation rate of a powder compact with a bimodal particle size distribution.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
equation for grain growth, and equations (16) and (19) can be used to formulate a rate equation for densi®cation. The most interesting ®nding of this numerical study is that the shrinkage and shrinkage rate between two particles is not signi®cantly aected by the size dierence of the two particles as long as the dierence is less than 50%. The densi®cation rate equation, equation (33), for powder compacts with bimodal particle size distribution, is directly based on this numerical conclusion and therefore can be used to verify the numerical ®ndings experimentally. AcknowledgementsÐThe authors wish to thank A. C. F. Cocks for his invaluable suggestions throughout this work. Parts of the numerical simulations were performed by ®nal year project students including A. Sem, H. D. Luong and C. Poth in the Department of Mechanical Engineering of the University of Surrey. This research is ®nancially supported by the EPSRC (grant GR/K78102), which is gratefully acknowledged. REFERENCES 1. Ashby, M. F., Acta. metall., 1974, 22, 275. 2. Swinkels, F. B. and Ashby, M. F., Acta metall., 1981, 29, 259. 3. Cocks, A. C. F., Acta. metall., 1994, 42, 2191±2210. 4. Du, Z. Z. and Cocks, A. C. F., Acta. metall., 1992, 40, 1981.
5. Patterson, B. R. and Benson, L. A., Prog. Powder Metall., 1984, 39, 215. 6. Ting, J.-M. and Lin, R. Y., J. Mat. Sci., 1994, 29, 1867. 7. German, R. M., Metall. Trans., 1992, 23A, 1445. 8. Coble, R. L., J. Am. Ceram. Soc., 1973, 56, 461. 9. Tanaka, H., J. Ceram. Soc. Japan, 1995, 103, 138. 10. Parhami, F., Cocks, A. C. F., McMeeking, R. M. and Suo, Z., private communication. 11. Pan, J. and Cocks, A. C. F., Acta. metall., 1995, 43, 1395±1406. 12. Sun, B., Suo, Z. and Cocks, A. C. F., J. Mech. Phys. Solids, 1996, 44, 559. 13. Herring, C., in The Physics of Powder Metallurgy, ed. W. E. Kingston. McGraw-Hill, New York, 1955. 14. Chuang, T. and Rice, J. R., Acta. metall., 1973, 21, 1625. 15. Chuang, T., Kagawa, K. I., Rice, J. R. and Sills, L. B., Acta. metall., 1979, 27, 265. 16. Svoboda, J. and Riedel, H., Acta. metall., 1995, 43, 1. 17. Pharr, G. M. and Nix, W. D., Acta. metall., 1979, 27, 1615. 18. Bross, P. and Exner, H. E., Acta. metall., 1979, 27, 1013. 19. Bouvard, D. and McMeeking, R. M., J. Am. Ceram. Soc., 1996, 79, 666. 20. Zhang, W. and Schneibel, J. H., Acta metall. mater., 1995, 43, 4377. 21. Pan, J., Cocks, A. C. F. and Kucherenko, S., Proc. R. Soc. A, 1997, 453, 2161. 22. Turner, C. D., Fundamental Investigations of the Isostatic Pressing of Composite Powders, Ph.D. thesis, Engineering Department, Cambridge University, 1994.
Acta mater. Vol. 46, No. 13, pp. 4671±4690, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00144-X 1359-6454/98 $19.00 + 0.00
A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES OF DIFFERENT SIZES BY SOLID STATE DIFFUSION J. PAN{, H. LE{, S. KUCHERENKO and J. A. YEOMANS School of Mechanical and Materials Engineering, University of Surrey, Guildford GU2 5XH, U.K. (Received 30 September 1997; accepted 29 March 1998) AbstractÐIn this paper the numerical scheme developed by Pan and Cocks (Acta metall. 43, 1395±1406, 1995) is used to simulate the co-sintering process of two spherical particles of dierent sizes by coupled grain-boundary and surface diusion. The numerical analysis reveals many interesting features of the cosintering process. For example, it is found that the shrinkage between the two particles is not aected signi®cantly by the size dierence of the two particles as long as the dierence is less than 50%. Based on the numerical results, empirical formulae for the characteristic time of the co-sintering process and for the shrinkage rate between the two particles are established. The empirical formulae can be used to develop constitutive laws for early-stage sintering of powder compacts which take into account the eect of particle size distribution. To demonstrate this, a densi®cation rate equation for compacts with bimodal particle size distributions is derived. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Sintering is a crucial step for the fabrication of a ceramic component from a powder compact. Two things occur during sintering: grain growth and densi®cation. The driving force for the microstructural changes is the excess free energy associated with the large free-surface area of the ®ne powders. The actual mechanism of matter redistribution depends on the material system, the particle size, the sintering temperature and the level of external pressure, if such pressure is applied. Coupled grainboundary and surface diusion is often the dominant mechanism for the sintering of a ®ne particle compact which is subjected to a moderate pressure [1, 2]. This is a mechanism where grainboundary diusion transports matter to the junctions between grain-boundaries and pore surfaces, and surface diusion redistributes that matter onto the pore surfaces. A typical example of sintering by this mechanism is the pressureless sintering of alumina powder with a particle size of about 3± 8 mm at 14008C. In this paper we concentrate on this coupled diusion mechanism only. Considerable eorts have been made to understand the sintering process. Constitutive laws have been developed that enable ®nite element analysis to be performed for the sintering process [3]. The analysis can predict the history of stress, strain, relative density and grain size at any location of a {To whom all correspondence should be addressed. {Current address: Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K.
component during the sintering process [4]. However, the current generation of densi®cation laws assume uniform particle size and ignore the fact that most of the commercial powders consist of particles with a wide range of sizes. There have been relatively few studies examining the eect of particle size distribution on sintering. Patterson and Benson [5] did an experimental study on the eect of powder size distribution on sintering. Ting and Lin [6] derived a shrinkage rate equation for powder compacts taking into account the eect of particle size distributions. Grain growth was also considered in their model. Ting and Lin did not investigate the detailed kinetics of the co-sintering process between particles of dierent sizes. It was simply assumed that the sintering rate equation between two particles of dierent sizes takes exactly the same form as that between two particles of single size. For bimodal powder mixtures, German [7] proposed a simple model for prediction of the sintered density vs mixture composition from the knowledge of the densi®cation of the large and small powders. Probably the ®rst theoretical study of the co-sintering process between a pair of spherical particles of dierent sizes was made by Coble [8] who used the approximiate geometric relationships for identical particles to derive an approach rate equation for the non-identical particles. Recently Tanaka [9] re-investigated this problem. Assuming that the two particles maintain their truncated spherical shape during the co-sintering process, Tanaka obtained the rate equations for the sintering and coarsening of the two-particle system. More
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
recently Parhami et al. [10] investigated a similar problem using a variational approach. Three degrees of freedom were used to de®ne the geometry of a representative unit of a row of particles of two dierent sizes. Using the classical Rayleigh± Ritz method, a numerical solution was obtained for the co-sintering process. It is important to realize that the selected degrees of freedom in these models limit the co-sintering process to a speci®c kinetic route which can be very dierent from the actual one. For example, it was assumed that the main parts of the two particles remain spherical during the sintering process. This is not quite correct since surface diusion is often not fast enough for the two particles to maintain the near-equilibrium shape. The accuracy of these models can only be found when they are compared with a full solution. The main purpose of this paper is to investigate the sintering kinetics of two spherical particles of
dierent sizes by means of computer simulation. Such a computer simulation was made possible by a numerical scheme developed recently by Pan and Cocks [11]. Ignoring the interaction between the pair of particles and the particles surrounding them, the two particles can be considered as a representative unit of a powder compact. The numerical analysis is shown to reveal many interesting features of the co-sintering process. Based on the numerical results, empirical formulae are established in order to describe the various aspects of the co-sintering process analytically. These empirical formulae can be used to develop constitutive laws for early-stage sintering taking into account the eects of particle size distributions. To demonstrate this, a densi®cation rate equation is derived for powder compacts with bimodal particle size distributions based on the empirical formula.
Fig. 1. The co-sintering process of two particles in contact with each other (schematic drawings).
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Experimental work is underway to verify the major ®ndings of this numerical study. 2. DRIVING FORCE FOR SINTERING
Figure 1(a) shows the representative unit that is considered in this work. The total free energy of the system is 1 E gs dAs ggb dAgb s
gb
where gs and ggb are the speci®c energies of the free surface and the inter-particle boundary (the grainboundary), respectively. The tendency to reduce E is the driving force for a shape change of the system. For the two-particle system, it is obvious that E reaches its minimum value when the two particles become one perfect sphere. There are several mechanisms by which matter redistribution can be achieved. These include viscous ¯ow, lattice diusion, evaporation and condensation, and coupled grain-boundary and surface diusion. As mentioned in the Introduction (Section 1), only the last mechanism is considered here. The shape evolution is controlled by the kinetic law as well as by the driving force. A comprehensive discussion about the roles played by driving forces and kinetic laws, respectively, in the microstructral evolution has been given by Sun et al. [12]. Where the grain-boundary meets the free surface, the equilibrium between the surface tensions of the two particles and the grain-boundary tension has to be maintained. As shown in Fig. 1(b), this requirement of equilibrium tends to ``bend'' the grainboundary towards the small particle with ends pinned at the junction. The grain-boundary then moves towards the smaller particle to ¯atten itself. The combination of these two mechanisms results in migration of the boundary towards the smaller particle as long as the junction itself moves. The ¯attening of the grain-boundary is assumed to be much faster than the movement of the junction. 3. THE KINETIC LAW
The usual linear kinetic law (Fick's law) is assumed for grain-boundary and free-surface diusion. The diusive ¯ux j, de®ned as volume of matter ¯owing across unit area perpendicular to the ¯ux direction per unit time, is assumed to depend linearly on the gradient of the chemical potential m of the diusing species: jÿ
Dd rm kT
2
where D is diusivity, d is the thickness of the layer through which the material diuses, k is Boltzmann's constant and T is the absolute temperature.
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Along the grain-boundary, the gradient of the chemical potential is directly related to the gradient of stress, s, acting normal to the grainboundary [13], i.e. m ÿOs 3 where O is the atomic volume. Along the free surface, the gradient of the atomic chemical potential is related to the gradient of the free-surface curvature [13], i.e m ÿOgs k
4
Here gs is the surface tension and k is the principal curvature of the surface. In equation (2), Dd should be replaced by Dgbdgb for grain-boundary diusion and by Dsds for free-surface diusion. The subscripts ``gb'' and ``s'' represent grain-boundary and free surface, respectively. 4. NUMERICAL SCHEME AND NONDIMENSIONALIZATION
The coupled grain-boundary and surface diusion problem is generally too dicult to solve analytically. Cavity growth and sintering are two opposite phenomena. Under many practical circumstances both processes can be controlled by the coupled diusion mechanism. For cavity growth, a steady state solution was obtained by Chuang and Rice [14] and later a self-similar solution was obtained by Chuang et al. [15]. For the sintering problem of uniform particles, Svoboda and Riedel obtained an analytical solution for the so-called ``small scale'' diusion problem [16]. For more general situations, numerical methods have to be used. Pharr and Nix [17] studied the cavity growth problem while Bross and Exner [18] studied the sintering problem using similar numerical methods at almost the same time. More recent studies of sintering using numerical analysis include the work by Bouvard and McMeeking [19] and Zhang and Schneibel [20]. These eorts have considerably improved our understanding of sintering kinetics. However, there is a common problem to all these previous studies: surface diusion was assumed to be symmetric about the grain-boundary and consequently the sintering of particles of dierent sizes cannot be studied using these numerical schemes since matter diuses from the smaller particle to the larger one when the two particles in contact are of dierent sizes. Recently, a general numerical method has been developed by Pan and coworkers [11, 21], which can be used to simulate microstructural evolution controlled by solid state diusion and grain-boundary migration. Using this numerical method, the evolution history of a prescribed network of grainboundaries with internal and external free surfaces can be followed. The grain-boundaries and the free
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
surfaces can take any arbitrary shape. As a demonstrating example of the numerical scheme, the co-sintering process of two cylinders of dierent radii was simulated [11]. In this work, the numerical scheme described in Ref. [11] is used with some straightforward modi®cations to consider the axisymmetric problem of the two spherical particles instead of the two cylinders considered previously. The underlying principle used in this study is exactly the same as that described in Ref. [11], i.e. the chemical potential at the junction between the grain-boundary and the free surface is taken as an unknown. Matter conservation in the vicinity of the junction is used to determine the chemical potential. As soon as this chemical potential is known, the central ®nite dierence scheme is used to determine the migration velocity of the free surface. The approach velocity between the two particles is obtained analytically for the diusion problem in the circular disk of contact between the two particles. The pro®le of the system is then updated using the direct Euler integration scheme and the entire procedure is repeated for a required number of time steps. Rather than repeat the details of the numerical scheme here, emphasis is placed on the physical merits of the numerical analysis. It proves convenient to discuss the numerical results in terms of non-dimensionalized groups of material properties that control the sintering kinetics. A reference ``strain rate'' is de®ned as: e_ gb
Dgb dgb O gs kT r42
5
where r2 is the initial radius of the large particle. In this paper, all the lengths are scaled by r2 and the various physical variables are non-dimesionalized in the following way: k kr2 , s
sr2 gs
j
j e_ gb r22
_ W_ W e_ gb r2 t e_ gb t
6
where W_ is the approaching velocity between the two particles and t is the time. The free energy E can be non-dimensionalized as E 7 E 2 dA s g gb dA gb r2 gs s gb
and dA dA 2 r2 Then equation (2) becomes jgb rs for the grain-boundary diusion and js D s rk
g gb
ggb gs
8
10
for the free-surface diusion, in which Ds ds D s Dgb dgb
11
Two non-dimensionalized groups of the material properties have emerged from the above analysis: D s , which represents the relative importance of the two diusion processes, and g gb , which determines the dihedral angle, C, through the equation cos C
g gb 2
12
The numerical results are presented using D s and C as the input material properties. To apply the results to any real material system, data required by equation (5) as well as D s and g gb need to be available so that equation (6) can be used to transform the non-dimensionalized results into those for the material system concerned.
5. OVERVIEW OF THE COMPUTER SIMULATION
The computer simulations presented cover a wide range of dierent combinations of r1/r2, D s , C, and s 1 , which represent the ratio of particle radii, the ratio of surface diusivity over grain-boundary diffusivity, the dihedral angle and the normalized average stress applied on the grain-boundary, respectively. The ratio of particle radii, r1/r2, is varied between 0.1 and 1.0 and D s is varied between 0.01 and 100. Three dierent values of the dihedral angle, i.e. C = 458, C = 608 and C = 758, and three levels of the applied stress are used. Table 1 summarizes all the dierent cases. In total about 150 simulations were performed covering the various dierent cases. Ideally, the numerical analysis should start from zero contact area between the two particles. Numerically, however, the analysis has to start from a small initial contact area. In the numerical
Table 1. An overview of the computer simulation r1/r2
in which
9
0.1, 0.2, 0.3, 0.5, 0.7, 0.9, 1.0
D s
C
s 1
0.01, 0.1, 1.0, 100.0
458, 608, 758
0.0, 5.0, 50.0
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
simulations, the radius of the initial contact area has been taken as 0.1r1. The total volume of the two particles should remain constant throughout the co-sintering process but numerical errors cause volume ¯uctuations during the simulation. In all the simulations performed, the maximum ¯uctuation of the total volume was within 4% of the initial volume of the small particle. This indicates the high accuracy of the numerical scheme. The co-sintering process can be divided into two distinct stages, i.e. the stages before and after the grain-boundary disappears. Once the grain-boundary disappears, the system becomes one particle
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which evolves eventually into a perfect sphere as shown in Fig. 1. The second stage, which is completely controlled by surface diusion, is of little practical interest since at this stage the interaction between the system and the surrounding particles becomes signi®cant. All the simulations were therefore terminated as soon as the grain-boundary migrates out of the system. 6. CHARACTERISTICS OF SHAPE EVOLUTION OF THE TWO-PARTICLE SYSTEM
One of the purposes of the numerical study is to understand how the two-particle system evolves.
Fig. 2. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =1.0 and C = 608. (a) t 0, (b) t 1:153 10ÿ2 , (c) t 21:95 10ÿ2 , (d) t 28:19 10ÿ2 .
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Based on such an understanding, a simpler model of the system can be proposed and approximate solutions can be obtained using the variational approach described by Parhami et al. [10]. It is obvious that the closer the sizes of the two particles, the greater is the in¯uence of the smaller particle on the shape evolution of the entire system. This can be seen by comparing the numerical result from the case of r1/r2=0.5 with that of r1/r2=0.1 as shown in Figs 2 and 3, respectively. In these examples C = 608 and D s =1. For r1/r2=0.5 the large particle changes its shape rapidly from a sphere into a bulb. For r1/r2 =0.1, however, the large particle manages to maintain its original shape.
Varying D s (the ratio of surface diusivity to grain-boundary diusivity) changes whichever of the two diusion processes dominates the co-sintering process. It is the slower process that controls the overall rate, for example, when D s =100 grainboundary diusion dominates and when D s =0.01 surface diusion dominates. Figures 4 and 5 show the cases for D s =0.01 and 100, respectively, which can be compared with Fig. 2 where D s =1. In these examples, C = 608 and r1/r2=0.5. It can be observed that although D s varies over a range of ®ve orders of magnitude, the pattern of the shape evolution remains similar. The grain-boundary diffusion-controlled case shows a slightly smaller grain-boundary during the entire process. These ob-
Fig. 3. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.1, D s =1.0 and C = 608. (a) t 0, (b) t 1:227 10ÿ4 and (c) t 2:9 10ÿ4 .
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
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Fig. 4. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =0.01 and C = 608. (a) t 0, (b) t 0:8078, (c) t 10:56 and (d) t 20:05.
servations are also true for other ratios of particle radii. The numerical results suggest that D s mainly in¯uences the rate of the process not the pattern of evolution. The dihedral angle has a dramatic eect on the pattern of the shape evolution. This is clearly demonstrated by comparing Fig. 6, where C = 458, with Fig. 2, where C = 608, for r1/r2=0.5 and by comparing Fig. 7, where C = 458, with Fig. 8,
where C = 608, for r1/r2=0.9. In these examples, D s =1.0. It can be seen that a smaller dihedral angle helps the small particle to maintain a more rounded shape throughout the co-sintering process. As mentioned in the Introduction (Section 1), in eort to model the co-sintering process, Tanaka [9] assumed that the two particles maintain their truncated spherical shape as matter is transferred from the small particle to the large one while Parhami et
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Fig. 5. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =100 and C = 608. (a) t 0, (b) t 0:6 10ÿ3 , (c) t 7:171 10ÿ3 and (d) t 9:687 10ÿ3 .
al. [10] modi®ed Tanaka's model by introducing a cylindrical disk between the two spheres. From the computer simulations presented above it can be seen that the actual shape evolution of the two-particle system is very dierent from those assumed by Tanaka and Parhami et al. In fact the two-particle system can be better approximated by a system that consists of two truncated spheres connected by a truncated cone, as shown in Fig. 9(a). The approxi-
mate system is completely determined by ®ve geometric parameters: the radii of the large and small truncated spheres, r1 and r2, the radii of the top and bottom sections of the truncated cone, r1 and r2, and the height of the truncated cone, h. Matter conservation requires that only four of the ®ve parameters are independent; the system therefore has four degrees of freedom. The system starts from h = 0 and r1=r2=r0, where r0 is the initial neck
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
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Fig. 6. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.5, D s =1.0 and C = 458. (a) t 0, (b) t 7:948 10ÿ2 , (c) t 65:49 10ÿ2 and (d) t 80:77 10ÿ2 .
size. By comparing the numerical results for all the dierent cases, it is found that for r1/r2 larger than 0.5, the junction between the truncated cone and the large truncated sphere can be regarded as a smooth one. As a consequence, the degrees of freedom of the approximate system can be reduced to three as shown by Fig. 9(b). For r1/r2 less than 0.5, the approximate model with four degrees of freedom, shown by Fig. 9(a), is more appropriate.
7. TIME TO DISAPPEARANCE OF THE GRAINBOUNDARY
A characteristic time describing the co-sintering process is the time taken for the inter-particle boundary (i.e. the grain-boundary) to migrate out of the system. In the following discussions, this characteristic time is referred as to td (which is nondimesionalized by e_ gb ). Once the grain-boundary has disappeared, grain growth is complete as far as
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Fig. 7. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9, D s =1.0 and C = 458. (a) t 0, (b) t 12:68 (c) t 15:25 and (d) t 16:39.
the two particles are concerned and it is experimentally dicult to distinguish the system from its surrounding particles in a powder compact. The dependence of td on relative diusivity D s and the ratio of particle radii r1/r2 is shown in Figs 10±12, for C = 458, C = 608 and C = 758, respectively. The interesting feature of the numerical results is that td depends linearly on r1/r2 on the
log±log plots. The numerical results can be best ®tted using the following empirical formula 4:63 ggb r1 13 td 0:15 D sÿ0:85 gs r2 which is plotted in Figs 10±12 using solid and dashed lines to compare with the numerical results. Using equations (5) and (6) we obtain
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Fig. 8. Computer-simulated co-sintering process of two particles in contact with each other: r1/r2=0.9, D s =1.0 and C = 608. (a) t 0, (b) t 0:7869, (c) t 5:242 and (d) t 5:84.
Dgb dgb 0:85 ggb kT Ogs Dgb dgb Ds ds gs 4:63 r1 r42 r2
td 0:15
14
Equation (14) suggests that the in¯uence of grainboundary diusivity on the characteristic time is much weaker than that of surface diusivity. This is expected since it is the local surface diusion of matter from the small particle to the large one that controls the grain-boundary migration. The grainboundary is pinned at the junction between the grain-boundary and the free surface, and can only
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Fig. 9. An approximate model of the two-particle system (a) using four independent degrees of freedom and (b) using three independent degrees of freedom.
migrate when the junction moves by surface diusion. Grain-boundary diusion has little in¯uence on this process. It does, however, control the neck growth and shrinkage between the two particles which will be discussed in the Section 8. Equation (14) also suggests that the in¯uence of the size of
the large particle on the characteristic time is much weaker than that of the small one. This is simply because the grain-boundary always migrates through the small particle. Equation (14) breaks down when r1/r2 approaches unity for which td should be in®nity in
Fig. 10. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines represent the empirical formula of equation (13). C = 458.
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
4683
Fig. 11. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results; the solid and dashed lines represent the empirical formula of equation (13). C = 608.
theory. In reality the symmetric con®guration of two identical particles is unstable and a small perturbation of the particle geometry or material prop-
erty can destroy the symmetry. This was observed in the numerical simulation for identical particles during which numerical errors destroy the sym-
Fig. 12. The time to disappearance of the grain-boundary at various values of the relative diusivity and ratio of particle radii. The symbols represent the numerical results. The solid and dashed lines represent the empirical formula of equation (13). C = 758.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
metry slightly and cause the grain-boundary to migrate towards one of the particles.
8. SHRINKAGE AND CONTACT SIZE
Based on their ®nite dierence studies, Bouvard and McMeeking [19] suggested the following empirical formula to describe the relationship _ and the radius between the approach velocity, W, of the contact area, x, between two identical particles W_ a b s1 r 15 r_egb x=r4 x=r2 gs where r is the radius of the particles, s1 is the average stress transmitted onto the contact area, a is a pure number which depends on D s and b is a pure number which is insensitive to D s and s1. Figures 13±15 present the numerical results from _ to x for various values of r / this study relating W 1 r2 and for s 1 =0 (Fig. 13), s 1 =5 (Fig. 14), and s 1 =50 (Fig. 15), respectively, where _ W_ W r2 e_ gb x and
x r2
s 1
s1 r2 gs
In these examples, D s =1 and C = 608. The numerical results can be ®tted using a modi®ed version of equation (15) r1 a b s 1 r2 W_ 0:5 1 r2 gs r2 e_ gb x=r2 4 x=r2 2 16 which is plotted in the ®gures using solid and dashed lines. Here a and b have been taken as 9 and 8, respectively, which is consistent with the values used by Bouvard and McMeeking [19]. It can be seen that the numerical results con®rm equation (15) for the case of uniform particles. As the size of the small particle decreases, the approach rate decreases. This is incorporated in the empirical formula given by equation (16) by introducing a factor 0.5(1 + r1/r2). The interesting feature of the numerical results is that the particle ratio does not have a signi®cant eect on the approach rate provided that r1/r2 is larger than 0.5. This is especially the case when stress is applied as shown by Figs 14 and 15. As the matter redistribution from the grainboundary to the neck is a local event in the early stages of the co-sintering process, the relative size of the two particles only comes into play at a later stage. The numerical results indeed indicate that W_ depends on r1/r2 in the later stages. When stress is
Fig. 13. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =0.
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
4685
Fig. 14. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =5.
applied, it tends to dominate the driving force for the co-sintering process and further weakens the in¯uence of r1/r2. From Figs 13±15 it can be seen that
equation (16) breaks down for r1/r2 less than 0.5. The numerical results can be better ®tted by introducing a dierent factor, 0.5(1 + (r1/r2))x for
Fig. 15. Relationship between the shrinkage rate and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (16). C = 608, D s =1 and s 1 =50.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
example, to equation (16) where x is an empirical exponent. No attempt to do so has been made here since such ®tting does not give any further insight into the process. Figure 16 shows the numerical results relating the shrinkage W to the radius of the contact area x between the two particles. Coble used a simple relationship between the two variables [8] w 1 x 2 17 r 2 r in which r is the initial radius of the particles. For two identical particles, the numerical results con®rm equation (17) when the neck radius is larger than 0.2r. For non-identical particles, the numerical relationship can be ®tted using a modi®ed version of equation (17) ÿz 2 W r1 1 x 0:5 1 18 r2 r2 d r2 in which d and z are empirical parameters. Equation (18) is plotted on Fig. 16 using z = 1.5 and d = 2.4 for comparision with the full numerical results. The empirical ®tting breaks down when the neck size is very small or when r1/r2 is less than 0.5. Figure 17 presents the numerical results relating the normalized shrinkage, W/r2, to the normalized time, t_egb , for three dierent values of r1/r2 and for D s =1 and C = 608. Again it can be seen that the size ratio of the two particles has almost no eect on the shrinkage. This observation is true for the
entire range of values of D s and C covered in Table 1. From Fig. 17, it can also be observed that W/r2 is linearly dependent on t_egb on the log±log scale which suggests the following empirical relationship 1=n W ÿ lt_egb r2
19
where l and n are empirical parameters. It is found that n is insensitive to D s and C and is within the range between 3 and 4. The empirical parameter l is found to be dependent on D s and C. For example, l = 0.006316, 0.4453 and 1.1874 for D s =0.01, 0.1, and 1.0, respectively when C = 608. At any ®xed normalized shrinkage, the shrinkage rate is linearly dependent on l. The numerical results suggest that as surface diusivity increases over three orders of magnitude relative to grainboundary diusivity, the shrinkage rate increases over the same orders of magnitude. Figure 18 presents the numerical relationship between W/r2 and t_egb covering D s =0.01±10, C = 458±758 and r1/r2=0.5±1.0. Equation (19) is plotted on top of the numerical results using n = 4 and l = 1.1874. Equation (19) is invalid for D s larger than 10, i.e. the linear relationship between W/ r2 and t_egb on the log±log scale breaks down when D s is larger than 10. This is shown in Fig. 19 for an extreme case of D s =100. Probably the most interesting numerical result presented in this section is that for a wide range of
Fig. 16. Relationship between the shrinkage and the contact radius at various values of the ratio of particle radii. The solid and dashed lines represent the empirical formula of equation (18). C = 608, D s =l and s 1 =0.
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
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Fig. 17. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608 and D s =1.
material parameters, the shrinkage and shrinkage rate between the two particles are not signi®cantly aected by the size dierence between the two particles. This numerical ®nding simpli®es the task of constructing densi®cation laws for powder compacts when taking into account the eect of size distributions. It means that the size distribution only in¯uences densi®cation (in the early stages) by in¯uencing the initial density and the number of contacts of a powder compact. The diusion kinetics between the particles are not aected at least in the early stages of the sintering process. This
conclusion is not valid if the size dierence between the particles is larger than 50%. 9. A DENSIFICATION RATE EQUATION FOR POWDER COMPACTS WITH A BIMODAL PARTICLE SIZE DISTRIBUTION
For a powder compact which consists of particles of only two dierent radii, Rl and Rs, for large and small particles, respectively, nl and ns are the number fractions of the large and small particles. There are three dierent types of contacts between the particles; nss, nll and nls represent the number frac-
Fig. 18. The shrinkage as a function of time. The shaded band represents the numerical results covering D s =0.01±10, C = 458±758 and r1/r2=0.5±1.0. The solid line represents the empirical formula of equation (19) using n = 4.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Fig. 19. The shrinkage as a function of time for various values of the ratio of particle radii. C = 608 and D s =100.
tions of contacts between small and small, large and large, and large and small particles, respectively. Turner [22] demonstrated that the number fractions of the dierent contacts can be expressed as 2 n l Rl nll nl Rl ns Rs ls
ÿ
2nl Rl ns Rs nl Rl ns Rs
nss
2
ns Rs nl Rl ns Rs
2
e_ ÿ
nsll nsls nsss W_ 2nsll Rl nsls Rl Rs 2nsss Rs
in which W_ is the shrinkage rate of a contact. Using equation (20) and equations (21)±(23), it can be shown that nsls V s 1 R s s nll 1 ÿ V s R 2s
25
2 1 V s 1 ÿ V s R 3s
26
and nsss nsll
20
Furthermore Turner found that the number fractions of the contacts in a string of two sizes of particle can be given as nsll
2nll Rl 2nll Rl nls Rl Rs 2nss Rs
21
nsls
nls Rl Rs 2nll Rl nls Rl Rs 2nss Rs
22
nsss
2nss Rs 2nll Rl nls Rl Rs 2nss Rs
23
where the superscript ``s'' denotes a string. In order to obtain the densi®cation rate equation, either equation (16) together with equation (18) or equation (19) can be used. For simplicity, the latter is used here, i.e. it is assumed that the normalized shrinkage rates for the three dierent types of contacts are the same. The line strain rate of the powder compact is simply
24
where R s =Rs/Rl and V s is the volume fraction of the small particles in the compact. Using equations (25 and 26), equation (24) can be rewritten as W_ 1 ÿ e_ ÿ w R s ,V s 2 Rl
27
in which ÿ w R s ,V s ÿ 2 ÿ ÿ R 3s 1 ÿ V s V s 1 ÿ V s 1 R s R s V s2 ÿ 2 ÿ ÿ 2 R 3s 1 ÿ V s 0:5V s 1 ÿ V s 1 R s R s V s2 R s 28 Identifying r2 in equation (19) as Rl and substituting equation (19) into equation (27) gives e_ ÿ
1 w R s ,V s l_egb 1=n t 1ÿn=n 2n
29
PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
Referring to the relative density of the compact as D and the densi®cation rate as D_ then 1 D_ D 3 ÿ3_e D D0
30
in which D0 is the initial density of the compact. Combining equation (29) with equation (30) gives 1 1=n 1ÿn=n 3 ÿ D 3ÿ D_ l_egb t w R s ,V s D0 D 2n
31
which can be integrated with respect to time giving 1ÿ
1 1=n 1=n D0 3 1 ÿ ÿ w Rs ,Vs l_egb t 2 D
32
Eliminating t from equations (31) and (32) and noticing equation (5) results in 0 11ÿn 1 1 ÿ 1 D D D_ 0 3C 3 B1 ÿ 33 Awn R s ,V s 4 @ A D D Rl D0 in which A is the only parameter that depends on material properties and temperature: Dgb dgb Ogs 3 21ÿn l 34 A 2n kT As discussed in Section 8, l is an empirical parameter which depends on Dsds/Dgbdgb and gs/ggb, and increases as Dsds/Dgbdgb increases. In practice A can be determined experimentally as a single par-
4689
ameter of the material which is advantageous as the constituent properties, especially the thermal diusivity, are dicult to measure accurately. It must be pointed out that equation (33) is only valid for the early stage of sintering during which the interaction between dierent contacts is insignificant and the contact number is not aected by density changes of the compact although it would be possible to release the second limitation. The conclusion that size dierence does not aect the shrinkage between two particles can be used to develop densi®cation rate equations for powder compact with a known size distribution. This has not been pursued here but rather equation (33) has been constructed to establish a macroscopic consequence of the numerical ®ndings so that experiments can be performed to verify the numerical results. ÿ Function w R s ,V s represents the dierence in densi®cation rate between a uniform powder com ÿ pact and a bimodal powder compact. w R s ,V s is plotted in Fig. 20 where the eects of volume fraction and size of the small particles can be seen clearly. 10. CONCLUSIONS
Particle size distribution can in¯uence the sintering process in two respects, i.e. grain growth and densi®cation. The two processes become strongly coupled in the intermediate stage of the sintering process. In the early stage, however, they can be treated separately. For early stage sintering, equation (14) can be used to formulate a rate
Fig. 20. Eect of the volume fraction V s and relative size R s of the small particles on the initial densi®cation rate of a powder compact with a bimodal particle size distribution.
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PAN et al.: A MODEL FOR THE SINTERING OF SPHERICAL PARTICLES
equation for grain growth, and equations (16) and (19) can be used to formulate a rate equation for densi®cation. The most interesting ®nding of this numerical study is that the shrinkage and shrinkage rate between two particles is not signi®cantly aected by the size dierence of the two particles as long as the dierence is less than 50%. The densi®cation rate equation, equation (33), for powder compacts with bimodal particle size distribution, is directly based on this numerical conclusion and therefore can be used to verify the numerical ®ndings experimentally. AcknowledgementsÐThe authors wish to thank A. C. F. Cocks for his invaluable suggestions throughout this work. Parts of the numerical simulations were performed by ®nal year project students including A. Sem, H. D. Luong and C. Poth in the Department of Mechanical Engineering of the University of Surrey. This research is ®nancially supported by the EPSRC (grant GR/K78102), which is gratefully acknowledged. REFERENCES 1. Ashby, M. F., Acta. metall., 1974, 22, 275. 2. Swinkels, F. B. and Ashby, M. F., Acta metall., 1981, 29, 259. 3. Cocks, A. C. F., Acta. metall., 1994, 42, 2191±2210. 4. Du, Z. Z. and Cocks, A. C. F., Acta. metall., 1992, 40, 1981.
5. Patterson, B. R. and Benson, L. A., Prog. Powder Metall., 1984, 39, 215. 6. Ting, J.-M. and Lin, R. Y., J. Mat. Sci., 1994, 29, 1867. 7. German, R. M., Metall. Trans., 1992, 23A, 1445. 8. Coble, R. L., J. Am. Ceram. Soc., 1973, 56, 461. 9. Tanaka, H., J. Ceram. Soc. Japan, 1995, 103, 138. 10. Parhami, F., Cocks, A. C. F., McMeeking, R. M. and Suo, Z., private communication. 11. Pan, J. and Cocks, A. C. F., Acta. metall., 1995, 43, 1395±1406. 12. Sun, B., Suo, Z. and Cocks, A. C. F., J. Mech. Phys. Solids, 1996, 44, 559. 13. Herring, C., in The Physics of Powder Metallurgy, ed. W. E. Kingston. McGraw-Hill, New York, 1955. 14. Chuang, T. and Rice, J. R., Acta. metall., 1973, 21, 1625. 15. Chuang, T., Kagawa, K. I., Rice, J. R. and Sills, L. B., Acta. metall., 1979, 27, 265. 16. Svoboda, J. and Riedel, H., Acta. metall., 1995, 43, 1. 17. Pharr, G. M. and Nix, W. D., Acta. metall., 1979, 27, 1615. 18. Bross, P. and Exner, H. E., Acta. metall., 1979, 27, 1013. 19. Bouvard, D. and McMeeking, R. M., J. Am. Ceram. Soc., 1996, 79, 666. 20. Zhang, W. and Schneibel, J. H., Acta metall. mater., 1995, 43, 4377. 21. Pan, J., Cocks, A. C. F. and Kucherenko, S., Proc. R. Soc. A, 1997, 453, 2161. 22. Turner, C. D., Fundamental Investigations of the Isostatic Pressing of Composite Powders, Ph.D. thesis, Engineering Department, Cambridge University, 1994.